Lecture note on
6th April (2022)

Magnetism (1)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 1

Basic Notions of Magnetism

Magnetic ﬁelds are well known as “ﬁelds” that generate remote (but actually proximity) forces as well as magnetic
ﬁelds or gravitational ﬁelds. Materials respond to magnetic ﬁelds on one hand, create magnetic ﬁelds on the other.
Such properties are called “magnetism.” Perhaps the most prominent “magnetism” for us is spontaneous magnetization
represented by permanent magnets. On the other hand, every material has some magnetic properties. Then what do wecall magnetism? What is the origin of magnetism? We will consider these problems in this half-year lecture. I do notthink I can give you sufﬁcient answers though I would like to try to give you some usuful hints to consider the problems.
In this chapter, we will have a short look at very basic notions in magnetism. I may skip some of the contents in thelecture notes in the real lectures due to the time limitations.

1.1 Electromagnetic ﬁelds in the vacuum and those with materials
We skip the very elementary electromagnetism, with which all of you are already familiar. First, we consider themagnetic properties of matter phenomenologically.

1.1.1 The Maxwell equaitons and magnetic moment
In classical theory, the electromagnetic ﬁeld in a vacuum is described by the Maxwell equations
ρ
,
ϵ0
∂B
∇×E =−
,
∂t
∇·E =

E01-1

(1.1a)(1.1b)

∇ · B = 0,

∇ × B = µ0 j + ϵ 0∂E∂t(1.1c)

.

(1.1d)

We adopt MKSA (SI) unit of system (Appendix 1A). The annotations of symbols may be skipped, but for conﬁrmation,
B is the magnetic ﬂux density with unit of [T] (Tesla) in SI. The unit [T] is the same as [Wb/m2 ], where we consider
the number of magnetic ﬂux [Wb] (Weber). The unit of magnetic ﬁeld intensity is [A/m], and we usually use symbol
H the quantity measured in this unit. In the vacuum, they are in a linear relation B = µ0 H with the coefﬁcient µ0(permeability of vacuum). Because µ0 µ0 has a physical dimension in MKSA unit system, thus B is a different quantity
from H to be strict. However µ0 = 4π × 10−7 [H/m] is a universal scalar and we often call B as “magnetic ﬁeld”, aswell as H in the vacuum. In materials, the situation changes.
In eq. (1.1c), E and B are not symmetrical even after tunings of coefﬁcients. The origin is ρ (charge density) in the rhs
of eq. (1.1a), and j (current density) in the rhs of eq. (1.1d). These come from the fact that the electric monopole exsits.
Though the possibility of the existence of magnetic monopole is not completely eliminated, there has been no convincing
report on the ﬁnding of magnetic monopole. At present, as in eqs. (1.1), we do not consider the existence of magnetic
monopole. In eqs. (1.1), hence, magnetic ﬁelds are created by electric currents and time-derivatives of electric ﬁeld as ineq. (1.1d). However, as we see later, electrons and some of nuclei have spin angular monments and associated magnetic
dipole moments. These produce dipole magnetic ﬁelds around them.
The concept of dipole magnetic ﬁeld can be introduced as
the shrinkage limit of a circular current(1B.2). On the other
hand, in correspondence with the electric dipole, introductionof ﬁctitious magnetic charges which always appear as a pairwith the same amplitude and the opposite sign, and the limitof shrinkage under the condition of keeping the product (magnetic charge)×(distance) constant. Then let us deﬁne magnetic moment as follows. We use [Wb] (weber) as the unit of“magnetic charge” in MKSA system in parallel with the unitof electric charge ([C]) = that of electric ﬂux due to the Gauss

Fig. 1.1 Pair of force moment on a magnetic dipole
in a magnetic ﬁeld.

theorem. We consider a magnetic dipole (before taking theshrinkage limit), which has magnetic charges ±qm with distance l. Let the dipole be placed in a uniform magnetic ﬁeld

H and has an angle θ to the ﬁeld (Fig. 1.1). The magnetic charges get the force qm H from the ﬁeld and the dipole get thepair of force
L = −qm lH sin θ = −(qm l/µ0 )B sin θ.

(1.2)

The quantities that depend qm and l only in the form of their product qm l, do not change with taking the limit. Hence wewriteµ ≡ qm l/µ0 ,(1.3)

and call µ magentic moment.
Because the couple coment in eq. (1.2) drives a rotation of the magnetic moment to the direction of θ = 0, it can be
expressed by a static magnetic potential ϕm as follows.
ϕm = −µB cos θ.

(1.4)

By generalizing the above to a vector representation, we get
L = µ × B,

ϕm = −µ · B.

E01-2

(1.5)

Fig. 1.2 (a) Schematic view of polar coordinate (r, θ, φ). (b) Schematic diagram of magnetic power force lines frommagnetic dipole.

1.1.2 Dipole interaction
Let a magnetic moment along z-axis be at the origin. We use the polar coordinate (r, θ, φ) in Fig. 1.2(a).
From the spherical ﬁelds that the magnetic charges create, it is easy to see that the magnetic ﬁeld has no component
along φ. It is well known that the magnetic ﬂux density (Br , Bθ ) along (r, θ) are
|µ| 2 cos θ ,
4πµ0 r3
|µ| sin θ

Bθ =
. 
3
4πµ0 rBr =(1.6)

We see they are inverse proportional to the cube of distance. The derivation can be ﬁned in Appendix 1B1. We can draw
the magnetic force line by connecting tangent lines of magnetic ﬁeld vectors and obtain, e.g., that in Fig. 1.2(b)*1 .
Next we consider two such magnetic moments µ1 , µ2 . As in Fig. 1.3, the vector going from µ1 to µ2 is written as r.
µ1 and µ2 form the potential in eq. (1.4) in the magnetic ﬁelds of eq. (1.6). Then the total potential of the two magnetic
moments is
U=

1
4πµ0 r3µ1 · µ2 −


3
(µ
·
r)(µ
·
r)
.
1
2
r2

(1.7)

The derivation is given in Appendix 1B2. The interaction between the moments expressed by eq. (1.7) is called dipoledipole interation. In the potential of eq. (1.7), the stable conﬁguration can be obtained by maximizing the amplitude of
the second term because the sign is minus and the coefﬁcient is larger than the ﬁrst one. |r| is canceled by the denominatorFig. 1.3 Classical interaction of two magnetic dipoles. (a) Stable conﬁguration. (b) Unstable conﬁguration.

*1 In this ﬁgure, it is not taken into consideration that the magnetic ﬁeld line density is proportional to the magnetic ﬁeld.

E01-3

and the numerator, hence the stable conﬁguration is obtained as in Fig. 1.3(a). On the ohter hand the conﬁguration in
Fig. 1.3(b) has a higher energy and is unstable. The effect of classical dipole-dipole interaction is generally much smaller
than the quantum mechanical exchange interaction. However, in some characteristic phenomena, it plays important roles.
A more fundamental problem of multiple magnetic moments is how we can view a set of magnetic moments from afar distance. The problem of taking a limt of moment-moment approach with a constant moment-distance product, is the
problem of multipole. However, simply looking from a distance, the problem is the same as that a set of various charges
from a distance can be treated as a point charge with the sum of the charges. This from the linearity of the Maxwell
equations. Then let {µ0 , µ1 , · · · } be the magnetic moments under consideration and we see the set as a moment whichis the sum
µ=

X

µi .

(1.8)

i

1.1.3 Magnetization of materials
Generally a magnetic ﬁeld induces a magnetic moment in materials. This phenomenon is called magnetization of amaterial. Assuming a uniform distribution of induced magnetic moments, the moment per unit volume is called magnetization or magnetic polarization. Now, we write the magnetization as M . Then from deﬁnition (1.3), we express theinduced magnetic moment as the sum of equal descrete moment µ = qm l/µ0 of concentration N (per unit cell) as
M=Xµ = N qm l/µ0 ≡ ρl/µ0 .

(1.9)

unitvol.

Here, ρ ≡ N qm is like a density of magnetic charges. In the naive model described in Fig. 1.4, small bar magnets withlength l = |l| are aligned. The magnetic charges of neighboring magnets cancel each other due to the zero distance andno magnetic charge exists inside the material naturally. The magnetic charges then appear just at the ends of the material.
When the end surfaces are taken perpendicular to the magnetic polarization, the surface magnetic charge density is qm
times s, the areal density of “rods” which are the serieses of bar magnets. We consider a slab with unit area and heightl, which should contain just one moment along the height, then the total number inside should be s. On the other handfrom the deﬁnition the number should be N l, which means s = N l. Therefore the areal density of magnetic charge σ atthe ends is given by
σ = qm s = qm N l = µ0 |M |,(1.10)

that is, the magnetic polarization is the same as the areal density of magnetic charges.
As considered above, the magnetic moment induced in the material (there is also spontaneous magnetization that occurs
without an external magnetic ﬁeld), that is, the magnetization generates a magnetic ﬁeld around the material. The ﬁeld
is in a far distance, that of the magnetic moment of LS|I| = V |M |, where S is the area of ends of the material, L is
the length. Thereofre, measurement of outer magnetic ﬁeld originated from the material can give M . In the discussion

Fig. 1.4 A naive model of magnetization of a material which is composed of many small magnets with length l andmagnetic charges ±qm .

E01-4

of magnetism, i.e., magnetic properties of materials, (apart from how far the above simple model can be used) what weshould consider in the ﬁrst place is how magnetic moments are induced in the material ?

1.1.4 Electromagnetic ﬁeld in the presence of materials
Since the magnetic moment is expressed as a circular current, the effect of the magnetic moment in a substance canalso be expressed as a current. As illustrated in Fig. 1.5, we take the coordinate r ′ inside the material and sum up theeffect of local moments to give



Z
Z
µ0
M′ × r
1
µ0
′
′
dv ′
=
−
dv
M
×
∇
4π mat
r3
4π
r

 mat
Z
1
µ0
=
dv ′ M ′ × ∇′.
4π matr

A=(1.11)

Here the integration volume is taken inside the material. The symbols with prime as M ′ means they are expressed as afunction of the coordinate (x′ , y ′ , z ′ ). Further partial integration givesZµ0∇′ × M ′A=
dv ′.
4π matr(1.12)

Then if we writejM ≡ ∇ × M ,(1.13)

from ∇ · jM = 0, we can view jM as a kind of electric current.
We add the true current of real charge j to the above “equivalent current” toobtain the vector potentioal as
µ0
A=
4π

Z

Then if we writeµ0
B =∇×A=
4π
Fig. 1.5 Illustration of vector potential formed by magnetic moments inside a material

dv ′
Z

′
j ′ + jM
.
r

dv ′

′
(j ′ + jM)×r,r3(1.14)(1.15)

we obtain the relation∇ × B = µ0 (j + jM ) = µ0 j + µ0 ∇ × M .

(1.16)H ≡ B/µ0 − M ,(1.17)∇ × H = j,

(1.18)

Deﬁning magnetic ﬁeld as
we ﬁndnamely H does not depend on the equivalent current of magnetic moment.
Now we consider the electromagnetic ﬁeld in the presence of a material. Electric ﬂux density D, and magnetic ﬂux
density B is given by
D = ϵ0 E + P(1.19a)B = µ0 (H + M ),(1.19b)where P is electric polarization. In the case of electric charge, true charge ρt exists other than the polarization charge ρp .
From ρp = −∇ · P we can write∇ · D = ρt ,(1.20)

which means the effect of polarization in taken into D. Further, if we express ∇ × H with D, we obtain the formula∂D,∂twhich is in the same form as (1.1d) and the effect of magnetization is included into H.
∇×H =j+E01-5(1.21)

M

M
paramagnetic
material

H

H
diamagnetic
material(a)(b)

Fig. 1.6 (a) Schematic diagrams of M-H curves in paramagnetic ad diamagnetic materials. (b) Schematic diagramof M-H curve in ferromagnetic material. Hc : coercive force, Mr : remanent magnetization, Ms : saturation magnetization.

1.1.5 M(B)-H curve
As above, the response of materials to magnetc ﬁeld H appears as magnetization M . When M is proportional to H,
we writeM = χH,(1.22)

where the coefﬁcient χ is called magnetic susceptibility. Then we write
B = (χ + µ0 )H ≡ µ∗ H,(1.23)and call µ∗ ≡ χ + µ0 magnetic permeability. µ̄ ≡ µ∗ /µ0 = χ̄ + 1 = χ/µ0 + 1 is called relative permiability.
Among such materials with linear response, we call paramagnetic material for those with χ > 0 and diamagnetic
material for those with χ < 0. And there are many materials such as ferromagnetic materials in which the linear relationin eq. (1.22) does not hold. These materials are very important particularly in the ﬁeld of magnetics – ﬁeld of application.
In such cases, behavior of magnetization is often presented in the form of M-H curve, in which magnetization M isplotted versus H. Figure 1.6 shows a schematic of M-H curve. In the case of linear response in eq. (1.22), the M-H curveshould be linear as shown in Fig. 1.6(a). On the other hand, Fig. 1.6(b) shows a schematic of ferromagnetic response.
With up-down sweeps of H, the response of M is strongly non-linear, and different behavior is observed for the directionof the sweeps, which phenomenon is called hysteresis. B-H curves are also adopted for giving the same information.

1.2 Measurement of magnetization
Here we introduce some of experimental methods for the measurement of magnetization before going into the theories.
Also we touch on the problem of demagnetizing ﬁeld, which is important for measurement.

1.2.1 Methods of magnetization measurement
Below we list some representative methods. There are two major ways for the measurement: (a) Measurement ofmagnetic ﬁeld caused by magnetization; (b) Variation in magnetization is detected as voltage caused by electromagnetic
induction. Examples of (a) are, vibrating sample magnetometer (VSM), superconducting quantum interference device(SQUID) magnetometer. An example of (b) is a pick-up coil type magnetometer.

E01-6

(a)

(b)

Fig. 1.7 (a) Principle of magnetization measurement in VSM. (b) Principle of magnetization measurement in AGM.
From https://www.toyo.co.jp/material/casestudy/detail/id=7003(a1) Vibrating sample magnetometer: Figure 1.7(a) shows the principle of magnetization measurement by VSM. Whena magnetized substance is spatially vibrated, the magnetic ﬁeld generated by the magnetization is not spatially uniform,
so it vibrates with time when viewed at a ﬁxed point in space. Then in a detection coil placed at the point the vibratingﬁeld procuces an alternating voltage though electromagnetic induction. In VSM the ditection of the AC voltage gives
measurement of magnetization.
In actual use of VSM, as annotated in the ﬁgure, the sample, in addition to the vibration, moves slowly through two
inversely-wound detection coils to produce a signal peak and dip, which are mirrored to each other. With this devicing,the signal offset can be eliminated.
(a2) Alternating-gradient magnetometer: AGM[1] utilizes the fact that a magnetic moment gets force in magnetic ﬁeld
gradient. As shown in Fig. 1.7(b), alternative current in coils attached to an external magnet provide a vibration in ﬁeld
gradient. Then the vibration in the force on the suspender is detected to give the magnetization. Force detection has
become extremely sensitive by the method using a laser and a cantilever, which is familiar with atomic force microscopes(AFM) (the piezo element is used in the ﬁgure), so extremely high sensitivity can be obtained.
(a3) SQUID magnetometer: Magnetic ﬂux piercing a superconducting ring is quantized by the unit of quantum ﬂux

Critical Current Jc (arb.)Φ0 ≡ h/2e ≈ 2.07 × 10−15 Wb. A superconducting quantum interference device (SQUID) is a superconducting ring

-1

0

1

2

3

4

5

Magnetic flux (Φ0)

(a)

(b)
Fig. 1.8 (a) Schematic diagram of critical current in a SQUID device as a function of magnetic ﬁeld. (b) Superconducting circuit diagram of a SQUID magnetometer.

E01-7with some (in the case of dc-SQUID, two) weak links. The superconducting critical current Jc through it largely oscillateswith magnetic ﬂux with the period of Φ0 as illustrated in Fig. 1.8(a). The unit Φ0 itself is already very small and theoutput of a SQUID is strongly non-linear as shown in Fig. 1.8(a), which fact enables us to measure orders of smallerchange in the magnetic ﬂux.
Because SQUID sensors should be placed in weak magnetic ﬁelds, a closed loop of superconductor is used as shownin Fig. 1.8(b). One end of the loop is magnetically coupled to a SQUID sensor through a coil. As in VSMs, a specimen
moves in counter-wound superconducting coil (in the ﬁgure the winding is − + +−), which form the closed loop. The
function ﬁtting to the lineshape gives precise value of magnetization. To get high S/N, the specimen should be suspended
by a uniform substance with small magnetic susceptibility. The upper limit of the ﬁeld is determined by superconducting
critical ﬁeld of the pick-up coil. Because the SQUID magnetometers have very high sensitivity, they are applied formeasurement of environmental magnetic ﬁeld or magnetic ﬁeld leakage from brains (magnetoencephalograph).
(a4) NV center magnetometer: Lattice defects formed as complex of a nitrogen (N) impurity and a vacancy (V) in
diamonds have quantum states in the bandgap. Combination of optical excitation/detection and electron spin resonancewith microwave provides highly sensitive detection of very local magnetic ﬁeld. S/N can be highly enhanced by, e.g.,combination of spin-rotation pulse sequences. In some cases, the sensitivity is as high as the replacement of SQUID
magnetometers.
(b1) Pick-up coil method: It is used that the response of the coil changes depending on the magnetic permeability inside
the coil. A specimen is inserted into a coil. When an AC magnetic ﬁeld is appled, a voltage is induced on the coil due tothe ﬂux variation. Integration of the signal gives the variation of magnetic ﬂux and thus the magnetization.

1.2.2 Effect of demagnetizing ﬁeld
One thing to note when measuring magnetization is the effect of demagnetizing ﬁeld. This appears as shape dependence, especially when measuring the magnetization of ferromagnets. In the magnetic charge model, when a material
with a ﬁnite size is placed in a magnetic ﬁeld and magnetized, magnetic charges (magnetic poles) appear at both ends ofthe sample, which creates a magnetic ﬁeld inside the sample in the opposite direction to the external ﬁeld. This is thedemagnetizing ﬁeld (Fig. 1.9(a)). The demagnetizing ﬁeld Hd is proportional to the magnetization M as
Hd = N

M
,
µ0

(a)

(1.24)(b)

Fig. 1.9 (a) Conceptual diagram of demagnetic ﬁeld. (b) Calculation of demagnetic ﬁeld coefﬁcient of plate-shaped sample.

E01-8

where N is called demagnetizing factor. The demagnetizing factor depends on the shape of specimen. For needle-likethin specimen, it is almost zero while large for thick and short ones.
As the simplest case, we consider the case of plate-like sample in Fig. 1.9(b). For magnetization M , the areal densities
of surface magnetic charge is ±M . From the symmetry, the magnetic force lines are perpendicular to the plane. Applying
the Gauss theorem to a cylinder containing both front and back of the plate, we know that no magnetic ﬂux comes out
because the total magnetic charge in the cylinder is zero. Here we again apply the Gauss theorem to another cylinder
which contains just one of the surface with a bottom of unit area to obtainZ
M
Hn ds = Hd =,µ0surfacewhich gives the demagnetizing factor N = 1. Demagnetizing factors for various shapes have been calculated.
When there is large demagnetizing ﬁeld, e.g., hysteretic M-H curve as in Fig. 1.6(b) is largely distorted. The following
example is introduced in ref. [2]. Permalloy (an alloy of Fe and Ni, symbol Py) has a very small coercive force 2 A/m (≈0.025 Oe), and under usual condition, the magnetization saturates at very small external ﬁelds. However, if we make asphere with Py, becase the saturation ﬁeld of Py is about 9.23×105 A/m and the demagnetizing factor of sphere is 1/3, thedemagnetizing ﬁeld amounts to 3.08×105 A/m (=3860 Oe). That is, saturation magnetization cannot be obtained unlessa magnetic ﬁeld of about 100,000 times that without a demagnetizing ﬁeld is applied. Normally, when submitting data asan M-H curve to a scientiﬁc paper, etc., it is necessary to correct the demagnetizing ﬁeld or to state that it has not beencorrected.

1.3 Classical theory of magnetization
Let us go into the physical mechanism of magnetic properties. Though magnetism still has many unsolved problems
the present understandings have been obtained with quantum mechanics. Within the classical theory, even elementary
understanding is difﬁcult. We see that in this section. Here we refer to “classical theory” that inside materials exists
a group of electrons which are classical particles with charge −e and mass m. And they create the equivalent current(1.13).

1.3.1 Classical treatment of paramagnetic moment
Before digging into how to deal with magnetism in classical mechanics and statistics, let us assume that there isalready a magnetic moment in matter that has a degree of freedom to change direction, and see the consequence. Matteris composed of atoms, and we will consider a model in which the electrons around the nucleus originally have a magnetic
moment due to orbital motion.
Let us consider a set of molecules in a magnetic ﬁeld along z-axis with ﬂux density B. Each molecule has independent
magnetic moment µ. The magnetic energy of a moment is U = −µ · B = −µB cos θ. In classical statistics, the averageof z-component in the moment over the is given by



Z

Z
U
U
exp −
⟨µz |µz ⟩ =
exp −
µz dΩ
dΩ
kB T
kB T



Z

Z
µB cos θµB cos θexp=expµ cos θdΩ
dΩ
kB T
kB T

 Z π


µB cos θ∂exp= kB Tlog 2πsin θdθ
∂B
kB T


 0

µB
kB T
= µ coth
−
,
kB T
µB

E01-9

(1.25)

where Ω is the solid angle. In high temperature approximation µB ≪ kB T , the average is⟨µz |µz ⟩
µ2 1
∼
.
B
3kB T

(1.26)

This indicates Curie law, that is the magnetic susceptibility is inversely proportional to temperature.

1.3.2 Classical theory of diamagnetism
We consider an electron moving along a circle with radius r in xy plane. We write the circle as Γ and the area as S,
and write down the integral form of the Maxwell equation (1.1b) asIZ∂
E · dl = −B · dσ.
∂t SΓ(1.27)

From this, the induced electromotive force for variation of magnetic ﬂux B is given as
2πrE = −


∂
Bπr2
∂t

∴E=−

r dB
.
2 dt

(1.28)The electron is accelerated by −eE to the tangential direction and the time derivative of angular momentum L isdL
r2 dB
= r × (−eE) = e
.
dt
2 dt

(1.29)

2

Then the shift 0 → B creates the angular momentum L = e r2 B. The velocity of electron also increases from 0 to v thenv = L/mr. As describe in Appendix 1B.2, the magnetic moment of this circular current is (the area of circle)×(crrent).
Then it isµ = SJ = πr2

ev
e r2
L e
= πr2
=
e B.
2πr
mr πr
2m 2

(1.30)

If we replace r, the distance between nucleous and electron, with the average x2 + y 2 x2 + y 2 av ,µ=−e2
x2 + y 2 x2 + y 2 av B.
4m(1.31)

1.3.3 Breakdown of classical theory of magnetism
In the above, seemingly reasonable results have been obtained even within classical mechanics. However, below, we
see the theory is actually broken down in a simple discussion. We introduce electromagnetic ﬁeld to simple single-particle
hamiltonian H = p2 /2m by changing energy E and momentu p as
E → E + eϕ,That is the hamiltonian is written asH=p → p + eA.

(1.32)

1(p + eA)2 − eϕ.
2m(1.33)

This way of introduction is justiﬁed by the derivation of canonical equationmdv= −e[E + v × B],
dt

which reproduces the Lorentz force.
We take symmetric gauge A = (B × r)/2. Then (1.33) is calculated as follows.
H=

p2
e
e2
+
(r × p) · B +(B × r)2 .
2m 2m
8m

E01-10(1.34)

From this, the magnetic dipole moment µm originates from the motion of electrom is calculated asµm = −∂Hee2=−(r × p) −(r × (B × r)).
∂B
2m
4m

(1.35)

Here, the ﬁrst term in rhs is called paramagnetic term, which is proportional to the angular momentum r × p. On theother hand, the second term is proportional to magnetic ﬂux density |B|, which indicates that the inductive electric ﬁeld
accelerates the electron and that this comes form the eddy current-like motion to cancel the external ﬁeld. Hence this iscalled diamagnetic term.
Let us move on to an N -particle electron system. The hamiltonian is written with writing electron-electron interaction
as V as

N 
X
1

HN =

n=1

2m


2
(pn + eA(rn )) − eϕ(rn ) + V (r1 , r2 , · · · , rN ).

(1.36)The partition function Z at temperature T is
N Z
Y
drn dpn −H/kB T
Z=
e
.
h3
n=1

(1.37)

Though it is classical, it has the Planck constant h because we need to calculate the number of states and that should bebased on a unit space in the r-p phase space.
Here we write πn = pn + eA(r), thenN Z
Y
drn dπn −H′ /kB T
e
,
h3
n=1

N  2
X
πn
− eϕ(rn ) + V (r1 , r2 , · · · , rN ),
H′ =
2m
n=1

Z=

(1.38)

which has no A in the expression. Hence the statistical average of the magnetic moment is naturally⟨µm |µm ⟩ = −1 ∂F
1 ∂ ln Z== ⟨µpara |µpara ⟩ + ⟨µdia |µdia ⟩ = 0.
N ∂B
N kB T ∂B(1.39)

That is, within the classical picture of electrons and classical statistics, the pramagnetic and diamagnetic terms cancel
each other and there shuold be no magnetism in this system. This is called Bohr-van Leeuwen theorem.

1.4 Spin and magnetic moment of electron
There are several factors that cause magnetism in quantum theory, one of which is that in quantum theory, an electron
has a spin and a spin magnetic moment. The question why a point charge like an electron can have an internal freedom
called spin and that also has a magnetic moment can be answered clearly by relativistic quantum mechanics. All of you
should have already learned about this in the undergraduate course. However we would like review here the beautiful
logics how this is derived[3].

1.4.1 Dirac equation
We consider quantum mechanics in the form of wavefunction and try to ﬁnd out a form which stands with the special
relativity. Schrödinger equation is a non-relativistic approximation and for the above purpose, we need to ﬁnd out theform which is invariant for Lorentz transformations.
The one-dimensional Schrödinger equation is obtained by replacing the energy and the momentum as
E → iℏ∂,∂tp → −iℏ

E01-11∂,∂x(1.40)

in the energy-momentum relation E = p2 /2m in Newtonian mechanics. Furthermore, the wavefunction in the wave
mechanics should have the meaning of probability amplitude and hence the differential equation for the wavefunction
should be that of single-derivtive in time[4]．
First the relativistic energy-momentum relation isE 2 = (pc)2 + (mc2 )2 .

(1.41)

If the transformation (1.40) is applied directly to the above we obtain Klein-Gordon equation but this is in the second
order in time and does not fulﬁl the condition for the wavefunction. If the differential equation is ﬁrst order in time andthe orders for time and space should be the same, the derivative on the space also should be the ﬁrst order. Then we writeXE=

αk pk c + βmc2 ,(1.42)

k=1,2,3and try to compromise the above with eq. (1.41). Taking the square of the lhs and the equation to be eq. (1.41), theconditions are α2 = 1, β 2 = 1,
 kαk αj + αj αk = 0 (k ̸= j),
αk β + βαk = 0.

(1.43a)(1.43b)(1.43c)In order to satisfy the above, we consider matrices for αk , β and the dimension should be at least 4×4.
Then the wavefunction should have four components.

The equation for the 4-component wavefunction

ψ = t (ψ1 , ψ2 , ψ3 , ψ4 ) (t means transpose) should be


X
∂ψ 
∂
iℏ
= −iℏcαk+ βmc2  ψ∂t∂xk(1.44a)k=x,y,z≡ HD ψ,HD = cαp + mc2 β.

To obtain a speciﬁc form of αk , β, we introduce the following Pauli matrices:








0 1
0 −i
1 0
1 0
σx =
, σy =, σz =
, I=
.
1 0
i 0
0 −1
0 1

(1.44b)(1.45)

These matrices have the following relations
σi σj = −σj σi = iσk ,σx2 = σy2 = σz2 = I.

(1.46)

Here for (i, j, k), (x, y, z) are assigned with cyclic rotations. From the above we reach a speciﬁc representation (Pauli
representation) as follows.


αk =

0
σk


σk
,
0


β=

I
0


0
.
−I

(1.47)

Pauli representaion is one of the possible representations and we can ﬁnd inﬁnite numbers of representation with unitarytransformation. Calculated results for the observables should be the same for all the representations. The four dimensions
correspond to the spin degree of freedom and the freedom of particle-antiparticle (isospin). These four freedoms exist on
the spatially point charge of electron-positron. In conclustion, this is the equation for the particle with spin 1/2 and witha ﬁnite mass.

1.4.2 Spin angular momentum
We consider a centeral force potential V (r) and write
H = HD + V (r).
E01-12(1.48)

The angular momentum
L=r×p(1.49)[L, H] = iα × p.

(1.50)

does not commute with the hamiltonian (1.48) as

α is a vector of components αk . We expand Pauli matrices to 4×4 dimension as


σk 0
(4)
σk =
,
0 σk
(4)

and write the vector of elements σx,y,z as σ. Then from the relation[σ, H] = −2iα × p/ℏ,

(1.51)

we deﬁne the total angular momentum J as
Total angular momentum


J =L+



ℏ
σ ≡ L + s,2

(1.52)
then we obtain[J , H] = 0,(1.53)

that means J is a constant of motion. Namely s ≡ (ℏ/2)σ is an observable which has characteristics of angularmomentum. This is the spin angular momentum. We reach the conclusion that though an electron is a point in thespace, it has an angular momentum as if it has a rotation.

1.4.3 Magnetic moment
We consider the Dirac equation in the presence of static electromagnetic ﬁelds. Just like eq. (1.32), we introduce ascalar and a vector potential to obtainiℏ
Now we rewrite it to







∂ψ
= [cα(p + eA) + βm − eϕ] ψ.
∂t


X

∂
 iℏ ∂ + eϕ − c
αj −iℏ
+ eAj
∂t
∂r
j
j=x,y,zThen operate(1.54)
− βmc2  ψ = 0.



X
∂
∂
iℏ + eϕ + c
αj −iℏ+ eAj + βmc2∂t∂r
jj=x,y,z(1.55)(1.56)

from the left. After some algebra by using the commutation relations of αj , β, we reach
"
#
2
∂
2
2
2 4
2
iℏ + eϕ − c (p + eA) − m c + icℏe(α · E) + iℏc e(αx αy Bz + αy αz Bx + αz αx By ) ψ = 0. (1.57)∂t
Here from
αx αy = iσz(4) ,we can write (1.57) as"∂iℏ + eϕ∂t

αy αz = iσx(4) ,αz αx = iσy(4)(1.58)#

2− c (p + eA) − m c + icℏe(α · E) − ℏc eσ · B ψ = 0.
2

2

2 4

E01-132(1.59)

To obtain steady state we writeand obtainhψ(r, t) = exp(−iϵt/ℏ)φ(r)(1.60)i2(ϵ + eϕ) − c2 (p + eA)2 − m2 c4 + icℏe(α · E) − ℏc2 eσ · B φ = 0.

(1.61)

Here we put ϕ = 0, E = 0. In non-relativistic approximation p ≪ mc, i.e., ϵ ≈ mc2 , hence we write ϵ = mc2 + δ andignore the power of δ/mc2 higher than the second order to obtain*21eℏ(p + eA)2 +σ · B φ = δφ.
2m2m(1.62)

Here we deﬁne the quantity called Bohr magneton asBohr magneton
µB ≡




eℏ
≈ 9.274 × 10−24 JT−1 ,2m(1.63)
then in eq. (1.62) the term related to the magnetic ﬁeld iseℏ
2
σ · B = µB σ · B = µB s · B.
2mℏ

(1.64)

This means that an electron has, with the spin angular momentum s, a magnetic moment −2µB s/ℏ

Appendix 1A Unit systems of electromagnetism
The unit system, or metrology is extremely important for human life even beyond the boundaries of science or theframework of scholarship. In usual systems, at ﬁrst place a small number of basic units are determined by some way, andthen other units are determined through universal laws of physics. Naturally, there are many possible ways to constituteunit systems. We should choose a unit system convenient for the problem that we need to tackle from many possible ones,
particularly in the case of electromagnetism. Many researchers, even among non-expert in unit, have their own opinions
for unit systems. Hence it is impossible to force everyone to choose one. In spite of such tendency, the SI (Système
International) unit system is deﬁned as an international standard in consideration of practicality, logical consistency, andhistorical continuity. I would like to adopt the SI unit system as far as I can, but here, I introduce major systems of unit inelectromagnetism in very short. If you are interested in the metrology, I would like to recommend Ref. [5]. Care should
be taken, though, that the committee meeing of the SI unit is held annually, and the unit may have big changes as did in
2019. It is necessary to refer to the web etc. for the latest deﬁnition. Here I summarize CGS-esu (electrostatic system ofunit) and MKSA (SI) system of unit very brieﬂy.

1A.1 CGS-esu
The Coulomb’s law should be written in the form
F = kqq1 q2.
r2(1A.1)In CGS electrostaticunit, kq is just 1 and with no physical dimension. Then the electric charge can be expressed by [L],[M], [T] as[Q] = [M 1/2 L3/2 T −1 ].

*2 This φ has four components and then we need to reduce the dimension to two with the non-relativistic approximation. But here we skip theprocedure for simplicity. The result here is the same for this but to derive spin-orbit interaction, this is indispensable.

E01-14

When the force between electric charges with the same amount at the distance of 1 cm is 1 dyn, the charge is 1 esu (CGSesu). There are three basic quantities and there is no factor of (4π)−1 in the Coulomb’s law (this means 4π appears in theMaxwell equation). Hence this unit system is classiﬁed to three components irrational unit system.

1A.2 MKSA unit system
Until a while ago, the current was introduced as the fourth basic quantity, the unit of the current was A (ampere), and
1 A was deﬁned from the force acting between the parallel conductors separated by 1 m in a vacuum with a length of
1 m as to be 2×10−7 N. However in the redeﬁnition in 2019, the fourth basic quantity becomes the charge, which is
introduced by determining the elementary charge e as 1.602176634×10−19 C (coulomb). Then the current is introduce
by the charge and the time (A·s =C). Because the number of basic quantities is four and kq has the factor (4π)−1 , the unit
system is classiﬁed to four components rational unit system.

Appendix 1B: Dipole ﬁeld and dipole inteaction
1B.1 Magnetic ﬁeld created by magnetic dipole
From eq. (1.1d), when there is no true current and no time-derivative of electric
ﬁeld, the rotation of magnetic ﬁeld ∇ × B is zero. In such a case, we can deﬁne magnetostatic potential ϕm as a scalar function of spatial coordinage r. The
magnetic ﬁeld is given by
H = −∇ϕm .

(1B.1)

We consider the magnetostatic potential ϕm for the situation that magnetic charges
of ±qm are placed at ±p = (0, 0, ±l/2) respectively.


qm
1
qm
−
ϕm (r) =.
4πµ0 |r − p| |r + p|

(1B.2)

We assume that l is sufﬁciently smaller than |r| (|r| ≫ l), and expand ϕm with thepower of l and take the ﬁrst order term to obtainϕm (r) =Fig. 1B.1 Magnetic charge
model of magnetic dipole.

qm lz
qm l cos θ=.
34πµ0 r
4πµ0 r2(1B.3)The last expression is on the polar coordinate (r, θ, φ).
We write l = (0, 0, l), and the magnetic moment as µ = qm l/µ0 , the magnetic ﬁeld originates from the magnetic
dipole isµ · r
1
.
∇
4πµ0
r3

(1B.4)

∇=

∂
1 ∂
1
∂
er +
eθ +
eφ
∂r
r ∂θ
r sin θ ∂φ

(1B.5)Br =|µ| 2 cos θ,
4πµ0 r3|µ| sin θ,
4πµ0 r3(1B.6)

B=−
We apply the polar coordinate representation of ∇to eq. (1B.3), and we reachBθ =which is nothing but eq. (1.6).

E01-15

1B.2 Dipole ﬁeld as shrink limit of circular current
In the text, we considered the ﬁctitious “magnetic charge” to make the physical picture simpler. The experiments so far
have shown that there is no isolated magnetic monopole like electric monopole. Then there is an opinion that we should
not use the magnetic charge even as a mathematical concept. On the other hand, there are researchers who positively use
the magnetic charge as an established physical concept [2], because there are many such physically established concepts
like the vector potential which is not an observable but a mathematical method.
This problem also relates to the view of electric-magnetic symmetry in theMaxwell equation (1.1). That is, so called the problem whether E-B formulation or E-H formulation. In the former, the magnetic ﬁeld is introduced along
the Viot-Savart law as formed by a current on a wire element while in the latter the ﬁeld is formed by magnetic charges through the Coulomb law. In CGS
unit system, these two ways of introduction do not give signiﬁcant difference.
However in MKSA unit system, this leads to the difference in the unit of magne-Jtization. The same problem appears when we introduce magnetic dipole throughSmagnetic charges. When we consider a pair of moment (1.2), the selection of
E-H correspondence or E-B correspondence results in the difference whether

Fig. 1B.2 Deﬁnition of magnetic
dipole by circular current.

eq. (1.3) gets µ0 or not. In this lecture we do not go into the problem to construct electromagnetic theory though we adopt E-B formulation for the unit ofmagnetization.

Now we consider a circular current J surrounding the area S in xy-plane as in Fig. 1B.2. The vector potential A is

ZZZ
I
I 
j
µ0 J
ds
µ0 J 11µ0
dv ==1 + 2 (R · s) + · · ·
A=
4π
r
4π
r
4π R
R
I
µ0 J 1≃(R · s)ds.
(1B.7)
4π R3
sx component of the integration over ds is
I X
I
I
X
(R · s)dsx =Ri si dsx =Ri si dsx ,i=x,y,zi=y,zand the term of Rx sx vanishes because sx goes back and forth over the integral
interval. On the other hand for sy dsx , sx dsy , as in the right ﬁgureIIsy dsx = − sx dsy = −S.
This gives

I

I(R · s)ds = −(R · ds)s,and from the identities of vector analysis we get
I
I
I
1
1
{(R · s)ds − (R · ds)s} =(s × ds) × R,
(R · s)ds =
2
2
where

S

1
2

(1B.8)

Is × dsis the vector perpendicular to the current plane and with the size of the area (S) of the circular current. Then we deﬁne avector µ as

 I

1
µ=J
s × ds ,2
E01-16(1B.9)

then we can writeA=µ0 µ × R.
4π R3(1B.10)

By using this equation and after some algebra, we obtain, e.g., x-component of B as
Bx =

∂Az
∂Ay
µ0  µ · r 
−
=−
∇ 3
.
∂y
∂z
4π
r
x

(1B.11)

Then the magnetic ﬁeld created by the circular current is given byB=−µ0 µ · r
∇ 3 .
4π
r

(1B.12)This is in accordance with eq. (1B.4), and now we know that the circular current is working as a magnetic dipole.

1B.3 Dipole interaction(Under construction)

References[1] P. J. Flanders. An alternating-gradient magnetometer (invited). Journal of Applied Physics, Vol. 63, No. 8, pp.
3940–3945, April 1988.
[2] S. Chikazumi and S. H. Charap. Physics of Magnetism (Wiley Series on the Science and Technology of Materials).
Wiley, 1 1964.
[3] Shinichiro Tomonaga. The Story of Spin. University of Chicago Press, 10 1998.
[4] J. J. Sakurai and Jim Napolitano. Modern Quantum Mechanics. Cambridge University Press, 9 2020.
[5] N. V. Raghavendra and L. Krishnamurthy. Engineering Metrology and Measurements. Oxford Univ Pr, 10 2016.

E01-17

Magnetism (2)Lecture note on
13th April (2022)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

1.4.4 Spin-orbit interaction
Another important effect stems from the Dirac equation is spin-orbit interaction (SOI). We have already seen that thefour components in the solution of the Dirac equation correspond to (freedom of particle-antiparticle)×(spin freedom).
In the Pauli representation (1.44b), (1.47), within non-relativistic limit of p ≪ mc, the upper two rows correspond to thesolution with positive energy, the lower two correspond to that with negative energy. However with growing p, the shifts
from the free particle become large and some mixing appears between the upper two and the lower. For example, we
consider a free dirac particle propagating along z-direction with up-spin for quantization axis of z-axis. Let tan 2θ betan 2θ =then the four component wavefunction isp,
mc(1.65)



cos θ
 0 

ψ↑ = ei(kz−ωt)  sin θ  ,0(1.66)

which indicates the increase of negative energy component with p. Inclusion of this leakage into the picture of particle
with two-component of spin creates the SOI as the correction.
The equation for stationary solution obtained from the treatment in (1.60) is given by(cα · p + βmc2 + V )φ = ϵφ,(1.67)

where σj are Pauli matrices defined in eq. (1.45), α is a vector of elements αk defined in eq. (1.47), β is defined ineq. (1.47). We write φ as a two-component vector of φA and φB as φ = t (φA φB ). Then eq. (1.67) is written in asimultaneous equation as follows.
σ · pφB = c−1 (δ − V )φA ,−1

σ · pφA = c(1.68a)(δ − V + 2mc )φB ,2(1.68b)where δ = ϵ − mc2 and σ is a vector of Pauli matrices as elements. φB can be erased virtually as
c−2 σ · p(δ − V + 2mc2 )−1 σ · pφA = (δ − V )φA .

(1.69)

We apply low energy expansion, in which we assume the kinetic energy of electron is sufficiently smaller than the rest
energy mc2 .
2 −1

c (δ − V + 2mc )
2

]
[
δ−V
1
+ ··· .
1−
≈
2m
2mc2

(1.70)The second term in rhs corresponds to (v/c)2 , where v is the electron velocity and we take up to this term. Substituting
this to eq. (1.69) leads to the equation of ψA . However, the normalization condition is ⟨φ|φ⟩ = ⟨φA |φA ⟩ + ⟨φB |φB ⟩ = 1,
which should be considered in solving eq. (1.68) for φA . The first order perturbative mixing in wavefunction is on thesecond order in the normalization and in eq. (1.70). Hence in eq. (1.70), we take the first term in rhs. We use thecommutation of σj and pi , σj† = σj and the following identity(σ · a)(σ · b) = (a · b) + i[σ · (a × b)],E02-1

In the presence of magnetic field B = ∇ × A, we replace p with p + eA to get
]
( 2)
[ 2
p + eℏσ · Bv|φA ⟩ = O.
⟨φB |φB ⟩ = ⟨φA |4m2 c2c2(

Then as the wavefunction
φa =p2 + eℏσ · B
1+8m2 c2)
φA(1.71)is enough for the approximation to order of (v/c)2 . From eq. (1.69), we reach the Pauli equation (approximation to orderof (v/c)2 ) as
[ 2
]
p
eℏ
eℏσ · p × E
eℏ2
p4
eℏp2
(eℏB)2
+V +
σ·B−
−
∇·E−
−
σ·B−
φa = δφa ,2m2m
4m2 c28m2 c28m3 c24m3 c28m3 c2(1.72)

where E = ∇V /e is the electric field. In the lhs of eq. (1.72), the third term is the Zeeman energy, the fourth is the
spin-orbit interaction.
If we write Bso ≡ p × E/2mc2 , the fourth SOI term looks like in the same form as the third term. Namely the spinfeels an effective magnetic field of Bso , which is sometimes called a spin-orbit effective field.

1.5 Magetism in quantum theory
In Sec. 1.3.3, we saw that the magnetism cannot be explained within the framework of classical theory. On the other
hand in Sec. 1.4, in the Dirac theory, which considers both quantum theory and relativity, an electron, though it is a point
charge, has a spin angular momentum and an associated magnetic moment. We will see, throughout this lecture for asemester, how the difficulty in the classical theory is solved. But here we see quickly how this spin angular momentum
produces magnetism in a simple model of the classical framework plus spin.
A Hamiltonian with magnetic field B is given by
]
∑[ 1
H=
(pn + eA(rn )2 + U (rn ) + gµB sn · B + V (r1 , r2 , · · · ),2m
n(1.73)

where a nucleus potential is written as U (rn ). In (1.52), we define s as to have the dimension of angular momentum andthen (ℏ/2)σ is s. But here we redefine s to fit the ordinary definition:
s:

ℏσ
σ
→ ,
2
2

(1.74)

which has the difference of a factor ℏ. The third term in the summation is the Zeeman term mentioned in Sec. 1.4. g iscalled g-factor, which is just t within the approximation of eq. (1.72) In the quantum electrodynamics, this is a bit larger
than 2, due to the effect of electromagnetic field. As we will see later, in some cases we need to include orbital angularmomentum into the form of g-factor (Landé g-factor). Here we do not include the spin-orbit interaction.
Just as before, we take symmetric gauge A(rn ) = (B × rn )/2 to obtain]∑ [ p2nH=+ U (rn ) + V (r1 , r2 , · · · )
2m
n

H0

(1.75a)(ln + gsn ) · B

H1(1.75b)e2 ∑ 2 2{r B − (B · rn )2 }8m n nH2 ,(1.75c)

+ µB

∑
n

+

where we write the orbital angular momentum in unit of ℏ as
ℏln ≡ rn × pn .
E02-2

(1.76)

And H0 , H1 , H2 are the terms of order zero-th, 1st, 2nd order of B respectively.
There are commutation relations between the coordinate and momentum operators:[rnα , pnβ ] = rnα pnβ − pnβ rnα = iℏδαβ(α, β = x, y, z).

(1.77)

And from the redifinition of eq. (1.74),[snα , snβ ] = isnγ(α, β, γ = x, y, z (cyclic)).

(1.78)(α, β, γ = x, y, z (cyclic)).

(1.79)The same for the orbital angular momentum as[lnα , lnβ ] = ilnγ
The magnetic moment is given as in eq. (1.35) byµ=−∑e2 ∑ 2∂H= −µB(ln + gsn ) −{r B − rn (rn · B)}∂B
4m n nn= −µB∑(ln + gsn ) −ne2 ∑(rn × (B × rn )}.
4m n(1.80)

In comparison with (1.35), naturally the term sn is the difference in this mathematical form.
In a naive thought, the relativistic quantum mechanics brings in the spin-angular momentum and magnetic moment,
and that leads to the explanation of the magnetism. It is true that the spin magnetic moment plays an important role but
the above is too naive of course. In quantum theory, the quantization in orbital angular moments, spatial distribution ofparticles, and also statistics of particles give big differences both in paramagnetic and diamagnetic terms, which lead tothe appearance of magnetic moment. What we have seen are the minimum knowledges to consider the magnetism. In thenext chapter let us first have a look on the magnetism of atoms and ions, in which the quantization gives crutial effects.
I have a little comment on the spin of particles, that is the nuclear magnetism. Representative nucleons are protonand neutron, which both are known to have spin 1/2 (the histry of finding these facts is described in Ref. [1]). Though
the concept of spin was naturally introduced by the Dirac equation, proton and neutron belong to hadron and are notdescribed by the Dirac equation. Actually the g-factors are very different from 2 that is the value in the Dirac equation.
Proton has a charge of +e and the magnetic moment can be naturally understood. On the other hand, neutron does nothave total charge and still has a magnetic moment. This is due to the inner structure, that is the inner charge distribution.
The spins of neucleons have opened up a huge field of magnetic detection of various phenomena. A representative ismagnetic resonance imaging (MRI), which is now an important inspection device for the protection of lives. And theneutron diffraction is widely used to explore magnetic structures taking advantage of electrical neutrality.

E02-3

Chapter 2

Magnetism of Localized Electrons

The goal of elementary quantum mechanics is the derivation of hydrogen atomic spectrum, that is the quantum confinement by spherical Coulomb potential. But the goal is at the same as the starting point of atomic/molecular spectroscopy
as well as magnetism. The problem of localized electrons as the basics of magnetism is, in other words, the many-bodyproblem in localized potentials. The subject of this chapter is how the magnetic moments arise from many-body systems.

2.1 Localized many-body problem
Let us consider the magnetism in localized electron systems like atoms and ions. When they are placed inside crystals,
we need to consider the effect of surrounding lattices. Then we write the hamiltonian as
HL = HL0 + HC + HSOI + HCF ,(2.1)

where HL0 is an ordinary spherical potential hamiltonian, HC is the Coulomb interaction among electrons, HSOI is
the SOI, and HCF is the crystal field caused by surrounding lattices. The amplitude of HCF greatly depends on thesituation. When it is large, sometimes we need to go back to the single-electron problem in the potential under the effectof ligands[2]．There is no that term in the case of gases.

2.1.1 Problem of spherical potential
HL0 is the sum of kinetic energy and the spherical potential Vsp (r) created by nucleus as
HL0 =

∑
j

[

]
p2j
+ Vsp (rj ) ,
2m

E02-4

(2.2)

which has basically the same form as hydrogen atom problem. Then the energy eigenvalue is indexed with (main quantum
number, azimuthal quantum number, magnetic quantum number)=(n, l, m). A specific form of wavefunction in the polar
coordinate representation isψnlm (r) = Rnl (r)Ylm (θ, φ),(2.3)

where Rnl (r) is a radial wavefunction, Ylm (θ, φ) is the spherical harmonic function. In the case that Vsp (r) is the
Coulomb potential, which is proportional to r−1 ,Rnl (r) = bnl ρl e−ρ/2 L2l+1
n+1 (ρ),

ρ≡

2 r
,
n a0

(2.4)

where bnl is the normalization constant, L2l+1
n+1 , an associated Laguerre polynomial, is a polynomial of n + l − (2l + 1) =n−l−1-th order, a0 is a quantity of length dimension, the Bohr radius in the case of Hydrogen atom[3]. The eigenenergies
are degenerated for l, m and written as the function of main quantum number n as
ϵnl = −

R∞
,
n2

R∞ =

me4
.
8ϵ0 h3 c(2.5)

In the above, the spin-degree of freedom is dropped. Hence here, as in many non-relativistic approximation, we take thespin degree of freedom σ into account by taking the direct product. HL0 can be written in the form of second quantization
as
HL0 =

∑

ϵnl

nl

∑ †
anlmσ anlmσ .

(2.6)

mσ

2.1.2 Larmor precession
To see the effect of quantization on the magnetic moment caused by the orbital angular moment, we consider the
Coulomb potential of atomic number Z,
Vsp (rj ) = −Ze2 1.
4πϵ0 rj(2.7)

The total orbital angular momentum ℏL is written as the sum of each orbital contributions:ℏL = ℏ

∑

li .

(2.8)

i

To concentrate ourselves on the effect of orbitals, we tentatively drop the spin (s) term and write the term in (1.75b) as
H1 = µB L · B = µB Lz B.

(2.9)

B is taken to be parallel to z-axis. Lz is azimuthally quantized by
Lz = M : −L, −L + 1, · · · , L − 1, L.
The electron energy in the magnetic field is
E = E0 + µB M B ≡ E0 + ℏωL M,

ωL ≡

µB B
eB
=
(Larmor frequency),ℏ2m(2.10)

where E0 is the value for zero magnetic field.
We will have a look at a motion of angular momentum L in magnetic field B along z-axis. Heisenberg equation of
motion for L is given by
1dL= [L, H0 + H1 + H2 ],dtiℏ(2.11)

with the hamiltonian in eq. (1.75). In the hamiltonian, the term H0 in zero-th order of B is HL0 in eq. (2.2) in thisproblem. The terms of higher order in B than 2 are ignored here and we drop H2 .
E02-5

B

L

Fig. 2.1 Schematics of Larmor precession. The electron angular momentum L precesses around the axis of magnetic field B with angularfrequency ωL .

From the axial symmetry of magnetic field B and the spherical symmetry of potential, Lz is a constant of motion,
which commutes with the hamiltonian. For the residual Lx , Ly , fromdLy
= ωL Lxdt(2.12)Ly (t) = L0 sin(ωL t + θ0 ),(2.13)dLx= −ωL Ly ,dt
we obtain
Lx (t) = L0 cos(ωL t + θ0 ),which means Lx , Ly undergo a circular motion with angular frequency ωL , and the total momentum precesses as illustrated in Fig. 2.1. This is called Larmor precession.
This is generalized into Larmor theorem[4], which tells that the motion of electronic systems in magnetic field B is
the same as that in the zero-magnetic field if we see that on the coordinate system rotating with angular frequency ωLunder the condition (i) the central force potential sits still, (ii) the axial symmetry around the magnetic field B, (iii) the
phenomenon up to the first order of magnetic field.
The above is for the orbital angular momentum. In the case of spin angular momentum, the expression of the magnetic
moment has g-factor, then the Larmor frequency has the g-factor asωL = g

eB
eB
≈
.
2m
m

(2.14)

Further, the expression ωL is also used for nuclear magnetic moment etc. The coefficient of B for ωL is often written asγ (that is ωL = γB), where γ is called gyromagnetic ratio. These are the basis of magnetic resonance experiment, andwe revisit the magnetic resonance experiments afterwords.

2.2 Magnetism of inert gas atoms and closed shell ion cores
At the beginning of chapter 1, I wrote that what we feel the magnetism is from the permanent magnet. In comparison
with that, it may be a bit hard to feel that the gases also have the magnetism. However, if the isolated atoms or molecules
have the magnetism, the gases of those particles naturally have magnetic moments. An example of important role playedby the gas magnetism, is the birth of stars, in which the interstellar magnetic field (∼ 10−10 T) is said to work for the
integration of gases as well as the gravitational force*1 . Another example is the trapping and cooling of laser cooled gases*1 However, according to recent observation by Alma telescope, it is deduced that the effect of magnetic field may be weaker than that ofgravitation[5]

E02-6

Z

Element

Susceptibility

2He−1.9 × 10−6

10Ne

−7.2 × 10−6

Tab. 2.1 Measured mole magnetic susceptibility of inert gases.

−6

18

Ar

−19.4 × 10

36

Kr

−28 × 10−6

54Xe

−43 × 10−6of neutral atoms. By adding a spatial distribution to the Zeeman split due to the magnetic moment by the magnetic field
gradient, atoms with the magnetic moment can be collected near one point in space (magnetic trap). Atoms with higher
kinetic energies tends to distribute positions far from the center of trap and the radio frequency wave rotates spins aroundthere and selectively expels out the high energy atoms, which leads to the cooling (evaporation cooling).
We consider the case, in which the orbitals (n, l) are important. The orbital angular momentum L, the spin angularmomentum S, and the total angular momentum J are written as
∑
∑∑
L=
lj =
⟨m|l|m′ ⟩nl a†mσ am′ σ ,
S=

∑
j

J=

∑

(2.15a)

σ mm′

j

sj =

∑ ∑ (σ)
m

σσ ′

2

σσ ′

a†mσ amσ′ ,jj = L + S.

(2.15b)(2.15c)j

In inert gases like He, Ne, Ar, or in ion cores like Ag+ , which have closed shell structures, the total orbital and spin
angular momentums are zero (L = S = 0) due to the cancelling between electrons. Then the paramagnetic term is zeroin these cases. This is, of course, an important information and this leads to the survival of the diamagnetic term (secondterm) in eq. (1.80). The magnetic moment due to this term isµdia = −

e∑
µB ∑
e2 ∑
[rn × (B × rn )] = −[rn × (ωL × rn )] = −[rn × (mvn )],
4m n
2 n
ℏ n

(2.16)

where ωL is a vector of the Larmor frequency produced with replacing B in eq. (2.10) with the vector B. Equation (2.16)
indicates that the diamagnetism arises from the outer product of momentum and coordinate created by the Larmor precession, that is, the angular momentum of the Larmor precession. Hence we call the diamagnetism as Larmor diamagnetism. We have calculated classically the magnetic moment induced by the magnetic field in zero-field and without
angular moment (L = 0) in Sec. 1.3.2. The quantum mechanical calculation of eq. (2.16) gives the same results asµd = −e2
e2 2⟨x2 + y 2 ⟩ B = −⟨r ⟩ B,4m
6m(2.17)

where we have used the fact that ⟨x2 + y 2 ⟩ = (2/3) ⟨r2 ⟩ in the radial wavefunction in eq. (2.4). The molar susceptibilityof inert gases of atomic number Z is thenχ=−NA Ze2 ⟨r2 ⟩.
6m(2.18)

This leads to the estimation of radius of atoms (ions) from the susceptibility measurement[6]. The experimental values
in Tab. 2.1 tells that⟨r2 ⟩ ∼ a2B .

(2.19)

2.3 Electronic state of magnetic ions
Next we proceed the most important example of the appearance of paramagnetic term, which is zero in inert gases.
That is, the magnetic ions, which have open shells and really big effects on the magnetism like ferromagnetism, etc.
E02-7

2.3.1 Ground multiplex – Hund’s rule –The orbital with angular momentum l can take the eigenvalue m of z-component as m = −l, −l + 1, · · · , l and withthe spin degree of freedom, the number of possible states is 2(2l + 1). The many-body state accommodating multiple
electrons can be indexed with L and S in eq. (2.15), which can be written as (L, S).
These are degenerated for HL0 in eq. (2.1) though the Coulomb interaction HC causes splitting in (L, S) state. Eachof them is degenerated by (2L + 1)(2S + 1) folds. These are called LS multiplex.
To consider which state is the ground state, we represent the Coulomb interaction as
⟨
⟩
1 ∑ ∑
e2
†
HC =
m1 m2
m3 m4 ama†aa,
1 σ1 m2 σ2 m3 σ3 m4 σ42 m ,··· ,m σ σ
4πϵ0 r1

4

1(2.20)

2

where l is not written. The bracket term in the coordinate representation is
⟨
⟩ ∫
e2
e2
m1 m2
um (r2 )um4 (r1 ).
m3 m4 = dr1 dr2 u∗m1 (r1 )u∗m2 (r2 )
4πϵ0 r4πϵ0 |r1 − r2 | 3(2.21)The number of this combination of (m1 , · · · , m4 ) is 54 ＝ 625 for d-orbital and much more for f -orbital. Then we needto consider the contribution of dominant terms. From eq. (2.21), the term of m1 = m2 = m3 = m4 is the largest and
⟨
⟩
∑
e2
m1 m1
m1 m1 a†m1 ↑ a†m1 ↓ am1 ↑ am1 ↓ = U0n̂m↑ n̂m↓ (n̂mσ = a†mσ amσ ).
(2.22)
4πϵ0 rm
In the rhs, the spin state is (↑, ↓) due to the Fermi statistics (the Pauli principle) on the spin-term in eq. (2.20). In consequence, the above represents the effect of the Coulomb repulsion between the electrons with spin (↑↓) accommodated in
the same orbital.
The next is the term of m1 = m4 ̸= m2 = m3 , given as
1 ∑U (m1 , m2 )n̂m1 n̂m2
2

(
n̂m =

m1 ̸=m2

∑

)
nmσ

.

(2.23)σThis represents the Coulomb repulsion between the electrons in different orbitals.
And the next is the case of m1 = m3 ̸= m2 = m4 . The contribution is written as
(
)
1
1 ∑ ∑
1 ∑
†
†
J(m1 , m2 )am1 σ1 am2 σ2 am1 σ2 am2 σ1 = −J(m1 , m2 )n̂m n̂m + 2sm1 · sm2 .
2
2
2 1 2
σ σ
m1 ̸=m21

m1 ̸=m2

2

Ion

Configuration

LS

3d2 4s2

Ti3+ , V4+

3d1

2

1/2

V

3

2

3d 4s

V

3+

3d

2

3

1

Cr

5

1

3d 4s

3+

3d

3

3

3/2

Mn

5

2

3d 4s

3d

4

2

2

Fe

6

2

3d 4s

3d

5

0

5/2

Co

7

2

3d 4s

3d

6

2

2

Ni

8

2

3d 4s

3d

7

3

3/2

Cu

10

1

3d 4s

3d

8

3

1

Zn

10

2

3d

9

2

1/2

Element

Configuration

Sc

1

3d 4s

Ti

2

3d 4s

Cr

,V

3+

, Cr

2+

, Mn

2+

Mn

3+

Fe

Co

2+

3+

, Fe
2+

Co

2+

Ni

2+

Cu

2+

E02-8

Tab. 2.2 Electronic configurations of 3dmetal ions, ground state
L and S derived from Hund’s rule.

(2.24)

Here, sm is the spin operator of orbital m, which is already defined in eq. (2.15b). We here write down it again as∑ (σ )
sm =a†mσ1 amσ2 .
(2.25)
2
σ1 σ2
σ σ
1

2

In eq. (2.24), the term 2sm1 · sm2 appears. At the first sight this is a bit strange. However, if we write the matrix
representation for two-electron spin space (2 × 2=4 dimensional), the lhs and the rhs are the same. Which means thisrepresentation is possible.
The matrix element J(m1 , m2 ) is called exchange integral. By using Fourier transformation of Coulomb potential,
we obtain∫e2
um (r2 )u∗m2 (r2 )
4πϵ0 |r1 − r2 | 1
[∫
]
∫
e2
= dr1 dr2 u∗m1 (r1 )um2
dq 2 eiq·(r1 −r2 ) um1 (r2 )u∗m2 (r2 )
ϵ0 q
∫
∫
2
2
e
= dq 2
dr1 u∗m1 (r1 )um2 (r1 )eiq·r1 > 0.
ϵ0 qJ(m1 , m2 ) =dr1 dr2 u∗m1 (r1 )um2

(2.26a)(2.26b)(2.26c)That is J(m1 , m2 ) is always positive. Hence the two spins in two orbitals tend to be parallel. Most naive explanation isthat the electrons in the same state try to avoid each other and as a result the Coulomb energy is lowered.
From the above, the ground multiplex can be determined from the following Hund’s rule.

Hund’s rule

1. Multiplex with maximum S
2. Multiplex with maximum L among those have maximum SThe LS multiplexes in Tab. 2.2 are determined from the Hund’s rule.

Appendix 2A: Second quantization method
We hve short review of second quantzation method, which will be frequently used. For the knowledges of the physicalmeanings, the mathematical proofs, you are recommended to refer to famous textbooks like Ref. [7, 8]. The explanationof the physical meaning in Ref. [1] also has good reputation.

2A.1 Creation and annihilation operators
In number representation of a homogeneous multi-particle system, a many-body state is represented by the numbersof particles {nj }, where j = 1, 2, · · · are the indices of single particle states. Let n be the vector of elements {nj } andwe write the state in number representation as|n⟩ = |n1 , n2 , · · · ⟩.

(2A.1)

When every single-particle state is empty, we call the many-particle state “vacuum” and write it as |0⟩. We consider
an operator that add a particle to the j-th state and call it a creation operator. Writing a creation operator as a†j , thedefinition is given bya†j |0⟩ = |1j ⟩.

(2A.2)

aj , the hermitian conjugate of a†j , decreases the number of particles at j-th state by one. The operation of aj on thevacuum produces zero. aj is called an annihilation operator.

E02-9

Henceforth we need to specify the statistics of homogeneous particles. In the case of fermions, a single state canonly be occupied with a single particle and the number cannot be increased with repetitive operation of a†j . Namely,(a†j )2 = (aj )2 = 0. Similar thought gives aj a†j |0⟩ = |0⟩, a†j aj |1j ⟩ = |1j ⟩. In consequence, the creation and annihilation
operators of fermionic particles satisfy the following anti-commutation relations.
[ai , aj ]+ = [a†i , a†i ]+ = 0,

[ai , a†j ]+ = δij ,(2A.3)

where [A, B]+ = AB + BA. Then applying the operator n̂j ≡ a†j aj on the state |n⟩ givesn̂j |n⟩ = nj |n⟩.

(2A.4)

This is understood by considering the cases of nj = 0, 1. This tells that n̂j is a number operator that has the eigenstatesof particle numbers in j-th state.
For bosons, creation and annihilation operators b†j , bj satisfy the following commutation relation[bi , bj ] = [b†i , b†i ] = 0,[bi , b†j ] = δij .

(2A.5)

bj |0⟩ = 0 is the same as fermions. Also the same for number operators.
n̂j |n⟩ = b†j bj |n⟩ = nj |n⟩.

(2A.6)

We can obtain the expression of |nj ⟩ by the creation operator and the vacuum as1|nj ⟩ = √ (a†j )nj |0⟩nj !

(2A.7)

after some algebra to calculate the normalization constant.

2A.2 Expression of operators
In the expression of second quantization, the particle statistics can be represented by the (anti-)commutation relations
between creation and annihilation operators. Hence for human, the second quantization expression is generally simpler
than the direct treatment of many-body wavefunctions like the Slater determinant. Thus in the discussion of many-bodyproblem, the second quantization is mostly used.
We consider the simplest case, in which a many-body operator is actually written as a sum of single-particle operators.
That can be expressed in the coordinate representation as
F(r1 , r2 , · · · ) =∑f (ri ).

(2A.8)

i

For fermions, by letting the Slater determinant be |ψ1,2,··· ⟩, a matrix element is expressed as⟨ψm1 ,m2 ,··· |F|ψn1 ,n2 ,··· ⟩ =∑⟨ψm1 ,m2 ,··· |f (ri )|ψn1 ,n2 ,··· ⟩,(2A.9)iwhich contains many complicated terms.
In second quantization, defining an operator
F =∑⟨m|f |n⟩ a†m an

mn∫

drϕ∗m (r)f (r)ψn (r),(2A.10)⟨ψm1 ,m2 ,··· |F|ψn1 ,n2 ,··· ⟩ = ⟨m|F |n⟩ ,(2A.11)⟨m|f |n⟩ =we can write the matrix element as

E02-10where |m⟩ are the number representation of corresponding states. The matrix elements and the operators representing
the statistics can be treated separately. This gives a better view of the calculation.
For multiparticle operators the same approach is possible. For example in the case of a two-particle operator g(r1 , r2 )we can writeG=

1 ∑⟨kl|g|mn⟩ a†k a†l an am .
2

(2A.12)klmn

References[1] Shinichiro Tomonaga. The Story of Spin. University of Chicago Press, 10 1998.
[2] Hans L. Schlafer and Gunter Gliemann. Basic Principles of Ligand Field Theory. John Wiley & Sons Ltd, 1 1969.
[3] Walter Greiner. Quantum Mechanics: An Introduction. Springer, 2 2009.
[4] Leon Brillouin. A theorem of larmor and its importance for electrons in magnetic fields. Phys. Rev., Vol. 67, pp.
260–266, Apr 1945.
[5] Patricio Sanhueza, Josep Miquel Girart, Marco Padovani, Daniele Galli, Charles L. H. Hull, Qizhou Zhang, Paulo
Cortes, Ian W. Stephens, Manuel Fernández-López, James M. Jackson, Pau Frau, Patrick M. Koch, Benjamin Wu,
Luis A. Zapata, Fernando Olguin, Xing Lu, Andrea Silva, Ya-Wen Tang, Takeshi Sakai, Andrés E. Guzmán, Ken’ichi
Tatematsu, Fumitaka Nakamura, and Huei-Ru Vivien Chen. Gravity-driven magnetic field at ∼1000 au scales inhigh-mass star formation. The Astrophysical Journal Letters, Vol. 915, No. 1, p. L10, June 2021.
[6] J. H. Van Vlek. The Theory Of Electric And Magnetic Susceptibilities. 08 2015.
[7] Richard P. Feynman. Statistical Mechanics: A Set Of Lectures (Frontiers in Physics). CRC Press, 8 2019.
[8] L. D. Landau. Quantum Mechanics (Non-Relativistic Theory) Course of Theoretical Physics , Volume 3, Third
Edition. Butterworth-Heinemann, 1 1981.

E02-11

Lecture note on
20th April (2022)

Magnetism (3)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

2.4 LS coupling
We explained Hund’s rule through estimation of Coulomb mutual interaction. In addition to these two terms (H0 , HC ),the hamiltonian (2.1) for localized electron system in solids, has HSOI and HCF . It is difficult to treat these simultaneously
and we need to treat the term with higher priority and to treat the other as perturbation. First, let us examine the casethe effect of HSOI is stronger than that of HCF . Within this framework, there are several approaches: (i) first the groundstate LS multiplex is picked and examine how that is split into sub-levels with HSOI (LS coupling approach); (ii) Singleelectron state with spin-orbit interaction is prepared firstly, then the Coulomb effect is taken into account via Hund’srule to find the ground state (j-j coupling). This and the next sections are devoted to these two approaches illustrated inFig. 2.2.

2.4.1 Effect of spin-orbit interaction on single-electron states
Here we revisit the single electron Hamiltonian for relativistic electron. The third term in eq. (1.72) is
−

e2 ℏ
e2 ℏ
eℏσ · p × E
=
−
σ
·
(p
×
∇V
)
=
ζ(r)s · l ≡ ξ(r)l · s,
4m2 c24m2 c22m2 c2(2.27)

where V (r) is a potential with spherical symmetry. We first write it as V (r) taking the origin at the center of the potential,
then define ζ(r) as ∇V = (r/r)(dV (r)/dr) ≡ ζ(r)r, and apply l = r × p for the above. Assuming it as the Coulomb
potential V (r) = −Ze2 /(4πϵ0 r), ξ is given byξ(r) =

Ze2
1,
2
2
2m c (4πϵ0 )r3(2.28)

which tells that the spin-orbit interaction is important for the atoms with large atomic number Z and for the atomic
orbitals with smaller r, namely the orbitals closer to the atomic core. As examples of representative atoms, ions that have

Fig. 2.2 Illustration of two representative approaches. In LS coupling (Russell-Saunders coupling) approach, theground state LS multiplex split by the Coulomb interaction is further split into the levels with J as the quantum
number by the spin-orbit interaction. In j-j coupling approach, spin-orbit interaction is considered at the singleelectronic states, then the Coulomb interaction in multi-electron states is taken into account through Hund’s rule.

E02-1

open shells, we have listed 3d transition metals and Lanthanoid with 4f open shells. The above thought suggests that theapproaches of “SOI-first” (both LS and j-j) are more appropriate for Lanthanoid than 3d transition metals.
The single-electron Hamiltonian with the SOI is then given by
H = H0 + Hso =p2+ V (r) + ξ(r)l · s.
2m

(2.29)The Hamiltonian (2.27) does not commute with l, s, which are no longer
constants of motion. This can easily be confirmed from (1.78) and (1.79). Onjthe other hand, commutation relations, e.g,l[l · s, ˆlz ] = iℏ(−ly sx + lx sy ),[l · s, ŝz ] = iℏ(−lx sy + ly sx ) = −[l · s, ˆlz ],tell that the total angular momentum
j =l+ss(2.30)

commutes with Hamiltonian (2.27), and is a constant of motion. j satisfies theFig. 2.3 Illustrates precessions ofl and s around j.

commutation relations in the same forms as (1.78) and (1.79). Hence it gets
directional quantization, and the eigenfunction is indexed by (j, m) as |j, m⟩(m = −j, −j + 1, · · · , j). |j, m⟩ can be obtained, e.g., in the form of expansion

with the eigenfunctions of H0 .
On the other hand, l2 , s2 commute with l · s, thus with H, making them as constants of motion. l · s is also written asl · s = (l + s) · s − s2 = j · s − s2 .
This form tells that s and l have the Zeeman-like term with j (a constant of motion) as magnetic field in eq. (2.9). Ina classical picture, l and s precess around j satisfying eq. (2.30) as illustrated in Fig. 2.3. The angular velocity of theprecession is proportional to the spin-orbit coupling strength ξ.
The eigenvalues of l · s are obtained from2l · s = (l + s)2 − l2 − s2 = j 2 − l2 − s2
as

[
]
1
3
[j(j + 1) − l(l + 1) − s(s + 1)]/2 =j(j + 1) − l(l + 1) −.
2
4

(2.31)(2.32)

Then the energy eigenvalues are given by
ϵnlj = ϵnl +

[
]
ηnl
3
j(j + 1) − l(l + 1) −,2
4

Fig. 2.4 Schematic diagram for splitting of l = 2 (i.e. 3d orbital) multiplet state with (2l + 1)(2s + 1) =10 fold degeneracy
by the SOI. η3d is the integrated value of eq. (2.34) for 3d-orbital.

E02-2(2.33)

Fig. 2.5 Schematic diagrams of how electrons arepacked according to Hund’s rules into a degenerate state with the same angular momentum quantum
number. (a) The number of electrons is less than or
the same as the number of orbitals. (b) The number of electrons is more than that of orbitals. The
sum of the orbital contributions corresponding to thespins represented by the arrows in the upper column
is zero.

(a)(b)

where ηnl is the integral of the radial wavefunction and ξ(r), namely
∫ ∞
ηnl =
ξ(r)Rnl (r)2 r2 dt.

(2.34)

0j can take values |l ± 1/2|, splitting the energy level indicated by (n, l) into two levels. Figure 2.4 shows the case of 3dorbital.

2.4.2 Spin-orbit interaction in multi-electron states
The SOI Hamiltonian for a multi-electron system, then can be written as∑∑∑
HSOI =
ξ(ri )li · si →ξi li · si → ξ
li · si .
ii(2.35)i

In the above we first replace ξ(ri ) → ξi because the radial part will be integrated out as in eq. (2.34). Next ξi comesout from the summation on i because in an LS-multiplet, the orbitals should have the same radial part. The Coulomb
repulsion splits the degenerated levels in the single-electron problem into LS-multiplets. In the previous section, Hund’srule has been derived as a way to find the state in which the effect of Coulomb repulsion is minimized. An LS-multiplet(L, S) still has (2L + 1)(2S + 1)-fold degeneracy due to the freedom of orbital and spin angular momenta.
In the presence of HSOI , just as in the single-electron problem, neither the total orbital angular momentum L nor the
total spin angular momentum S (defined in (2.15)) commutes with the Hamiltonian, that is, they are not constants ofmotion. The total angular momentum defined as
J =L+S(2.36)

is a constant of motion.
The whole energy level structure is importtant in the discusstion of localized electron systems. For the magnetism,
the ground state is particulary important. As noted, in the LS-coupling approach, we first consider the “ground state”
LS-multiplet, and examine how this is split by HSOI , find the ground state among them. When the number of electons n
is smaller than that of degenerated orbitals 2l + 1 for the orbital angular momentum l, in the ground state every orbital
accomodates 0 or 1 electron. Then according to Hund’s rule, in the ground state all electron spins are in parallel. Namelya spin dose not depend on orbital index i:
si =

1
1
S=
S
n
2S

(n ≤ 2l + 1).

(2.37)

Substituting the above into eq. (2.35), we can write
HSOI = ξ

∑
i

(
)
∑
ξ
li · si = ξ
li · s =L · S ≡ λL · S.
2Si(2.38)In the case of n > 2l + 1, as shown in Fig. 2.5(b), though the electron spins are still in parallel for 2l + 1 electrons, thecorresponding orbitals take all possible values of ml , and the summation over li vanishes in (2.38). Hence the residual
E02-3

Elements

Electronic

Electronic

Configuration

Configuration(Lanthanoid)

atom R

ion R

3+

2

5d6s

La

Ground state
L
0

2

Ce

4f 5d6s

Pr

3

2

4f 6s

Nd

4

2

4f 6s

Pm

5

2

4f 6s

Sm

6

2

4f 6s

Eu

7

2

4f 6s

Gd

7

4f 5d6s

Tb

9

2

4f 6s

Dy

10

2

4f 6s

Ho

11

2

4f 6s

Er

12

2

4f 6s

Tm

13

2

4f 6s

Yb

14

2

4f 6s

Lu

14

2

2

4f 5d6s

4f

1

4f

2

4f

3

4f

4

4f

5

4f

6

4f

7

4f

8

4f

9

4f

10

4f

11

4f

12

4f

13

4f

14

3
5
6
6
5
3
0
3
5
6
6
5
3
0

S
0
1/2
1
3/2
2
5/2
3
7/2
3
5/2
2
3/2
1
1/2
0

J

multiplet
1

0

gJ

S0

0

5/2

2

F5/2

6/7

4

3

H4

4/5

9/2

4

I9/2

8/11

5

I4

1/5

H5/2

2/7

0

7

F0

0

7/2

8

S7/2

2

6

7

F6

3/2

H15/2

4/3

5

I8

5/4

I15/2

6/5

6

3

H6

7/6

7/2

2

F7/2

8/7

1

0

4
5/2

15/2

6

6

8
15/2

0

4

S0

Tab. 2.3 The electron configuration of the lanthanoid ion and the basis multiplex of the ion. Spectroscopic symbols
are also listed for the ground state. A spectroscopic symbol 2S+1 LJ expresses a multielectron state indexed by(L, S.J). 2S + 1 and J are given in Arabic numbers, and L is given from the correspondence 0, 1, 2, 3, · · · withsymbols S, P, D, F, · · · .

n − (2l + 1) spins and orbitals actually contribute the SOI. Those effective spins are in anti-parallel states written as −s,where s is defined in eq. (2.37). The above discussion is summarized in the following form *1 .
[(2l+1 )
( n
) ]
∑
∑
ξ
HSOI = ξli · s −li · s = − L · S = −λL · S.
2S
i=1

(2.39)

i=2l+2In the above we have found that the SOI on the ground state LS-multiplet found from Hund’s rule can be expressed in
the same form as that on a single-electron state by using (L, S, J ). Then as in the case of single electron problem, thesplit states of the ground LS-multiplet with (2L + 1)(2S + 1)-fold degeneracy can be indexed by J, the eigenvalue of J .
Possible value of J are from the definition,
J = |L − S|, |L − S| + 1, · · · , L + S.

(2.40)The expectation value of L · S is obtained just as in (2.31) and (2.32). The result is
L·S =

1 2
1
(J − L2 − S 2 ) = [J(J + 1) − L(L + 1) − S(S + 1)].
2
2

(2.41)

Then from eq. (2.38) and eq. (2.39), the ground state is the state for J = |L − S| in the case of n ≤ 2l + 1, and that is thestate for J = |L − S| in the case of n > 2l + 1.
The above is applied to obtain the multiplet ground states thus obtained for lanthanoid are listed in Tab. 2.3. These
atoms often constitute (compound) ionic insulators. In many cases, 5d, 6s electrons on the outer shells and also a 4f
electron are emitted. In such ions, the outmost shell composed of 5s, 5p electrons is closed just as Xe while 4f electrons
on the open shell still exist inside the outmost shell. This situation is advantageous for localized electron system in*1 Such simplification of operator form can be generalized into the method of “operator equivalent” by Stevens[1]．

E02-4spherical potential because the effect of HCF is comparatively weak[2]. The SOI has strong effect and the LS-coupling
approach is a good approximation in most cases with some exceptions[3].
The eigenstates with quantum numbers J and M , composed of (L, S)-multiplet is formally written as|J, M ⟩ =∑⟨L, Ml ; S, Ms |J, M ⟩ |L, Ml ; S, Ms ⟩,(2.42)

Ml Ms

where ⟨L, Ml ; S, Ms |J, M ⟩ are Clebsch-Gordan or Wigner coefficients.
So far we have assumed the second quantization formalism and estimated the matrix elements on the assumption of“independent calculation of matrix elements and particle statistics.” We can write, e.g., the second quantization representation of HSOI on the orbital (n, l), as
HSOI =∑λnl (mσ, m′ σ ′ )a†mσ am′ σ′ ,mm′ σσ ′λnl (mσ, m′ σ ′ ) ≡(2.43a)(σ )Zeﬀ e2 ℏ2 ⟨r3 ⟩′⟨m|l|m⟩·.
nl
2m2 c2 (4πϵ0 )2 σσ′(2.43b)

Here for the spherical potential, we take the effect of screening by core electrons into account by changing Z into aneffective atomic number Zeﬀ to get V (r) = −Zeﬀ e2 /(4πϵ0 r).

2.5 j-j coupling
As mentioned in the beginning of the previous section, the j-j coupling approach starts from the single-electron statesin which the SOI is already included as in Fig. 2.2. In the thought similar to the derivation of Hund’s rule in Sec. 2.3.1, theground state in which the effect of Coulomb interaction is taken into account, is obtained for the electron configurationthat maximizes the quantum number J.
Let us see the case of Pr3+ ((4f )2 ) as an example. f -electron has l = 3, then j = 3 ± 1/2 = 5/2, 7/2. The single
electron ground state is thus j = 5/2. The electron configuration with the maximum J is j = 5/2, 3/2 then Jmax = 4.
This value of J agrees with that in the LS-coupling approach (Tab. 2.3). This state in number representation is obtained
as|J, M ⟩ = |4, +4⟩ = a†+5/2 a†+3/2 |0⟩.

(2.44)

From eq. (2.42), the creation operator of j = 5/2 and jz = m state, a†jz can be represented by the creation operator a†msof states (l = 3, m, s) asa†jz =∑
m,s√⟨3, m; 1/2, s|5/2, jz ⟩ a†ms =7 + 2jz †ajz +1/2↓ −14√

7 − 2jz †ajz −↑ .
14(2.45)

The way of approximation is different for j-j coupling and LS-coupling. Hence they give different ground states.
Although according to the report ref. [4], they have large overlaps to each other. Care should be taken for the following.
As will be seen in the next section, in LS-coupling paramagnetic moment expressed as µ = µB (L + gS) is not parallel
to J = L + S due to the g-factor. Hence µ precesses around J and the coefficient is given as the average. In j-j couplingµ is given as the sum of electronic magnetic moments. Therefore the g-factor is the same as that of single electron.
Namely l in the discussion of LS-coupling is replaced with j, thus (2j + 1)/2(j + 1) for n ≤ 2j + 1 and (2j + 1)/2J
for n > 2j + 1.

E02-5

2.6 Paramagnetism generated by magnetic ions
As mentioned, HCF is important for magnetic ions with an open-shell structure of 3d electrons, but before proceeding
to the theoretical examination, we will apply the LS-coupling approach to paramagnetism and examine to what extent it
can explain the experiements.

2.6.1 Free local moment and Curie law
The LS-coupling approach is a tool to find the ground state of magnetic ions in crystals. Hence we examine the behaviorof magnetization in a simple model of magnetic moments free to each other.
The Hamiltonian in the first order of mgnetic field is from eq. (1.75b)
H1 = µB (L + gS) · B.

(2.46)On the other hand, in the ground multiplet state in the LS-coupling, J is the angular momentum quantum number. The
main term thus should be in the form of
H1 = gj µB J · B,

(2.47)

where the coefficient gJ includes various effects. Comparison of these two, gJ J = L + gS, J = L + S, we reachLandé g-factor
gJ =1 + g g − 1 L(L + 1) − S(S + 1)−.
2
2
J(J + 1)

(2.48)



i This gJ is called Landé g-factor. gJ is listed in Tab. 2.3 for lanthanoid. As already noted, this expression is obtained
within the narrow space of ground state in LS-coupling.
The expectation value of magnetic moment −gj µB J is
M = ⟨−gj µB Jz ⟩ = −[ ()]
Tr[gj µB Jz exp(−gj µB Jz B/kB T )]∂−gj µB Jz B
= kB Tlog Tr exp,
Tr[exp(−gj µB Jz B/kB T )]∂B
kB Twith taking the field along z-axis. The partition function is calculated as
[
) ]
(
1
1
(
) sinhgJ µB J +B
−gj µB Jz B
2kB T
2Tr exp=.
kB Tsinh(gJ µB B/2kB T )From the above, the magnetization per a single ion M is obtained as()
gJ µB JB
M = gJ µB JBJ,
kB T(2.49)(2.50)(2.51)

where BJ (x) is called Brillouin function given by
Brillouin function
BJ (x) =

2J + 12J + 1
1
x
coth
x−
coth
.
2J
2J
2J
2J

(2.52)
The functional behavior is drawn in Fig. 2.6.
In the case of x ≪ 1 (weak field, high temperature), the Brillouin function can be approximated as BJ (x) ∼ (J +1)x/3J, then we reach the Curie law:χ=dM
J(J + 1)= (gJ µB )2
,
dB
3kB T

which is also obtained in the classical theory.
E02-6(2.53)

Fig. 2.6 Brillouin function(2.52) for
J = 1/2, 3/2, ∞.

2.6.2 Comparison with experiments: magnetization curve
Figure 2.7 shows well-known experimental results, in which the magnetizations of alums (sulfate) with various magnetic ions are successfully fit to the Brillouin function. A systematic response to expected values of J is observed.

Fig. 2.7 Fittings of the Brillouin function to the magnetic field and temperature dependencies of magnetization ofvarious paramagnetic salt (sulfates (alum) with various magnetic ions). Left: Cr (J = 3/2). Annotation of ”Langevin”
means the Langevin function corresponding to J = ∞ in the Brillouin function. Right: The same experiments for Cr(J = 3/2), Fe (J = 5/2), Gd (J = 7/2). From [5].

E02-7

Configurationionp (exp.)gJ [J(J + 1)]1/2

2[S(S + 1)]1/2

3d1

V4+

1.8

1.55

1.73

V

3+

2.8

1.63

2.83

V

2+

3.8

0.77

3.87

3+

3.7

0.77

3.87

4+

4.0

0.77

3.87

2+

4.8

0

4.90

3+

5.0

0

4.90

2+

5.9

5.92

5.92

Fe

3+

5.9

5.92

5.92

Fe

2+

5.4

6.7

4.90

2+

4.8

6.63

3.87

2+

3.2

5.59

2.83

2+

1.9

3.55

1.73

Configurationionp (exp.)gJ [J(J + 1)]1/2

2[S(S + 1)]1/2

4f 1

2

Ce3+

2.5

2.54

2.56

2

3

H4

3+

Pr

3.6

3.58

3.62

4f 3

4

I9/2

Nd3+

3.8

3.62

3.68

4f

5

6

H5/2

3+

1.5

0.84

1.53

4f

6

7

3+

3.6

0.00

3.40

4f

7

8

3+

7.9

7.94

7.94

4f

8

7

3+

9.7

9.72

9.7

4f

9

3+

10.5

10.65

10.6

3+

10.5

10.61

10.6

3+

9.4

9.58

9.6

3+

7.2

7.56

7.6

3+

4.5

4.54

4.5

2

D3/2

2

3

3d

3

3d

4

F2

F3/2

Cr

Mn
4

5

3d

D0

Cr

Mn
5

3d

6

S5/2

6

3d

5

7

4

3d

8

3d

4f

F9/2
3

3d

9

D4

2

F4

D5/2

6

F5/2

F0

S7/2
F0

H15/2

4f

10

5

4f

11

4f

12

3

4f

13

2

4

I8

I15/2
H6

F7/2

Mn

Co
Ni

Cu

Sm
Eu

Gd
Tb

Dy
Ho
Er

Tm
Yb

Tab. 2.4 Comparison of effective Bohrmagneton number p (eq. (2.55)) obtainedin experiments, that given by the LScoupling approach (2.54), and that by“spin-only” model for 3d transition metal
ions. The data are taken from [6], [7].

Tab.
2.5 Comparison of effective
Bohr magneton number p (eq. (2.55)) obtained in experiments, that given by the
LS-coupling approach (2.54), and thatby “spin-only” model for 4f lanthanoid.
The data are taken from [6], [7].

2.6.3 Effective Bohr magneton number
Another check of the theory is the effective Bohr magneton number p defined from the inverse proportionality constant
C to temperature from the Curie law. In eq. (2.53) we write√
p = gj J(J + 1),
then p in the theory isp=(2.54)√
3kB C/µB .

(2.55)In Tab. 2.4 and in Tab. 2.5, we compare effective Bohr magneton number p (eq. (2.55)) obtained in experiments, thatgiven by the LS-coupling approach (2.54), and that by “spin-only” model for 3d transition metal ions and 4f lanthanoid
respectively.
In the case of lanthanoid in Tab. 2.5, the experimental values of p is well explained by gJ
3+

3+

√

J(J + 1) obtained fromthe LS-coupling other than Eu , Sm . On the other hand, in the case of 3d transition metals, the experiments differ so√√much from gJ J(J + 1), rather they are close to “spin-only” 2 S(S + 1). That looks as if L = 0 and the phenomenon
E02-8

Fig. 2.8 (a) Crystal structure of K2 NiF4 , in which Ni ion has an octahedron coordination. (b) Crystal structureof Cs2 CuCl4 , in which Cu ion has a tetrahedron coordination. (c) Schematic view of octahedron coordination. (d)Schematic view of tetrahedron coordination.

is called “quenching of orbital angular moment.” In 3d electron systems, the difference from the LS-coupling is so largethat it is also difficult to explain with j-j coupling.
As mentioned above, the 4f electron system is relatively inside the atomic structure even though the shell is open, theinfluence of the surroundings is small, and the approximation of the lone electron system is relatively small. Accordingly,
it is indicated that the breakdown of LS-coupling approach in 3d electron systems is considered to be due to the effect of
HCF . Therefore, next, let us consider incorporating the influence of the crystal field from the beginning.

2.7 Magnetic ions in crystal fields
Magnetic atoms (atoms of open electron shells) exist as positive ions in many insulators with strong paramagnetism.
Such ions are surrounded by negative ions. An example in Fig. 2.8(a) is K2 NiF4 , in which structure Ni ion is surrounded
by F ions in octahedron coordination. Figure 2.8(c) shows a schematic view. As annotated, the surrounding ions ormolecules are called ligand. Another example in (b) is Cs2 CuCl4 , in which structure Cu ion is surrounded in tetrahedroncoordination. These two are the representative coordination.

2.7.1 Level splitting by octahedron coordination
Let us consider the octahedron coordination in Fig. 2.8(c) in a simplest way. We take the coordinate origin at the ion,and ligand ions are at (±R, 0, 0), (0, ±R, 0), (0, 0, ±R). The vectors pointing them are written as Ri = (R, θi , φi ) andthe potential generated by the ligands is written asvc (r) =

∑
i

∑
Zi e2
Ze2
√
=
.
|r − Ri |r2 + R2 − 2Rr cos ωi
i

E02-9

(2.56)

Here we take CGSesu for the simpler form of Coulomb potential. In the polar coordinate, (θi , φi ) are(π/2, 0), (π/2, π/2), (0, 0), (π/2, π), (π/2, 3π/2), (π, 0)(2.57)

for i = 1 ∼ 6.
We assume that the averaged distance of 3d electrons from the nucleus is sufficiently shorter than R. Then vc can be
expanded with r/R asvc (r) =∞ ( )k∑ Ze2 ∑
r
i

R

R

k=0

Pk (cos ωi ),(2.58)

where Pk (cos ωi ) are Legendre functions defined asdn

1[(x2 − 1)].

(2.59)∞∑1=
Pn (x)tn .
1 − 2xt + t2
n=0(2.60)Pn (x) =2n n! dxn

Apparently these are the expansion coefficients of√From an identity of spherical harmonic function Ykm (θ, φ),Pk (cos ωi ) =

k
∑
4π
∗
Ykm (θ, φ)Ykm(θi , φi ).
2k + 1(2.61)

m=−k

Note ωi is a function of r. This results in dropping of i from the argument in the first Ykm . Here we define two functions:√√
4π Ze2 ∑4π
Tkm ≡Ykm (θi , φi ), Ckm ≡Ykm (θ, φ),(2.62)k+1
2k + 1 R
2k + 1iwith which vc (r) is written asvc (r)∞ ∑k∑rk Tkm Ckm (θ, φ).

(2.63)

k=0 m=−k

From the symmetry of positions (2.57), Tkm = 0 for odd m. Others are
√
]
(π)
2
Ze2 [
Tk0 =
+
Θ
(π)
,
Θ
(0)
+
4Θ
k0
k0
k0
2k + 1 Rk+1
2
√
(π) (
8
Ze2
mπ )
Tkm =
Θ
1
+
cos
(m: even),km
2k + 1 Rk+1
2
2

(2.64a)(2.64b)where Θ(θ)km are defined as
Ykm (θ, φ) = Θkm (θ)eimφ .

(2.65)

From the form of Θ(θ)km , Tkm ’s are also zero for odd k.
From above, we reach the expression to the fourth order of k:
[
]
√
6Ze2
2
5
4
vc (r) =+ Der C40 (θ, φ) +(C44 (θ, φ) + C4−4 (θ, φ)) ,
R
5
14
where
D=

35Ze
.
4R5

(2.66)(2.67)

Let vc b(r) be the potential without the energy shift term (the first term in eq. (2.66)). We also restore vc b(r) into theexpression in cartesian coordinate, then obtain()
3 4
vcb (r) = eD x + y + z − r .
5
4

4

E02-104(2.68)

Appendix 3A: Isolated electron systems and point group
3A.1 Definition of group
When the operation ∗ is defined between any elements of a set A and the following conditions are satisfied, we call A
a “group” with respect to ∗.
1. ∀a1 , a2 ∈ A {a1 ∗ a2 ∈ A} (closed for the operation *)2. ∀a1 . a2 , a3 ∈ A {(a1 ∗ a2 ) ∗ a3 = a1 ∗ (a2 ∗ a3 )} (associative law).
3. ∃E ∈ A {∀a1 ∈ A {a1 E = Ea1 = a1 }} (existence of unit element).
−1−1
4. ∃a−1
1 ∈ A {∀a1 ∈ A {a1 a1 = a1 a1 = E}} (existence of the inverse element)．

When elements ai , aj of a group A are written as ai = aaj a−1 , where a is also an element of A, we say ai and aj areconjugate to each other. Generally a group is classified into classes, which are sets of conjugate elements.
An element of a group ai has a corresponding square matrix D(ai ) and the operation a ∗ b = c is projected to
D(a)D(b) = D(c). We call D(ai ) as a representation of group A. There are infinite number of representations. Whena square matrix S transfers as D′ (ai ) = S −1 D(ai )S, we call D′ (ai ) an equivalent representation to D(ai ). The direct
sum of D(1) and D(2) is defined as∀ai : D(ai ) =( (1))
D (ai )
0,0
D(2) (ai )which is expressed as D = D(1) ⊕ D(2) . The above is summarized as “D is reduced to D(1) and D(2) by equivalent
conversion with S.” Expressions that cannot be further reduced are called irreducible expressions. Expression of anelement a can be written as Dij (a) = ⟨χi |a|χj ⟩ by using a basis |χi ⟩ of expression. Equivalent conversion matrix S is abasis transformation matrix. When a is an operator, the matrix representation with basis |χi ⟩ is χ-expression of a.

3A.2 Symmetry operations of point group
A set of symmetry operations around a point in space is called a point group.
E: Identical operation
Cn:

Rotation of 2π/nC2′:π rotation around two-fold axis perpendicular to the principal axis. Written as C2′ or U2 and calledI:

Umklappung.
Space inversion (r → −r)σ:

MirroringICn:

Circumference. Space inversion after rotation of 2π/n.

Sn:

Improper rotation. Mirroring after rotation of 2π/n.

(Continue to next time)

Appendix 3B: Clebsch-Gordan coefficient
Students must have learned about coupling of angular momentum in elementary quantum mechanics. But here wehave a short review. Even in classical mechanics, additive quantities should be summed up in the coupled system. In thecase of angular momentum, they should be summed up as vectors. In quantum mechanics, the operator of total angularE02-11momentum is written as the sum of angular momentum operators. However in quantum mechanics, due to the directional
quantization, eigenfunctions and eigenvalues are not the sums of each eigenfunction and eigenvalue.

3B.1 General treatment
Let ĵ1 , ĵ2 be independent angular momentums, and consider simultaneous eigenstates |ji2 , mi of operators ĵ12 , ĵiz
(i = 1, 2). We write a wavefunction of the coupled system asψ = |j12 , m1 ⟩|j22 , m2 ⟩.

(3B.1)The total angular momentum Jˆ of the coupled system isJˆ = ĵ1 + ĵ2 .

(3B.2)

However, ψ in eq. (3B.1) is not an eigenstate of, e.g., Jˆ2 apparently. From the independence of ĵ1 , ĵ2 (thus they
commute each other), and from the general properties of angular momentum, [Jˆ2 , Jˆz ], [ĵ 2 , ĵ 2 ] are apparently zero, and12

the following commutation relation can be proven.
[Jˆ2 , ĵα2 ] = [Jˆz , ĵα2 ] = 0.

(3B.3)

Simultaneous eigenstates of Jˆ2 , Jˆz , ĵ12 , ĵ22 can be obtained from basis transformation from eq. (3B.1) as
|J

2

j1
∑

, M, j12 , j22 ⟩ =

j2
∑

CjJM
|j12 , m1 ⟩|j22 , m2 ⟩,1 m 1 j2 m 2(3B.4)

m1 =−j1 m2 =−j2where CjJM
are called Clebsch–Gordan coefficients.
1 m 1 j2 m 2

3B.2 Coupling of two spins
For general way to find CjJM
, refer to textbooks. Instead we obtain them for the simplest case of j1 = j2 = 1/2,
1 m 1 j2 m 2
namely two spins. We take ŝi (i = 1, 2) for the two spin operators and write the total spin operator asŜ = ŝ1 + ŝ2 .
Because ŝ1 commutes with ŝ2 ,First, for Ŝz we seeŜ 2 = ŝ21 + 2ŝ1 · ŝ2 + ŝ22 ,(| ↑1 ⟩| ↑2 ⟩Ŝz| ↓2 ⟩| ↑2 ⟩(3B.5)

Ŝz = ŝ1z + ŝ2z .

) ()| ↑1 ⟩| ↓2 ⟩| ↑1 ⟩| ↑2 ⟩0=.
| ↓1 ⟩ ↓2 ⟩0−| ↓1 ⟩| ↓2 ⟩(3B.6)(3B.7)

Therefore the eigenvalues of Ŝz are ±1, 0 with two-fold degeneracy. The “size” of S, S is then 0 or 1. Next, generally
for the operator ŝ of spin 1/2
2

ŝ

(
1
4 0

3
= ŝ2x + ŝ2y + ŝ20 =0
1)(3B.8)

holds. In Ŝ 2 ŝ21 + ŝ22 are 3/2ŝ1 · ŝ2 ŝ1 · ŝ2 ŝ1 · ŝ2 ŝ1 · ŝ2 ŝ1 · ŝ2 ŝ1 · ŝ2 , Residual ŝ1 · ŝ2 are calculated as follows. For simplicity we use expression| ↑1 ⟩| ↓2 ⟩ = | ↑↓⟩.

(ŝ1 · ŝ2

| ↑↑⟩
| ↓↑⟩

| ↑↓⟩
| ↓↓⟩

)
=

1
4

(

.| ↑↑⟩
2| ↑↓⟩ − | ↓↑⟩E02-12)−| ↑↓⟩ + 2| ↓↑⟩.
| ↓↓⟩Because neither| ↑↓⟩ nor | ↓⟩| ↑⟩ is not eigenstates, this part should be diagonalized by transformation. The results are
simplyχ± ≡ | ↑↓⟩ ± | ↓↑⟩.

(3B.9)

This is easily confirmed asŝ1 · ŝ2 χ+ = (1/4)χ+ ,ŝ1 · ŝ2 χ− = (3/4)χ− .

(3B.10)From the above, the eigenvalues of Ŝ 2 are 0, 2, the eigenfunctions are
1|1, 0⟩ = √ (| ↑↓⟩ + | ↓↑⟩),
2
1|0, 0⟩ = √ (| ↑↓⟩ − | ↓↑⟩)
2

|1, −1⟩ = | ↓↓⟩,

|1, 1⟩ = | ↑↑⟩,.

(3B.11)

Here we use S as index, not S(S + 1). The state of S = 1 is called spin triplet state, S = 0 is called spin singlet state.
Then we obtain follows as Clebsch-Gordan coefficient in eq. (3B.4),
1−1
11= C1/2,−1/2,1/2,−1/2= 1,
C1/2,1/2,1/2,1/2

√
00
10
10
= C1/2,−1/2,1/2,1/2= C1/2,1/2,1/2,−1/2= 1/ 2,
C1/2,1/2,1/2,−1/2√
00
C1/2,−1/2,1/2,1/2= −1/ 2.

References[1] K W H Stevens. Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions.
Proceedings of the Physical Society. Section A, Vol. 65, No. 3, pp. 209–215, March 1952.
[2] Yoshichika Ōnuki. Physics of Heavy Fermions. WORLD SCIENTIFIC, September 2017.
[3] J. H. Van Vleck. The Theory Of Electric And Magnetic Susceptibilities. 08 2015.
[4] Takashi Hotta. Microscopic approach to magnetism and superconductivity of f-electron systems with filled skutterudite structure. Journal of the Physical Society of Japan, Vol. 74, No. 4, pp. 1275–1288, 2005.
[5] Warren E. Henry. Spin paramagnetism of cr+++ , fe+++ , and gd+++ at liquid helium temperatures and in strong
magnetic fields. Phys. Rev., Vol. 88, pp. 559–562, Nov 1952.
[6] Figgis. Introduction to Ligand Fields. Krieger Pub Co, 6 1966.
[7] 上村洸, 菅野暁, 田辺行人. 配位子場理論とその応用 (物理科学選書). 裳華房, 6 1969.

E02-13

Lecture note
27th April (2022)

Magnetism (4)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Let us continue the discussion of octahedral ligand filed. We begin with the cartesian coordinate expression of thepotential.

(

)
3 4
vcb (r) = eD x + y + z − r .
5
4

4

4

(2.68)

In eq. (2.1), the above potential should be HCF , which now has more priority than HSOI for present. First we consider
the single-electron problem in the central force plus octahedral potential. Since our problem is on the 3d electrons in an
open shell, we restrict ourselves to the space generated on the basis of d-orbital wavefunctions. Also it can be known
from the group theory, that will be introduced afterwards, the potential in (2.68) does not split s and p orbitals, as can beeasily guessed by considering the symmetry of px,y,z .
In order to diagonalize this potential within d-orbitals (l = 2) ϕnlm : (n, l, m) = (n, 2, m = 2, 1, 0, −1, −2), in an
orthodox way, we write down the secular equation and then obtain the eigenvalues and the eigenvectors[1]. But here, we
deduce the functional forms of diagonalization basis functions from a simpler thought[2]. The radial part is common ford-orbitals (rather for principal quantum number n). The parts of spherical harmonic functions are√5
Y20 (θ, φ) =(3 cos2 θ − 1),16π√
15cos θ sin θe±iφ ,Y2±1 (θ, φ) = ∓8π√
15Y2±2 (θ, φ) =sin2 θe±2iφ .
32π(2.69a)(2.69b)(2.69c)This directional dependences in cartesian coordinate are quadratic in (x, y, z) as r cos θ = z, r sin θ cos φ = x,r sin θ sin φ = y. Since the non-spherical part in the potential (2.68) is given as an even function of (x, y, z), offdiagonal elements of the matrix representation on the (partial) basis of the quadratic functions yz, zx, xy are zero byintegrating out the odd functions. Residual quadratics are x2 , y 2 , z 2 . For vanishing off-diagonal terms in the matrix
representation for x4 + y 4 + z 4 , we need to take differences between (x2 , y 2 , z 2 ). Possible independent candidates arex2 − z 2 , y 2 − z 2 though they are not orthogonal. Hence with orthogonalization we reach 3z 2 − r2 , x2 − y 2 .
These directional dependencies can be obtained in the linear combination of eq. (2.69) as
√
i
15 yz
ϕξ = √ (ϕn21 + ϕn2−1 ) =Rn2 (r),4π
r22Fig. 2.9 Eigenstates of d-electron in an octahedral ligand field (eq. (2.70)). The surfaces of (absolute value of
wavefunction)=(constant) are drawn with shading.

E04-1(2.70a)Fig. 2.10 Level splitting of d-orbital by an octahedral ligand potential. D, q are defined in eq. (2.67) and eq. (2.71)
respectively. The uniform increase of levels by the constantterm is also drawn.

1
ϕη = − √ (ϕn21 − ϕn2−1 ) =2iϕζ = − √ (ϕn22 − ϕn2−2 ) =
2

√
√

15 zx
Rn2 (r),4π r2(2.70b)15 xy
Rn2 (r).
4π r2(2.70c)And√

5 3z 2 − r2
Rn2 (r),16π
r2√
5 x2 − y 2
1ϕv = − √ (ϕ322 + ϕ32−2 ) =Rn2 (r).
16π r22

ϕu = ϕ320 =(2.70d)(2.70e)The “shapes” of these wavefunctions are expressed in Fig. 2.9 as shaded surfaces of (absolute value of direction-dependent
part in the wavefunction)=(a constant).
We restrict ourselves to 3d electrons. Let q be
q=

2e
2e 4
⟨r ⟩ =
105
105∫|R32 (r)|2 r4 (r2 dr),(2.71)and the group of three wavefunctions ϕξ , ϕη , ϕζ and the group of two wavefunctions ϕu , ϕv have the eigen energies
ϵ1 = −4Dq,

ϵ2 = 6Dq(2.72)

respectively. The states corresponding to ϵ1 and ϵ2 are called T2g , Eg respectively named after the point groups. The
single-electron orbitals belonging to those are called t2g (or dϵ) orbital and eg (or dγ) orbital. The level splitting ofd-orbital is illustrated in Fig. 2.10. The shift between t2g and eg is roughly explained from the shapes of wavefunctions
drawn in Fig. 2.9 as follows. The three orbitals of t2g have zero amplitude when one of (x, y, z) is zero, avoiding thepositions of ligands, hence decrease the Coulomb energy. On the other hand, eg orbitals elongate to the directions ofligand, enhancing the Coulomb energy.
The angular moments of these orbitals are zero. For example, ⟨ϕζ |lz |ϕζ ⟩ gets +2 from ϕ322 , −2 from ϕ32−2 , andthe total is zero. Similarly ⟨l2 ⟩ = 0. This is the result of linear combination, in which the sum of the orbital angularmomentum vanishes. In other words, the eigenstates are the standing waves for the octahedral potential, naturally havezero angular momentum. Very important conclusion of this analysis is that the result explains the experimental results in
Tab. 2.4, in which the effective number of Bohr magneton appears to be as if the orbital angular moment vanishes.

2.7.2 Ground states of multiple electrons
Next we should consider the electron configuration in these t2g , eg orbitals along with the Hund’s rule. Here we need tocompare the crystal (ligand) field splitting 10Dq and the energy gain of the exchange integral (eq. (2.20c)) by following
E04-2

Fig. 2.11 Two possible electron accommodations of t2g and eg orbitals. There are high-spin states and low-spinstates for electron numbers more than n = 4.

the Hund’s rule. When the former surpasses the latter, the configuration of more accommodation in t2g orbital with lower
total spin is favored. Such a multiple-electron state is called low-spin state.
Figure 2.11 shows possible spin configurations (high- and low-spin states) for n = 4 ∼ 7. For n = 8, 9, t2g orbitals
are fully occupied, and there is a single possible configuration. In the example of an iron ion Fe2+ in a hemoglobin,
the existence of two possible (high, low) spin states brings about a dramatic effect. Four nitrogen atoms of porphyrin
and imidazole nitrogen at the end of the globin protein polypeptide chain are coordinated to this iron ion, which is not a
regular octahedron coordination. However we still take t2g , eg orbitals as the basis since the splitting due to the further
lowering in the symmetry is not very large. When the complex structure does not have an oxygen atom, namely a fivecoordination state, the electrons are in a high-spin state (t42g e2g ) while under the coordination of an oxygen atom (one of
two in a molecule), the splitting 10Dq between t2g -eg becomes larger and the state transits to the low-spin state t62g . The
transition can be detected by, e.g., the electron spin resonance.
The ground state of t2g or eg still has 3-fold or 2-fold degeneracy. When the octahedral structure gets a distortion alongz-axis, the degeneracy is lifted and generally the energy ofground state lowered. Hence when the energy lowering of theground state surpasses the energy increase due to the lattice
distortion, the lattice-distorted state is favored and the symmetry lowered. This phenomenon is called the Jahn-Teller
effect and observed, e.g., in CuSiF6 6H2 O salt. Or in somecases, the crystal field effect is coupled to the lattice vibration yielding the effect called dynamic or vibronic Jahn-Teller
effect.

2.7.3 Van Vleck paramagnetism
In the above, on the paramagnetism of 3d transition metals and 4f lanthanoid, we have seen that, the former can beunderstood by considering the effect of ligand field while the latter can be fairly understood by considering the SOI via
LS coupling (or j-j coupling) approach. However we still have a problem in the latter. As we see in Tab. 2.5, there are
big discrepancies between the theory and the experiment for Eu3+ , Sm3+ . The problem was theoretically solved by Van
Vleck, and the phenomenon is called Van Vleck paramagnetism. The above figure is from the Nobel lecture[3] given by

E04-3

Van Vleck, which demonstrates that his theory quantitatively explains the discrepancies in the simple theory.
Let us see how the theory works in the case of Eu3+ . The electron configuration is 4f 6 , the electron number is lessthan 2l + 1 = 7, then we see the coefficient of L · S is positive from eq. (2.38). Namely L and S couple as anti-parallel.
The ground state is 7 F0 , which means J = 0. In the excited states, possible total angular momenta are J = 1, 2, · · · , 6.
From eqs. (2.38),(2.41),
HSOI = λL · S =λ[J(J + 1) − S(S + 1) − L(L + 1)].
2

(2.73)The splitting width of LS-multiplet (L, S are fixed, J is the index of split states) is∆ELS = ELS (J) − ELS (J − 1) = λJ.

(2.74)The energy difference between the ground state with J = 0 and the first excited state with J − 1 is λ, which is as small as
200∼300 K. Hence even at low temperatures, when there is a finite external magnetic field, the term HSOI causes mixingof the state J = 1 with the J = 0 ground state, which gives rise to a finite magnetic moment.

2.8 Symmetry and degeneracy of quantum states
In the above we have solved a specific problem of d-electrons in an octahedral potential. In more general treatment,
we should apply regular perturbation theory. However, the thoughts from the symmetry of the system is very helpful
in the calculation since we can largely decrease the amount of calculation[4]. The symmetry of a system is defined bysymmetry operations. A symmetry operation is a transformation in a space of some degrees of freedom. Examples
are rotations, parallel transformations, mirror reflections, etc. in the coordinate space. When a system is invariant undera symmetry operation, the system has the symmetry for the operation. The total symmetry of a system is defined as aset of symmetries possessed by the system. In the following we will have a short look at a general method to know thedegeneracy of quantum states from the symmetry of the system.

2.8.1 Symmetry operations in point groups
We use group theory for such discussions. In Appendix 3A, we have seen that symmetry operations constitute a group,
an element of which has a corresponding matrix of representation. Thus the symmetry of a system can be specified by
the corresponding group. The symmetry groups that have correspondence to spatially localized systems are called point
groups. Particularly in crystals, the constraint of discrete translational symmetry restricts the number of possible point
groups called “crystal point groups” to 32 listed in Tab. 2.6.
We won’t go deep into mathematics though the group theoretical knowledges of symmetry operation are indispensablefor the researchers of crystallography, symmetry-sensitive properties like multi-ferroics. Embarrassing in symmetrygroup theories is that similar symbols are
The problem with handling a group of symmetric operations is that they are all confusing with similar symbols tovarious concepts, and the symbols are different and more confusing depending on the style of the mathematicians. In
physics, Schönflies and Mulliken symbols have been used, and we also follow that here but recently “international
standard” symbols are also frequently used. At present I cannot find the way to describe the expressions in a beautifullyorganized manner.
R

When a set of functions Aφ = {φ1 , φ2 , · · · } is transformed back to itself by a symmetry operation R, that is, A −→Aφ′ = {φ′1 , φ′2 , · · · } = A , A can be a representation basis of R and the corresponding matrix is given as
Dij (R) = ⟨φi |R|φj ⟩ .

E04-4(2.75)

system
triclinic
monoclinic

orthorhombic

tetragonalrhombohedral

hexagonal

cubic

icosahedral

Schönflies

Hermann-Mauguin symbol

examplessymbolfull

abbreviated

C1

1

1

Ci , (S2 )1̄

1̄

Al2 SiO5

C1h , (S1 )

m

m

KNO2

C2

2

2

C2h

2/m

2/m

C2v

2mm

mm

D2 , (V )

222

222

D2h , (Vh )2/m2/m2/m

mmm

C4

4

4

S4

4

4

C4h

4/m

4/m

D2d , (Vd )

4̄2m

4̄2m

C4v

4mm

4mm

D4

422

42

D4h

4/m2/m2/m

4/mmm

C3

3

3

AsI3

C3 , (S6 )3

3

FeTiO3

C3v

3m

3m

D3

32

32

Se

D3d

32/m

3m

Bi, As, Sb, Al2 O3C3h , (S3 )

6

6

C6

6

6

C6h

6/m

6/m

D3h

62m

62m

C6v

6mm

6mm

ZnO, NiAs

D6

622

62

CeF3

D6h

6/m2/m2/m

6/mmm

T

23

23

NaClO3

Th

2/m3

m3

FeS2

Td

43m

43m

ZnS

O

432

43

β-Mn

Oh

4/m32/m

m3m

NaCl, diamond, Cu

C5

5

5

C5i , (S10 )

10

10

C5v

5m

5m

C5h , S5

5

5

D5

52

52

D5d

52/m

5/m

C80

D5h

10̄2/m10̄2/m

C70

I

532

532

I, Ga

CaWO4

Ih

TiO2 , In, β-Sn

Mg, Zn, graphite

C60

E04-5

Tab.
2.6 Crystal systems, Schönflies symbols,
Hermann-Mauguin symbols
and examples of materialsof 32 crystal point groups.
From Ref. [4].

Symmetry operation

Rotation axis

Number of operation

E

Identical transformation

C4

π/2 rotation around 4-fold axis

x, y, z3

C2 = C42
C43

π rotation around 4-fold axis

x, y, z33π/2 rotation around 4-fold axis

x, y, z3

C2

π rotation around 2-fold axis

(0,1,1), (1,0,1), (1,1,0)

6

1(0,1,−1), (−1,0,1), (1,−1,0)C32π/3 rotation around 3-fold axis

(1,1,1), (1,1,−1), (1,−1,1), (−1,1,1)4C32

4π/3 rotation around 3-fold axis

(1,1,1), (1,1,−1), (1,−1,1), (−1,1,1)

4

Tab. 2.7 Symmetry operations octahedral group (O group).

Here, if D(R) can get further block diagonalization by unitary matrix S (i.e., replacement of basis) as
D1 (R)0
D2 (R)
SD(R)S −1 = 
,
..
.
0

(2.76)

D(R) is reducible and can be expressed as a direct summation of D1 (R), D2 (R), · · · as
D(R) = D1 (R) ⊕ D2 (R) ⊕ · · · .

(2.77)On the other hand, if such block diagonalization is impossible, the representation is irreducible. Expression of a reducible
representation as a direct summation of irreducible representations, is called reduction.
Irreducible or reducible namely the possibility of block diagonalization cannot be judged from the simple diagonalizability of matrix. Often adopted is the use of character table. Character of representation is the trace of representation
matrix, which is invariant for the changing of basis. Hence a character for reducible representation is the sum of characters for irreducible representations in the direct summation. From this the reduced form in the direct summation can be
deduced.
Table 2.7 lists the symmetry operations that keep an octahedral system invariant. Those operations constitute a groupcalled octahedral group, of which the symbol is O. As in Tab. 2.6, it belongs to the cubic symmetry. The octahedral
complex systems with magnetic ions also have the space-inversion symmetry in addition to O. Hence to be strict, weneed to consider Oh group. Although here for simplicity we consider O group and it is known that the level-splitting
behavior is the same for Oh . Group O has 24 symmetry operations as the elements as listed in this table.
O

E

8C3

3C2 = 3C42

6C2′

6C4

Γl=0

A1

1

1

1

1

1

Γl=1

T1

3

0

−1

−1

1

Γl=2

E + T2

5

−1

1

1

−1

Γl=3

A2 + T1 + T2

7

1

−1

−1

−1

Γl=4

A1 + E + T1 + T2

9

0

1

1

1

Γl=5

E + 2T1 + T2

11

−1

−1

−1

1

Γ1

A1

1

1

1

1

1

Γ2

A2

1

1

1

−1

−1

Γ12

E

2

−1

2

0

0

′
Γ15

T1

3

0

−1

−1

1

′
Γ25

T2

3

0

−1

1

−1

E04-6

Tab.
2.8 Symmetry operations ingroup O (topmost low), and characters.
The representations in the upper low areon bases of eigenfunctions of orbital angular momentum. Those in the lower low
display irreducible representations.

Next we consider the representations of these symmetry operations. A simple choice of representation basis is the
eigenfunctions in a system of a spherical potential. They can have the orbital angular momentum as the quantum number,
namely s, p, d, f, · · · . The representations based on them are listed in the upper band in Tab. 2.8. The symbols Γl=0 ∼Γl=5 are the representations on the orbital angular momentum l. However they are not generally irreducible. The symbols
of irreducible representations are named in the following way. A, E, T are for 1, 2, and 3-dimensional representations.
The suffices do not have strict rule, being numbered starting from O group. In the table, the irreducible representations
A1 , A2 , E, T1 , T2 of group O and the characters are listed in the lower band. From the comparison of these characters,
we know that Γl=0 , Γl=1 are irreducible. This fact indicates that s and p orbitals do not split in an octahedral potential.
For l ≥ 2, Γl are reduced as indicated in the table. When we consider the split of d-orbital, we added suffix g as T2g and
Eg . Actually we need to consider group Oh including the space-inversion operation i. The symmetry operations are thedirect product with i and the number of i operations brings the difference between even number (gerade, g in Germany)and odd number (ungerade, u). The irreducible representation also needs the suffix of g (even).

2.8.2 Symmetry operation and degeneracy
Let R be one of such symmetry operation and a function φ transform as
φ′ = Rφ.

(2.78)

Assume an operator O is transformed by R to O ′ , then the operation of O ′ on the transformed function Rφ should be theresult of operation R on the O-operated original φ. Therefore
O ′ Rφ = ROφ = ROR−1 Rφ.
That is, O is transformed to ROR−1 . Now assume H is invariant for the symmetry operation R.
RH R−1 = H ,∴ [R, H ] = 0.

(2.79)Let ϕ an eigenfunction of H with an eigenvalue of E, that is H ϕ = Eϕ. ThenH Rϕ = RH R−1 Rϕ = REϕ = ERϕ,

(2.80)

which indicates that ϕ′ = Rϕ is an eigenstate of H with the same eigenvalue E. If ϕ and ϕ′ are independent to eachother, they are degenerate eigenfunctions. This degeneracy is based on the symmetry of the system. Other kinds ofdegeneracy is called “accidental” and usually lifted by some factors in real systems.

Fig. 2.12 Level splitting of 5-fold l = 2 states in various symmetries expressed by crystal point groups. The numbers
attached to the levels indicate the degree of degeneracy. From Ref. [4].

E04-7

Let {ϕi } be eigenfunctions with degree of degeneracy d with eigenvalue E of H . The representation matrix of R in
this space is given by
Dij (R) = ⟨ϕi |R|ϕj ⟩ .
That is,
Rϕν =

d
∑

(2.81)Dµν (R)ϕµ .

(2.82)µ=1

Here D(R) must be irreducible. Otherwise, e.g., assume D(R) is block diagonalized as()
D1 (R)
0
D(R) = D1 (R) ⊕ D2 (R) =.
0
D2 (R)
Then the basis is transformed by the transformation matrix for the reduction as {ϕi } → {χi }. We can thus divide the
(1)

(2)

basis {ϕi } into {ϕi } and {ϕi } which belong to D1 (R) and D2 (R) respectively. They are not transformed to eachother by symmetry operation, hence the degeneracy is accidental.
From the above discussion and by the character tables of irreducible representation, we can deduce how the degeneratestates under consideration split when the symmetry of the system changes. Further we can know the shape of wavefunction from the basis of representation, hence infer the order of energy levels. Thus obtained level splitting of d-orbitals invarious ligand field potentials is illustrated in Fig. 2.12.

2.9 Experiments and applications of localized spins
We have studied magnetic atoms or ions, which have large localized magnetic moments based on the measurement ofmagnetic susceptibility and the crystal structures. Here we have a look at fundamentals of magnetic resonance – a very
important experimental method for magnetism. As an application of paramagnetic salt, we have short visit to magnetic
refrigeration.

2.9.1 Magnetic resonance
In Sec. 2.1.2, we have mentioned that the magnetic moment tilted in a static magnetic field causes Larmor precession.
In the method of magnetic resonance (MR), by applying an oscillating magnetic field at the Larmor frequency andby observing the resonance, we obtain not only the information of the magnetic moment itself, but also that of theenvironment surrounding the moment. It is needless to say the academic and social importance.
Let J be the total angular momentum of an isolated electron system. The Zeeman term in a static magnetic field B0
along z axis is given by
H1 = gJ µB J · B0 ,(2.47)

which is just the same as eq. (2.47). Heisenberg equation of motion is written asdJi= [gJ µB J · B0 , J ],dtℏ

(2.83)

to which the commutation relation
Jy Jz − Jz Jy = iJx ,Jz Jx − Jx Jz = iJy ,Jx Jy − Jy Jx = iJz(2.84)

is applied to obtain
dJ
gJ µB
=
B0 × J .
dt
ℏ

E04-8

(2.85)

Fig. 2.13 Left: Precession of angular momentum J around the in-plane magnetic field B1 /2 on the rotating coordinate. Right: Illustration of the total motion tracing the end-point of J from the static coordinate. Under realistic
conditions, the rotation around z-axis is much faster.

This expresses the precession with Larmor frequency
ωL = gJ

eB0.
2m(2.86)

Now if we observe this precession from the coordinate rotating around z-axis with frequency ωL , the precession iscanceled out and the momentum looks as if it sits still. Namely in this coordinate, the effect of static field along z axis
vanishes and virtually zero-field state is realized.
An oscillation magnetic field B(t) = B1 cos(ωt) (on the non-rotating coordinate) perpendicular to the static field B0
can be written as the sum of rotating field as
B(t) =

B1[exp(iωt) + exp(−iωt)].
2(2.87)

When ω ≈ ωL , from the rotating coordinate the ω-component almost sits still while the −ω-component is rotating with
2ω. We take rotating wave approximation, in which the latter is ignored. Now on the rotating coordinate, a static fieldof B1 /2 is applied in xy-plane and the angular momentum starts precession around this field with frequency
ω1 = gJ

eB1.
4m(2.88)

This is the basic process of magnetic resonance.
Figure 2.13 illustrates the motion of angular momentum under a magnetic resonance, in which the end-point draws aspiral. Note that under realistic conditions, the rotation around z-axis is much faster.

References[1] C.J. Ballhausen. Introduction to Ligand Field Theory (Advanced Chemistry). McGraw-Hill Inc.,US, 12 1962.
[2] Joachim Stöhr and Hans Christoph Siegmann. Magnetism: From Fundamentals to Nanoscale Dynamics (Springer
Series in Solid-State Sciences, 152). Springer, 9 2006.
[3] J. H. Van Vleck. Quantum mechanics: The key to understanding magnetism. Science, Vol. 201, No. 4351, pp.
113–120, July 1978.
[4] Mildred S. S. Dresselhaus. Group Theory: Application to the Physics of Condensed Matter. Springer, 10 2010.

E04-9

Lecture note
11th May (2022)

Magnetism (5)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

In the last lecture, for magnetic resonance, we introduced the concept of ”observation from rotating coordinates” . The
time evolution of the angular momentum was treated in the Heisenberg picture, and the results were the same as thatobtained in the classical picture as in eq. (2.83). Then in the following, for simplicity, we consider classical equations of
motion for macroscopic magnetic moment M and introduce energy and phase relaxations phenomenologically[1].
dMz
M0 − Mz= γ[M × H]z +
,
dt
T1
dMx,y
Mx,y
= γ[M × H]x,y −.
dt
T2(2.89a)(2.89b)

T1 , T2 are longitudinal and transverse relaxation times respectively, or energy and phase relaxation times. We here use H
instead of B to avoid confusion due to the existence of M though the use of B is not the problem at all. The deﬁnitionof gyromagnetic ratio should be changed from eq. (2.14) so as to have consistent dimension. We consider a static ﬁeld
H0 along z and a rotating ﬁeld H1 /2 in xy-plane with an angular frequency (−)ω. The total ﬁeld is
H =(

H1
H1
cos ωt, −sin ωt, H0 ).
22(2.90)The magnetic moment derived from the angular momentum gets minus sign due to the sign of electric charge (eq. (1.80)).
Hence the rotation is clockwise for positive magnetic moment M . The equations of motion are thus,
dMx
Mx
H1
= γ[My H0 + Mz
sin ωt] −
,
dt
2
T2
dMy
H1
My
= γ[Mzcos ωt − Mx H0 ] −
,
dt
2
T2
M0 − Mz
dMz
H1= γ[−Mx H1 sin ωt − Mycos ωt] +
.
dt
2
T1

(2.91a)(2.91b)(2.91c)Fig. 2.14 The plot of eq. (2.96). The
real part (χ′ (ω), blue line) and the imaginary part (χ′′ (ω), red line) of the complex susceptibility around a magnetic
resonance in the presence of relaxations.

E05-1

Fig. 2.15 Typical setup of electron
paramagnetic resonance (electron spinresonance) experiment.The circle markwith numbering indicates a circulator of
microwave. A microwave enters via terminal 1 through 2 to the cavity with asample and the reﬂected wave propagates
through 3 to a diode. The detected signal is sent to a phase sensitive detector(PSD). From [2].

We introduce the coordinate system (x′ , y ′ , z ′ ) (z ′ = z) rotating around z-axis clockwise with angular frequency ω toobtainMx′ = Mx cos ωt − My sin ωt,(2.92a)My′ = Mx sin ωt + My cos ωt.

(2.92b)By using eq. (2.92), we rewrite eq. (2.91) into the kinetic equations of (Mx′ , My′ , Mz ). Under the conditions:
dMy′
dMx′
=
= 0 (stationary state),dt
dt
Mz ≃ M0 = χ0 H0 (oblique angle is small),(2.93)

we obtain the following solutions.
(ω0 − ω)T2 H1 /2
1 + (ω0 − ω)2 T22 + γ 2 (H1 /2)2 T1 T2
H1 /2
My ′ = χ 0 ω 0 T 2.
1 + (ω0 − ω)2 T22 + γ 2 (H1 /2)2 T1 T2

Mx ′ = χ 0 ω 0 T 2(2.94a)(2.94b)

ω0 is Larmor frequency ωL in eq. (2.86) with B = µ0 H0 .
The solution expresses a state with a comparatively large relaxation, thus the moment gets large friction in rotation
around H1 . Hence the stationary state has a small angle to z-axis. From the rest system,
Mx = χ′ (ω)H1 cos ωt + χ′′ (ω)H1 sin ωt,′′′

My = −χ (ω)H1 sin ωt + χ (ω)H1 cos ωt.

(2.95a)(2.95b)

In case γ 2 H12 T1 T2 ≪ 1 (large relaxation), the real and imaginary parts of the susceptibility are given byχ 0 ω0(ω0 − ω)T2
T2,
2
1 + (ω0 − ω)2 T22χ 0 ω01
χ′′ (ω) =,T2
2
1 + (ω0 − ω)2 T22
χ′ (ω) =(2.96a)(2.96b)respectively, that is χ = χ′ − iχ′′ . They are plotted against ω0 − ω in Fig. 2.14.
In the experiments, the specimens are placed in coils or cavities of resonators and the variations of resonance characteristics are detected. An increase in energy dissipation in a resonator results in a widening and lowering of the resonancepeak *1 . Figure 2.15 illustrates a typical setup of experiments. A microwave enters via terminal 1 through 2 to the cavity
with a sample and the reﬂected wave propagates through 3 to a three port connection for interference and then to a detector (diode). The signal is then sent to a phase sensitive detector (PSD). Variation of resonant frequency in microwave
cavity is not so easy in most cases, so usually it is ﬁxed. The frequency of input microwave is tuned to the resonance.
*1 This can be understood, e.g, by inserting a resistor into an LC resonator circuit in a model for calculation.

E05-2

Instead, the Larmor frequency ωL in eq. (2.86) of the sample is swept by the external magnetic ﬁeld. In eq. (2.96), thiscorresponds to the variation of ω0 , and anyway the resonance curve is obtained. ω0 in the coefﬁcients of eq. (2.96) gives
the ﬁrst order slow variation while the resonance generally has a sharp lineshape and the distortion due to the coefﬁcientis ignorable. The imaginary part in eq. (2.96), χ′ (ω) (eq. (2.96b)) represents the energy dissipation, which shows a peakstructure as the red line in Fig. 2.14.
Coils for the ﬁeld modulation is drawn in Fig. 2.15. In many experiments, ﬁeld modulation with frequencies around
100 kHz is applied for improvement of signal to noise ratio by taking PSD signal. Such signals appear as frequencyderivative of eq. (2.96b).

2.9.2 Hyperﬁne structures in electron paramagnetic resonance
Owing to the large magnetic moment of electrons, electron spin resonance (ESR)or electron paramagnetic resonance(EPR)has high sensitivities among various magnetic resonance. Atomic wavefunctions of electron have ﬁnite amplitudes
in the neighbors of nucleuses. When a nucleus has a ﬁnite spin, the exchange interaction (which will be introduced later)
between the nuclear spin I and the electron total angular momentum J brings about hyperﬁne interaction, which canbe expressed as
HHF = AI · J .

(2.97)A is a parameter of the strength in the interaction. Since the Hamiltonian in eq. (2.97) has the same form as the spin-orbit
interaction eqs. (2.38), (2.39), we consider the total angular momentum F = I +J , and write the simultaneous eigenstateof H, F 2 , Fz as |F, MF ⟩. Then just as eq. (2.41), the eigenvalue of HHF is given by
HHF |F, MF ⟩ = A

F 2 − I2 − J2
F (F + 1) − I(I + 1) − J(J + 1)
|F, MF ⟩ = A|F, MF ⟩.
22(2.98)The naming of “hyperﬁne” interaction comes from the further splitting of absorption levels, that already split by HSOI .
We do not have enough time to go into very important ﬁeld and I recommend the readers who have strong interest in thisﬁeld to refer to the journal “Hyperﬁne Interactions” *2 .

2.9.3 Electron paramagnetic resonance in paramagnetic salts
EPRs are ﬂourishing in the paramagnetic salts that have 3d transition metals or 4f lanthanoids as sources of magnetic
moments. They have comparatively high moment densities (depending on the impurity concentrations), and importantly
they are insulating that allows high frequency electromagnetic wave to go into the bulk in the absence of the skin effect.
Here we introduce an example of analyzing EPR data. Because EPR not only is sensitive but also has high resolution
bringing rich information on the localized electronic states. For the actual analysis, we thus need a higher order approximation. In the case of 3d transition metals, we have discussed that the effect of orbital angular moment disappears dueto the strong effect of ligand ﬁelds. Conversely, in 4f lanthanoid, we have given the priority to the spin-orbit interaction.
However, these are the ﬁrst order approximation and in the next step, we need to consider the SOI in the former, and the
crystal ﬁeld splitting in the latter. We skip the hyperﬁne interaction, which is not important for nucleus without spin, but
consideration of the SOI on crystal ﬁeld split levels is indispensable. For that we introduce the concept of effective spin
Hamiltonian, which is common in this ﬁeld. The effective spin Hamiltonian has no orbital operator but gives the sameanswer as the original one, if we restrict the problem to the effect of the SOI on the orbital levels that diagonalize thecrystal ﬁeld Hamiltonian[3].

*2 https://www.springer.com/journal/10751

E05-3

Let {φ0 , φ1 , · · · , } be an orbital basis that diagonalizes Horb = H0 + HCF , and in ket format |n⟩o . We write
the energy eigenvalues as o ⟨n|Horb |n′ ⟩o = En δnn′ , the total spin of the basis as S, the spin wavefunction as{ϕ−2S , ϕ−2S+1 , · · · , ϕ2S }, ket-expression |m⟩s .
The perturbation Hamiltonian of the SOI and the Zeeman effect is
H′ = λL · S + µB (L + ge S) · H,

(2.99)

where ge is g-factor of electron. We write a wavefunction in an expanded form as∑∑Ψ=
anm φn ϕm =anm |n⟩o |m⟩s .
nm(2.100)nm

Then the eigenvalue equation of the total Hamiltonian isHΨ = (Horb + H′ )Ψ = EΨ.

(2.101)

Here e.g., the orbital part is integrated out in o ⟨l|H′ |n⟩o but this still has the spin part as operator(s). Then we deﬁne aspin Hamiltonian as a second-order perturbation formula, in which the orbital part is integrated out. If the unperturbed
ground state has degeneracy, we change the basis according to the perturbation theory, but there should be no confusion
in using the symbols like |n⟩o . We also restrict the transition matrix elements to the ones with the ground state |0⟩o . Thus
we obtain *3
H̃ = o ⟨0|H′ |0⟩o +∑ o ⟨0|H′ |n⟩ o ⟨n|H′ |0⟩o

E 0 − Enn̸=0

The introduction of ligand ﬁeld quenches the orbital angular momentum.
o|0|L|0⟩o = 0 to giveo.

(2.102)

Namely the diagonal terms vanish′o ⟨0|H |0⟩o = ge µB S · H.

(2.103)′o ⟨0|H |n⟩o = o|0|L|n⟩o · (λS + µB H),(2.104)H̃ = ge µB S · H − (λS + µB H)Λ(λS + µB H),(2.105)In the same way, fromwe obtain

where Λ is a tensor given byΛij =∑ o ⟨0|Li |n⟩ o ⟨n|Lj |0⟩oEn − E 0n̸=0o(i, j = x, y, z).

(2.106)

These elements are not necessarily zero and the effect of spin-orbit interaction appears in the second order perturbation.
The same view is given by considering that the vanishing expectation value of orbital angular momentum comes from theway of superposition and the cancellation does not work for off-diagonal elements. Equation (2.105) is expanded asH̃ = µB Sge (1 − λΛ)H − λ2 SΛS − µ2B HΛH.

(2.107)

The ﬁrst term is written in the Zeeman form asg̃ = ge (1 − λΛ).

(2.108)This is an expansion of g-factor to a tensor form. Taking the principal axes of the tensor to x, y, z, we can write thesecond term in the following form.
[−λ SΛS = D
2

S(S + 1)Sz2 −3]
+ E(Sx2 − Sy2 ),(2.109)*3 You may not be satisﬁed with this treatment. In that case refer to a tutorial review [4]. A textbook in Japanese [5] gives rigorous treatment basedon the projection operator method.

E05-4

Fig.
2.16 Level
splittingof3+3
Cr (3d , S=3/2) in weak crystal ﬁeld approximation and Zeeman
splitting by external ﬁeld. E=0 is
assumed. Expected three ESR peaksignals for microwave frequency νcorresponding to the red arrows areillustrated.

where Λ determines coefﬁcients D and E in proportion to λ2 , called axial and orthorhombic coefﬁcients respectively *4 ．
The third term does not cause level-splitting being a small effect of orbital magnetic moment induced by external ﬁeld.
With dropping this term we reach[H̃ = µB S g̃H + D

S(S + 1)
Sz2 −

]

3

+ E(Sx2 − Sy2 ).

(2.110)As a simple example, let g̃ be isotropic, E = 0, and S = 3/2. The spin states active for EPR are four states of
Sz = ±1/2, ±3/2. At H = 0 they degenerate into two with the distance of 2|D| from eq. (2.110). Figure 2.16 illustrates
such situation. The LS multiplet (3d)3 is split as in the ﬁgure in weak ﬁeld approximation given in App. 5A. The groundFig. 2.17 EPR spectra of BaTiO3 doped with
1% Cr3+ and Fe3+ . Blue and red lines are data
for magnetic ﬁeld parallel and perpendicular to caxis respectively. The data are offset for clarity.
Peaks (look like dissipation because of derivative
taking in experiment) assigned as a, b, and c correspond to a, b, and c in Fig. 2.16. The microwave
is in X-band (9-10 GHz), for which the value is
not given. The annotations in blue letters indicate
absorptions of S =3/2 state in Cr3+ , and S=1
state in Cr dimers. Those in black letters indicate
absorptions of S = 5/2 in Fe3+ . From [6].

*4 Repeated usage of D or E for different quantities inevitably causes confusion.

These jargons are commonly used in many research papers
and sometimes we ﬁnd such symbols appear in a single diagram or table with different meanings. In this ﬁeld we are often forced to express
complicated concepts with simple and clear symbols, and at present we ﬁnd no other better way. The readers are requested to judge from thecontext.

E05-5

Ion

Crystalg|D| (cm−1 )|E| (cm−1 )Fe3+

BaTiO3

2.000

0.022

0.0079another report

2.003

0.0987

1.975

0.046

Cr3+

BaTiO3h-BaTiO3

H1 gz =

1.9797

H1 gx,y =1.9857

H2 gz =

1.9736

H2 gx,y =1.97560.0055

Tab. 2.9 Values of g, |D|, |E| for Fe3+
and Cr3+ in BaTiO3 obtained from EPR
lines in Fig. 2.17.

0.105
0.3220state is A2 (S = 3/2). The external ﬁeld split the two into four as in the ﬁgure. With microwave of frequency νcorresponding to the red arrows in the ﬁgure, absorptions at positions a, b, and c assuming equal populations of levels at
high-enough temperatures.
Cr3+ (d3 , 4 F ) often takes the states close to the above ideality. Figure 2.17 shows an EPR experiment on Cr3+ , inwhich they doped 1% Fe3+ and Cr3+ into cubic state of BaTiO3 . Three main peaks corresponding to a, b, and c inFig. 2.16 are observed[6]. The lineshape did not change with the direction of magnetic ﬁeld indicating that g̃ is isotropic
though ﬁnite E is deduced from the relative peak positions. The data indicate doped Cr3+ is in tetragonal state. The
absorption peaks for S = 1 are from Cr dimers. The parameters are shown in Tab. 2.9.
BaTiO3 also has hexiagonal phase h-BaTiO3 . Cr3+ impurities in h-BaTiO3 show very different EPR absorption lines.
Even this spectrum can be explained by the spin Hamiltonian of eq. (2.110) with an anisotropic g̃ and parameters shownin the table[7].

2.9.4 Magnetic refrigeration
Magnetic refrigeration (MR) is a cooling method that can exert cooling ability in a wide temperature range by selecting
a magnetic moment system (working substance) from room temperature to ultra-low temperature. The MR with paramagnetic salt was once main player in creating very low temperature till 1960’s, then in the range from 0.01 to 1 K theposition was replaced by the dilution refrigerators[8]. On the other hand, below few mK, once Pomeranchuk cooling
was used, but then the main shifted to nuclear demagnetization.Even in regions other than low temperatures, the MR
are now used for various purposes. Exploration of cooling substances spans not only paramagnetic materials but also

Fig. 2.18 Concept of magnetic refrigeration. Left: Illustration of temperature variation of entropy in a magnetic
system in zero and ﬁnite magnetic ﬁelds. The green arrow indicates ﬁrst isothermal application of magnetic ﬁeld.
The brown arrow shows adiabatic demagnetization. The right ﬁgure illustrates evolution of spin distribution in thesequence of isothermal ﬁeld application and adiabatic demagnetization in a simple two-level spin system.

E05-6

Fig. 2.19 Entropy in eq. (2.113) is expressed as afunction of α for J = 1/2, 3/2, 5/2.

those have phase transitions caused by interactions between spins and magnetic ﬁeld. In recent years, regardless of its
pros and cons, the possibility of liquefying and transporting hydrogen has been explored, and development research using
magnetic refrigeration for cooling is also being conducted[9].
The principle of MR is simple as follows. A free spin system has random directions in spin and the maximum entropy
at zero ﬁeld. With the isothermal magnetization, the spins are aligned along the magnetic ﬁeld by the Zeeman effect. The
process reduces the entropy. Hence a heat is discharged to the environment. Then during the adiabatic demagnetization,
the entropy is kept constant. In the spin distribution, there is no increase in the population to excited states in spite ofdecrease in the distance between the ground and the excited states. This is a low temperature state. With connecting thiscoolant to a material to be cooled, a heat is transferred from this material to the coolant. The process is summarized inFig. 2.18.
Let S be the entropy of system composed of paramagnetic ions with total angular momentum J. At zero ﬁeld, S =kB NA ln(2J + 1) for 1 mol. Let us write the variation in S with isothermal magnetization up to B as ∆S(B, Ti ). Thenwith the ﬁnal temperature Tf after the adiabatic demagnetization, ∆S(B, Ti ) is given by
(
)
∫ Ti
Cm
∂S
∆S(B, Ti ) = S(0, Ti ) − S(B, Ti ) =dT, Cm = T.
T∂T B=0Tf(2.111)

Cm is the speciﬁc heat at zero ﬁeld.
The magnetization M and the entropy S are calculated just as eq. (1.25) as[()]
2J + 12J + 11αgµB B
M = NA gµBcothα − coth
, α≡
,
2
2
2
2
kB T
[
]
[
]
S
α
α 2J + 1
2J + 1sinh[(2J + 1)α/2]= coth −α cothα + ln
.
NA kB
2
2
2
2
sinh α/2(2.112)(2.113)Figure 2.19 shows the results for J =1/2, 3/2, 5/2. Since α ≈ 1.344B (B in T) for g = 2, T =1 K, the most of entropy
in a free spin system at T =1 K can be removed by cycling of B =2 T.
The spins in paramagnetic salts maintain a relatively large entropy even at low temperatures where other degrees of
freedom are quenched, such as lattice vibration, so they are used to obtain extremely low temperatures based on low
temperatures. However, e.g., if we choose a material with a phase transition induced by magnetic ﬁeld, a large decreasein entropy is expected even at high temperatures. This can be understood by a Maxwell relation
(
)
(
)
∂S
∂M
=
,
∂B T
∂T B
E05-7

(2.114)

Fig. 2.20 Schematic diagramof active magnetic refrigeration (AMR). From Akiko Saito,
Toshiba Review Vol.62, 78 (2007)(in Japanese).

in which magnetic ﬁeld variation of entropy is proportional to temperature derivative of magnetization.
Gd (Curie temperature 289 K) is frequently used as a cooling material, which can have a cooling power at aroundroom temperature.Further research and development of working materials and methods are underway with the aim of
putting them into practical use near room temperature. For the use of MR under continuous inﬂow of heat, a continuous
cooling method is active magnetic refrigeration (AMR) illustrated in Fig. 2.20. Magnetization/demagnetization is doneby rotation of tables, to which permanent magnets are ﬁxed. The turn table scheme realizes successive on/off of magnetic
ﬁeld and continuous ﬂow of refrigerant realizes continuous cooling.
There are many proposals and exploration of cooling substance, e.g., nano-particles of ferromagnet that have giant
magnetic moments, materials with meta magnetic transitions (ﬁrst order transition from Pauli paramagnet to intinerant
electron ferromagnet). If time permits we will revisit such transitions. They are already beginning to be used and somecommercial machines are appearing.

E05-8

Chapter 3

Magnetism of conduction electrons

Magnetic levitation of graphite
https://sci-toys.com/scitoys/scitoys/magnets/pyrolytic graphite.html

In the previous chapter, the focus has been on the paramagnetism of insulators including magnetic ions. The magnetismthat appears due to the phase transition caused by the interaction between electron spins will be investigated in thefollowing chapters. Here we have a short look on the magnetism of conduction electrons other than such cooperative
phenomena.

3.1 Pauli paramagnetism
3.1.1 Conduction electrons
Electronic states in crystals are described by energy bands separated by energy gaps on the energy axis due to discrete
spatial translational symmetry. In a system of free electrons, due to the fermionic constraint, there exists the Fermi level
determined by the number of particles and the band structure. The orbital states described by Bloch functions haveup-down spin degeneracy at zero ﬁeld. In metals, the Fermi levels place inside energy bands (conduction bands), andwe measure the Fermi energy EF from the bottom of conduction band. At room temperatures EF ≫ kB T , and the
distribution is Fermi degenerated.
In many actual metals, the electronic structures are complicated due to, e.g., multiple bands in k-space, etc. However,
here, to explore the essential properties, we assume a single band with single band in the center of k-space with asingle effective mass. Further we ignore the mutual electron interactions, which must exist in real systems. Such a noninteracting model well describes behavior of electrons in actual metals in most cases. This gets theoretical support fromLandau’s Fermi liquid theory[10, 11].

3.1.2 Magnetic response by electron spins
To see the response of electron spins, we write the Hamiltonian in second-quantized form as
H =

∑
kσ

∑ †
1
Ek c†kσ ckσ + gµB B
σckσ ckσ .
2
kσ

E05-9

(3.1)

kz

kz

E+dE
ky

E

ky

kx
2p/L

kx

(a)

(b)

Fig. 3.1 (a) Red points represent eq. (3.2)
in k-space. (b) To calculate the density ofstates, we count the numbe of possible points
in the shell between E and E + dE.

c†kσ is a creation operator of an electron with wavenumber k, spin σ (σ = ±1 for ↑↓). The ﬁrst term is the kinetic energy,
the second is the Zeeman energy.
We take periodic boundary condition in a cube with a side of L. kL = 2nπ (n: integer) along a side givesk=2π(nx , ny , nz ) (nx , ny , nz : integers).
L

(3.2)

The discrete points in k-space are displayed in Fig. 3.1(a). The point density is (L/2π)3 . Let ρ(E) the energy densityof states per volume and per spin. ρ depends on the spin direction in ferromagnets but is common in paramagnets with
time-reversal symmetry. In three dimensional free electron systems, Ek = ℏ2 k 2 /2m with the effective mass m. The
wavenumbers which give a constant E for Ek are, in k-space, the points expressed in (3.2) and on the sphere with radius√kE = 2mE/ℏ. To obtain ρ(E), we count such points within the shell between E and E + dE and divide by dE as inFig. 3.1(b). Thus we obtain
1
ρ(E) = 3
L

(

L
2π

)3

2 dkE
=
4πkE

dE

1
π 2 ℏ3√mE.
2

The expectation value of the magnetic moment is given by
[ (
)
(
)]
gµB ∑
gµB ∑
gµB B
gµB B
†
−
σ ⟨ckσ ckσ ⟩ =f Ek −− f Ek +
,
2
2
2
2
kσ

(3.3)

(3.4)

k

where
f (E) =

1exp[(E − µ)/kB T ] + 1

is the Fermi distribution function with µ as the chemical potential. Then the magnetization M is given by
)
(
)]
[ (
∫
gµB ∞
gµB B
gµB B
M=
− f Ek +.
dEρ(E) f Ek −2 0
2
2
To determine µ, we write the electron concentration as Ne and use
[ (
)
(
)]
∫ ∞
gµB B
gµB B
Ne =
dEρ(E) f Ek −+ f Ek +
.
2
2
0

(3.5)

(3.6)

(3.7)

Thus determined µ is the Fermi energy EF described in Sec. 3.1.1.
We then obtain the spin magnetic susceptibility of conduction electrons at T = 0 asPauli paramagnetic susceptibility
χPauli =( gµ )2B

2[2ρ(EF )].


(3.8)

The spin susceptibility of Fermi-degenerated free electrons is constant and proportional to the density of states at the
Fermi level. This is called Pauli paramagnetism. This result holds for general complicated band structures as long as
the system can be treated as a Fermi liquid.
E05-10

3.2 Landau diamagnetism
As we have seen as the Bohr-van Leeuwen theorem, that a classical system of charged particles cannot have magnetism.
In real systems of charged particles, even without orbital quantization by local potentials or spins, the quantization causes
magnetism. Let us have a look on that.

3.2.1 Landau quantization
Landau quantization is universally observed in the systems of free charged particles under magnetic ﬁeld. Particularly
in two-dimensional systems, it results in an extraordinary phenomenon called quantum Hall effect. Here we study its
effect on orbital magnetism. We directly treat the orbital motions and write the Hamiltonian asH =1 ∑(pi + eA)2 .
2m i

(3.9)

We take Landau gauge A = (0, Bx, 0) to express the magnetic ﬁeld B along z-axis. m is the electron rest mass.
Schrödinger equation for a single electron is
[
(
)2 ]
ℏ2
∂
∂2
∂2
eB
−
+ 2+
−i
x
ψ = Eψ.
2m ∂x2∂z∂yℏ(3.10)

The coordinate operator in the lhs of (3.10) is only x. Thus we assume plane wave along y and z, namely ψ =exp[i(ky y + kz z)]u(x). By substituting this to (3.10), we obtaine[()2 ] ()ℏ2 d 2 uℏ2 kz2eB−+ ky −
x u = E−u.
2m dx2ℏ
2m(3.11)

This is in the form of one-dimensional harmonic oscillator with the center coordinate
xc = ℏky /eB.
The frequency of oscillator ωc is

mωc2
(eB)2
=
2
2m

∴ ωc =(3.12)

eB.
m

This is called cyclotron frequency. The energy eigenvalues are()ℏ2 kz21ℏ2 kz2E(n, kz ) =+ n++ (2n + 1)µB B
ℏωc =
2m
2
2m

(3.13)(n = 0, 1, 2, · · · ).

(3.14)The motion along z-axis is free-electron like but in xy-plane the kinetic energy is discretely quantized. This is called
Landau quantization. The energy levels of harmonic oscillator indexed by n in (3.14) are called Landau levels.
It seems a bit strange that the solutions are strongly anisotropic although the system is uniform in xy-plane. This is
the result of selection of Landau gauge to solve the equation. The symmetrically localized solutions can be obtained, e.g.
by taking the symmetric gauge. These are degenerated for the same n, and with superpositions we can have eigenstateswith various distributions in xy-plane. In semi-classical approach, the Landau quantization can be viewed as the quantization of cyclotron motion through the spatial localization. However there still exists the freedom in taking the center ofcyclotron motion, that gives large degree of degeneracy and results in the eigenstates with various outlook.

E05-11

3.2.2 Orbital diamagnetism
We consider the normalization of eigenstates in the form of exp[i(ky y + kz z)]u(x) in the cube with side length L.
z-direction is the same as free electrons. From kz = (2π/L)nz (nz = 0, ±1, · · · ), Ez = ℏ2 kz2 /2m, possible number of√kz for kinetic energy along z-direction less than Ez is 2L 2mEz /h. Also for y, because the form is plane wave, fromthe periodic boundary condition
ky =

2π
ny
L

(ny = 0, ±1, ±2, · · · ).

(3.15)On the other hand, ky relates with the center xc of harmonic oscillator along x-direction as (3.12), then when xc is in theregion [−L/2, L/2],
−

L
ℏ
ℏ 2π
L
≤
ky =
ny ≤
2
eB
eB L
2

∴ |ny | ≤

eBL2
.
4πℏ

This means the degree of degeneracy of a single Landau level in xy-plane is eBL2 /h.
From the above, Ω(E), the number of states below the total energy E isΩ(E) =

n∑
max
√
L3 √
8meB
E − (2n + 1)µB B,
2
h
n=0

(

where
nmax = int

E − µB B
2(3.16)).

Because the density of states is given by dΩ/dE, the Free energy of the system is∫dΩ
F = N µ − 2kB T
ln{1 + exp[−(E − µ)/kB T ]}dE,
dE(3.17)(3.18)

where we consider the spin degree of freedom 2. The integral part is partially integrated as follows.
∫

(
)
∫
dΩ
1
exp[−(E − µ)/kB T ]ln{1 + exp[−(E − µ)/kB T ]}dE = − Ω(E) −dE
dE
kB T 1 + exp[−(E − µ)/kB T ]
]
∫ [∫
d
1
1
Ω(E)dEdE=
kB T
dE 1 + exp[(E − µ)/kB T ]
√
∫ n∑
max
d
1 2 8m eBL3
1
[E − (2n + 1)µB B]3/2=
dE.
kB T 3h2dE
1+exp[(E− µ)/kB T ]n=0

From µB = eℏ/2m we can rewrite F as∫
F = Ne µ − A

ϕ(E)

d
1
dE,
dE 1 + exp[(E − µ)/kB T ](3.19a)where
16L3 3/2
m (µB B)5/2 ,3π 2 ℏ3
(
)]3/2
n∑
max [
E
1
.
ϕ(E) =
− n+
2µB B
2
n=0
A=

Taking the limit B → 0, it becomes

F = Ne EF − Aϕ(EF ).

(3.19b)(3.19c)(3.20)

Here we use the asymptotic form for x ≫ 1
n∑
max [
n=0

(
)]3/2
2
1
1
≈ x5/2 − x1/2 + · · · .
x− n+
2
5
16
E05-12(3.21)

This is obtained by applying the Euler–Maclaurin formula to F (y) = (x − y)3/2 as follows.
n0∑

F (n + 1/2) ≈∫ n0 +1

dyF (y) −

0

n=0

1 ′
[F (n0 + 1) − F ′ (0)]242
1
≈ x5/2 − x1/2 .
5
16Hence we obtain the asymptotic expansion
2
ϕ(E) =
5

(

E
2µB B

)5/2

1
+
16

(

E
2µB B

)1/2
+ ··· ,

(3.22)

which gives the free energy as
F = const. −L3
ρ(EF )(µB B)2 + · · · .
3(3.23)

Then we obtain the orbital diamagnetic susceptibility of free electrons asLandau diamagnetic susceptibility
2χLandau = − ρ(EF )µ2B .
3(3.24)This is called Landau diamagnetism.
Taking the effective mass to the free electron rest mas, χLandau has the opposite sign to χPauli and the magnitude is
1/3. The total susceptibility is thenχ = χPauli + χLandau =4
ρ(EF )µ2B .
3(3.25)

Appendix 5A: Weak crystal ﬁeld approximation
In the lecture on 3d transition metal ions, we consider the approximation of a strong crystal ﬁeld (ligand ﬁeld), thatis, the one-electron states of 3d electron in the crystal ﬁeld are considered ﬁrst, and ﬁll them with electrons to consider
multi-electron states. On the other hand, there is also a method that ﬁrst assumes the LS multiplex term as we did in thecase of 4f lanthanoid ion, and then considers how this multi-electronic state splits in the crystal ﬁeld using point grouptheory. This is called weak crystal ﬁeld approximation. In Fig. 5A.1, we show how the state dn splits in an octahedral
potential.

Fig. 5A.1 Split of dn state in an octahedral crystal potential in weak crystal ﬁeld approximation.

E05-13References[1] Charles P. Slichter. Principles of Magnetic Resonance (Springer Series in Solid-State Sciences) by Charles P.
Slichter(2010-12-01). Springer, 1653.
[2] Marcus A. Hemminga and Lawrence Berliner. ESR Spectroscopy in Membrane Biophysics (Biological Magnetic
Resonance Book 27) (English Edition). Springer, 8 2007.
[3] M. H. L. Pryce. Proc. Phys. Soc., Vol. A63, p. 25, 1950.
[4] Mohammad Mostafanejad. Basics of the spin hamiltonian formalism. International Journal of Quantum Chemistry,
Vol. 114, No. 22, pp. 1495–1512, June 2014.
[5] 上村洸, 菅野暁, 田辺行人. 配位子場理論とその応用 (物理科学選書). 裳華房, 6 1969.
[6] Sundar R. Bairavarasu, Matthew E. Edwards, Medury D. Sastry, Tatiana Kukhtareva, Holger M. Jaenisch, Rastgo H.
Hawrami, Dimitrios Lianos, and Manmohan D. Aggarwal. Photo-EPR studies of photorefractive batio3 heavily
doped with cr3+ : evidence of photo-induced dissociation of cr3+ dimers. In Ruyan Guo, Shizhuo S. Yin, andFrancis T. S. Yu, editors, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device
Applications. SPIE, September 2007.
[7] R Böttcher, E Erdem, H T Langhammer, T Müller, and H-P Abicht. Incorporation of chromium into hexagonalbarium titanate: an electron paramagnetic resonance study. Journal of Physics: Condensed Matter, Vol. 17, No. 17,
pp. 2763–2774, April 2005.
[8] O. V. Lounasmaa. Experimental Principles and Methods Below 1K. Academic Pr, 6 1974.
[9] Koji Kamiya, Koichi Matsumoto, Takenori Numazawa, Shinji Masuyama, Hiroyuki Takeya, Akiko T. Saito, Naoya
Kumazawa, Kazumi Futatsuka, Keigo Matsunaga, Tsuyoshi Shirai, Suguru Takada, and Teruhito Iida. Active
magnetic regenerative refrigeration using superconducting solenoid for hydrogen liquefaction. Applied Physics
Express, Vol. 15, No. 5, p. 053001, January 2022.
[10] L. D. Landau. The theory of a fermi liquid. Sov. Phys. JETP, Vol. 3, pp. 920–925, 1957.
[11] Piers Coleman. Introduction to Many-Body Physics. Cambridge University Press, 11 2015.

E05-14

Lecture note
18th May (2022)

Magnetism (6)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

3.2.3 de Haas-van Alphen effect
When we derived Landau orbital diamagnetic susceptibility, we took the small ﬁeld limit, in that the Landau level
splitting ℏωc is much smaller than EF and applied an asymptotic formula. When the magnetic ﬁeld goes up to a comparatively large region *1 , the Landau quantization gives a dramatic effect on the magnetization. That is the oscillationof magnetization called the de Haas-van Alphen effect. In solids, the orbital diamagnetism strongly depends on the band
structure particularly around the Fermi surface. This is in strong contrast to the spin paramagnetism. Hence the relation
between the Pauli paramagnetism and the Landau diamagnetism shown in the previous subsection generally does nothold in solids. This sensitivity of diamagnetism to the band structure is applied for exploration of band structures.
We rewrite eq. (3.19) asFℏωc= µ − 3/2
ne
E
F

Z ∞
dE
0




3/2 
1
∂f
ℏωc
,
E− n+
−
2
∂E
n=0
X

(3.26)

where ne = Ne /L3 . Also, the relation 2µB B = ℏωc is used to restore ℏωc in preparation for the change in the effective
mass. The summation over n should be taken for positive arguments in the paretheses (· · · ). On the other hand, the energy
derivative of Fermi function −∂f ∂E approaches a delta function for T → 0. We guess, therefore, the magnetization
varies largely for the magnetic ﬁeld where the Landau levels (n+1/2)ℏωc coincide EF . This oscillation of magnetization
against the magnetic ﬁeld is called de Haas-van Alphen effect, dHvA effect.
The density of states of a one-dimensional system is givenbyℏ2 k 2
Ek =
,
2m

1 L
ρ1d (E) =L 2πr 2 −1
ℏ k
1
m
=
.
m
2πℏ 2E

By counting all these states, we get the total density of states
per spin as1
ρ(E) =
2πℏ

r

mX
1
p
.
2 n=0 (E − (n + 1/2)ℏωc(3.27)The summation over n should be taken for E > (n+1/2)ℏωc .
The density of states in (3.27) is divergent at E/ℏωc as shown
Fig. 3.2 Density of states is plotted against energy
measured by Landau level spacing ℏωc .

in Fig. 3.2. With increasing magnetic ﬁeld, the position of
EF decreases due to the increase in ℏωc . When EF passes a
position of divergence, a rapid rearrangement in electron pop-

ulation occurs and the thermodynamic functions including the magnetization get rapid changes.
To see how one can get the information of the Fermi surface, refer to Appendix 6B. In the lecture, I will introduce an
example of experiment on Thallium-based high-Tc cuprate[1].
*1 In usual metals, ℏωc of “strong magnetic ﬁeld” at an ordinary level does not go up to the level of Fermi energy. Hence even in this treatment,

we still use the condition ℏωc ≪ EF

E06-1

3.3 Orbital diamagnetism in graphene, graphite
What we have seen above is the magnetic response of the free electron system. The Pauli paramagnetism comes froma small shift of the spin bands on the energy axis. Hence we can guess that the same formula is applicable for band
electrons in metals as long as the band has no anomaly at the Fermi level.
On the other hand, the orbital diamagnetism largely gets the effect of the band structure. Even in the low ﬁeld limit,
which is the constraint for the Landau diamagnetism, the magnetic response of Bloch electrons is a difﬁcult problem. The
theoretical frameworks from the viewpoint of linear response (Kubo formula)[2] and so forth have long been tried. The
ﬁeld is still active and there have been reports on theoretical developments aiming at application to graphene from thisdepartment[3, 4, 5]. Here I would like to introduce shortly a characteristic example of graphene and multilayer graphene(thin graphite). In the beginning of the present chapter, I have introduced the fact that a graphite has a very large negative
orbital susceptibility. This is due to its characteristic band structure.

3.3.1 Orbital diamagnetism in graphene
We are familiar with graphite, e.g., as a material used in cores of pencils. It is said to be the most stable thermodynamically as elementally crystal of carbon. It’s crystal structure is a stack of honeycomb planes as shown in Fig. 3.3. A single
atomic layer of the graphite is called graphene, which can be extracted by exfoliation, grown by CVD, or thermalizationof SiC.
Carbon atoms in a plane of graphene are strongly connected to the neighbors by covalent bonds with no buckling.
Hence graphene conduction electrons in pz orbitals can be treated as a complete two-dimensional orbital system. And
just at EF in pure graphenes, the linear dispersions form crossing points called Dirac points as shown in Appendix 6C.
Hence the band is massless and gapless. The orbital susceptibility of graphene has long been calculated. In the simplest
model[6], it is given by (in cgs unit)χ(EF ) = −g v g s e 2  e 2
δ(EF ),
6πc(3.28)

where gv = 2 is the orbital degeneracy represented as K and K’ points in k-space, gs = 2 is the spin degeneracy, c is the
speed of light. Here EF is measured from the Dirac points, and the susceptibility in (3.28) is inﬁnite when EF is at the
Dirac points, and is zero elsewhere.
This can be roughly interpreted as follows. Let an electron be in a cyclotron motion (thus not at Dirac points) under amagnetic ﬁeld B. When B changes, an electric ﬁeld E is created as in eq. (1.28). In sec. 1.3.2, E accelerates the electron
resulting in the diamagnetism. In the present case, E gives an increase in the kinetic energy of the electron though that
does not enhance the velocity due to the linear dispersion. Hence the diamagnetic susceptibility is zero. However, when

Fig. 3.3 Illustration of graphite crystal structure. Honeycomb sheets made of carbon covalent bonds are stacked
with an in-plane half lattice constant shift by layer. That is,
the same 2-dimensional lattice appears alternatively, which
structure is called AB-stacking. The sheet-to-sheet is connected by the van der Waals coupling.

E06-2

the increase in the energy goes over a Dirac point, the velocity jumps, e.g., from −ve to +ve . The acceleration is inﬁnite
giving the divergence of the susceptibility at the Dirac point.
In the lecture I would like to introduce an experiment, in which the authors claim they have observed such peculiar
orbital diamagnetic susceptibility of graphene.

3.3.2 Diamagnetic susceptibility of multi-layer graphene
The extraordinary diamagnetism that causes magnetic levitation, and the strong magneto-optical response have long
been known, but clear interpretations have not been given yet. It has been said that we need to consider a more practicalmodel, in which experimental details like inhomogeneity in the ﬁeld[7].
In order to treat the space distribution of magnetic ﬁeld, we consider the Fourier expansion in x: B(r) = B(q) cos qx,of the ﬁeld along z-axis onto 2-dimensional system in xy-plane. The current distribution in response to B(r) generally
has the form of jy (r) = jy (q) sin qx[9]. The q-component of magnetization m(r) is obtained from the two-dimensional
current as この２次元電流から，磁化 m(r) の q 成分は
jy = −c∂m→ m(r) = m(q) cos qx∂xm(q) = −jy (q).
cq

The susceptibility χ(q) is deﬁned as m(q)/B(q). Then the response current of graphene to B(r) = B(q) cos qx is
jy (r) = −g v g s e2 v
B(q) sin qx.
16ℏc(3.29)The diamagnetic inductive current by this responding current is from Amperé’s law
Bind (r) = −αg B(r),αg =2πgv gs e2 v
≈ 4 × 10−5 ,16ℏc2(3.30)

which does not depend on the space distribution wavenumber q. Therefore any spatial distribution of magnetic ﬁeld
causes the inductive magnetic ﬁeld of −αg times the original ﬁeld and the theorem of superposition leads to the total
inductive ﬁeld of −αg times the total original ﬁeld.
As an example, let us place a magnetic charge qm in the region z > 0. The graphene at z = 0 partially screens the ﬁeld
to create the mirror charge −αq qm seen from z < 0, namely the ﬁeld caused by the mirror charge is superposed to theoriginal ﬁeld. The same for z > 0 and the ﬁeld created by the mirror charge of αq qm in z < 0 region is overlapped to theoriginal ﬁeld. The force, which the original magnetic charge in Fig. 3.4(a) gets from the graphene, is calculated from themirror charge in Fig. 3.4(c).
Let us consider the case a permanent magnet approaches a graphene. The magnet is a half-inﬁnite cylinder with aradius a having the edge magnetic charge density σm . When the edge reaches the graphene, d = 0, the force given to2
the magnet per area is 2παg σm. In the case of Neodymium magnet, σm can be around 1 T. The force is then 0.16 dyne,

which is surprisingly large for a single atomic layer.

Fig. 3.4 Mirror magnetic charge
caused by a graphene.
Thegraphene is illustrated as the blackline at z = 0. (a) Magnetic ﬁeld
by a magnetic charge. (b) Inducted ﬁeld in z < 0, (c) and inz > 0. From [8].

E06-3

Fig. 3.5 (a) Band structures and (b) orbital diamagnetic susceptibilities χ(EF ) for single to four layer graphenes. Γis the width of energy ﬂuctuation caused by impurities. From [8, 10].

In an ordinary 2-dimensional metal with an effective mass m∗ , the susceptibility is a constant χ0 = −e2 /(24πm∗ c2 )in the long wavelength limit. Let us take an example of GaAs 2-dimensional electrons with m∗ = 0.067m0 . Then theratio of repulsive force fc to that of graphene fg is fg /fc ≈ a/(0.01 nm). The radius is in cm order then the ratio goesup to 9-digits.
Next we go to multi-layer graphenes, which can be seen as thin ﬁlms of graphite[10]. They have the AB stacking as
in Fig. 3.3 with a weak interlayer coupling with the coupling energy of 0.4 eV. The structure with more layers than two,can be treated as a repetition of the AB stack and the calculation can be reduced to the case of the bilayer graphene. A
single-layer graphene has gapless linear dispersion both at K and K’. Also in a bilayer graphene, a pair of bands has azero-gap with ﬁnite effective masses, the other pair has a gap due to the inter-layer coupling. An N = 2M layer graphene
has M sets of bilayer-type bands. When N = 2M + 1, a set of single-layer bands is added.
Figure 3.5 shows such calculation of the band structures and the orbital diamagnetic susceptibilities for single tofour layer graphenes. The band with a Dirac point appears for an odd number of N giving a large contribution to thediamagnetic susceptibility. For an even N , that contribution disappears. However the further increase in the number oflayers results in the increase in the diamagnetic susceptibility. This increase is thought to lead to the large diamagnetismin graphite.

E06-4

Chapter 4Interaction between SpinsFighting tops
https://www.youtube.com/watch?v=2WaU7NDOHLQ

We have looked for “entities” which work as magnetic dipoles in materials and found in the quantum theory, thatthe spins and the orbitals are working as magnetic dipoles namely magnetic moments in various forms. However themacroscopic moments never appear without applying external magnetic ﬁeld due to the randomness in the direction of
microscopic moments. In ferromagnetic materials the microscopic moments – let us call them spins – are aligned in thesame direction at zero magnetic ﬁeld and at ﬁnite temperatures. This fact indicates that, as we have seen in the magnetic
refrigeration, the entropy is largely reduced and there should be a large decrease in the internal energy to compensate
that. In other words, there should be some interaction that decreases the total energy. Let us examine such possibilities.

4.1 Exchange interaction
In chapter 1, we introduced the classical interaction between two magnetic dipoles (spins) µ1 , µ2 , namely the moments
feel their magnetic ﬁeld each other. Let r12 be the vector connecting the two moments, then the potential is given by *2µ0 µ 1 · µ 2(r1 · r12 )(r2 · r12 )U (µ1 , µ2 , r12 ) =
−3
.
(4.1)
3
5
4π
r12
r12
It is easy to guess from the analogy of bar magnets that the stable conﬁguration is that the two spins are in-line. However
this has the quantitative problem. Let µ1,2 be 5µB , r12 be 200 pm (typical lattice constant), then U is about 2 K. Hence
this interaction cannot explain real ferromagnets, which keep alignments of spins above room temperature quantitatively.
Based on the quantum theory, Heisenberg showed the possibility of far-stronger mutual interaction between spins, thatcomes from a characteristic quantum effect[11]. It is now known through various researches that the direct exchangeinteraction, which Heisenberg claimed to be the source of ferromagnetism, cannot explain real ferromagnetism in materials. However, the concept of exchange interaction is still used in the present understandings. Here let us introduce thenaive direct exchange interaction.

*2 In eq. (1.7), we used E-H formulation. Here we restore it to E-B formulation (i.e., SI unit system).

E06-5

4.1.1 Heitler-London approximation
In the ground state of Hydrogen molecule, two electrons with ↑ and ↓ are accommodated in the bonding orbital. Hencethe spin is zero and the orbital diamagnetism appears (molecular orbital approximation, MO). However, here we consider
the case that the inter-atom distance is larger and instead of considering bonding and anti-bonding orbitals, we consider
atomic orbitals φa , φb and the two-body wavefunction is composed with keeping the Fermi statistics (Heitler-London
approximation). In this approximation, the wavefunction of two electrons is written in the form of Slater determinant:1 φa (r1 )χa (s1 )Ψ= √N φa (r2 )χa (s2 )φb (r1 )χb (s1 ),φb (r2 )χb (s2 )(4.2)

where N is a normalization constant, (ri , si ) are space and spin coordinates of i-th electron. si corresponds to zcomponent of spin and can take one of two values ±1/2. We write the state of the spin pointing +z as α(s), in whichχ(s) isχ(1/2) = 1,

χ(−1/2) = 0.

(4.3)

χ(−1/2) = 1.

(4.4)

On the other hand, the −z pointing state β(s) is given asχ(1/2) = 0,Pauli’s exclusion principle is fulﬁlled asΨ(r1 , s1 ; r1 , s1 ) = 0,

Ψ(r1 , s1 ; r2 , s2 ) = −Ψ(r2 , s2 ; r1 , s1 ).

(4.5)

Ψ takes argument of spin functions (χa , χb ), and we can classify Ψ by the spin states (α, β) as {Ψαα , Ψαβ , Ψβα , Ψββ }.
We take this as a basis and consider the expression of the interaction Hamiltonian Hint . As an example of the matrix
elements, ⟨αα|Hint |αα⟩ has two terms:
XZ⟨αα|Hint |αα⟩ =dr1 dr2 Ψ∗αα Hint Ψααs1 ,s2Z=

dr1 dr2 φ∗a (r1 )φ∗b (r2 )Hint φa (r1 )φb (r2 ) −Z

dr1 dr2 φ∗a (r1 )φ∗b (r2 )Hint φb (r1 )φa (r2 ),(4.6)

where we take N = 2, ⟨φa |φb ⟩ = 0. The second term in rhs of (4.6) is a matrix element between the states in which theelectrons are exchanged, and called exchange integral. This is essentially the same as J(m1 , m2 ) in eq. (2.26). In Ch.2,
we derived Hund’s rule from this integral. The direct exchange interaction is essentially the same. We write the ﬁrst andthe second term in rhs of eq. (4.6) as Kab and Jab . The 4×4 matrix elements are as follows.
αα

αβ

βα

ββ

αα

Kab − Jab

0

0

0

αβ

0

Kab

−Jab

0

βα

0

−Jab

Kab

0

ββ

0

0

0

Kab − Jab

(4.7)

This can be easily diagonalized and the eigenfunctions are
Ψαα






1
√ (Ψαβ + Ψβα ) (s1 + s2 = 1),

2


Ψββ

1
√ (Ψαβ − Ψβα ) (s1 + s2 = 0).
2

(4.8)

The three states for s1 + s2 = 1 are spin triplet and the one for s1 + s2 = 0 is spin singlet. The eigenenergy of theformer is Kab − Jab and that of the latter is Kab − Jab . The spin states thus give the difference in the energy. When
E06-6

Jab is positive, the spin parallel state has a lower energy giving ferromagnetic interaction while the interaction is antiferromagnetic when Jab is negative. The situation can be viewed as the exchange integral creates the interaction between
the spins and we call the interaction exchange interaction. Though Jab appears from the integration of orbital states, thesymmetry of the electronic states requires the interaction between spins.
Then we look for an effective Hamiltonian composed of spin operators just like the spin Hamiltonian introduced in
Ch.2 to analyze EPR experiments. First we introduce spin operators sa and sb , which operate on the states a and b
respectively (don’t get confused with s1 , s2 ). Because they commute each other,2sa · sb = (sa + sb )2 − s2a − s2b = S 2 − s2a − s2b .

(4.9)⟨↑↑ |S 2 | ↑↑⟩ = S(S + 1) = 2, S 2 |↑↓⟩ = 0,


1 1
3
2
2
sa = s b =+1 = ,
2 2
4(4.10)

Here we use(4.11)and calculate the elements of operator (1 + 4sa · sb )/2 to obtain
3
1
1= =⇒ (1 + 4sa · sb ) = +1,
4
2
2
3
3
1
(↑↓) → 2sa · sb = 2 − 2 × = − =⇒ (1 + 4sa · sb ) = −1.
422(↑↑) → 2sa · sb = 2 − 2 ×

Therefore we can adopt
1
Hint = Kab − Jab (1 + 4sa · sb )2

(4.12)as an effective Hamiltonian.
Then we expand the above concept to the interaction between general spins Si indexed by i and formally extract the
spin part to obtain Heisenberg HamiltonianHeisenberg Hamiltonian
H = −2X

Jij Si · Sj .

(4.13)

⟨i,j⟩





This is an important basics for us to treat various phenomena originated from the interaction between spins.
For exchange integral, when the interaction is the Coulomb repulsion,
Jab =

e2
4πϵ0

Z

dr1 dr2 φ∗a (r1 )φ∗b (r2 )1
φb (r1 )φa (r2 )r12(4.14)is always positive, which can be proven as eq. (2.26). And Jab can go up to 0.1 eV depending on the way of estimation,
hence might explain the room temperature ferromagnetism. In the above we see that the Coulomb repulsion causes astrong ferromagnetic interaction between spins in Heitler-London approximation. This is called direct exchange interaction. However, here we must notice that if we adopt the molecular orbital method, the interaction in the form of
Heisenberg Hamiltonian can be derived with negative J, that is the anti-ferromagnetic interaction. This can be moreeasily understood from the general theory that in a general Schrödinger equation, the ground state has no degeneracy andthe wavefunction has no node[12]. Hence the orbital part of wavefunction is symmetric resulting in the antisymmetricspin part.

E06-7

4.1.2 Exchange interaction in the presence of tunneling
As an improvement of Heitler-London approximation, we consider the electronic state, in which two electrons areaccommodated in a single atom as a possible conﬁguration. In this treatment, a small amplitude of wavefunction
1 φa (r1 )χa (s1 )Ψ′ = √
N ′ φa (r2 )χa (s2 )φa (r1 )χ′a (s1 )φa (r2 )χ′a (s2 )(4.15)

is overlapped to the one in (4.2). The transition of Ψ → Ψ′ means tunneling of an electron φb χb → φa χ′a . Hence forsuch a superposition to occur, ⟨Ψ|H |Ψ′ ⟩ ̸= 0. For that, due to Pauli’s exclusion principle, χa and χb should be the spins
opposite to each other. When sa and sb are anti-parallel, this hopping process leads to the energy decrease
Wab = −

1| ⟨Ψ′ |H |Ψ⟩ |2∆E(4.16)

in the second order perturbation. For parallel spins, there is no such energy decrease. We thus write this part formally1(1 − 4sa · sb )Wab ,2(4.17)

which is in the same form as eq. (4.12). In summary we reach
′
Hint
=

1
(−Jab + Wab ) − 2(Jab + Wab )sa · sb .
2

(4.18)

Wab is negative and when Jab + Wab < 0 the total interaction becomes antiferromagnetic. We have conﬁrmed HeitlerLondon approximation overestimates the ferromagnetic interaction.

Appendix 6A: Aharonov-Bohm phase and degeneracy of Landau levels
I would like to introduce the way to memorize the degree of Landau level
degeneracy (the number of states per unit area). Of course this is just an
example, and you can ﬁnd your own way. In the existence of magnetic ﬁeld,
the momentum ℏk is modiﬁed toe ℏk → ℏk + eA = ℏ k + A .
ℏThis gives the phase evolution θ when a plane wave exp[ik · r] propagatesin space from a point P1 to P2 asZ P2 
θ12 =

k+
P1

e 
A · dr(I) =ℏZ P2

k · dr(I) +

P1

e
ℏ

Z P2
P1

(k)

(A)

A · dr(I) = θ12(I) + θ12(I) .

(6A.1)The sufﬁx I means the path signed as I in the ﬁgure. The ﬁrst term in the rhs is the ordinary kinetic phase and the second
term expresses the effect of magnetic ﬁeld. The latter is called Aharonov-Bohm (AB) phase.
In path II, the kinetic phase differs from that in path I by the difference in the length. The AB phase in path II is also
different from that in I. This can be understood by considering the route going back from P2 to P1 on path II. That is
#
"Z
Z P2
I
Z
P2
e
e
h
e
Φ
∆θAB =
A · dr(II) =(6A.2)
A · dr(I) +A · dr =rotAdσ = 2π , ϕ0 = .
ℏ P1
ℏ
ℏ
ϕ
e
0
P1
S
Here ϕ0 is called ﬂux quantum, which has the form of the ratio of h to e. This is an easy-to-memorize form. A physicalmeaning is that 2π of AB phase is given to an electron going around this amount of ﬂux, and this is the condition ofquantization in the electron loop.
E06-8

The number of states in the magnetic ﬁeld of magnetic ﬂux density B in two-dimensional system is given by the ﬂux
per unit area (i.e. B) divided by the ﬂux quantum. That is,
N (B) =

B
eB
eBm
m
m
=
=
= 2 (hωc ) =(ℏωc ).
h/e
h
hm
h
2πℏ2

(6A.3)

Hence we can see the number of states per Landau level eB/h and the two-dimensional density of states m/2πℏ2 in avery clear way.

Appendix 6B: Contribution of k-slab to the dHvA effect
There are various equivalent way to explain the principle of the dHvA effect. In the explanation from the density ofstates as in the text, it is rather difﬁcult to see the effect contains the information of k-space. Hence here we explain
the effect in the way given in Ref. [13]. Here we consider free electrons with an isotropic effective mass for simplicity.
Electrons are free in z-direction (ﬁeld direction) and Landau-quantized in xy plane as in eq. (3.14). Hence we deﬁnequasi-Fermi energy as the kinetic energy in xy-plane for a ﬁxed kz , that isEF′ ≡ EF −

ℏ2 kz2.
2m(6B.1)

And we treat the system as a set of two-dimensional electrons with the Fermi energy of EF′ under Landau quantization.
In k-space, each kz is assigned to such a virtual two-dimensional system.
Then we consider a slab with thickness δkz in k-space corresponding to a kz . We call the region k-slab. The densityof states in a k-slab “per magnetic ﬂux density” ξ is given by
1 L
e
eB 1
ξ=
δkz
=
δkz
L 2π
h B
4π 2 ℏ



δkz 1
=
2π ϕ0


,

(6B.2)

where we also use things mentioned in App. 6A. I have put a comment on this quantity as a formula in the last parentheses.
ϕ0 ≡ h/e is ﬂux quantum introduced in App. 6A. At absolute zero, the Landau levels corresponding to the integer q
which satisﬁes (q + 1/2)ℏωc ≤ EF′ are occupied. Let us write the maximum integer in q as qmax and the number ofoccupied Landau levels is qmax + 1. Then the number of electrons belong to this k-slab isne (kz ) = (qmax + 1)ξB

(6B.3)

per unit area in the real space.
With increasing B, ne increases linearly in accordance with (6B.3), and when B exceeds the value determined by thecondition
qmax +

1E′
mEF′ 1
= F =
,
2
ℏωc
ℏe B

(6B.4)

qmax decreases by one and ne discretely decreases. Namely, ne (kz ) oscillates periodically against 1/B and the amplitude(the amount of dropping at the condition (6B.4)) increases with B as ξB though the center of the oscillation is the electron
concentration of virtual 2-dimensional electrons (let us write it ne0 ) in the k-slab before the application of magnetic ﬁeld.
The behavior is drawn in Fig. 6B.1. As shown in the ﬁgure, the electronic states of number eB/h at zero ﬁeld are assigned
to a Landau level (App. 6A). In the oscillation, at the magnetic ﬁeld ne (B) = ne0 , where the electron concentration hits
the center, (qmax + 1)(eB/h) should be equal to ne0 . We write qmax + 1 at such points as ν, the value of magnetic ﬁeld
as Bν . Then they are in the relation
Bν =

1 ne0
2π ne0
=
ϕ0
ξ ν
δkz
ν

E06-9

(ν = 1, 2, · · · ).

(6B.5)

Fig. 6B.1 Illustration of relations between the Landau level distance ℏωc , the two dimensional electron concentration
ne , and jumping at which qmax changes. The lower shows corresponding energy diagrams (occupied energy levels inzero ﬁeld is in orange color.

And the boundaries between different values of qmax are
Bν± =

mEF′
1
.
ℏe ν ± 1/2(6B.6)The energy Uν0 of electrons in the k-slab corresponding to this Bν is expressed asUν0 = ξBν ℏωc

ν−1
X
q=0

1
q+
2


+ ne

ξBν ℏωcν 2ℏ2 kz2ℏ2 kz2=
ν + ne2m
22m=ℏωcν 2h2ℏ2 kz2ℏ2 kz2
ne0 =
ne0 .
ne0 +
n2e0 +
2ξBν
2m
2mδkz
2m

(6B.7)

This does not depend on the magnetic ﬁeld and is the same as the sum of energies in k-slabs below EF . In the region in
the magnetic ﬁeld [Bν+ , Bν− ], this expression holds with replacing ne0 with ne . We thus consider the quantityU=h2ℏ2 kz2+ (ne0 − ne )EF′ .
n2e + ne2mδkz
2m

(6B.8)In the rhs, the ﬁrst two terms are the extension of (6B.7) and the energy of electrons in k-slab. In the third term, theenergies coming in/out the k-slab at EF′ . This term keeps U continuous even at the magnetic ﬁeld of eq. (6B.6), where
ne gets a gap. Now we write EF′ ash2 ne0eBν 2π h ne0=.
m δkz e Bνm δkz

(6B.9)h2h2(n2e − n2e0 ) + EF′ (ne0 − ne ) =(ne − ne0 )2 ,
2mδkz
2mδkz

(6B.10)EF′ = ℏωcν ν = ℏThen the variation in U is
δU = U − Uν0 =which is more than or equal to zero.

The contribution of electrons in this k-slab to the magnetization isδM = −
dne /dB is from eq. (6B.3)

∂U
h2
dne
=−
(ne − ne0 )
.
∂B
mδkz
dB

(6B.11)dne
E′= νξ ≃ F ξ.
dBℏωc

(6B.12)

E06-10Then the contribution is obtained asδM ≃ −

EF′(ne − ne0 ).
B(6B.13)

While B in the denominator increases linearly, ne − ne0 strongly oscillates resulting in the oscillation of magnetic
susceptibility.
In order to see the behavior of the total magnetization, the contribution in eq. (6B.13) should be summed up over kz .
For that, ﬁrst δM is expanded in a Fourier series against the axis of B −1 as follows.
δM = δkz∞
X

Ap sin px,

x = 2π

p=1

EF′
.
ℏωc

(6B.14)

We apply a mathematical identity for −π < x < π
x=e∞X(−1)n−1nn=1

and rewriteδM = −

sin nx,(6B.15)X
1
1
sin pxξEF′ x = EF′
(−1)p
.
2π
π
p
p

(6B.16)

Therefore the expansion coefﬁcients are obtained as
Ap =

1 ′
ξ
eEF′
EF (−1)p.
= (−1)p
pπ
δkz
4pπ 3

(6B.17)

From the above, the summation over kz can be written in the form of integration:M=e X (−1)p4π 3 pp

Z kF−kF

dkz · EF′ sin



pπ
ℏωc


EF −

ℏ2 kz22m

.

(6B.18)Here, though the magnetic ﬁeld is comparatively strong, we assume the condition ℏωc ≪ EF still holds. Then in the
integrand in eq. (6B.18), EF′ varies in the section [−kF , kF ] as a parabola with the maximum at kz = 0. On the other
hand, the sine function rapidly oscillates against kz . As a result the integration cancels out other than the region aroundkz = 0, where dEF′ /dkz ∼ 0. Hence EF′ outside the sine function can be replaced with EF . Further, applying the identity
Z ∞
cos
0

π 2
x dx =
2

Z ∞
sin
0

1
π 2
x dx = ,2
2(6B.19)the integration in (6B.18) is calculated to be

EF

ℏωc m
2p

1/2


sin

2πpEF
π
−
ℏωc
4


.

(6B.20)

Then we can write down the magnetization as
M=



2πEF
EF e3/2 (ℏB)1/2 X (−1)p
π
sin
p
−
.
4π 3
ℏωc
4
p3/2
p

(6B.21)

The above discussion is for an ideal metal with a spherical Fermi surface. But this can be extended to general Fermi
surfaces. Even in the general case, the dHvA oscillation is dominated by the region where dEF′ /kz ≈ 0. Hence themagnetic ﬁeld angle dependence of the dHvA oscillation (amplitude, period, etc.) gives detailed information on the
Fermi surface.

E06-11

Appendix 6C: Band structure of graphene
One of the ways to form a two dimensional electron system is to utilize two-dimensional crystals (two-dimensional materials). Graphene is the representative two-dimensional material. Graphene provides a good expample for the applicationof tight-binding calculation and we would like to see how the things go in a practical (though simplest) example.
The crystal structure of single-layer graphene is show in Fig. 6C.1(a), which is a simple honeycomb structure of carbon
atoms. The diamond drawn in the ﬁgure is the unit cell and the primitive lattice vectors and the primitive reciprocal lattice
vectors are written as
√
a1 =


3a/2
,
a/2

a2 =

 
0
,
a


b1 =

√ 
4π/ 3a
,
0


b2 =

√ 
−2π/ 3a.
2π/a(6C.1)

Henceforce we calculate the electronic states of graphene under simplest approximation. Because the approximationis rough, quantitative comparison with experiments is difﬁcult. However, the results help understanding properties ofgraphene, e.g. the Dirac points appear at the Fermi level in pure graphene. Carbon belongs to group-IV and the outmost
electrons exist in the orbitals 2s, 2px , 2py , 2pz . It is easy to see that these orbitals form sp2 -hybrids and the electronic
states separate to σ-electrons (sp2 ) and π-electrons (pz ). σ-electrons form the honeycome through covalent bonding andthe energy bands lie at low energy region. Then the electronic states placed around the Fermi level are π-electrons. Hence
we consider Schrödinger equation on π-electrons on the honeycomb lattice.
We write the equation asψ = H ψ,(6C.2)

and as Fig. 6C.1(a), we separate the lattice sites to A-sites and B-sites on different sub-lattices. We consider a kind of

y

ky

A

a2

b2
K
B

a1

x

M
K

b1

G

(a)

(b)

Fig. 6C.1 (a) Two dimensional cryatal structure of graphene. Carbon atoms form a honeycomb lattice. It can be also
viewed as an overlap of two face-centered square lattices placed at A and B positions. (b) Reciprocal lattice of (a).
b1 , b2 are the primitive reciprocal lattice vector corresponding to a1 , a2 . The centtral point of the ﬁrst Brillouin zoneis Γ-point and as other points with high symmetries, K-point and M-point are indicated in the ﬁgure.

E06-12

kx

y

a
a ö
æ
, ÷
çè 2 3 2 ø

d2
d1

a
a ö
æ
,ç÷
2 ø
2 3
è

x
æ
ç
è

a
ö
,0 ÷
3
ø

d3
Fig. 6C.2 Vectors indicating three directional couplings between nearest neighbor carbon atoms.

tight-binding approximation between the two-sites. That isψ = ζA ψA + ζB ψB ,X
ψA =exp(ikrj )ϕ(r − rj ),(6C.3)(6C.4a)

j∈A

ψB =

X

exp(ikrj )ϕ(r − rj ),(6C.4b)j∈Bwhere ϕ(r) is atomic wavefunction of π-electrons, rj are the lattice points. Here we write the matrix elements of the
Hamiltonian between the each sub-lattice wavefunctions as∗
HAA = ⟨ψA |H |ψA ⟩, HBB = ⟨ψB |H |ψB ⟩, HAB = HBA= ⟨ψA |H |ψB ⟩.

(6C.5)

And the number of atoms in the system is 2N , that is⟨ψA |ψA ⟩ = ⟨ψB |ψB ⟩ = N.

(6C.6)

Let ⟨ψA |ψB ⟩ be zero. We substitute (6C.3) to (6C.2). The condition of have non-trivial (ζA , ζB ) givies the cecularequation

Lastly

HAA − N E
HBA

HAB
= 0.
HBB − N Ep
E = (2N )−1 HAA + HBB ± (HAA − HBB )2 + 4|HAB |2 ≡ hAA ± |hAB |,(6C.7)(6C.8)

where we have used HAA = HBB , which comes from the symmetry, and we use lower cases for the quantities per atomwith being devided by (2N )−1 .
HAB =

X

exp [ik(rj − rl )] ⟨ϕ(r − rl )|H |ϕ(r − rj )⟩r .

(6C.9)

l∈A,j∈B

We further approximate that the off-diagonal matrix elements of H just exist between the nearest neighbor sites. For the
calculation we take the atom indicated as A in Fig. 6C.1(a) as the center atom. The vectors from A to the nearest neighboratoms 1, 2, 3 are di (i = 1, 2, 3) respectively. As is apparent from the ﬁgure,




kx a
kx
ky
ky
kx
k · d1 = √ , k · d2 = − √ +a, k · d3 = − √ −
a,
2
2
3
2 3
2 3

(6C.10)where a = |a1 | = |a2 |. The terms ⟨ϕ(r − rl )|H |ϕ(r − rj )⟩r should be equal due to the symmetry and we write it as ξ.
Consequently the residual resonant integral from the crystal structure is the repetition of the above and3X
hAB = exp(ik · dj ) ξ.
j=1

E06-13(6C.11)

E

kx
ky
Fig. 6C.3 Left: surface plot of eq.(6C.12). The ﬁgure shows theappearance of Dirac points, where the vertices of energy corns crashat the K-point. Upper: Schematical drawing of a Dirac point.

Substituting eqs.(6C.10), (6C.11) into eq.(6C.8), we get the following expressio for the energy.
s
√
3kx a
ky a
ky a
cos
+ 4 cos2.
E = hAA ± ξ 1 + 4 cos
2
2
2(6C.12)The second term is the perturbation from the nearest neighbor resonant integral, which vanishes at K-point in the reciprocal space(kx , ky ) =

0, ±

4π
3a




,

2π
2π
√ ,±
3a
3a




,

2π
2π
−√ , ±
3a3a
.

(6C.13)

We write ky = 4π/3a and around kx = 0(one of the K-points), eq. (6C.12) can be approximated as4π
E kx ,3a√≈ hAA +

3ξa|kx |.
2(6C.14)

Namely, at the K-point the upper band has a lower pointed shape. Because the same for the lower band and as a result,at the K-point, as shown in Fig. 6C.3, the band structure called Dirac point, which has no energy gap, no effective mass,
appears.
Equation (6C.12) is for a very simpliﬁed model. Just like a cosine band appeared in the tight-binding model in onedimension, the model itself does not have realistic meaning. However the model tells that the reason why we have theDirac points at K-points is that the exsittence of three equivalent resonant integrals in eq. (6C.11). The inference holds forthe band calculation with any level precision since it is based on the symmetry. That meas the K-points in real graphene
are really Dirac points.

References[1] P M C Rourke, A F Bangura, T M Benseman, M Matusiak, J R Cooper, A Carrington, and N E Hussey. A detailed
de haas-van alphen effect study of the overdoped cuprate tl2 ba2 cuo6+δ . New Journal of Physics, Vol. 12, No. 10, p.
105009, October 2010.
[2] Hidetoshi Fukuyama. Theory of orbital magnetism of bloch electrons: Coulomb interactions. Progress of Theoretical Physics, Vol. 45, No. 3, pp. 704–729, March 1971.

E06-14

[3] Masao Ogata and Hidetoshi Fukuyama. Orbital magnetism of bloch electrons i. general formula. Journal of the
Physical Society of Japan, Vol. 84, No. 12, p. 124708, December 2015.
[4] Masao Ogata. Orbital magnetism of bloch electrons: II. application to single-band models and corrections tolandau–peierls susceptibility. Journal of the Physical Society of Japan, Vol. 85, No. 6, p. 064709, June 2016.
[5] Masao Ogata. Orbital magnetism of bloch electrons: III. application to graphene. Journal of the Physical Societyof Japan, Vol. 85, No. 10, p. 104708, October 2016.
[6] J. W. McClure. Diamagnetism of graphite. Phys. Rev., Vol. 104, pp. 666–671, Nov 1956.
[7] Bogdan Semenenko and Pablo Esquinazi. Diamagnetism of bulk graphite revised. Magnetochemistry, Vol. 4, No. 4,
p. 52, November 2018.
[8] 越野幹人. グラフェンにおける巨大な軌道反磁性 (最近の研究から). 日本物理学会誌, Vol. 65, No. 1, pp. 21–25,
2010.
[9] Mikito Koshino, Yasunori Arimura, and Tsuneya Ando. Magnetic ﬁeld screening and mirroring in graphene. Phys.
Rev. Lett., Vol. 102, p. 177203, Apr 2009.
[10] Mikito Koshino and Tsuneya Ando. Orbital diamagnetism in multilayer graphenes: Systematic study with theeffective mass approximation. Phys. Rev. B, Vol. 76, p. 085425, Aug 2007.
[11] W. Heisenberg. Mehrkörperproblem und resonanz in der quantenmechanik. Zeitschrift für Physik, Vol. 38, No. 6-7,
pp. 411–426, June 1926.
[12] L D Landau and E. M. Lifshitz. Quantum Mechanics: Non-Relativistic Theory (English Edition). ButterworthHeinemann, 12 1981.
[13] Charles Kittel. Quantum Theory of Solids 2E Rev P. WILEY, 3 1987.

E06-15

Lecture note
25th May (2022)

Magnetism (7)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Lat week, we show that a ferromagnetic exchange interaction appears in Heitler-London approximation though a
hopping of electrons between the sites cases anti-ferromagnetic exchange interaction. The exchange interaction due
to the transfer of electrons is a kind of kinetic exchange interaction, which often gives anti-ferromagnetic interaction
though not always *1 . Also, we found that the spin-spin interaction is always anti-ferromagnetic in a pure two-atom-model*2 . From the numerical simulations so far, they have found no realistic example in which the direct exchange interaction

causes ferromagnetism. However we extracted the concept of direct exchange interaction from the HL approximationand inferred that some “exchange interaction” is working in any magnetic materials. We thus introduced Heisenberg
Hamiltonian (Heisenberg model). You may wonder why this kind of change in the related freedoms is possible. This canbe understood in the context of quantum entanglement as stated in Appendix 7A. In the appendix, it is shown that thequantum entanglement is not a pure mathematical notion but a real physical phenomenon, useful in the experiment.

4.1.3 Hubbard model
The mechanism of anti-ferromagnetic exchange interaction in electron transfer can also be seen in the following simple
model. Let us consider two sites (i, j) and write the electronic states as |n; m⟩ (“;” separates the parameters of two states).
Electron hopping between the two sites is taken into account and the hopping operator is written as t(a†iσ ajσ + h.c.). Dueto the Pauli’s exclusion principle, two electrons occupying a single site are limited to spin up-down pair. Such states havehigher energy due to the on-site repulsion and can exist as intermediate states during hopping process. We write thisincrease in the energy in the intermediate states as U , namely
U = E(|0; σ, −σ⟩) − E(|σ; −σ⟩).

(4.19)The hopping process, however decreases the energy in the second perturbation and the amplitude of decrease is about|t|2 /U . The above simple model is called Hubbard model, in which the hopping amplitude with the nearest neighboris t and the electron gets on-site repulsion U for double occupation. Here we introduce the simplest two-site model but
extended ones are one of the most important model in exploration of magnetism. With the notation niσ = a†iσ aiσ , themodel can be expressed in Hamiltonian form as
X †H =t(a1σ a2σ + a†2σ a1σ ) + U (n1↑ n1↓ + n2↑ n2↓ ).

(4.20)σ=↑↓We limit the electron number as ne = 2 then the available states are11|↑↓; 0⟩ , |0; ↑↓⟩ , |↑; ↑⟩ , √ (|↑; ↓⟩ + |↓; ↑⟩), |↓; ↓⟩ , √ (|↑; ↓⟩ − |↓; ↑⟩).
2
2
The operators of spin at each site, total spin, and total electron number, are deﬁned asXXX † σ 
aiσ′ , S =aiσsi , N =
niσ ,
si =
′
2 σσ
′
i=1,2
i,σ

(4.21)(4.22)

σσ*1 Superexchange interaction, which we will see in the next section, is also a kind of kinetic exchange interaction. But sometimes it gives ferro-magnetic interaction[1, 2].
*2 Then you might ask how about the Hund’s rule?

This is a natural question long been addressed. Actually many textbooks on the physicalchemistry say that “Hund’s rule is empirical rules that cannot be mathematically proved.” In large scale numerical calculations, it turned out thatthe disturbance of nuclear-potential screening by core electrons has larger effect than the Pauli exclusion principle. This is an example and there
are various new ﬁndings[3, 4]. Anyway we need more study in this ﬁeld.

E07-1

No.

S

Sz

E

1

0

0

U


2

1+

1
a

1−

1
a


3

4

1

+1

Eigenstate




U
2
U
2

1
√ (|↑↓; 0⟩ − |0; ↑↓⟩)
2
r
√
1+a
1−a
(|↑↓; 0⟩ + |0; ↑↓⟩) +|0, 0⟩
2
2
r
√
1+a
1−a
|0, 0⟩ −(|↑↓; 0⟩ + |0; ↑↓⟩)
2
2

0

Tab. 4.1 Eigenenergies of twoelectron, two-site Hubbard modeleq. (4.20). a is deﬁned as a−2 =1 + (4t/U )2 .

|1, +1⟩

50|1, 0⟩

6−1|1, −1⟩

which commute with H and S 2 , Sz , N can be good quantum numbers. For N = 2, we can compose 6 eigenstatescommon for these operators from (4.21). The eigenergies of them are listed in Tab. 4.1 with deﬁnition of a−2 = 1 +(4t/U )2 . When the Coulomb repulsion is sufﬁciently larger than the hopping effect t/U ≪ 1, from this table and from
calculation similar to that for the direct exchange interaction, the effective spin Hamiltonian is obtained as


1
4t2
Heﬀ = −J s1 · s2 −,, J =−
4U(4.23)

which again gives an anti-ferromagnetic exchange interaction.

4.2 Superexchange interaction
There are many magnetic materials, in which negative ions (anions) with closed shell electronic structures exists between magnetic positive ions. Figure 4.1(a) shows the crystal structure of perovskite-type KFeF3 , which is an antiferromagnet with the Néel temperature (explained later) 173 K. In the structure, F−1 ’s exist between Fe2+ ’s The interaction mechanism between such magnetic ions was proposed long time ago as superexchange interaction. As named,
it is a kind of exchange interaction. In the starting point, the negative ions do not have spin and the exchange effect ﬁrst
causes spins in negative ions and then the created spins have interactions with magnetic ions. Hence the effect should
be second order perturbation. Figure 4.1(b) shows the model, in which a part of electrons shift from a negative ion toneighboring magnetic ions. The transfer results in the appearance of a little spin on the negative ion that has the exchangeinteractions with neighboring magnetic ions.

Fig. 4.1 (a) Cristal structure of anti-ferromagnet KFeF4 in perovskite structure. (b) Schematic diagram of superexchange interaction.

E07-2

The unperturbed state of negative ions is symmetric for spins, but the hopping to the magnetic ions is asymmetricdue to the asymmetry of the magnetic ions. Hence a spin created on a negative ion reﬂects the direction of spin on
the magnetic ion. The exchange interaction of this created spin with another magnetic ion thus results in the effective
exchange interaction between magnetic ions. The above exchange interaction mechanism mediated by non-magnetic
negative ions is called superexchange interaction[5].

4.2.1 Conditional change in sign and amplitude
Though the superexchange is a kind of kinetic exchange interaction, it is not necessarily anti-ferromagnetic because
it is a three site problem*3 . The sign and the amplitude of the superexchange interaction depend on the conditions
such as coupling angle of magnetic ion-negative ion-magnetic ion. The dependence was half-empirically summarized as
Goodenough-Kanamori’s rules.
Consider the combination of d-orbital (i, j), and write the interaction asX−
2Jeﬀ (i, j)si · sj .

(4.24)

i,j

For simplicity, we assume Hund’s rule on each magnetic ion, namely the total spin S1 =Pi s1i ，S2=Pj s2j ismaximized. Let n be the number of electrons in the open shell of each ion, then si = S/n. The interaction is written
again as in the form
−2J12 S1 · S2 .

(4.25)The half empirical rules on the amplitude and the sign are called Goodenough-Kanamori’s rules[6, 7]. Here I just quote
what Kanamori himself describe: (after telling the difﬁculty in the calculation of J12 ) “However, about the amplitude
and the sign of J12 , rather vague rules exist, which can be theoretically explained and show good agreements withexperiments[8].”For example, when two magnetic ions and an anion line up in a straight line, the interaction between magnetic ionsof the same species is anti-ferromagnetic (J < 0). When one of d orbitals has electrons more than 5 and less than 5

Fig. 4.2 Examples of Goodenough-Kanamori’s rules. (a) Magnetic ions and an anion line up in a straight line (180◦ ).
(b) Bonding bends at an anion by 90◦ . +. − represent the signs of orbitals (phases). The line diagrams below theorbital drawings show electron hoppings between the orbitals. ◦’s and ×’s represent possibilities of hoppings. From
[8].
*3 Conversely, the two involved interaction is anti-ferromagnetic, the total interaction should be ferromagnetic.

E07-3

in the other, the interaction is ferromagnetic (J > 0). When the arrangement has a 90◦ bending at an anion as in (b),the tendency is inverse to (a), namely the interaction is ferromagnetic for the same species magnetic ions but the case ofd5 . However there are various cases and exceptions. For details, refer to the original paper by Kanamori[7], or review
articles[5].

4.3 s-d exchange interaction
In the superexchange interaction, we have considered insulating crystals without conduction electrons. Here on thecontrary, consider the case of metallic crystals with many conduction electrons which interact with the spins on magnetic
ions. Such situation occurs, e.g., small amount of magnetic ions like Fe or Mn are doped as impurities into non-magnetic
metals.

4.3.1 Conduction electrons around local moments
Let S be the total spin operator of a magnetic impurity, s be the spin operators of conduction electrons. We write thematrix element of electron scattering by the local moment as
Hscatt = −2Jkk′ S · s.

(4.26)In the long wavelength approximation (2π/k is longer than the radius of scattering center), Jkk′ = J (const.). In otherwords, the interaction is δ-function (point-contact) type and with taking the position of impurity at the origin r = 0 theabove is approximated by
Hscatt = −2Jδ(r)S · s.

(4.27)We represent the conduction electrons as s, localized electrons at magnetic ion as d and call the interaction s-d exchangeinteraction. I hope we can have time to go into a many body effect caused by this interaction later. From the conduction
electrons, the interaction in eq. (4.27) is equivalent to the δ-function like effective magnetic ﬁeld 2JSδ(r)/(ge µB ) at theorigin with the direction of S. The Fourier transform of the ﬁeld isZ
2Jδ(r)dq√ Bq eiq·r .
Beﬀ (r) =·S =g e µB(2π)3 V

(4.28)

We write the magnetization of conduction electrons as m(r) and the susceptibility χ(q) in the wavenumber space isdeﬁned byZ
m(r) =

χ(q)Bqdq√ .
(2π)3 V

(4.29)For simplicity we consider the free electron model. We treat eq. (4.27) as a perturbation to the plane wave to obtain
Z
eik·r
JS
eik·r
dq
√ ,
φk (r) = √ ±(4.30)VE(k + q) − E(k) (2π)3 V
Vwhere the double sign ± reﬂects the sign of the inner product of S and s. Then we can writeZ 
g e µB ∗1
ge µB JS1dq∗
mk (r) =.
(φk− φk− − φk+ φk+ ) = −
+
eiq·r
2
2
V
E(k + q) − E(k) E(k − q) − E(k)(2π)3(4.31)
With summing up the above over k and from eq. (4.28), (4.28), we reach the expression
Zg 2 µ21
1dk
χ(q) = e B+
2V k≤kF E(k + q) − E(k) E(k − q) − E(k) (2π)3
2kF + q3N (ge µB )2 1
4k 2 − q 2
.
=
1+ F
log
8
EF 2
4qkF
2kF − qE07-4

(4.32)

Fig. 4.3 Plot of f (x) in eq. (4.33). The region
is limited to x > 0 as f (x) is an even function.

The last part in the parenthesis (· · · ) is written as
f (x) = 1 +1 − x2
1+x
log
,
2x
1−x

(4.33)

where x = q/2kF *4 . The function is plotted in Fig. 4.3.
To calculate m(r) from eq. (4.29) and χq in eq. (4.32), we need the integral




Z
Z
1
q
q
2 ∞
iq·r
F (r) =
dqe f
q sin(qr)f
=
dq
2π
2kF
r 0
2kF


Z ∞
q
1
q sin(qr)f
=
dq.
r −∞
2kF

(4.34)In the application of partial integral, we need to apply Cauchy’s principal value since df /dx diverges at x = 1 as shownin Fig. 4.3. We also use the identities
Z ∞sin[2kF r(1 ± x)]
dx = π,1±x−∞
to obtain
F (r) = −16πkF3

Z ∞cos[2kF r(1 ± x)]
dx = 0,
1±x−∞

2kF r cos(2kF r) − sin(2kF r).
(2kF r)4(4.35)(4.36)

Then we ﬁnally reach the expression of the local magnetization m(r):m(r) = −3 N ge µB F (r)J
Sz ,
32π 2
EF

(4.37)

where we take z-direction to that of S and the expectation value Sz is used.

4.3.2 RKKY interaction
In eq. (4.37), the dependence on r is from F (r) in eq. (4.36), the r-dependent part of which is plotted in Fig. 4.4. The
decay with r is associated with an oscillation.
If we have another magnetic ion within the decay length, the conduction electrons interact with it by s-d exchangeinteraction and as a result, a kind of exchange interaction between the magnetic ions is established. This is called RKKY
interaction *5 . The interaction is estimated as
Z
3N J 2− m(r)Beﬀ (r − R)dr =F (R)S1z S2z ,16π 2 EF

(4.38)*4 The notation is confusing with the Fermi distribution function (the function itself is close!). But we do not have so many good symbols and thisis a custom notation.
*5 Capital letters of Ruderman-Kittel-Kasuya-Yosida [9, 10, 11].

E07-5

Fig. 4.4 Plot of r-dependent part ofeq. (4.36), that is −(x cos x − sin x)x4
(x = 2kF r). It decays with r to zero with an
oscillation.

where R is the vector connecting the two magnetic ions, and the spins of them are written as S1 and S2 . This equation
is obtained from the expectation value m(r) and thus expressed in the product of expectation values along z-axis. It
is an exchange interaction and we can replace S1z S2z with S1 · S2 to modify it to an RKKY Hamiltonian for quantum
mechanical calculations.
As in eq. (4.38), the RKKY interaction oscillates with the period about (2kF )−1 and decays as R−3 . However when the
distance between the ions is short, namely the second ion is close to the origin in Fig. 4.4, the interaction between the ionsis always ferromagnetic because the s-d interaction works twice. De Gennes pointed out the possibility of ferromagnetism
in diluted magnetic alloys[12] via this RKKY ferromagnetic interaction. However it is still unclear weather such systems
really exist or not.

4.4 Double exchange interaction
The double exchange interaction mechanism was proposed by Zener[13] for the explanation of ferromagnetism in
Mn perovskite type compound magnets. A typical example is aMnO3 , which is an insulating anti-ferromagnet due to thesuperexchange interaction. However, with replacing a part of La with Ca, the material La1−x Cax MnO3 (0.2< x <0.4)
shows metallic conduction and transits to a ferromagnet.
Such a system can be modelized as Fig. 4.5. 3d electron levels are split by the octahedral potential of perovskite
to t2g orbitals and eg orbitals. t2g orbitals have lower energy and strongly localized while eg orbitals spread over theneighbors, being hybridized with 2s, 2p orbitals to form a band. In LaMnO3 , each Mn ion is 3+ and in the high spinstate with a single electron in an eg orbital. In spite of the formation of a band, the system is an insulator because of
the on-site Coulomb repulsion U for the hopping to the neighboring eg orbitals (due to the anti-ferromagnetic order, theelectron spins do not disturb the hopping). The situation is close to the Hubbard model introduced in Sec. 4.1.3, and such
insulators caused by the electron correlation are called Mott insulators.

Fig. 4.5 Schematic diagram of double exchange interaction.

E07-6

With replacing a part of La(3+) with Ca(2+), a part of electrons in Mn is transferred to Ca resulting in the appearance of
Mn4+ . In Mn4+ , the eg orbital is empty making the transfer of electrons via oxygen atoms possible (sometimes described
as “hole doping” in the vocabulary of semiconductor physics). This leads to the breaking of electron correlation thatsustains the insulating manner and the metallic conduction appears. In the metallic state, when an eg electron shifts tothe next Mn ion, the energy is lower for aligned spins of t2g electrons due to Hund’s rule in ions. And the higher transfer
of electrons causes lowering of the kinetic energy resulting in the ferromagnetism. As can be seen in the above scenario,
this can also be viewed as a kinetic exchange interaction.
So far we have seen superexchange, RKKY, and double exchange interactions. At the ﬁrst glance they are so different,
but it is difﬁcult to say weather they are essentially different. The naming of the interactions comes from the humanwish to understand the complicated behavior of spins in simple views. It is at a higher level than the fundamental lawof physics and sticking to the classiﬁcation of interactions is not a meaningful idea. Particularly in the case of spins, theinteraction is due to the magic of entanglement as in Appendix 7A. We must keep in mind that sometimes the magic goesaway.

4.5 Anisotropic exchange interaction
In many cases actual electrons in crystals have anisotropy reﬂecting complicated band structures. Such anisotropies inorbitals are reﬂected to spins through the spin-orbit interactions. Hence in treating the exchange interactions, we shouldtake into account the anisotropy. We thus express the exchange interaction between sites i and j with a tensor Jij . Thenthe Hamiltonian of the spin system is given by
H =X

HA (Sj ) −j

1 Xt
Si Jij Sj ,2

(4.39)i̸=jwhere t a is the transpose of a. The anisotropic energy on site is written as HA . We write the tensor elements indices asµν
Jijand separate (µ, ν) to symmetric and anti-symmetric parts as

J µν =X
1 µν
[(J + J νµ ) + (J µν − J νµ )] ≡ K µν +ϵµνξ Dξ .
2(4.40)

ξ=x,y,zϵµνξ is the complete anti-symmetric tensor (Levi=Chivita symbol). The site indices (i, j) are omitted. The second termin the rhs, the anti-symmetric part of eq. (4.39) is expressed as



X
(DM)
ξ
Hij
= t Siϵµνξ Dij
Sj = Dij · (Si × Sj ).
(4.41)

ξ

Dij

(DM)

= −Dji and this anti-symmetric exchange interaction Hij

is called Dzyaloshinskii-Moriya (DM)interaction[14, 15].
The DM interaction is important in the discussion of magnetic anisotropy of anti-ferromagnets, weak ferromagnets. In
resent researches, the DM interaction is important in the magnetic interaction between thin ﬁlms and in other spintronics
ﬁeld.

E07-7

Chapter 5

Theories of magnetic insulators

Ferrite core transformer
There are many magnetic insulators as ferrites (AFe2 O4 , A=Mn, Co, Ni, Cu, Zn, · · · ). They are very advantageous forthe high frequency use in the absence of eddy current and of skin effect. Now they are indispensable in high frequency
techniques. Also since they do not deteriorate such as rust, the ferrites are mostly used for the magnets on blackboards
etc., the magnetic sheets that can be used with cut *6
In the previous chapter we have introduced the Heisenberg model, which is one of the human-friendly models for
dealing with magnetism. In insulators, superexchange interactions work between localized spins, and a situation close to
the Heisenberg model may be realized in essence. Here we will see what kind of phenomena the model includes.

5.1 Molecular ﬁeld theory
We consider mean ﬁeld theory, which is the most basic approach in many body problems. In the ﬁeld of magnetism,
the approach is also called molecular ﬁeld approximation.

5.1.1 Ferromagnetic Heisenberg model
We consider the Heisenberg model in eq. (4.13) with the magnetic ﬁeld of magnetic ﬂux density B:
H = −2JXSi · Sj − µ

⟨i,j⟩

X

B · Si ,(5.1)

iwhere the sum on ⟨i, j⟩ is taken for the nearest neighbors. The interaction is ferromagnetic, i.e., J > 0, µ is taken as
positive. In the mean ﬁeld approximation for site i, the surrounding spins are replaced with averaged magnetic moments.
Heﬀ (i) = −2J

X⟨Si+δ ⟩ · Si − µB · Si = −µBeﬀ · Si .

(5.2)

δ

δ is for nearest neighbors. The effective ﬁeld Beﬀ in eq. (5.2):µBeﬀ = 2J

X⟨Si+δ ⟩ + µB,δwhich is also called molecular ﬁeld.

*6 Long time ago, the ferrites were also used in magnetic tapes.

E07-8

(5.3)

Here we use the expression (2.51) for magnetization M per single ion with replacing gJ µB → µ, J → S, Brillouinfunction BJ → BS to obtain
M = µSBS

µS
kB T



2αz J
B+
M
µ2


.

(5.4)

αz is the number of neighboring sites and the direction of B is taken to z-direction. Equation (5.4) is a self-consistentequation for M .
We use the expansion of coth in Brillouin function BS and obtain
BS (x) =S+1
1 [(S + 1)2 + S 2 ](S + 1) 3x−x + ··· .
3S
90
S3

(5.5)

Then eq. (5.4) is expanded as

1−



2
2αz J
1
1
2αz J
2
2
[(S
+
1)
+
S
]
χ
M
+
M 3 = χ0 B.
0µ2
90(kB T )3µ2

(5.6)

χ0 = µ2 S(S + 1)/3kB T represents the Curie law in eq. (2.53).
When (5.6) has a non-zero solution of M (spontaneous magnetization) for B = 0, the system can have a ferromagnetic
state. The condition is the coefﬁcient of the ﬁrst order in M in the lhs of (5.6) becomes zero. This gives the ferromagnetictransition temperature (Curie temperature) TC as
k B TC =2
S(S + 1)αz J.
3

(5.7)

The susceptibility for T > TC is obtained from eq. (5.6) by considering the ﬁrst order term as

−1
2αz J
S(S + 1)
χ = χ0 1 −
χ
,
= µ2
0
µ2
3kB (T − TC )(5.8)

which diverges as (T − TC )−1 . This behavior is called Curie-Weiss law.

Appendix 7A: Quantum entanglement and quantum dot experiments
Although the magnetic-mediated spin-to-spin interaction is very weak, we found that a strong spin-to-spin interaction
occurs due to the relationship with the orbit. Then the spin Hamiltonian model is introduced. As above, in quantum
theory, the freedoms in an interaction model can be exchanged under some conditions. This is on the concept of quantum
entanglement, which is widely used in the quantum information ﬁeld.

7A.1 Quantum entanglement and effective Hamiltonian
The readers are already familiar with the concept and here a brief introduction is given. Let us consider two systems
with two-dimensional orthogonal basis {|1⟩ , |2⟩} and {|p⟩ , |q⟩}. We write the wavefunctions in the systems as |ψ⟩ =a1 |1⟩ + a2 |2⟩, |ϕ⟩ = ap |p⟩ + aq |q⟩. When these two have no relation between them, a state of the combined system(of course the combined system can be considered even if there are no relation) is written as a direct product of the two
wavefunctions:|Ψn ⟩ = |ψ⟩ ⊗ |ϕ⟩ = a1 ap |1⟩ |p⟩ + a1 aq |1⟩ |q⟩ + a2 ap |2⟩ |p⟩ + a2 aq |2⟩ |q⟩ .
E07-9(7A.1)

On the other hand, we consider the case when the wavefunction of combined system is written in the form√|ξ⟩ = (|1⟩ |p⟩ + |2⟩ |q⟩)/ 2.

(7A.2)In this state, assume we perform measurement on |ψ⟩ and obtain e.g. the result 1. Then without observation on |ϕ⟩ andwithout direct interaction between the two systems, the state of |ϕ⟩ is determined to p by the above measurement. Thissituation is described as |ψ⟩ and |ϕ⟩ are entangled. This was clearly pointed out in the famous EPR paper[16, 17]. In
particular the state in eq. (7A.2) is called a maximally entangled state and the operations on the systems are unseparable.
Further, we write another maximally entangled state√|ζ⟩ = (|1⟩ |q⟩ + |2⟩ |p⟩)/ 2,(7A.3)and consider the case the basis space is limited to {|ξ⟩ , |ζ⟩}. We assume the Hamiltonian in the system {|1⟩ , |2⟩}

h11 h12Hn =.
(7A.4)
h21 h22Then⟨ξ|Hn |ξ⟩ = h11 + h22 ,⟨ξ|Hn |ζ⟩ = h12 + h21 ,⟨ζ|Hn |ζ⟩ = h11 + h22 .

Hence if we can prepare an operator Ha working on {|p⟩ , |q⟩} and gives the same matrix as Hn
h11 h12Ha =,h21 h22(7A.5)(7A.6)

Hn and Ha give the same results in the basis space {|ξ⟩ , |ζ⟩}.
As seen above, the concept of “effective Hamiltonian” works when the basis space is limited to entangled states of two
freedoms. In such states, operations on one system is equivalent to those on the other system. Hence they are working
as “operators” very differently but equivalent. In quantum physics, though not always explicitly, the concept of quantum
entanglement is used in various situations.

7A.2 Quantum entanglement and observation
Quantum entanglement is not only a tool for theory, but also used widely in experiments, conﬁrming that the EPR
paradox is not a mere thought experiment.
The electron paramagnetic resonance is introduced as an experiment to observe the Larmor precession of spins inmagnetic ﬁeld. The traditional way to detect the resonance is to detect the lowering in the Q-value of resonator due to theabsorption of energy in electromagnetic wave by the spin system at the resonance. To give a change in the characteristics
of a macroscopic resonator, an ordinary experiment in microwave needs at least 1010 spins[18]. In this method, in otherwords, the magnetic ﬁeld caused by the magnetic dipoles of spins are detected though the interaction in eq. (4.1). The
signal is naturally tiny and it is hopeless to detect single spin precession.
Let us consider “what is measurement.” A possible answer is that a measurement is to create an entanglement between
the freedoms of an object and those of something human can directly distinguish. As a system for measurement we
consider {|↑⟩ , |↓⟩} and {|A⟩ , |B⟩} as another system which human can directly distinguish. Then the measurement is tocreate a maximally entangled state1Ψ = √ [|↑⟩ |A⟩ + |↓⟩ |B⟩].
2(7A.7)The system {|A⟩ , |B⟩} is readily integrated out and the measurement is accomplished when this Ψ is created *7 .
Now, then, in the case of a spin, instead of entanglement of spin with a photon through the magnetic moment, entanglement with other freedom with much larger effect might make the detection of single spin possible.
(to be continued).
*7 In the terms of Schrödinger’s cat problem, dead/alive of the cat is determined before the box is opened.

E07-10References[1] Høgni Weihe and Hans U. Güdel. Quantitative interpretation of the goodenough-kanamori rules: a critical analysis.
Inorganic Chemistry, Vol. 36, No. 17, pp. 3632–3639, August 1997.
[2] Zhishuo Huang, Dan Liu, Akseli Mansikkamäki, Veacheslav Vieru, Naoya Iwahara, and Liviu F. Chibotaru. Ferromagnetic kinetic exchange interaction in magnetic insulators. Phys. Rev. Research, Vol. 2, p. 033430, Sep 2020.
[3] 本郷研太, 小山田隆行, 川添良幸, 安原洋. フント則の起源は何か?(最近の研究から). 日本物理学会誌, Vol. 60,
No. 10, pp. 799–803, 2005.
[4] 佐甲徳栄. ヘリウム様原子におけるフントの第一規則の起源 （解説）. 日本物理学会誌, Vol. 68, No. 6, pp.
358–365, 2013.
[5] Jacques Curély. The microscopic mechanisms involved in superexchange. Magnetochemistry, Vol. 8, No. 1, p. 6,
December 2021.
[6] John B. Goodenough. Theory of the role of covalence in the perovskite-type manganites [La, m(II)]Mno3 . Phys.
Rev., Vol. 100, pp. 564–573, Oct 1955.
[7] Junjiro Kanamori. Superexchange interaction and symmetry properties of electron orbitals. Journal of Physics andChemistry of Solids, Vol. 10, No. 2-3, pp. 87–98, July 1959.
[8] 金森順次郎. 磁性 (新物理学シリーズ 7). 培風館, 1969.
[9] M. A. Ruderman and C. Kittel. Indirect exchange coupling of nuclear magnetic moments by conduction electrons.
Phys. Rev., Vol. 96, pp. 99–102, Oct 1954.
[10] Tadao Kasuya. A theory of metallic ferro- and antiferromagnetism on zener’s model. Progress of Theoretical
Physics, Vol. 16, No. 1, pp. 45–57, July 1956.
[11] Kei Yosida. Magnetic properties of cu-mn alloys. Phys. Rev., Vol. 106, pp. 893–898, Jun 1957.
[12] P.G. De Gennes. Polarisation de charge (ou de spin) au voisinage dúne impureté dans un alliage. Journal de
Physique et le Radium, Vol. 23, No. 10, pp. 630–636, 1962.
[13] Clarence Zener. Interaction between the d-shells in the transition metals. ii. ferromagnetic compounds of manganesewith perovskite structure. Phys. Rev., Vol. 82, pp. 403–405, May 1951.
[14] Igor Dzyaloshinsky. A thermodynamic theory of weak ferromagnetism of antiferromagnetics. Journal of physics
and chemistry of solids, Vol. 4, No. 4, pp. 241–255, 1958.
[15] Tôru Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Physical review, Vol. 120, No. 1,
p. 91, 1960.
[16] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered
complete? Phys. Rev., Vol. 47, pp. 777–780, May 1935.
[17] Giuliano Benenti, Giulio Casati, Davide Rossini, and Giuliano Strini. Principles of Quantum Computation andInformation: A Comprehensive Textbook. World Scientiﬁc Publishing Company, 1 2019.
[18] Patrick Bertrand. Electron Paramagnetic Resonance Spectroscopy: Fundamentals. Springer, 2 2020.

E07-11

Lecture note
1st June (2022)

Magnetism (8)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Last week we introduced the Heisenberg model for magnetic insulators. In mean ﬁeld (molecular ﬁeld) approximation, the power
series expansion of the Brillouin function (eq. (5.6)) gives the equation, which leads to the Curie-Weiss law:χ=µ2 S(S + 1)
1.
kB
T − TC(5.8)

In the region T < TC , the solutions of M ̸= 0 exists for B =0, i.e., the spontaneous magnetization appears. In the vicinity of
TC , the term of M 3 in eq. (5.6) is included to give the spontaneous
magnetization as√
M =µFig. 5.1 Schematic drawing of susceptibility
χ above TC and spontaneous magnetization M
below TC for a ferromagnet.

S(S + 1)10√
3 (S + 1)2 + S 2

√
1−

T
.
TC

(5.9)

On the other hand, at T ≪ TC , we use the asymptotic expression()]( x ) [ 2S + 1
1
2S + 1
BS (x) ∼ 1 − exp −
+
exp −
x
(5.10)SS
S
S

for x ≫ 1. The ﬁrst two leading terms give
[

(

3 TC
M = µ S − exp −
S+1 T

)]
,

(5.11)

which approaches the perfect magnetization µS with T → 0. The temperature dependences of χ and M obtained above
are summarized schematically in Fig. 5.1.

5.2 Phenomenology of ferromagnetic transition: the GL theory
The above simple results still contains characteristic features of cooperative phenomena. For example, in eq. (5.8),
T ≈ TC , we can write
χ∝

1
TC
=1+
+
1 − (TC /T )
T

(

TC
T

(

)2
+

TC
T

)3
+ ··· ,

which expresses the following process: the effective ﬁeld by the neighboring sites gives the excess polarization proportional to TC /T , while the neighboring sites get the same excess polarizations that give the feedback of (TC /T )2 . This
series continues inﬁnitely. The series reaches the radius of convergence at T = TC and the spontaneous magnetization
appears there.
We know that even such a simple model includes a mechanism of the appearance of ferromagnetism. Then, here,
we have a look on a very general properties of phase transitions and try to ﬁnd the correspondence with the mean ﬁeld
approximation.

E07-1

Fig. 5.2 Illustration of power expansion formula for the free energy F by M eq. (5.14). (a) Case a ≥ 0. (b) casea < 0. In (c), a varies continuously through 0, and F is plotted in a wireframe.

5.2.1 Free energy
Here we introduce the Ginzburg-Landau (GL) theory of phase transition[1, 2]. This was constructed to understandsuperconductivity and superconducting transition phenomenologically. The GL theory, however, can be applied to a wide
range of phase transition phenomena, and naturally have hugely been affecting the study of critical phenomena[3, 4]. In
the theory, the free energy of the system is a function of physical quantities. In an equilibrium, the free energy should be
at a minimum for the parameters “adjustable” by the system such as magnetization. In other words, the parameters takethe values that make the free energy take a minimum. Let F be the free energy per spin, then we consider the functional
form of F (M ), where M is the magnetization per spin.
In order to consider the symmetry of the system, we turn off the magnetic ﬁeld in the Hamiltonian in eq. (5.1), which
reduces the symmetry. Now we perform a kind of symmetry operation of spin inversion on all sites, namely∀i Si → −Si .
For this operation, the Hamiltonian in eq. (.1) with B = 0 is invariant. Accordingly F is unchanged. On the other hand,from the deﬁnition,
M = ⟨Si ⟩ → ⟨−Si ⟩ = −M ,(5.12)

that is the parameter M is inverted. The above inference leads toF (M ) = F (−M ),(5.13)

namely F is an even function of M . Therefore we can expand F to the power series of small M (hence close to thetransition) to the forth order as
F (M ) = F0 + aM 2 + bM 4 .

(5.14)First in eq. (5.14), to have a stable point of F at ﬁnite M , b should be positive (b > 0). Under this condition, a positivea (a ≥ 0) always gives M = 0 as the stable point of F as in Fig. 5.2(a). For a < 0, two ﬁnite values of M give energy
minima, hence are stable points, lower than the energy for M = 0 as in Fig. 5.2(b). The equation which gives the stable
points is∂F= 0 = 2aM + 4bM 3 = 2M (2bM 2 + a),∂M(5.15)

which is in the same form as in eq. (5.6), thus is the same equation. This is sometimes called “magnetic equation ofstate.” As in Fig. 5.2, the system is paramagnetic for (a) a ≥ 0, and ferromagnetic for (b) a < 0. We now ﬁnd that a is
a parameter: i) which determines F (M ); ii) which has no anomaly at zero, the transition point of M . Therefore a is a“relevant” parameter for the transition (in a sense, a parameter that drives the transition). a must vary in the ﬁrst order forE07-2

thermodynamic parameters like temperature, pressure, etc. Figure 5.2(c) shows such continuous change in F (M ) withthe variation of a, and the appearance of stable points other than M = 0 at the transition point a = 0. This indicates that
we are considering a second order phase transition, which does not have latent heat at the ferromagnetic transition. If wetake temperature T as the relevant parameter of the transition, a should be in the ﬁrst order of T . As the expression of a,which is zero at the transition point and in the ﬁrst order in T , we can write a = k(TC − T )/TC . Then a ﬁnite solutionof M0 for eq. (5.15) is given by√√ak(TC − T )M0 = − =.
2b2bTC(5.16)

5.2.2 Spontaneous symmetry breaking
In the region T ≤ TC , F (±M0 ) are the thermodynamically stable solutions, in which ⟨M ⟩ = M0 or −M0 . These
correspond to the spontaneous magnetization of ferromagnets, which was introduced in the ﬁrst lecture *1 . For the
expansion of eq. (5.14), we have used the symmetry of free energy (5.13) deduced from the symmetry of Hamiltonian(5.1) for the symmetry operation ∀i : Si → −Si . In the region T < TC , M = 0 is unstable and one of stable solutions±M0 is realized. Due to (5.13), the symmetry operation does not change the free energy, but now M is the parameter
determining the state of the system. That means the operation changes the state. The situation is summarized thatthe symmetry of realized state is broken while that of the system (Hamiltonian) is kept. Such a phenomenon is called
spontaneous symmetry breaking. The concept was introduced by Yoichiro Nambu[5, 6, 7] from the analogy of the
BCS theory (and the Bogoliubov theory) for superconductivity and the mechanism for the appearance of particle mass. It
is one of the basic concepts in physics, has been widely applied to a variety of phenomena under active research. There
are many textbooks including the one for general public written by Nambu himself[8, 9, 10, 11].
The continuous spatial symmetry in the original system with random direction of spins is broken in the state withspontaneous magnetization M0 , in which the spins are in order pointing a single point in space. The parameter thatappears at the critical point and represents the order of the state is called order parameter.

5.3 Critical exponent
F in the presence of spontaneous magnetization M0 is given as a function of temperature as
F (T ) = F0 + aM02 + bM04 = F0 −a2
k 2 (TC − T )2
= F0 −.
4b4bTC2(5.17)

Then in T ≤ TC , the speciﬁc heat C is given by
C = −T

∂2F
k2 T
=
.
∂T 2
2bTC2

(5.18)On the other hand in T ≥ TC , C = 0 because M0 = 0 and
F (T ) = F0 . Then the speciﬁc heat has a jump of
∆C =

k2
,
2bTC

(5.19)

at T = TC which is illustrated in the left.
Now when a small magnetic ﬁeld is introduced, in the lowest order approximation, we can replace the term of external*1 As also introduced in the lecture, in practice, with zero-ﬁeld cooling from above the Curie temperature, we cannot observe macroscopic sponta-

neous magnetization due to the formation of magnetic domains, which build up magnetic circuit and conﬁne the magnetic ﬂux inside.

E07-3

ﬁeld in Hamiltonian (5.1) with −BM , where M is the magnetization. That is we add the ﬁrst order term as

Then fromF (M ) = F0 + aM 2 + bM 4 − BM.

(5.20)∂F= 0 = 2aM + 4bM 3 − B,∂M

(5.21)

we get M 3 ∝ B just at the critical point T = TC because a = 0.
So far we have obtained the expressions for magnetization M , susceptibility χ and speciﬁc heat C in the forms of{
B 1/δ
(T = TC ),M∝(5.22a)(TC − T )β (T < TC ),{χ∝(T − TC )−γ′(TC − T )−γ{C∝(T − TC )−α′(TC − T )−α(T > TC ),(T < TC ),(5.22b)(T > TC ),(T < TC ).

(5.22c)As above, we pick up a relevant parameter, which drives a phase transition in the system, and consider the shift fromthe critical value. The power indices of the “shift” in the functional expressions of physical quantities are called criticalexponents. This is particularly important concept in the second order phase transitions. The above symbols α, β, γ, δ,· · · are habitually used in the ﬁeld of magnetism and statistical physics. There is an anomaly of jump for the speciﬁc heatin eq. (5.22c), we apply the forms of critical exponent separately for T < TC and for TC < T . In both cases the main
term is a constant in the expression of T − TC , hence α = α′ = 0.
The critical exponents depend on symmetry, dimension, range of interaction, way of approximation, etc. of the model.
On the other hand, the variations in the system parameters do not change the exponents. The property is expressed as
the critical exponents have universality. Further, we can classify the theoretical models (including approximations) ofphase transitions with the set of the values of critical exponents. This classiﬁcation called universality class, depends, as
we saw in the introduction of the GL theory, often symmetry of the system. The universality class is also determined bygeneral properties of the system such as the spatial dimension, the range of interaction. The next table summarizes thevalues of critical exponents in the mean ﬁeld theory.
Critical exponent

α

β

γ

δ

Mean ﬁeld approximation

0

1/2

1

3

Though we have introduced the concept of universality class, the model examined is only the mean ﬁeld theory of
Heisenberg model. We would like to have a short look at the other models.

5.4 Theoretical models of ferromagnet (localized spins)
The theoretical models of magnetic materials are the big stage for statistical physics. In the above we consider the
Heisenberg model as ferromagnetic insulators. In the Heisenberg model, the spin variable has three components Si =(Six , Siy , Siz ). In the XY model, the spin components are limited to two, that is Si = (Six , Siy ). In the Ising model, thespin has a single component. In the Heisenberg model and the XY model, the spin degree of freedom takes a continuous
value while in the Ising model it is quantized to the two values.

E07-4

5.4.1 XY modelFirst we pick up a direction for “angle zero,” then because the spins in this model are in the two-dimensional plane, we
can assign an angle ϕi for each site i measured from the angle zero. Then the angle between the spins at sites i and j isϕi − ϕj . Accordingly the Hamiltonian of the XY model can be written as∑

H = −Jcos(ϕi − ϕj ).

(5.23)⟨i,j⟩

In the two dimensional XY model, there is no long range order due to the Mermin-Wagner theorem though it has anothertype of phase transition, in which the order parameter decreases not with exponential damping but with power of distance
damping. The transition is called Berezinskii-Kosterlitz-Thouless (BKT) transition[12]. The BKT transition is causedby excitations called vortices, in which the spins form rotation structures. They have two possible directions of rotation,
and we distinguish them with the naming vortex and anti-vortex. An attractive force works between a vortex and ananti-vortex, which form a vortex pair bound state. The bound states are more stable than free unbound vortices. In the
low temperature phase all vortices are bound into pairs. With increasing temperature the number of vortex pairs increases
and at the transition point free unbound vortices appear due to the weakening of attractive force by screening. This canbe taken as a two-dimensional melting transition.
It is easier to realize the XY model (5.23) in superconducting Josephson networks than in spin systems[13]. A Josephson network is an arrangement of superconducting islands and junctions connecting them. They can be prepared by e.g.,lithography, or growth of granular ﬁlms. We can write the phase of superconducting order parameter on each plaquette(island) i as ϕi *2 . Then the summation of Josephson energy is written in the form of (5.23). Also two-dimensional vortices mentioned above appear in a thin ﬁlm of superﬂuid on a plate (the ﬁlm ﬂow effect). Hence observation of the BKTtransition has been reported in such systems.

5.4.2 Ising model
The name of the Ising model comes from Ernst Ising, who showed the solution of this model in the case of nearest
neighbor interaction[14]. It can be expressed by the Hamiltonian:H = −J∑Si Sj − h

∑

⟨i,j⟩

Si ,

(5.24)iwhere i, j are indices of the lattice. Si is the Ising spin on i, which takes values ±1. In the second term µB is written
as h for simplicity. The Ising model may be the most known model of magnetic materials. The model is so simple, andoverall, not only the solution by Ernst Ising for one-dimensional model, but also the rigorous solution of two-dimensionalmodel in the absence of magnetic ﬁeld[15] are the base of study for various physics in this system.
The critical exponents of these models are listed in the following table[16]. What is written using the decimal point isthe value obtained by the computation with the Monte Carlo method.
Model (Universality class)

α

β

γ

δ

2D Ising

0

1/8

7/4

15

3D Ising

0.115

0.324

1.239

4.82

3D XY

−0.01

0.34

1.32

4.9

3D Heisenberg

−0.11

0.36

1.39

4.9

Mean ﬁeld approximation

0

1/2

1

3

*2 This quantity is not gauge invariant, not an observable. However the phase difference appears in the Hamiltonian is a gauge invariant observable.

E07-5

5.5 Anti-ferromagnetic Heisenberg model
Next we consider the case when the interaction is anti-ferromagnetic (J < 0) in the Heisenberg model:∑∑
H = −2J
Si · Sj − µ
B · Si .
⟨i,j⟩

(5.1)

i

We consider a two dimensional square lattice with nearest neighbor interaction. When the system is in the antiferromagnetic order, the classical ground state of two-dimensional square lattice Heisenberg model is a Néel ordering
state, in which the neighboring spins are in anti-parallel order. We divide the entire crystal lattice into A and B sublattices, and consider a state in which the spins are parallel in each lattice. In the treatment of ferromagnetic Heisenberg
model, we ﬁrst applied an external magnetic ﬁeld to give direction to the isotropic space*3 . In the anti-ferromagnetic
case, we take a similar method. This time as in the right panel of Fig. 5.3, we need to prepare the ﬁeld that changes the
direction alternatively with site[17]. Anyway the alternative ﬁeld will be set to zero in the ordered state. We consider
the starting state as the moments are alternatively aligned with the alternative ﬁeld with oblique angle due to the external
magnetic ﬁeld as illustrated in the right panel of Fig. 5.3.
Let Bu be the external constant ﬁeld, ±Bs be the site-alternative ﬁeld. The ﬁelds on the two kinds of sites are}
BA = Bu + Bs ,(5.25)BB = Bu − Bs .
The effective Hamiltonian of the molecular ﬁeld approximation is∑Heﬀ (i) = −2J⟨Si+δ ⟩ · Si − µBA · Si(i ∈ A),(5.26a)(j ∈ B).

(5.26b)δHeﬀ (j) = −2J∑⟨Sj+δ ⟩ · Sj − µBB · SjδThe averaged magnetic moments at the two sites are
MA = µ ⟨Si ⟩ = Mu + Ms
MB = µ ⟨Sj ⟩ = Mu − Ms}.

Fig. 5.3 Left: Illustration of Néel anti-ferromagnetic order. Right: Drawing of “seeds ﬁeld” to set the spins aroundthe anti-ferromagnetic order. From [17].
*3 In theories, without such “seeds” ﬁeld, the system continues to take the unstable solution.

E07-6(5.27)

We here deﬁne the Brillouin “vector” function as⃗ S (x) = BS (x) x .
Bx
Then the self-consistent equation is written as *4 ,
{
[
]}
µS
2αz J
⃗
Mu + Ms = µS BS
Bu + Bs +(Mu − Ms ) .
kB Tµ2⃗ S (x) ∼ (S + 1)x/3S, we write
Above the critical temperature T > TN , from B
[
]
2αz J
Mu + Ms = χ0 Bu + Bs +
(M
−
M
)
.
u
s
µ2

(5.28)(5.29)(5.30)The deﬁnition of χ0 is in eq. (5.6).
Then uniform susceptibility χu , and sublattice susceptibility χs are given by
(
)−1
Mu
2αz J
χu = lim= χ0 1 −χ0,
Bu →0 Bu
µ2
(
)−1
Ms
2αz J
χs = lim= χ0 1 +χ0.
Bs →0 Bs
µ2

(5.31a)(5.31b)

Because J < 0, χu does not diverge. On the other hand χs diverges at Néel temperaturek B TN =2
S(S + 1)αz |J|.
3

(5.32)

Hence with Bs → 0, we have sublattice spontaneous magnetization Ms .
From the expansion around the spontaneous magnetization Ms ,
[
(
)
(
)(
)]
µS −2αz JµS −2αz J
d
µ2
⃗
Ms +
BS
Ms
Bu .
Mu + Ms = µS BS−Mu −kB T µ2dMs
kB T µ22αz J
From the ﬁrst term in the rhs, we obtain the self-consistent equation for Ms as()µS −2αz J
Ms = µSBS
Ms .
kB T µ2(5.33)(5.34)

By taking derivative of both sides with Ms we know
1 = µS

d
BS
dMs

(

)
µS −2αz JMs .
kB T µ2

Then from the second term in the rhs of eq. (5.33), we obtain
Mu = −Mu −

µ2
Bu .
2αz J

(5.35)

And from Mu = −µ2 Bu /4αz J, the uniform susceptibility is obtained as
µ2
Mu
=−
.
Bu →0 Bu−4αz Jχu = lim
The above results are illustrated in Fig. 5.4.

*4 αz = 4 in the present caseE07-7

(5.36)

Fig. 5.4 Illustration of (uniform) susceptibility and sublattice spontaneous magnetization in anti-ferromagnetic
Heisenberg model.

References[1] V.L. Ginzburg and L.D. Landau. Zh. Eksp. Teor. Fiz., Vol. 20, p. 1064, 1950.
[2] Vitaly L. Ginzburg. On superconductivity and superﬂuidity (what i have and have not managed to do), as well ason the‘physical minimum’ at the beginning of the 21st century. ChemPhysChem, Vol. 5, No. 7, pp. 930–945, July
2004.
[3] Shang-keng Ma. Modern Theory Of Critical Phenomena (Advanced Books Classics). Routledge, 5 2018.
[4] 西森秀稔. 相転移・臨界現象の統計物理学 新物理学シリーズ. 培風館, 11 2005.
[5] Yoichiro Nambu. Quasi-particles and gauge invariance in the theory of superconductivity. Phys. Rev., Vol. 117, pp.
648–663, Feb 1960.
[6] Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. i. Phys. Rev., Vol. 122, pp. 345–358, Apr 1961.
[7] Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. ii. Phys. Rev., Vol. 124, pp. 246–254, Oct 1961.
[8] 南部陽一郎. クォーク第 2 版: 素粒子物理はどこまで進んできたか (ブルーバックス). 講談社, 2 1998.
[9] 南部陽一郎. 素粒子論研究 (わが研究の思い出) (⟨ 特集 ⟩ 日本物理学会のあゆみ). 日本物理学会誌, Vol. 32, No. 10,
pp. 773–778, 1977.
[10] Y. Nambu. Broken Symmetry: Selected Papers of Y. Nambu (World Scientiﬁc Series in 20th Century Physics). World
Scientiﬁc Pub Co Inc, 10 1995.
[11] PHILIP W. ANDERSON. BASIC NOTIONS OF CONDENSED MATTER PHYSIC. TAYLOR & FRANCIS, 2 2019.
[12] V N Ryzhov, E E Tareyeva, Yu D Fomin, and E N Tsiok. Berezinskii – kosterlitz – thouless transition and twodimensional melting. Physics-Uspekhi, Vol. 60, No. 9, pp. 857–885, September 2017.
[13] Shingo Katsumoto. Single-electron tunneling and phase transitions in granular ﬁlms. Journal of Low Temperature
Physics, Vol. 98, No. 5-6, pp. 287–349, March 1995.
[14] Ernst Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik, Vol. 31, No. 1, pp. 253–258, February
1925.
[15] Lars Onsager. Crystal statistics. i. a two-dimensional model with an order-disorder transition. Phys. Rev., Vol. 65,
pp. 117–149, Feb 1944.
[16] H W J Blöte, E Luijten, and J R Heringa. Ising universality in three dimensions: a monte carlo study. Journal of
Physics A: Mathematical and General, Vol. 28, No. 22, pp. 6289–6313, nov 1995.
[17] 上田和夫. 磁性入門 (物性科学入門シリーズ). 裳華房, 単行本, 9 2011.

E07-8

Lecture note
8th June (2022)

Magnetism (9)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

In the last lecture, we stopped at the molecular ﬁeld approximation of anti-ferromagnetic Heisenberg model. The
lecture note was at the beginning of ferrimagnetism. I would like to add the treatment of parallel magnetic ﬁeld tospontaneous (internal) magnetization of an anti-ferromagnetic state. And would like to change some notations in theferrimagnetic part. Hence, here, I would like to rewrite the ferrimagnetic part.

5.5.1 Parallel susceptibility
Next we consider the case that the external magnetic ﬁeld is parallel or anti-parallel (collinear) to the spontaneous
magnetizations of sublattices. The Heisenberg model is completely isotropic. If we assume that the system always takes
the lowest energy state, then the magnetization should rotate to be perpendicular to the external magnetic ﬁeld. However
real materials usually have some magnetic anisotropies that lock the directions of moment. We thus consider the case theexternal ﬁeld is collinear to the spontaneous magnetization. The effective ﬁelds in A and B sublattices are written as
Beﬀ (A) = B + Bsub (A),

(5.37a)Beﬀ (B) = B + Bsub (B).

(5.37b)

Because the vectors are collinear, we do not use vector symbols here. Then as is due course, we write down a set ofself-consistent equations for magnetizations as
)]
(
µS
2αz J
hMA i = µSBS
hMB i ,
B+
kB T
µ2
(
)]
[
2αz J
µS
B+
hMA i ,
hMB i = µSBS
kB T
µ2
[

(5.38)

where BS (x) is the Brillouin function. With solving the above and from the relation *1 ，χ∥ = lim

B→0

M A + MB,
B

Fig. 5.1 Left panel: Schematic diagram of temperature dependent susceptibility in the molecular ﬁeld approximationof the anti-ferromagnetic Heisenberg model. The susceptibilities for magnetic ﬁeld perpendicular (χ⊥ ) and parallel(χ∥ ) to the spin polarization. Measured susceptibilities of (a) GdNiGe3 , (b) MnF2 . From [1].
*1 As is the case of spontaneous ferromagnetic magnetization, we need to solve the equation numerically.

E09-1(5.39)

Fig. 5.2 Examples of spin conﬁgurations in metal oxides, ﬂuorides. (a) and (b) shows the spin-directions at magnetic
ions. (c) also shows the positions of F-atoms. (a) MnO. From [2]. Red broken lines show the sheets of aligned spins.
(b) NiO. From [3]. (c) MnF2 . From [4].

we obtain the parallel magnetization.
From eq. (5.38), in the limit of T → 0, MA = −MB = µS as in the case of ferromagnetism, χ∥ → 0. On the other
hand, χ∥ = χ⊥ at T = TN The susceptibility, thus has a large anisotropy below TN as shown in the left panel of Fig. 5.1.
Though the situation of perpendicular ﬁeld to the sublattice magnetization cannot be realized in the Heisenberg model, asmall anisotropy may enable it. In many anti-ferromagnets, such properties have been really observed. Figure 5.1(a), (b)show examples of GdNiGe3 , MnF2 , which are claimed to be close to the anti-ferromagnetic Heisenberg model.

5.5.2 Antiferromagnetic insulators
So far, we have discussed the magnetic susceptibility of antiferromagnetic materials with a very simple twodimensional Heisenberg model. As mentioned in the section on superexchange interactions, oxides and ﬂuorides ofmagnetic metals are often antiferromagnetic. Figure 5.2 shows examples of spin conﬁgurations in anti-ferromagnetic
insulators. As in the Heisenberg model, neighboring (though with intermediate negative ions) spins at magnetic ionshave opposite directions. As can be seen in the ﬁgure, actual “sublattices” can be taken as two-dimensional spin-aligned
sheets. In such a case, the structure is a kind of magnetic superlattice.
In an anti-ferromagnetic ordered state, the spins have a periodic structure with a larger volume than that of lattice (unitcell). This is sometimes called a spin-superlattice. For this situation, we can apply a concept called magnetic unit cell,which is the unit of periodicity including the spin conﬁguration. These two kinds of unit cell lengths are indicated inFig. 5.2(a).
χu in eq. (5.31a) does not diverge because J < 0. Instead it shows the temperature dependence
χu ∝

1
,
T +θ

(5.40)

which is different from the Curie law. This θ is called Weiss temperature.
The Néel temperatures and the Weiss temperatures of typical anti-ferromagnets are liste in Tab. 5.1.

Fromeqs. (5.31,32), these two kinds of temperatures should be symmetric to 0 K. Of course, a simple model even without
anisotropy should give results far different from the reality. However, there is some rough correlation between them.

E09-2

Material

Lattice-type of magnetic ionsNéel temperature (K)Weiss temperature (K)

MnO

fcc

116

610

MnS

fcc

160

528

MnTe

hexagonal

307

690

MnF2

bct

67

82

FeF2

bct

79

117

FeCl2

hexagonal

24

48

FeO

fcc

198

570

CoCl2

hexagonal

25

38

CoO

fcc

291

330

NiCl2

hexagonal

50

62

NiO

fcc

525

∼ 2000

Cr

fcc

308

Tab. 5.1 Néel temperatures and Weiss temperatures of typical anti-ferromagnets.

5.5.3 Spin ﬂop transition and metamagnetism
Consider a general material with susceptibility χ. With increasing the external magnetic ﬁeld, the energy gain due tothe magnetization is∫ B
Em = −0

M (B ′ ) dB ′
= −χ
µ0
µ0

∫ B
0

χ
B ′ dB ′= − 2 B2.
µ0 µ0
2µ0

(5.41)In the region T < TN , χ⊥ > χ∥ as shown in Fig. 5.1, hence the energy is lower for the magnetic ﬁeld perpendicular tothe sublattice magnetization. As mentioned in the beginning of “parallel susceptibility,” with increasing a parallel ﬁeld,
the energy difference overcomes the anisotropic energy K at a certain point, at which the sublattice magnetizations rotate
to the direction perpendicular to the magnetic ﬁeld. This is called spin ﬂop transition. The critical ﬁeld is obtained fromχ⊥ − χ∥ 2
Bc = K,
2µ20√

as
Bc = µ02K.
χ⊥ − χ∥(5.42)(5.43)

After the transition, the ﬁeld also gives an oblique angle to sublattice magnetizations as shown in Fig. 5.3(a). In theprocess of increasing ﬁeld, the total magnetization increases with the ﬁeld proportionally and saturates at the ﬁeld ofcomplete polarization.
In conventional anti-ferromagnets, this critical ﬁeld is too large to reach in many of laboratories. Recently, however,there have been many reports on the spin ﬂop transition in nano-ferromagnets or in molecular ferromagnets. Figure 5.3(b)shows such an example of a crystal composed of a polymer {[Mn2 (bpdo)(H2 O)4 ][Nb(CN)8 ] · 6H2 O}n . It has Néelpoint at TN =15 K, which is comparatively low. We observe a clear spin ﬂop transition at around 0.6 T at 1.8 K.
So far we have considered the case of nearest-neighbor-only exchange interaction. That is, only inter-sublattice interaction is considered and intra-sublattice interaction is ignored. In reality, the super-exchange interaction often works
between spins in a sublattice (intra-sublattice interaction). In some cases, an anti-ferromagnetism at zero ﬁeld is realized
by a small difference in inter-sublattice anti-ferromagnetic interaction and intra-sublattice ferromagnetic interaction. In
such a case, increasing the external ﬁeld lowers the energy of magnetic moments parallel to the ﬁeld. And at a certain

E09-3

Fig. 5.3 (a) Illustration of spin ﬂop transition (red line) and metamagnetism (blue line). (b) Spin ﬂop transition
appeared in a polymer anti-ferromagnet {[Mn2 (bpdo)(H2 O)4 ][Nb(CN)8 ] · 6H2 O}n . The inset shows the susceptibility at a low (0.1 T) ﬁeld. It is an anti-ferromagnet with TN =15 K, with the spin-easy axis along a-axis. From [5].
Lower panel shows the molecular structure with axes b (pointing up), c (pointing left), a (coming up from this paper).
(c) Examples of metamagnetic transition in a Mott insulator Sm0.5 Ca0.25 Sr0.25 MnO3 [6].

ﬁeld, the whole system goes to a ferromagnetic, which phenomenon appears as a sudden or steep increase in magnetization up to the saturation. This is called meta-magnetism[6]. An example is shown in Fig. 5.3(c). Meta-magnetism often
has strong temperature dependence. If we ﬁx the magnetic ﬁeld close to the critical ﬁeld, the temperature also drives ameta magnetic transition, which gives a very large (∂M/∂T )B . From eq. (2.114), this is very advantageous for magnetic
refrigeration. And now the application of meta-magnetism to high-efﬁciency magnetic refrigeration is active([7] is an
example from a helimagnetism).

5.6 Ferrimagnetism
The most typical material of magnetic insulators is ferrite, which I
mentioned in the beginning of this chapter.

5.6.1 Magnetism in ferrite
In the ferromagnetism of ferrite (AFe2 O4 , A=Mn, Co, Ni, Cu, Zn,· · · ), the spin-alignment is a mixture of anti-ferro and ferro types. Because the sublattices have different magnetic moments, they do not cancel
out. As a result a total ﬁnite spontaneous magnetization appears. Suchmagnetism is named ferrimagnetism after ferrite.
In the unit cell of spinel-type ferrite, there are 16 Fe3+ , 8 M2+ , 32
O2− . Spin magnetic moments at Fe ions are mostly cancelled by antiferromagnetic interaction, and spins at M2+ survive, causing the ferrimagnetism. The expected magnetic moments of ferrite and experimental values along with this statement are listed
below showing a good agreement.

E09-4

Materials

MnFe2 O4

FeFe2 O4

CoFe2 O4

NiFe2 O4

CuFe2 O4

Moment (Theory)

5µB

4µB

3µB

2µB

1µB

Moment (Exp.)

5.0

4.2

3.3

2.3

1.3

TN (K)

783

848

793

863

728

A bit detailed discussions on the magnetism of ferrite can be found in ref. [8] (in Japanese), or in refs. [9, 10]. For theapplication of ferrite magnetism, refer to refs. [11, 12]. Ferrite is extremely important in industrial application. They show
various magnetism depending on species of M, crystal types and shapes of samples. Even now, research is extremely
active, and many review papers for each individual ferrite can be found even in the last few years.

5.6.2 Molecular ﬁeld approximation of ferrimagnetism
We here use a Heisenberg model with unbalanced sublattices A and B.
BA = αMA + (−γ)(−MB ) = αMA + γMB ,(5.44a)BB = γMA + βMB ,(5.44b)where we consider not only inter-sublattice exchange interaction but also intra-sublattice interaction. The imbalance istaken into account by the difference between α, β. The inter-sublattice interaction is γ (must be common).

5.6.3 Magnetization below the Néel temperature
The set of self-consistent equations for magnetizations MA and MB in sublattices is from molecular ﬁeld approximation(5.44) as[]µSA(αMA + γMB ) ,
kB T
]
[
µSB
(γMA + βMB ) ,MB = µSB BSB
kB T

MA = µSA BSA(5.45a)(5.45b)where the Brillouin function is written as BS (x). Though µ = gµB may be different for sublattices if g-factors aredifferent, we here assume they are common for simplicity.
To obtain MA , MB , thus the total magnetization M = MA − MB , we need to solve eq. (5.45) numerically.
In such compensated ferrimagnetism, magnetizations show complicated temperature dependences below Néel temperature due to the differences between SA and SB , α and β. As an example, compensated ferrimagnetism is displayed in

(a)

(b)
Fig. 5.4 (a) Conceptual scheme of compensated ferrimagnetism. (b) Compensated ferrimagnetism appeared in themagnetization of amorphous alloy Gd-Co-Mn. From [13].

E09-5

Fig. 5.4. In conceptual diagram in Fig. 5.4(a), A-sublattice has a larger lowest temperature magnetization (SA > SB ).
On the other hand, B-sublattice has a larger intra-sublattice interaction (β < α), then the growth of magnetization MB
below Néel is faster with lowering the temperature. Hence in a certain temperature region lower than TN , MB > MA
holds and as indicated by the broken line MA − MB is negative though in the measurement, magnetization M parallelto the ﬁeld always has lower energy and the green line is observed. With lowering of temperature, MA increases andthe total magnetization disappears at the point of MA = MB . But with further lowering of temperature, MA > MBand the magnetization reappears. As a whole, it becomes a curious temperature dependence as shown by the green line.
Figure 5.4(b) shows such a temperature dependence of an amorphous alloy Gd-Co-Mo[13]. In “complete compensation
type” ferrimagnetism, there is a difference in intra-interaction but the sublattice magnetic moments are the same and thetotal magnetization disappears at lowest temperatures[14].

5.7 Helimagnetism
It is not alway possible to separate a spin system with anti-ferromagnetic interaction into a small numbers of magnetic
sublattices. Also as considered in the previous section, ranges of interactions may span over more than single (magnetic)
lattice constant. Let us consider helimagnetism, that appears in such a complex system in a Heisenberg model. It has aspiral-like spin conﬁguration, which shows considerable difference from parallel/anti-parallel (collinear) conﬁgurations
so far considered. In the treatment of anti-ferromagnetism, ﬁrstly the anti-ferromagnetic ground state (Néel ordered state)is given. Then the molecular ﬁeld is considered based on the state. This time, we have a look on the process to ﬁnd out
the ground state[15].

5.7.1 Classical Heisenberg model
Here the exchange interaction potential J depends on the combination of sites (i, j) and a site-dependent magnetic
ﬁeld Bi is working on each site.
H =−∑Jij Si · Sj − µ

∑

⟨i,j⟩

Bi · S i .

(5.46)

i

However we put Bi = 0 for a while. And in the ﬁrst place, to see that a helical spin conﬁguration can be stable, we
consider a classical Heisenberg model, in that the spins are treated as classical vectors.
To look for an ordered stated, we assume an ordered state and perform Fourier expansion as
1 ∑hSi i = √hSq i exp(iq · ri ).
N q
Then| hSi i |2 = S 2 =1 ∑hSq i · hSq′ i exp(i(q + q ′ ) · ri )N′

(5.47)(5.48)

q,q

Now the expectation value of Hamiltonian can be written as∑∑
hH i = −Jij hSi i · hSj i = −Jq hSq i · hS−q i ,
⟨i,j⟩

where
Jq =

(5.49)q∑

Jij exp[−iq · (ri − rj )](5.50)

jis the Fourier transform of the interaction potential. Taking the sum of both sides of eq. (5.48)on subscript i, the righthand side is∑
1 ∑∑hSq i · hSq′ i exp(i(q + q ′ ) · ri ) =hSq i · hSq′ i δq,−q′ .
N i
′
′
q,q

q,q

E09-6

Then
N S2 =∑

hSq i · hS−q i .

(5.51)q

This works as a condition fulﬁlled by the classical solution.
In the Heisenberg model, Jij = Jji and also is real, Jq should be an even function of q. We then assume that Jq takes
the maximum (i.e., has a repeating structure with a ﬁnite period), and let ±Q be the wavenubers that give the maxima to
Jq . When Q = K − Q with an inverse lattice vector K, the system is in classical anti-ferromagnetic state, and out of
our scope here. Then though it is a bit bold, under the condition (5.51), we assume
hSQ i 6= 0,

hS−Q i 6= 0,(others) = 0.

(5.52)

Then eq. (5.48) can be written as follows:
N S 2 = hSQ i · hSQ i exp(2iQ · ri ) + hS−Q i · hS−Q i exp(−2iQ · ri ) + 2 hSQ i · hS−Q i .

(5.53)

Because the sum in the rhs of eq. (5.51) should be taken for q = ±Q, in thepresent case 2 hSQ i · hS−Q i, that just corresponds to the third term in eq. (5.53).
From the above we get
hSQ i · hSQ i = hS−Q i · hS−Q i = 0.

(5.54)

This condition is, for example, for hSQ i,
Re[hSQ i] = a, Im[hSQ i] = b 7−→ |a|2 − |b|2 = 0, a · b = 0,(5.55)

that is, the amplitude of the real and the imaginary parts are the same and theyshould be orthogonal. Then, we can write√N
hSQ i =
S(u − iv),
2

(5.56)

where u and v are orthogonal unit vectors. This leads to the expectation value of spins in the ground state:
hSi i = S[u cos(Q · ri ) + v sin(Q · ri )].

(5.57)In this spin conﬁguration, the spin rotates along and around Q-axis in the plane stretched by (uandv). The conﬁguration is called helical spin structure. Though the structure is affected by crystal anisotropy in real materials, thetheoretical Heisenberg model is isotropic and the plane of (u, v) can be taken to any direction.

5.7.2 Molecular ﬁeld approximation
Based on the classical ground state, we apply the molecular ﬁeld approximation, by introducing the site-dependent
magnetic ﬁeld
Bi = Bq [u cos(q · ri ) + v sin(q · ri )](5.58)

into eq. (5.46). We write the averaged spin as
hSi i = mq [u cos(q · ri ) + v sin(q · ri )].

(5.59)

Then along with molecular ﬁeld procedure, the effective Hamiltonian at site i is given by
Heﬀ (i) = −(2mq Jq + µBq )[u cos(q · ri ) + v sin(q · ri )] · Si .
Then as is the course, a self-consistent equation is given by[]S
mq = SBS(2mq Jq + µBq ) .
kB T
E09-7

(5.60)(5.61)

This has a just same form as that in the ferromagnetism. If we deﬁne “helical magnetization” as µmq , we obtain thehelical magnetization in paramagnetic states as
(
)−1
µmq
2Jq
= χ0 1 − 2 χ0.
Bq →0 Bqµχq = lim(5.62)

The critical temperature for the appearance of helical order TQ is given byk B TQ =2
S(S + 1)JQ .
3(5.63)

5.7.3 Observation of helimagnetism, skyrmion excitations
In the above, a helimagnetism in a Heisenberg model is considered. In real materials, it is said that NiBr2 or β-MnO2
may have situations close to the model. There have been found many materials with helimagnetisms. Holmium (Ho)has a helimagnetism originated from the RKKY interaction. The Dyaloshinsky-Moriya (DM) interaction often causes
helimagnetism.
Here I would like to introduce an experimental method called “Lorentz microscope” for observing real space imageof magnetic structure, and observation of helimagnetism and related magnetic phenomena. One of the most powerful
methods to observe spin conﬁgurations is the neutron diffraction. Actually one of the motivations for ﬁnding helimagnetism was an anomalous neutron spectrum of β-MnO2 , etc. Despite the powerfulness of neutron diffraction in detecting
periodic structures, it has difﬁculties in catching local real space images. Lorentz transmission microscope is one of themethods to observe local images *2 .
Figure 5.5 shows the principle of Lorentz microscope, which utilizes the bending of electron beams by Lorentz force
from the magnetic ﬁeld in samples. Complete re-focusing of electrons, however, restore the bending resulting in no-

Fig. 5.5 Left panel: Illustration of electron beam lines in a Lorentz microscope. Focused electron beams go thougha sample, and re-focused by electron lenses for forming an image on a screen. Three right panels: When the sample
has inner magnetic ﬁeld, the Lorentz force gives curving on the electron beams. Left in right: When the screen is justat the focal level, the bending are restored to have no contrast. Center in right: When the screen is a bit far, the overfocusing results in a contrast. Right in right: When the screen is closer, the under focusing also results in a contrast.
From Li-cong et al. Ch. Phys. B 27, 066802 (‘18).

*2 There are many others like traditional observation of distribution of magnetic powders, micro MOKE with utilizing the Kerr rotation, magnetic

force microscope, which detects magnetic ﬁeld gradient, scanning SQUID, etc.

E09-8

Fig. 5.6 Left: Cubic ϵ-FeSi (B20-type) structure without centrosymmetric point. A: Illustration of helimagnetism.
B: Expected contrast in Lorentz micrograph for an observation in the angle of z-axis. C: False colored real space
imaging of helimagnetism in Fe1−x Cox Si at 20 K by a Lorentz microscope. D: Magnetic moment along z-axis
deduced from the above[16].

contrast. Hence as illustrated in the right two panels in Fig. 5.5, the beams are a bit defocused to have a contrast, of which
the intensity is reversed by the direction of defocusing.
Figure 5.6 shows an example of observing helimagnetism. The sample is Fe1−x Cox Si in a non-centrosymmetric cubic
ϵ-FeSi (B20) structure. Because of the lack of inversion symmetry, a term in similar form as an electric ﬁeld appears inelectric effective Hamiltonian and causes strong spin-orbit interaction. This leads to a strong DM interaction that creates
the helimagnetism.
When one observes a helimagnetic spin ordered state in a side-view, as in Fig. 5.6A, B, the magnetization is modulated

Fig. 5.7 Left: Experimentally obtained phase diagram of Cu2 OSeO3 . At low temperatures, low magnetic ﬁelds, itshows a helimagnetism, which is overtaken by a phase with skyrmion excitations with increasing the magnetic ﬁeld.
With further increase of the ﬁeld, it changes to a ferrimagnetism. Right: A: Unit cell structure of Cu2 OSeO3 . B: Cu
spin conﬁguration in the ferrimagnetic phase. C ∼ G: Lorentz microscope images. At low magnetic ﬁeld, a stripe
due to helimagnetism, and at middle ﬁelds images of skyrmion are observed. H: shows schematic view of a skyrmion
spin conﬁguration. From ref. [17].

E09-9

wavy in space. This corresponds to a Fresnel conﬁguration of Lorentz microscope and actually in the image, a striping
contrast is observed as in Fig. 5.6C.
Figure 5.7 shows skyrmion excitations observed with a Lorentz microscope. This is observed in Cu2 OSeO3 , which
does not have inversion symmetry in lattice as shown in Fig. 5.7A, and the DM interaction causes a helimagnetism as in
the phase diagram at low temperatures and at low magnetic ﬁelds. The helimagnetic phase is detected as stripe imagesby a Lorentz microscope as in C, F. With increasing the magnetic ﬁeld, a phase with skyrmion excitation appears as in
D, G, E. They are observed to be aligned periodically. Further increase in the ﬁeld drives the system into a ferrimagnetic
phase, of which spin conﬁguration is shown in B.

5.8 Spin wave
The phase transitions with appearance of spontaneous magnetizations like ferromagnetic transition are an example of
spontaneous symmetry breakings. They are also associated with the appearance of excited states called spin wave. Let
us have a look on them.

5.8.1 Ferromagnetic spin wave
Here we need to consider dynamical properties of spins in ordered states. For that in ferromagnetic Heisenberg model(5.1), we consider the time evolution of operator Si by applying the Heisenberg equation of motion as
ℏ

∑
dSi
1
= [Si , H ] = −2J
Si+δ × Si − µB × Si ,dt
i(5.64)

δ

where δ is taken for the nearest neighbor of i. This calculation can be followed, with the use of commutation relation[S α , S β ] = iS γ , (α, β, γ) = (x, y, z; cyclic), e.g., as[Six , Six Sjx + Siy Sjy + Siz Sjz ] = [Six , Siy Sjy ] + [Six , Siz Sjz ] = i(Siz Sjy − Siy Sjz ) = i(Sj × Si )x .
Equation (5.64) is in the form of equation of motion for precession around the effective ﬁeld (the external ﬁeld plus thenearest neighbor interaction). Then we can foresee that the precessions are chained through the exchange interaction andforms a wave propagates over the spins. To see that we need to consider higher order approximation than the “averaged”
ﬁeld.
Then we consider the Fourier transform1 ∑Si exp(−iq · ri ),Sq = √
N i

Jq =

∑

J exp[−iq · (ri − ri+δ )],(5.65)

δand with substituting the inverse Fourier transformation into eq. (5.64), we represent (5.64) in the Fourier transformation
as
ℏ

2 ∑
dSq
= −√
Jq′ Sq′ × Sq−q′ − µB × Sq .
dt
N ′(5.66)q

With the above, we extract a wave from the precessions of spins. By taking z-axis along magnetic ﬁeld B, hS0 i =√N Sez has by far the largest expectation value in a ferromagnetic state. Hence in the ﬁrst term of rhs of (5.66), we
ignore the terms other than hS0 i to obtainℏdSq= −[2(J0 − Jq )S + µB]ez × Sq .
dt

E09-10(5.67)

In each component


dSqx

ℏ
= [2(J0 − Jq )S + µB]Sqy ,



dt


dSqy
ℏ
= −[2(J0 − Jq )S + µB]Sqx ,dt dSqz

 ℏ
= 0.
dt

(5.68)For Sqx and Sqy these are harmonic oscillator equations. By comparison of this with
Sqx + iSqy ∝ exp[−iϵq t/ℏ],(5.69)

we obtain the energy ϵq of precession in q space as
ϵq = 2(J0 − Jq )S + µB.

(5.70)

Because the precession is in q-space, this represents a propagating wave in real space.

5.8.2 Holstein-Primakoff transformation
As is the case of Larmor precession, the equation of motion derived quantum mechanically has the same form asclassical one. Then next we consider quantization of this wave. For a spin operator S, we write an eigenfunction |mi ofSz with eigenvalue m (m = −S, −S + 1, · · · , S − 1, S). The operation of up-down operators S± = Sx ± Sy gives√}
S(S + 1) − m(m + 1) |m + 1i ,√S− |mi = S(S + 1) − m(m − 1) |m − 1i .
S+ |mi =(5.71)Let us express a spin operator with creation and annihilation operators a† , a of bosons. For that we take the state
Sz = S, i.e., |Si as the vacuum of boson, and |S − ni as n boson state. Namely,a |Si = 0,
Then with n̂ = a† a, we can formally write

1|S − ni = √ (a† )n |Si .
n!

Sz = S − n̂,√
S+ = 2S − n̂ a,√S− = a† 2S − n̂.

(5.72)(5.73)

The above is called Holstein-Primakoff transformation.

Appendix 9A: Various “magnetism”
There are many ways to classify magnetisms and the classiﬁcation itself is not very important. For example, as a
macroscopic phenomena, there is “ferromagnetism”. However this contains a wide variety of magnetisms including“all-aligned” simple magnetism, ferrimagnetism, canted anti-ferromagnetism, etc. If we count for metastable conﬁguration, the number of classes is huge, and the classiﬁcation brings little knowledge. Here I would like to introduce somemagnetisms to have smooth talks with experts on magnetism.

E09-11

9A.1 Paramagnetism
So far we have seen the paramagnetism by local moments and that by itinerant electrons (Pauli paramagnetism). The
Pauli paramagnetism usually have smaller contribution to the susceptibility and for the detection, a combination withother methods is usually required.
There are the cases that a ﬁnite number of spins form an ordered state, which does not spread over the system. Such asystem generally shows paramagnetism with a huge magnetic moment, which is called superparamagnetism. The behavior of susceptibility resembles those of anti-ferromagnetism or spin-glass. Distinction of these is sometimes difﬁcult.

9A.2 Diamagnetism
As we saw in the Landau diamagnetism in metals, usually diamagnetism originates from orbital motion of electrons. It
sometimes becomes very large reﬂecting peculiar band structures as we saw in the section of graphite. Since water also
has a large diamagnetism, various things including water cause magnetic levitation in a very large magnetic ﬁeld. As a
bit special example, the superconductors have perfect diamagnetism (the Meissner effect).

9A.3 Ferromagnetism
As mentioned at the beginning of this section, there can be various deﬁnitions of ”ferromagnetism.” It often refers to thecase where itinerant electrons exist like metal and their spins become imbalanced to generate spontaneous magnetization.
Also ferrimagnetism is often called ”ferromagnetism.”

9A.4 Anti-ferromagnetism
すでに見たように，磁気副格子内でスピンの向きが揃っているが，副格子のモーメントが互いに反転しているために全体としては自発磁化を持たないように見える (帯磁率が発散しない) ものを反強磁性と呼ぶ．

9A.5 Ferrimagnetism
As in anti-ferromagnets, neighboring moments have antiparallel alignment. However because there are unbalances inthe size of moments or the numbers of magnetic sublattices, total spontaneous magnetization appears in ferrimagnetism.
Oxide ferromagnets like ferrites, garnets are this type.

9A.6 Canted anti-ferromagnetism
This type also has anti-ferromagnetic interactions though the moments in magnetic sublattices are not completely
inverted, but canted. Some of ferrite. Total spontaneous magnetization is generally small (weak ferromagnetism).

9A.7 Helimagnetism
The magnetic moments are arranged helically in space, and total spontaneous magnetization vanishes. On the other
hand, ”chirality” occurs depending on the winding direction of the spiral, which causes various phenomena. In somecases, a topological excitation called “skyrmion” appears, and they form a lattice. This kinds of materials have been
attracting attentions in these decades.

E09-12

9A.8 Spin density wave
The state in which the spin density and direction are spatially distributed in a wavy manner is called the spin density
wave (SDW). There are also anti-ferromagnetic SDW without total magnetization, and ferrimagnetic SDW with a totalmagnetization.

9A.9 Spin glass
When the localized magnetic moments exist randomly in space and the interaction between the moments is also random, the angles of the moments are randomly frozen, as in a glass state (amorphous state) in which atoms are randomly
aggregated. This is called spin glass. It is found in dilute magnetic alloys containing magnetic atoms as impurities. In
ferromagnetism and anti-ferromagnetism, there are only a few stable states of free energy, but in spin glass, there area large number of metastable points. The Nishimori quantum annealing theory is built on the mathematical similarity
between the relaxation from such metastable points to the true ground state by quantum tunneling (annealing) and a kindof optimization problem. This is the basics of modern quantum annealing computation[18]．The behavior of the magnetism is similar to that of anti-ferromagnetic materials. When cooling in a magnetic ﬁeld, the temperature dependence
becomes weaker on the lower temperature side than the spin glass transition point, and in zero magnetic ﬁeld cooling,
the magnetism disappears near zero degrees. As the temperature rises, the magnetization also rises and joins the coolingvalue in the magnetic ﬁeld at the transition point.

References[1] D. C. Johnston. Magnetic susceptibility of collinear and noncollinear heisenberg antiferromagnets. Phys. Rev. Lett.,Vol. 109, p. 077201, Aug 2012.
[2] W. L. Roth. Magnetic structures of mno, feo, coo, and nio. Phys. Rev., Vol. 110, pp. 1333–1341, Jun 1958.
[3] Sergio M. Rezende, Antonio Azevedo, and Roberto L. Rodrı́guez-Suárez.

Introduction to antiferromagnetic

magnons. Journal of Applied Physics, Vol. 126, No. 15, p. 151101, October 2019.
[4] Z. Yamani, Z. Tun, and D. H. Ryan. Neutron scattering study of the classical antiferromagnet MnF2 : a perfecthands-on neutron scattering teaching courseSpecial issue on neutron scattering in canada. Canadian Journal of
Physics, Vol. 88, No. 10, pp. 771–797, October 2010.
[5] Dawid Pinkowicz, Michał Rams, Wojciech Nitek, Bernard Czarnecki, and Barbara Sieklucka. Evidence for magnetic anisotropy of [NbIV(CN)8]4- in a pillared-layered mn2nb framework showing spin-ﬂop transition. Chemical
Communications, Vol. 48, No. 67, p. 8323, 2012.
[6] Sanjib Banik, Kalipada Das, Tapas Paramanik, Niranjan Prasad Lalla, Biswarup Satpati, Kalpataru Pradhan, and Indranil Das.

Huge magnetoresistance and ultrasharp metamagnetic transition in polycrystalline

sm0.5 ca0.25 sr0.25 mno3 . NPG Asia Materials, Vol. 10, No. 9, pp. 923–930, September 2018.
[7] Noriki Terada and Hiroaki Mamiya. High-efﬁciency magnetic refrigeration using holmium. Nature Communications, Vol. 12, No. 1, February 2021.
[8] 近角聡信. 強磁性体の物理 (上): 物質の磁性 (物理学選書 4). 裳華房, 単行本, 10 1978.
[9] A Broese van Groenou, P.F Bongers, and A.L Stuyts. Magnetism, microstructure and crystal chemistry of spinel
ferrites. Materials Science and Engineering, Vol. 3, No. 6, pp. 317–392, February 1969.
[10] Chemistry310 at Penn State University.
2021.

Chemistry libretexts; 8.7: Spinel, perovskite, and rutile structures,https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Book%3A_

E09-13

Introduction_to_Inorganic_Chemistry_%28Wikibook%29/08%3A_Ionic_and_Covalent_
Solids_-_Structures/8.07%3A_Spinel_Perovskite_and_Rutile_Structures

also

https://en.wikibooks.org/wiki/Introduction_to_Inorganic_Chemistry/Ionic_and_
Covalent_Solids_-_Structures#_8.6_Spinel,_perovskite,_and_rutile_structures.
[11] 電気学会マグネティックス技術委員会（編）. 磁気工学の基礎と応用. コロナ社, 10 2013.
[12] Alex Goldman. Modern Ferrite Technology. Van Nostrand Reinhold, 12 1990.
[13] R. Hasegawa, B. E. Argyle, and L-J. Tao. Temperature dependence of magnetization in amorphous gd-co-mo ﬁlms.
In AIP Conference Proceedings, Vol. 24, p. 110. AIP, 1975.
[14] Rolf Stinshoff, Ajaya K. Nayak, Gerhard H. Fecher, Benjamin Balke, Siham Ouardi, Yurii Skourski, Tetsuya Nakamura, and Claudia Felser. Completely compensated ferrimagnetism and sublattice spin crossing in the half-metallic
heusler compound mn1.5 fev0.5 Al. Phys. Rev. B, Vol. 95, p. 060410, Feb 2017.
[15] Akio Yoshimori. A new type of antiferromagnetic structure in the rutile type crystal. Journal of the Physical Societyof Japan, Vol. 14, No. 6, pp. 807–821, June 1959.
[16] Masaya Uchida, Yoshinori Onose, Yoshio Matsui, and Yoshinori Tokura. Real-space observation of helical spinorder. Science, Vol. 311, No. 5759, pp. 359–361, January 2006.
[17] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura. Observation of skyrmions in a multiferroic material. Science, Vol.
336, No. 6078, pp. 198–201, April 2012.
[18] 西森秀稔, 大関真之. 量子アニーリングの基礎 (基本法則から読み解く物理学最前線 18). 共立出版, 5 2018.

E09-14

Lecture note
15th June (2022)

Magnetism (10)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Last week, we have seen the Holstein-Primakoff transformation. However, we have skipped some elementary explanation on the classical view of the spin wave, and here I added some explanations in Appendix 10A.
Why we need to introduce the method of Holstein-Primakoff transformation? Because the variable in the Hamiltonian
is now spin, which differs from the canonical variables in classical mechanics such as a spatial coordinate of an electron,and the ordinary general method of quantum ﬁeld theory cannot be applied. Various problems in the Holtein-Primakoff
way were pointed out, e.g., in [1]. A particularly problematic is the “extension” of functional space for n over 2S in
the form of eq. (5.72). It is proven that the operators of physical quantities have no matrix element between the originalfunctional space and the extended space[2]. However we need to be careful that the proof is for exact theories and someapproximations may create some elements. We can escape the problem in treating small n cases.
In the quantization of the Heisenberg model by the Holstein-Primakoff method, the quantized bosons have mutualinteraction due to the non-linear term in eq. (5.73). In order to ignore the interaction and to treat it as the sum of harmonic
oscillators, we apply the following approximation. The expansions of eq. (5.73) with n̂:()
√a†j ajŜj+ = 2S 1 −+ · · · aj , 


4S
)
(

√
a†j aj
+ ··· , Ŝj− = 2Sa†j 1 −
4S(5.74)

are substituted to the Heisenberg model, to get∑∑
H = −2
Jij Ŝi · Ŝj = −2
Jij {Ŝiz Ŝjz + (Ŝi+ Ŝj− + Ŝi− Ŝj+ )/2}
⟨i,j⟩

= −2

∑

⟨i,j⟩

[
Jij S

2

1
1
− S(n̂i + n̂j ) + S(a†i aj + a†j ai ) + n̂i n̂j − a†i a†j aj aj − a†j a†j aj ai + · · ·
4

⟨i,j⟩

4

where n̂i = a†i ai . We take the terms to quadratic of ai , a†i to reach∑H = −2Jij [S 2 − S(n̂i + n̂j ) + S(a†i aj + a†j ai )].

],(5.75)(5.76)⟨i,j⟩

The result is the same if we take the terms with S in them.
We deﬁne the Fourier transform of a†j , aj as
1 ∑aj exp(iq · r), 
aq = √



N j
1 ∑


a†q = √
aj exp(−iq · r). N j(5.77)

With substituting the above, the Hamiltonian is ﬁnally given by∑∑
H = −2
Jij S 2 + 2[J0 − Jq ]Sa†q aq⟨i,j⟩= E0 +∑qℏωq a†q aq ,(5.78)qwhich is in the form of a set of spin waves without mutual interaction. In this way, we can take into account the interaction
systematically with taking the higher order terms one by one. Such quantized spin waves as bosons are called magnons.
E10-1

5.8.3 Magnon approximation of low energy excitations
As we saw in the derivation of eq. (5.76), the approximation of harmonic oscillators corresponds to the ignorance of
the interaction between magnons. The approximation is not good for many-magnon excitation at high temperatures. We
thus consider the contributions of magnons to physical quantities at low temperatures. With taking the magnetic ﬁeld
along z-axis, the magnetization is
⟨
M =µ

∑

⟩
Siz

= µSN − µ∑i⟨a†i ai ⟩ = µSN − µ(n(ϵ) =expn(ϵq ),(5.79)

q

i

where

∑

ϵ
kB T

)−1
−1

(5.80)

is the Bose distribution function. From eq. (5.78), ℏϵq = 2S(J0 − Jq . We assume the exchange interaction J works only
between nearest neighbors. We here consider a square lattice. Let a be the distance of nearest neighbors and q vector bealong a lattice direction. Then
[
(
)]
(qa)2
= 2SJ(qa)2 .
ℏϵq = 2S(J0 − Jq ) = 2SJ{2 − [exp(iqa) + exp(−iqa)]} ≃ 2SJ 2 − 2 1 −2(5.81)

From the above, with use of the low temperature asymptotic form of the Bose distribution function, M is given by[( )()3/2 ]3kB T
M = µN S − ζ,(5.82)
28πJSwhere ζ(x) is the Riemann’s ζ-function and ζ(3/2) ≈ 2.61.
Next, we consider the speciﬁc heat at low temperatures. The internal energy is obtained from the low temperatureasymptotic form of the Bose function and the dispersion relation as
)5/2
( )(
kB T
5
,
U = E0 +n(ϵq ) = E0 + 12πJSN ζ
2
8πJS
q
∑

(5.83)

from which the speciﬁc heat is obtained as
15∂U=
N kB ζ
C=
∂T
4

( )(
)3/2
5
kB T
.
2
8πJS

(5.84)

5.8.4 Anti-ferromagnetic spin wave
Next we proceed to the anti-ferromagnet. As in Sec. 5.5, we consider A and B sublattices with antiparallel magnetizations. We assume ferromagnetic Holstein-Primakoff transform can be applied to A-sublattice. Then a magnon
propagation in A-sublattice affects spins in B-sublattice and causes propagation of precession around z-axis. However in
B-sublattice the direction of the effective ﬁeld is inverse and the direction of precession should be inverse. Consequently,
the system can be described as coexistence of two-types of magnons with a mutual interaction. Then we need to consider
another kind of bosons in B-sublattice. The vacuum should be the inverse of that in A-sublattice and |0⟩B = |−S⟩ Thenfor site j in B-sublattice j (j ∈ B), we introduce the transformSjz = −S + b†j bj ,



√

†
†
Sj+ = bj 2S − bj bj ,

√



†
Sj− = 2S − bj bj bj .
E10-2

(5.85)

As in the case of ferromagnet, (5.73) and (5.85) are substituted into the anti-ferromagnetic Heisenberg model. Then
taking to quadratic of boson operators,
H = −αz |J|N S 2 + 2|J|S∑(a†i ai + b†j bj + ai bj + a†i b†j ),(5.86)⟨i,j⟩

where i ∈ A, j ∈ B. Let the Fourier transform of ai , bj be written as
√

2 ∑

ai =
aq exp(−iq · ri ), 


N q

√

2 ∑


bq exp(−iq · rj ), 
bj =

N q

(5.87)

then the Hamiltonian is re-written as
H = −αz |J|N S 2 + 2αz |J|S∑[a†q aq + b†q bq + γ(q)(a†q b†q + aq bq )],(5.88)q

where γ(q) is deﬁned asγ(q) = αz−1∑

exp(−iq · ρ)

(5.89)

ρwith ρ a vector connecting interacting two spins.
For the diagonalization of the Hamiltonian, we introduce the Bogoliubov transformation (aq , bq ) → (αq , βq ) as}
aq = cosh θq αq − sinh θq βq† ,(5.90)
bq = cosh θq βq − sinh θq αq† .
(αq , βq ) satisfy the following boson commutation relations,[αq , αq† ] = 1,

[βq , βq† ] = 1,

[αq , βq ] = [αq† , βq† ] = 0.

(5.91)The Hamiltonian reads
H = −αz |J|N S 2 + 2αz |J|S∑[(cosh 2θq − γ(q) sinh θq )(αq† αq + βq† βq + 1)q

− 1 − (sinh 2θq − γ(q) cosh 2θq )(αq βq + αq† βq† )].

(5.92)For the last off-diagonal term to vanish, we should choose the parameter θq assinh 2θq / cosh 2θq = tanh 2θq = γ(q).

(5.93)

Hence the diagonalized Hamiltonian is given by∑ √√H = −αz |J|N S 2 + 2αz |J|S[( 1 − γ(q)2 − 1) + 1 − γ(q)2 (αq† αq + βq† βq )].

(5.94)q

In eq. (5.94), the ﬁrst two terms without operator represent the ground state energy. The ﬁrst term is the energy ofNéel ordered state. The second term can be interpreted as the zero-point motion energy of magnons. The classicalNéel ordered state is not the quantum mechanical ground state unlike the ferromagnetic case. Namely the original
Sjz = S, S − 1, · · · , −S states are hybridized though the anti-ferromagnetic interaction and the perturbation decreases
the ground state energy. Accordingly, the expectation value of spin size diminishes from the full-size of S. The amountof the decrease is

2 ∑1 ∑⟨Sjz ⟩ = S −sinh θq = S −
N q
N q

(
√

1
1 − γ(q)2)−1 .

(5.95)The table below shows the decreases in spin size (∆ = S − ⟨Sjz ⟩), and the variation in energy (ϵ is deﬁned as E0 =N |J|αz S(S + ϵ)), calculated for some simple lattice structures[3].
E10-3Lattice

Square

∆

0.917

ϵ

0.158+0.0062S−1

Simple Cubic

Body Centered Cubic

0.078

0.05930.097+0.0024S−1

0.073+0.0013S −1

The anti-ferromagnetic magnons expressed by the last two terms, degenerate reﬂecting the equivalency of A- andB-sublattices. The dispersion relation is obtained fromϵq = 2αz |J|S√

1 − γ(q)2 .

(5.96)

γ(q) deﬁned in eq. (5.89) can be calculated, e. g., for simple cubic lattice with a unit cell size a, the dispersion in thelong wavelength limit is given by√
ϵq = 2 2αz |J|Saq,(5.97)

which is linear in q.
Speciﬁc heat is a quantity to be compared with experiments. As in the case of ferromagnet, we consider the internalenergy which is given in the case of simple cubic lattice by
U = E0 +

π2
N
15

(

k T
√ B
2 2αz |J|S

)3
kB T,

(5.98)

where E0 is the ground state energy given by the ﬁrst two terms in eq. (5.94). The calculated speciﬁc heats are summarized
in the following table including the cases of 1D, 2D, etc.[3]．
LatticeE0−
αz |J|N S 2
C
N kB
∆S

1D Chain−1

1+0.363S
(
)
2π
kB T
3 2αz |J|S
Diverge

2D Square Lattice−1

1+0.158S
(
)2
14.42
kB T
π
2αz |J|S
0.197

3D Simple Cubic
1+0.097S −1
(
)3
√ π2
kB T
4 3
5 2αz |J|S
0.078

The original purpose of considering magnons is to treat thermal ﬂuctuation at low temperatures correctly. Particularly
for anti-ferromagnets, there are real materials close to the theoretical models. Hence it is important whether the simple
models can explain such experiments. Figure 5.8 shows the crystal structure and the measured speciﬁc heat of an organic

Fig. 5.8 Crystal structure and speciﬁc heat of κ-(BETS)2 FeX4 . Left: (a) Molecular structure of BETS. (b) Crystal
structure viewed from b-axis. BETS molecules are in line with alternative oblique angles. (c) Crystal structure viewed
from c-axis. Right: Low temperature speciﬁc heat of a sample with X=Br. The contribution from phonons with T 3
dependence is indicated by a black line. The inset shows Cp T −1 as a function of T 2 [4].

E10-4

anti-ferromagnet κ-(BETS)2 FeBr4 (BETS = bis(ethylenedithio)tetraselenafulvalene)[4]. This material is not an insulator
having metallic conductivity. At low temperatures, it undergoes an anti-ferromagnetic transition at TN =2.47 K, andfurther, a superconducting transition at 1.1 K. The crystal shown in the left panel is composed of stacking of comparatively
small molecules. Electrons in π-bonds spread over ab plane, which is stacked along c-axis. The anti-ferromagnetism
originates from 3d-electrons in Fe with S =5/2. There are Jdd , which is direct interaction between d’s, and Jπd , whichis mediated by π.
The right panel in Fig. 5.8 displays the measured speciﬁc heat, which shows very sharp increase at TN . This corresponds
to the jump at TC in the ﬁgure in Sec. 5.3. The inset shows Cp /T as a function of T 2 . An ordinary metal has a speciﬁcheat Cm = AT + BT 3 from an electron contribution (∝ T ) and a lattice contribution (∝ T 3 ). This is written as
Cm /T = A + BT 2 and expressed as a line in this plot. The contribution of electrons is negligibly small as known fromthe ﬁtting at high temperatures. The heat capacity shows T 2 -like variation in the region lower than TN , as it shows a line.
This seems to be in accordance with the result of 2D speciﬁc heat in the above table. In the paper, however, the authors
claim that the results are in accordance with a theory on 1D anti-ferromagnetic chain.

5.8.5 Nambu-Goldstone theorem and spin wave
In the section of ferromagnetic transition in Heisenberg model, we have visited the concept of spontaneous symmetry
breaking (SSB). In typical continuous phase transitions, a symmetry should be spontaneously broken and at the sametime an order appears.
Nambu-Goldston theorem is summarized as follows:
Nambu-Goldstone theoremWhen a symmetry of a physical system is spontaneously broken, there is an excitation with zero energy (gap) in thelong wavelength limit.

2

Sometimes it is described as “excitation with zero-mass.” From the still energy E = mc , the two descriptions areequivalent.
This can be intuitively understood in the case of ferromagnetic transition. In the phenomenology in Sec. 5.2, as (a)→(b)in Fig. 5.2, the SSB caused by appearance of two minima in the free energy F (M ). For example, two-dimensional
Heisenberg model is isotropic and in the SSB state, the free energy of the state (M0 cos θ, M0 sin θ) does not dependon θ, in other words, the state can freely go around on the yellow line in the ﬁgure. The motion on the yellow line isNambu-Goldstone mode (NG mode) in the present case. This corresponds to the rotation of macroscopic magnetization

Fig. 5.9 Left: Free energy F of a system with rotational symmetry. The lowest F is obtained at M (magnetization)=0 on (Mx , My ) plane. Right: A spontaneous symmetry breaking yields an order parameter (magnetization). The states with minimum F exist continuously, between which the system can transit without energy (NambuGoldstone mode).

E10-5and can be seen as the limit of long wavelength. Classically the magnon wavelength is determined by the phase shift ofprecessions between neighboring sites. In the long wavelength limit, the phase shift is zero, and the spins are rotating
coherently. Hence we can say the magnon is the NG mode in the present case.
The dispersion relation of ferromagnetic magnon is quadratic in q as eq. (5.81). On the other hand in the antiferromagnetic magnons, the dispersion is linear as in eq. (5.96). Sometimes the latter is called type-A and the former iscalled type-B NG mode. On the other hand Nielsen-Chadha[5] called ℏω ∝ k 2n+1 as type-I, ℏω ∝ k 2n as type-II.
As is well known, it originated from Yoichiro Nambu’s idea for the acquisition of particle mass on the superconducting
BCS theory as a model, and from there, the standard theory of elementary particles makes great progress. And also in thecondensed matter theory, it is one of the central concepts as introduced by Anderson in the book “basic notions”[6]. Even
though there are many open questions even in the basics of the NG mode. Surprisingly there are many important ﬁndings´ NG theorem, the number of broken symmetries NBS
and progresses recently. Here I introduce an example. In the na’ive
and that of NG modes NNG should be the same(NBS = NNG ). However that does not hold in many simple examples. In
the case of 3D ferromagnetic transition, the rotation symmetries of two axes are broken, hence NBS = 2 though simple
ferromagnetic magnon NG mode number is one, i.e., NNG = 1.
For this problem, based on the pioneering works by Nielsen-Chadha[5] and by others, Watanabe-Murayama[7], andHidaka[8] reached the satisfactory answer in 2012 independently. This can be viewed as a generalization of the NG
theorem. To say it very short, let NI and NII be the number of type-I and type-II NG modes respectively, then
NI + 2NII = NBS .
For the detail see the review paper[9].

5.9 Experiments on magnons
As we saw in the above, magnons are elementary excitations from the ground state, considered to calculate macroscopic quantities of magnetic materials at ﬁnite temperatures. However recently, the concept of magnon goes beyond theframework. The researches are prosperous on the wave and the particle like manners, soliton physics or Bose-Einsteincondensation in the high-density non-linear region where the original spin wave approximation does not hold. The birthis given to the word “magnonics” and application to quantum information processing is seriously considered[10]. You
can ﬁnd many reviews[11] and textbools[12].
Here I would like to introduce rather old results, which are now the basis of present studies, however.

5.9.1 Measurement of magnon dispersion relation by neutron scattering
Neutron scattering has long been used as a means of measuring microscopic magnetic structures. It can be said that it isstill the most powerful reliable experimental method with atomic resolution. Inelastic scattering was used, in particular,
for magnon dispersion measurements.
We write the interaction between the magnetic moment µe of electrons in an atom and that of a neutron as

whereHint = −µe · Bn ,(5.99)(r)
Bn = rot µn × 3
r(5.100)is the magnetic ﬁeld of neutron magnetic moment µn , r is the vector connecting the atom and the neutron. When aneutron is scattered by such an interaction asℏk −→ ℏ(k − q),E10-6(5.101)

Fig. 5.10 (a), (b) Magnon dispersion relations calculated for MnF2 , FeF2 incorporating anisotropic ﬁeld[12]. Rightpanel: Magnon dispersion relation measured in MnF2 by time of ﬂight method of neutron scattering[13].

the variation in the energy is∆E =ℏ2(−2k · q + q 2 ).
2M

(5.102)

In very short, pulses of neutron with appropriate energy is applied to the sample, and the measurements of the energy(momentum) change and the scattering angle give certain information. This is (in the case of ∆E ̸= 0) the inelastic
scattering of neutron. On the other hand in the case of elastic scattering (∆E = 0), the wave nature of neutrons thediffraction is important.
To obtain the dispersion relation of magnons, the inelastic scattering of neutron is mostly used. Figure 5.10 shows themagnon dispersion relation of MnF2 obtained by neutron inelastic scattering. For the above information of (5.101) and(5.102), time-of-ﬂight (TOF) method was utilized[13]. The result shows a good agreement with the result of molecularﬁeld approximation with the effect of anisotropy.

5.9.2 Bose-Einstein condensation of magnons
In the process of introducing magnons with creation/annihilation operators, we deﬁned the vacuum |0⟩F as |S⟩, i.e., thestate of Sz = S, and |n⟩F as |S − n⟩. As mentioned, while we do not have negative n state from the deﬁnition, repetitive
operation of creation operator creates inﬁnite number of n states. However in reality, Sz can take only down to −S. This
physical space of function does not have any matrix element with the extended space as long as the theory is exact (noapproximation). This means that the magnons, though their creation/annihilation operators satisfy bosonic commutation
relations, the condition that “a single state can accommodate an inﬁnite number of particles” is not fulﬁlled. In this sense,
the statistics of magnon is not complete Bose statistics but a para-statistics[14].
A typical phenomenon appeared in the system of bosons is the Bose-Einstein Condensation (BEC). The BEC isvery shortly introduced in Appendix 10B. Superﬂuidity of helium, BEC in laser-cooled neutral atomic gases are therepresentatives. Even for these phenomena, the interaction between the particles exists and they are not jus the same as
the simple BEC described in the Appendix. Also, the superconductivity, in which weakly bound fermion pairs condensate
is a similar phenomenon. Even in the case of magnons, though they obey para-statistics, they can be viewed as “bosons
with hard cores” and there is a possibility that a similar phenomenon occurs. It is still difﬁcult, though, for magnons to
fulﬁll the basic condition of BEC that (averaged de Broglie wavelength)=(averaged particle distance) because the particle
density decreases with lowering the temperature, as is guessed from the calculation of the magnetization in eq. (5.79).
Then an experiment was carried out, in which a large number of magnons are excited by microwave and a non-equilibrium
BEC took place[15]. They studied the Brillouin scattering of light by magnons and observed anomalous narrowing of thelinewidth of the resonance.

E10-7

(a)

(b)

Fig. 5.11 (a) Temperature dependent magnetization of TlCuCl3 in magnetic ﬁelds[16], (b) Temperature variation of
the intensity of Bragg reﬂection (1,0,−3) in neutron diffraction[17]. As indicated in the right axis, this is proportional
to the square of vertical magnetic moment per site.

Let us see an experimental observation of BEC in thermal equilibrium[16, 17]. Due to the above restriction, thesituation is rather special. The material is a compound of TlCuCl3 in chemical formula. Two magnetic ions form pairs(dimer) with an antiferromagnetic coupling. There are singlet |0, 0⟩ and triplet (|1, −1⟩ , |1, 0⟩ , |1, 1⟩) as the states of thepair, in which the single is the ground state due to the antiferromagnetic coupling. There is an energy gap (spin gap)
between the ground state and the ﬁrst excited state. With applying magnetic ﬁeld, the energy of |1, 1⟩ is lowered andthe ﬁeld driven phase transition takes place for the ferromagnetism to appear. In this system, the magnon appears whenthe energies of |0, 0⟩ and |1, 1⟩ are close as the propagation of |1, 1⟩ states to the neighboring sites. As known from thisexample, the propagation of magnons is the equivalent to that of spin angular momentum and that brings a spin current.
For the behavior of magnetization in such a system without magnon BEC, a theory has been presented [18] and the hightemperature behavior is well explained. It predicts for the the magnetization transverse to the spontaneous magnetizationthat it should be constant for temperature. However in the experiment, as in Fig. 5.11(a) the magnetization once decreases
with decrease of temperature but the dependence is inverted around the transition point and increases. The magnon-BEC
theory explains the experiment that an order grows in the mixture of |0, 0⟩ and |1, 1⟩ and the transverse magnetization
appears as an average (does not cancel out)[16]. Furthermore as in Fig. 5.11(b) in neutron diffraction, it was conﬁrmedthat such an order actually grows[17]. From the above, the observation of magnon BEC is claimed.

Appendix 10A: Collective motion of spins
Because a macroscopic number of spins are bound to in a ferromagnetic state, the motion can be described as acollective motion. On the other hand, as lattice vibrations in lattice formation, a kind of collective motion from themagnetic ground state can exist and quantization as phonons is possible.
E10-8

10A.1 Collective motion of magnetization
For a ferromagnetic total magnetization
S=

∑

Si ,

(10A.1)i

we apply Heisenberg equation of motion asiℏ∂S= [S, H ],∂t

(10A.2)

which is formally the same as eq. (2.11) and represents the Larmor precession. A resonance experiment as EPR for a
total magnetization is possible and called as ferromagnetic resonance (FMR). From FMR we get various informationon the ferromagnetism and the spin wave.
Next we consider the case that the phase of the precession has a constant shift between neighboring spins. The above
motion of the total magnetization can be considered as the long wavelength limit of this motion. The magnetization andthe external ﬁeld direction is taken to z. This situation is expressed as
Six = A cos(ω0 t + θi ),Siy = A sin(ω0 t + θi ),(10A.3)

with a shift of θi with i. Let us use complex numbers for spins. And the Fourier transform and the inverse transform1 ∑Sjz exp(−iq · rj ),Sqx = √N j
are introduced.
We apply (10A.2) to Heisenberg Hamiltonian H = −2Jiℏ

1 ∑Sjx = √
Sqx exp(iq · rj )N q∑

(10A.4)⟨i,j⟩ Ŝi · Ŝj to obtain the following.

4i ∑
∂Sqx
=√ J
Siy Sjz exp(−iq · ri ){1 − exp[iq · (ri − rj )]}∂t
N(10A.5a)4i ∑∂Sqy
Six Sjz exp(−iq · ri ){1 − exp[iq · (ri − rj )]}.
= −√ J∂t
N(10A.5b)

⟨i,j⟩

iℏ

⟨i,j⟩

Fourier transform of J is written as
Jq =∑J exp[iq · (ri − rj )].

(10A.6)j

In the above, the sum over j and i can be taken anywhere because the interaction only depends on ri − rj . Further,
the sum only over the nearest neighbor because the interaction is assumed to work only for nearest neighbors. Here weapproximate Sjz by S, which corresponds to small angle approximation. Then
∂Sqx
= 2[J0 − Jq ]SSqy ,
∂t
∂Sqy
ℏ
= −2[J0 − Jq ]SSqx .
∂tℏ(10A.7a)(10A.7b)This is the same as eq. (5.68) but with B = 0.

Fig. 10A.1 Schematic diagram showing a constantphase shift between neighboring spins.

E10-9Appendix 10B: Bose-Einstein condensation
The Bose-Einstein Condensation (BEC) *1 is called a phase transitoin that is not due to the interaction between freedoms(quantum statistical phase transition). Though phase transitions caused by interaction beween some freedoms can be
intuitively understood, there are different types of phase transitions, in which the transitions are caused as the results ofcompetition between various factors. A representative is BEC.
In the case of bosonic systems, in spite of the absence of “force” betwen the particles, there exists the tendency forthem to occupy the same quantum state originating from their statistical property. Let us see that for the case of two
particles. We write a solution of the wave equation for two particles as ψ(x1 , x2 ). For the composition of wavefunctionsof the system Ψ(x1 , x2 ) that reﬂects the statistical property of bosons, the symmetrization of ψ results in1
Ψ(x1 , x2 ) = √ [ψ(x1 , x2 ) + ψ(x2 , x1 )] .
2

(10B.1)Hence the probability of ﬁnding the system at (x1 , x2 ) is|Ψ(x1 , x2 )|2 =]1[|ψ(x1 , x2 )|2 + |ψ(x2 , x1 )|2 + ψ(x1 , x2 )∗ ψ(x2 , x1 ) + ψ(x1 , x2 )ψ(x2 , x1 )∗ .
2

(10B.2)This reveals that the last two interference terms intensify the probability of ﬁnding the system under the condition of
x1 = x2 . Let us write the de Broglie wavelength as λ and the averaged distance between the particles as l. Then at low
temperatures λ ∼ l, this tendency of bosons makes many of them to occupy the state of k = 0, which behavior leads to
BEC. The above discusstion is expressed as
Ek =

∆p ∼

p2
= kB T,
2M√∴λ=M kB Thh∼√.
∆p
M kB T(10B.3)√
λ elongates as 1/ T with lowering the temperature. And with growing of the overlapp between the single particle
wavefunctions makes them undistinguishable and the symmetrization of the wavefunction cause the condensation to theground state in the phase space (r, p). The phase transition to the condensate at a certain temperature is BEC.

10B.1 Bose-Einstein condensation of ideal gas
Let us consider spin 0 ideal Bose gas. For the Bose distributionf (ϵ) =

1(β ≡ (kB T )−1 )e(ϵ−µ)β − 1(10B.4)

we deﬁne the point of µ = 0 as follows. At T = 0, from (10B.4) all the particles fall into the ground state, there wedeﬁneµ(T = 0) = 0.

(10B.5)

At ﬁnite temperatures, let N be the number of particles in the system:N=∑f (ϵi ).

i

*1 The acronym of BEC is applied to both Bose-Einstein Condensation and Bose-Einstein Condensate. In actual use, the confution is not serious.

E10-10∫

In the usual case we can writeN→f (ϵ)D(ϵ)dϵ.

(?)

Here the number of particle at the ground state N0 should be
N0 =

1
1
kB T
kB T
∼
=−
→µ∼−
.
e−µβ − 1−µβµN0(10B.6)

If we calculate the particle distribution on this line, for three dimensional ideal gas

Therefore

V m3/2N=√
2π 2 ℏ3

ϵ(k) =ℏ2 k 2
m3/2 V √then D(ϵ) = √ϵ.
2m
2π 2 ℏ3∫ ∞√0(mkB T )3/2√dϵ=Ve(ϵ−µ)β − 1
2π 2 ℏ3ϵ(10B.7)∫ ∞√0ex−α − 1xdx,

where x ≡ ϵβ and α ≡ µβ. We write the deﬁnite integral term as I(α), then I is
√ ( )
∫ ∞ √
x
π
3
I(0) =dx=ζ
∼ 2.6,x−1e2
20(10B.8)(10B.9)which decreases with increasing of the absolute value of α < 0. Then, in this logic, with T → 0 the maximum numberof N determined from (10B.8) goes to zeo. It is apparent that we have dropped something from the counting. That is, ofcourse, the macroscopic number of particles fall into the ground state.
From Eq. (10B.8),√

I(α) =

2π 2 ℏ3 N.
(mkB T )3/2 V

When this excessds (10B.9) at low temperatures the anomaly (increase in the particle number at the ground state.) occurs.
This critical temperature Tc is
T < Tc ≡

]2/3
[
2πℏ2
N
.
mkB ζ(3/2)V(10B.10)Here l ≡ (V /N )1/3 is the average distance between the particles and Eq. (10B.10) is interpreted asl=h√∼ λ(T = Tc ).
ζ(3/2) 2πmkB Tc(10B.11)This conﬁrms the statement that the BEC takes place when the average de Broglie wavelength is comparable with theaverage particel distance.
Below Tc , we add the number of ground state particles N0 to Eq. (10B.8):√∫ ∞V m3/2ϵN=√dϵ + N0 .
(ϵ−µ)β − 123e2π ℏ 0From Eq. (10B.6), N0 becomes a macroscopic number fro T < Tc , then µ = 0. Therefore[]( )3/2 ]
[
∫ ∞ √
ϵ
T
V m3/2V (mkB T )3/2√
I(0) = N 1 −.
N0 = N − √dϵ = N 1 −ϵβ − 1
2
3
2
3
e
N
T
2π ℏ 02π ℏc(10B.12)(10B.13)This is just like a spontaneous magnetization rapidly grows to ﬁnite values below the critical temperature in the ferromagnetic transition.
The total energy of the system for T < Tc is calculated as∫ ∞ 3/2
ϵV m3/2E=√dϵ2π 2 ℏ3 0 eβϵ − 1√ ( )∫ ∞ 3/2x
3 π
5
ここで T < Tc では
dx
=
ζ
より
x−1
e
4
2
0
( )(
3
5
m )3/2
V (kB T )5/2 .
E= ζ
2
2
2πℏ2
E10-11(10B.14)(10B.15)

C v / Nk B

2

1

0

1
T/Tc

2

Fig. 10B.1 Speciﬁc heat at constant volume of three dimensional ideal Bose gas as a function of temperature. Tc
is the critical temperature of the BEC.

Then the heat capacity at constant volume is calculated as( )(5m )3/2
15
5/2
Cv =
ζ
V kB T 3/2 .
4
2
2πℏ2

(10B.16)

Cv shows a cusp at Tc indicating that this is the phase transition.

10B.2 Bosonic stimulation
Here we have a look at bosonic stimulation for N particles, which is, though, essentially the same as what has been
mentioned on the case of two particles in Sed. ??. As we have seen, the bosonic stimulation works as if it is a driving
force in BEC or laser oscillation. Let us consider a identical boson system the case a particle in state φini gets perturbationand transitions to other single particel state φﬁn . Now the problem is the difference in the transition probabilities to thestate occupied with N particles and to the empty state. We write the initial state as(i)

ψ+ (r1 , · · · , rN +1 ) = √N∏
1
R̂m,N +1 det(+) {φi (rj )}φini (rN +1 ).
∏(N + 1)N ! l nl ! m=1(10B.17)

The symbol det(+) represents permanent, which is obtained by making the signs of all the terms into +. The ﬁnal
(f )state ψ+ is obtaned by exchanging φini with φﬁn . Let the matrix elements of perturbation Hamiltonian be a, i.e.
⟨φﬁn |Ĥ1 |φini ⟩ = a.
(f )

(i)

Assuming that φi (i ≤ N ) is orthogonal to φﬁn , among ⟨ψ+ |Ĥ1 |ψ+ ⟩, number of terms that give non-zero a is
∏
(f )
(i)
(N +1)N ! l nl !. This is equal to the sqare of the denominator in normalization constant. Then ﬁnally ⟨ψ+ |Ĥ1 |ψ+ ⟩ =a.
On the other hand, assuming all of φi (i ≤ N ) are φﬁn , we can write(i)ψ+ = √N∏

1(N + 1) m=1

R̂m,N +1 φﬁn (r1 ) · · · φﬁn (rN )φini (rN +1 ).

(10B.18)

All of the N ! terms in det(+) are φﬁn (r1 ) · · · φﬁn (rN ) and devided by N ! in the denominator of normalization constantto 1. However the ﬁnal state is(f )ψ+ = φﬁn (r1 ) · · · φﬁn (rN )φﬁn (rN +1 ).

(10B.19)√
Then we get ⟨φﬁn |Ĥ1 |φini ⟩ = a N + 1, and from the Fermi’s golden rule, the transition probability should be N + 1
times larger.
E10-12

References[1] 高橋康. 物性研究者のための場の量子論 1 (新物理学シリーズ 16). 培風館, 10 1974.
[2] Silvia Viola Kusminskiy. Quantum Magnetism, Spin Waves, and Optical Cavities (SpringerBriefs in Physics).
Springer, 2 2019.
[3] Ryogo Kubo. The spin-wave theory of antiferromagnetics. Phys. Rev., Vol. 87, pp. 568–580, Aug 1952.
[4] Shuhei Fukuoka, Satoshi Yamashita, Yasuhiro Nakazawa, Takashi Yamamoto, Hideki Fujiwara, Takashi Shirahata,
and Kazuko Takahashi. Thermodynamic properties of antiferromagnetic ordered states of π − d interacting systemsof κ − (BETS)2 FeX4 (x = Br, Cl). Phys. Rev. B, Vol. 93, p. 245136, Jun 2016.
[5] H.B. Nielsen and S. Chadha. On how to count goldstone bosons. Nuclear Physics B, Vol. 105, No. 3, pp. 445–453,
March 1976.
[6] PHILIP W. ANDERSON. BASIC NOTIONS OF CONDENSED MATTER PHYSIC. TAYLOR & FRANCIS, 2 2019.
[7] Haruki Watanabe and Hitoshi Murayama. Uniﬁed description of nambu-goldstone bosons without lorentz invariance. Phys. Rev. Lett., Vol. 108, p. 251602, Jun 2012.
[8] Yoshimasa Hidaka. Counting rule for nambu-goldstone modes in nonrelativistic systems. Phys. Rev. Lett., Vol. 110,
p. 091601, Feb 2013.
[9] Yoshimasa Hidaka and Yuki Minami. Spontaneous symmetry breaking and nambu–goldstone modes in open classical and quantum systems. Progress of Theoretical and Experimental Physics, Vol. 2020, No. 3, March 2020.
[10] H.Y. Yuan, Yunshan Cao, Akashdeep Kamra, Rembert A. Duine, and Peng Yan. Quantum magnonics: Whenmagnon spintronics meets quantum information science. Physics Reports, Vol. 965, pp. 1–74, June 2022.
[11] Philipp Pirro, Vitaliy I. Vasyuchka, Alexander A. Serga, and Burkard Hillebrands. Advances in coherent magnonics.
Nature Reviews Materials, Vol. 6, No. 12, pp. 1114–1135, July 2021.
[12] Sergio M. Rezende. Fundamentals of Magnonics (Lecture Notes in Physics). Springer, 8 2020.
[13] G. G. Low, A. Okazaki, R. W. H. Stevenson, and K. C. Turberﬁeld. A measurement of spin-wave dispersion in
MnF2 at 4.2 k. Journal of Applied Physics, Vol. 35, No. 3, pp. 998–999, March 1964.
[14] 高橋康. 物性研究者のための場の量子論 2 (新物理学シリーズ 17). 培風館, 4 1976.
[15] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin. Boseeinstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature, Vol. 443, No.
7110, pp. 430–433, September 2006.
[16] T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka. Bose-einstein condensation of dilute magnons in tlcucl3 .
Phys. Rev. Lett., Vol. 84, pp. 5868–5871, Jun 2000.
[17] Hidekazu Tanaka, Akira Oosawa, Tetsuya Kato, Hidehiro Uekusa, Yuji Ohashi, Kazuhisa Kakurai, and Andreas
Hoser. Observation of ﬁeld-induced transverse néel ordering in the spin gap system TlCuCl3 . Journal of the
Physical Society of Japan, Vol. 70, No. 4, pp. 939–942, April 2001.
[18] Masashi Tachiki and Takemi Yamada. Spin ordering and thermodynamical properties in spin-pair systems undermagnetic ﬁelds. Progress of Theoretical Physics Supplement, Vol. 46, pp. 291–309, 1970.

E10-13

Lecture note
22nd June (2022)

Magnetism (11)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

In the last week we saw an example of magnon BEC in thermal equilibrium observed in a bit special material *1 . Here
we would like to see an experiment on the BEC under quasi-equilibrium condition.
The experimental setup is shown in the left panel of Fig. 5.11[1]. A microwave pulse is applied to an YIG thin ﬁlm
for the generation of low energy (∼ 100 mK) magnons through a parametric process. A photon of microwave has a verysmall momentum, with which the excitation of a single magnon is difﬁcult. However as shown in the inset, it is possible
to excite two magnons with the same momentums but in opposite directions. Therefore, the excited magnons have almost
the half of the energy of applied microwave. The concentration and the energy distribution of magnons are measured
by Brillouin scattering of a laser light. The right panel of Fig. 5.11 shows the results. With increasing the power ofmicrowave from 4 W to 5.9 W, the number of magnons increased rapidly with keeping the width of distribution. The
observed distribution is much broader than the actual one as demonstrated in (d), in which the result of measurement witha higher resolution is displayed. This is claimed as the observation of BEC.

Fig. 5.11 Left: Experimental setup to observe a magnon BEC in quasi-equilibrium. A microwave is applied througha strip line. It excites a number of low energy magnons through a parametric process shown in the inset. A laser
beam is applied for measurement of magnon distribution with the Brillouin scattering. Right: Time evolution ofenergy distribution of magnons after a microwave excitation pulse. Black closed dots are for 5.9 W of the power ofmicrowave. Open ones are for 4 W. The red curve in (d) is with 50 MHz resolution measurement.

5.9.3 Ferromagnetic (Antiferromagnetic) resonance
As noted in Appendix 10A.1, ferromagnetic resonance (FMR) is a resonance absorption by the Larmor precession of amacroscopic magnetization. It can also be seen as the long wavelength limit of magnons. In an analysis of an experimenton a system with magnetic order, we need to consider complicated experimental details such as demagnetizing ﬁeld,
which was explained in the ﬁrst lecture. Let us write the free energy of the system F as


∑
∑
∑
∑
F =
λij Mi · Mj −Mi Ki,j Mj −Mi · H − N
Mj  ,
⟨i,j⟩

i,j

i

= −λM 2 + M · KM − M · (H − NM ),(5.103a)j(5.103b)*1 Dimer magnetism is not so rare, but it often has a very high critical ﬁeld of spin gap closing, that situation makes experiments difﬁcult.

E11-1

where eq. (5.103a) is for a paramagnetic case, and eq. (5.103b) is for a ferromagnetic case with macroscopic magnetization. Though we are mainly using B, which is mostly used in experiments, here, we use H to avoid confusion due toexistence of a spontaneous magnetization. In the rhs of eq. (5.103a), the ﬁrst term is the exchange interaction, the second:magnetic anisotropy, the third: Zeeman with demagnetizing effect (N (tensor), Sec. 1.2.2). Then the effective magnetic
ﬁeld working on M other than H isHeﬀ = λM − (K + N )M ,(5.104)in which λM can be dropped from the equation of motion because it is always parallel to M . Then the equation of
motion is then given by
1 dM= M × (H − KM − NM ),γ dt(5.105)where γ is the gyromagnetic ratio. This becomes a very complicated form in general experiments. However, in case thatthe sample shape has an axis of rotational symmetry taken along the easy axis of magnetization, and the external ﬁeld isalso along the easy axis, the resonance frequency ω is given by√ω = γ (H + (Kx − Kz + Nx − Nz )M )(H + (Ky − Kz + Ny − Nz )M ),(5.106)

where z-axis is taken along the ﬁeld.
Also in antiferromagnets, ferrimagnets, similar resonances take place due to the Larmor precession of macroscopic
magnetization. In order to treat the situation we consider the resonance conditions of magnetizations M1 , M2 *2 ofmagnetic sublattices. The effective ﬁelds Heﬀ1 , Heﬀ2 for M1 , M2 respectively, areHeﬀ1 = −λM2 + K11 M1 + K12 M2 + N(M1 + M2 )(5.107a)Heﬀ2 = −λM1 + K21 M1 + K22 M2 + N(M1 + M2 ).

(5.107b)

In antiferromagnetic case, because M1 = −M2 , the last terms in the two equations of eq. (5.107) vanish. And in thetensor of anisotropy, the followings hold.
K11 = K22 ,K12 = K21 .

(5.108)

If the anisotropy is uniaxial, the anisotropic energy FA is
FA = −

K1(cos2 θ1 + cos2 θ2 ),2(5.109)where θi are the angles between the primary axis and the magnetization Mi . The anisotropy tensor isKzz = −

K1
,
|M1 |

(others) = 0,

(5.110)where the primary axis is taken to z. The resonance conditions, then, are given as follows[2].
√ω±= 2λK1 + (K1 /|M1 |)2 ± H,γ√ω+
= B 2 − 2λK1γHere, Hc =√H ≤ Hc ,(5.111a)

H > Hc .

(5.111b)2λK1 is the critical ﬁeld of the spin-ﬂop transition. When the anisotropic ﬁeld K1 /|M1 | is much smaller

than the exchange ﬁeld λ|M1 |, we can write the condition as√ω±= 2λK1 ± H,γ

H ≤ Hc .

In the case of ferrimagnet, the complexities are unavoidable. Refer to [3, 4], if necessary.
*2 So far we have used MA , MB , which way makes the subscripts awkward. We will use 1, 2 instead, for a while.

E11-2(5.112)

Fig. 5.12 Spin wave resonance (Walker mode) observed ina permalloy ﬁlm with thickness of 560 nm. The horizontal
axis is magnetic ﬁeld in unit of Oe. The vertical axis is absorption strength for microwave of frequency 8.89 GHz. The
boundary condition leaded to the observation of the odd number modes from n=7 to 13 in eq. (5.114). The absorption lines
at higher ﬁelds were not counted by some reason. From [5].

5.9.4 Spin wave resonance(A)FMR is a resonance of total magnetization and can be seen as the limit of long wavelength. For thin ﬁlm samples,standing wave modes perpendicular to the ﬁlms are expected. The spectra of such standing wave mode is discrete. Thosemodes are also called Walker modes *3 . From the simplest dispersion relation (5.81) obtained in Secs.5.8.2, 5.8.3, theresonance frequency isωk = γH +2SJ(ka)2 .
ℏ

(5.113)

Here γ is the gyromagnetic ratio deﬁned for H- representation.
Now, we consider the case that spins are ﬁxed at the surfaces due to some strong magnetic anisotropy. From theboundary condition that the surfaces should be the nodes, the condition of resonance is
k=

nπ
,
L

n = 1, 2, · · · ,(5.114)

where L is the ﬁlm thickness[6]. In a comparatively simple case as the easy axis is perpendicular to the ﬁlm, various
information can be obtained by, e.g. changing the ﬁlm thickness. This is called spin wave resonance (magnon resonance). Figure 5.12 shows an example of spectrum. In the era of theoretical proposal and experimental conﬁrmation, themethod was frequently used for direct measurement of exchange interaction coefﬁcient J. In recent nano-sized magnets,
various conﬁnement shapes have been tested. In many cases, the analysis needs the help of numerical calculations[7].
A resonant motion of macroscopic magnetization can be viewed as a condensate of magnons and this can exists because they are bosons. The phenomenon is, however, different from the BEC, in which the boson system spontaneously
condensates breaking the gauge symmetry. On the other hands, resultant phenomena are similar in that many quantaoccupy a single quantum state and the condensate behaves as a classical wave. This is, for example, the same for electromagnetic waves, which we are using for communication in daily life. There is no problem in treating these as classicalwaves, but quantum mechanically, multiple photons are occupying a single quantum state and it is not too much to say
they are forming (near) coherent states. This has long been pointed out, e.g., in introductory publications on quantum
mechanics[8]. Accordingly, the fact that such condensates of magnons can be created at room temperatures with nonadiabatic methods, is not very surprising. But then, magnon condensate also shows quantum phenomena like quantum
tunneling of macroscopic magnetization, or quantum entanglement with photons, and so on. This means we need to beaware of what phenomena we are observing in studying them.
In the lecture, we will see some examples of classical wavy manner in magnon condensates.

*3 Because the wave equation has the form of Walker equation.

E11-3

5.10 Renormalization group and scaling theory of phase transitions
Let us be back to general theory of phase transition. Magnetic phase transitions are the best subject for guessing theconcept of scaling.
A huge number of review articles and books have been published on the renormalization group (RG), which is a major
theme for physics as a whole. Anyone interested in this issue should read, after all, a bible review[9] in this area. As
mentioned in the title of this paper, the original motive for developing the renormalization group theory is the Kondo effectdiscovered in dilute magnetic alloys and the Kondo problem (this is also a major subject for whole physics) raised by
Kondo’s theory. In that sense, the RG is closely related to the magnetism. I recommend as a basic textbook ref. [10, 11],as a standard for experts ref. [12] *4 . Among the commentaries on the Internet, ref. [13] seems to be careful and easy tounderstand. Also a review ref. [14] can be taken in arXiv. In the lecture I would give a super digest explanation.

5.10.1 Correlation function
For the scaling and RG, an important concept is the correlation function. So far we considered the case the magnetization is uniform in space. Even the ﬂuctuations, considered as magnons, have spatially uniform amplitude. However inreality, as we observed in the movie of numerical calculation of Ising model, with lowering the temperature, orderingsin spins locally appear (spin cluster), the size of which grows on approaching the critical point. If we observe the phenomenon from the view point of spatial ﬂuctuation and in the Fourier space, the characteristic wavelength of ﬂuctuation
grows to diverge at the critical point.
Let us treat the above process as follows. We consider a local magnetization density m, weakly depending on r asm(r). Let m be a local order parameter. By the same logic in Sec. 5.2.1, the free energy density f at r isabf (m(r, ∇m(r)) = f0 + m2 + m4 + c|∇m|2 − hm,
2
4(5.115)where in the rhs, the argument r is omitted. h is the local ﬁeld. Free energy F is represented in the functional form as∫
F {m(r)} =dr ′ f (m(r ′ ), ∇′ m(r ′ )).
(5.116)V

The partition function is[]
F {m(r)}Dm(r) exp −
.
kB T

∫
Z=

(5.117)

Here Dm(r) is a functional integral, which means taking the sum over all possible m(r). This is a well known concept
in the Feynman path integral[15, 16]. Because the probability density of realization of distribution m(r) is[]
1
F {m(r)}p{m(r)} = exp −
,
Z
kB T

(5.118)the statistical average of a physical quantity A is given by
1
hAi =
Z

∫

[

]
F {m(r)}Dm(r)A exp −.
kB T(5.119)Then as we did for eq/ (5.16), we assume the temperature dependences of a and b in eq. (5.115) asa = α(T − TC )(α > 0),b = const. (> 0).

*4 I recommend solving the problems. One of my friends in particle ﬁeld solved all of these. I myself tried some, but felt difﬁcult.

E11-4(5.120)c is also assumed to be constant. We write the correlation function of ﬂuctuation in the order parameter asg(r) = h(m(0) − hm(0)i)(m(r) − hm(r)i)i = hm(0)m(r)i − hm(0)i hm(r)i .

(5.121)In the region T > TC , the average of the order parameter itself is zero, thus the second term is zero. The Fourier
expansion representation of m(r) is

1 ∑mk exp(ik · r),m(r) = √V k(5.122)where V is the system volume. Because m(r) is real, m−k = m∗k should hold. In hm(0)m(r)i, the term exp(±ik · r)
can be expressed as(mk + m−k )(mk eikr + m−k e−ikr ) = 2|mk |2 e−ikr + 2|m−k |2 eikr .

By using this and from the translational symmetry of the system, we writeg(r) =

1 ∑h|mk |2 i exp(−ik · r).
V(5.123)kThen we obtain
F = V f0 +∑k|mk |2(a)
b+ ck 2 +2
4V∑

mk1 mk2 mk3 mk4 ,(5.124)

k1 +k2 +k3 +k4 =0

where we take the zero ﬁeld limit h → 0. In the region T > TC , we can ignore the last 4-th order term in (5.124). The
weight function (5.118) now gives the Gaussian distribution[]′)
2 ∑ (a
1
(r)2
(i)2
2
exp −
+ ck (mk + mk ) ,
Z
kB T
2

(5.125)kwhere we drop the ﬁrst constant term in eq. (5.124). In the last parentheses, the term |mk |2 is written in a real-imaginaryseparated form as(r)Re[mk ] = mk ,(i)

Im[mk ] = mk .

∑′k means taking the sum over independent k, which give a half of the sum in (5.124). Hence the factor 2 isgiven. From the above we can write the average of |mk |2 as

The symbolh|mk |2 i =kB T.
a + 2ck 2(5.126)

Substituting the above into eq. (5.123), g(r) is expressed as′

1 ∑ kB T
g(r) =
e−ik·r = kB T
V
a + 2ck 2k∫ ∞0e−ik·r d3 k
kB T exp(−r/ξ)=,2ck 2 + a (2π)3
8πd
r

√
ξ=

2c
.
a

(5.127)

The result indicates that the correlation is damped exponentially on the distance in the region T > TC . ξ in eq. (5.127)
is called correlation length. Considering the temperature dependence in eq. (5.120), the correlation length varies as(T − TC )1/2 in the vicinity of TC . While ξ diverges at TC , the correlation of ﬂuctuation decrease with the distance asg(r) ∝ r−1 .
In the region T < TC , a long range order appears resulting in the difference between the correlation of ﬂuctuation(5.121) and that of the order parameter itself, that isg̃(r) = hm(0)m(r)i .

(5.128)This g̃(r) is thus ﬁnite for r → ∞. This is equivalent with the appearance of long range order[17]. As for g(r), it goes
zero at r → ∞.

E11-5

5.10.2 Scaling relations
We introduced the concept of critical exponent in Sec. 5.3. In introducing the scaling relations, we review the “routine”
notations. The relevant parameters are the temperature t ≡ (T − TC )/TC , and the external magnetic ﬁeld h. The
behaviors in the critical region are
Speciﬁc heat :Magnetization (order parameter) :Susceptibility :Correlation length :C ∼ |t|−α ,m ∼ |t|β(t < 0)m∼h(t = 0),1/δχ ∼ |t|ξ ∼ |t|

−γ

−ν

,

.

As we saw in the previous subsection, these anomalous behaviors around the critical point t = h = 0, originate from thedivergence of correlation length, namely the divergence of the size of clusters with a short range order.
The correlation function of ﬂuctuation (5.121) of the system with spatial dimension of d is generally written asg(r) ∼exp(−r/ξ).
rd−2+η

(5.129)Here η is one of the critical exponents and in eq. (5.127) (i.e., the GL theory) η = 0. The GL theory also gives
α = 0, β = 1/2, γ = 1, δ = 3, ν = 1/2, η = 0.

(5.130)

The GL theory is a phenomenology. The critical exponents depend on the “universality” of the system as we already saw.
These critical exponents have the following relations.
γ = (2 − η)ν,(5.131a)

α + 2β + γ = 2,

(5.131b)

β + γ = βδ.

(5.131c)

Among them, eq. (5.131b) and eq. (5.131c) are called scaling relation, which can be derived from the scaling ansatz
explained in the following. We have taken t and h as the relevant parameters of the transition, but the ansatz tells the
behavior of the system in the vicinity of transition is determined by single relevant parameter h/|t|∆ . Here ∆ is called
gap exponent.
We assume that the anomalous part of the free energy fs can be expressed with the use of a function f± (x)()
hfs ∼ |t|2−α f±,|t|∆(5.132)around the critical point. ± in f± correspond to T ≷ TC . For h = 0, t can move without affecting h/|t|∆ . The factor|t|2−α is then attached to have the consistency with the speciﬁc heat. Now the magnetization and the susceptibility are∂fs′∼ |t|2−α−∆ f±(0) ∼ |t|β (t < 0)
∂h
∂2f
′′
(0) ∼ |t|−γ .
χ ∼ − 2 ∼ |t|2−α−2∆ f±∂hm(h = 0) ∼ −(5.133)(5.134)

These lead to the relations
β =2−α−∆(5.135)−γ = 2 − α − 2∆.

(5.136)∆=β+γ(5.137)

Then erasing

E11-6

from the above, a scaling relation (5.131b) is obtained.
′
Next to see the behavior of magnetization at t = 0, we need to explore the asymptotic behavior of f±for h/t∆ → ∞.

We assume the asymptotic form as′f±(x) ∼ xλ±(x → ∞).

(5.138)Then the asymptotic behavior of eq. (5.133) is′m ∼ |t|β f±(

For m to be ﬁnite for t → 0,
λ± =

h
|t|∆

)
∼

hλ ±
|t|∆λ± −β

.

β
β
=
.
∆
β+γ

(5.139)(5.140)From this relation and the relation and the deﬁnition of the critical exponent m ∼ h1/δ ,β+γ,β(5.141)2 − α = dν,(5.142)δ=which is eq. (5.131c). There is also a relation

with the spatial dimension d, and is called hyperscaling relation.

5.10.3 Renormalization group
As approaching the critical point, the correlation length of ﬂuctuation ξ gets longer. Within the space of this length,
some order is growing. Taking a distance x shorter than ξ (1 < x  ξ), we can assume that the system is uniform ina space with a size of x. We thus average physical quantities within the size x and then the unit of length is changed tox (i.e., x 7→ 1). This is called coarse graining. With this operation, the correlation length is transformed to ξ/x and thesystem looks being drew apart from the critical point. Similarly, the parameters of the system get various modiﬁcations
by this operation. This operation is called renormalization group (RG) transformation with scaling factor x.
Let R(x) be such an operation, then, e.g, the transform of H → H ′ can be written as
H ′ = R(x)H .

(5.143)

Sequential operation of R(x) and R(x′ ) is the same as a single RG transform with scaling factor of xx′ . That is
R(x′ )R(x) = R(x′ x).

(5.144)The RG transformation thus fulﬁls the associativity forming a semigroup. It is called then renormalization group.
Generally there is no inverse transformation from once coarse-grained system to the original ﬁne-grained system. Hencea set of such transformations is not a group but semigroup.
Figure 5.13(a) show an example of an Ising model on a two-dimensional square lattice, in which four spins are averaged
into one as
sq =

1∑
sqi .
4 i

Then a new lattice in Fig. 5.13(b) appears. The scaling factor is√(5.145)

4 = 2. The averaging makes the spins to be able to takethe value of ±1, and the model is no longer an Ising model. Also the range of interaction is modiﬁed. Accordingly, withrepetition of RG transformation, the system moves in the parameter(s) space. The invariants with RG transformation are
the symmetry of the order parameter and the spatial dimension. The parameter(s) space is then characterized by them.

E11-7

Fig. 5.13 (a) Illustration of an Ising model on two-dimensional square lattice. (b) Schematically shows a renormalization group transformation of averaging four spins in (a) as a single spin. (c) Schematic illustration of a ﬂow
diagram. A is a stable ﬁxed point. B is an unstable ﬁxed point.

Now we take the scaling factor as a continuous variable. Then the system transitions in the parameter space become
continuous. Figure 5.13(c) shows an example of such continuous “ﬂows” of system in parameter space by RG transformation. This kind of diagram is called ﬂow diagram. If a system is in the high temperature side of the criticaltemperature at the starting point, the RG transformation drives it to the completely disordered state. If the starting point
is in the opposite lower side of TC (t = 0), it will transition to the perfectly ordered state. Perfectly ordered/disorderedstates are invariant for RG transformation, hence they form ﬁxed points in the ﬂow diagram as illustrated in Fig. 5.13(c).
They are called stable ﬁxed points, an example of which is indicated as A in Fig. 5.13(c). They collect the ﬂow lines.
On the other hand, just on the critical point, the correlation length of ﬂuctuation diverges and the system on it does notchange with RG transformation. Therefore the critical point is a ﬁxed point. However, an inﬁnitesimally small shift in
the parameter causes repelling of ﬂow lines from the critical point. Hence it is called unstable ﬁxed point. In Fig. 5.13(c),B is an example of such an unstable ﬁxed point.

5.10.4 Derivation of scaling ansatz
We again take t (temperature) and h (magnetic ﬁeld) as the relevant parameters of transition. We write a RG transformation with a scale factor x ast′ = g1 (t, h),(5.146a)(x)h′ = g2 (t, h).

(5.146b)(x)

In the vicinity of the ﬁxed point t = h = 0 (critical point), we try to expand the above in the formt′ ' Λ11 (x)t + Λ21 (x)h,h′ ' Λ21 (x)t + Λ22 (x)h.

(5.147)

There is no constant (0-th order) term because it is a ﬁxed point. We further know that there is no linear coupling betweent and h because the sign of h is reversed with reversing the sign of the order parameter while t does not change with that.
Namely Λ12 (x) = Λ21 (x) = 0. Therefore the association law (5.144) gives(Λ11 (x))n = Λ11 (xn ),(Λ22 (x))n = Λ22 (xn ).

(5.148)This should hold for any natural number n and any x (> 1). This means Λ11 (x) and Λ22 (x) are power functions of x.
Λ11 (x) = xλ1 ,Λ22 (x) = xλ2 .

E11-8(5.149)

Now we apply a RG transform with scaling factor x, n-times on a system that has the starting point at (t, h). We
assume the ﬁnal state temperature t0 = xnλ1 t is far enough from the critical point. Because the correlation length of the
ﬂuctuation isξ(t)= xn =
ξ(t0 )(tt0)−1/λ1,(5.150)and from the critical exponent deﬁnition ξ ∼ |t|−ν , we obtaine ν = λ−1
1 .
On the other hand in d-dimensional system, the free energy density f (t, h) becomes xd times by the RG transformation.
Therefore

xnd f (t, h) = f (xnλ1 t, xnλ2 h) = f (t0 , (t/t0 )−λ2 /λ1 h).

We see t0 as a constant, then with an appropriate function f± (x) we write
( )
h
d/λ1
−λ2 /λ1dνf (t, h) = t
f± (th) = t f± ∆tThis is showing the scaling ansatz.

E11-9

∆=

λ2
.
λ1

(5.151)(5.152)

Chapter 6

Magnetism of itinerant electrons

The mechanism of ferromagnetism in common metals such as iron, cobalt, and nickel is thought to be signiﬁcantly
different from what we have seen so far.

References[1] Evgeny Y. Tsymbal and Žutić Igor, editors. Spintronics Handbook, Second Edition: Spin Transport and Magnetism:
Volume One: Metallic Spintronics (Spintronics Handbook: Spin Transport and Magnetism Book 1) (English Edition). CRC Press, 5 2019.
[2] T. Nagamiya, K. Yosida, and R. Kubo. Antiferromagnetism. Advances in Physics, Vol. 4, No. 13, pp. 1–112, January
1955. Over 100 pages review.
[3] Roald K. Wangsness. Magnetic resonance in ferrimagnetics. Phys. Rev., Vol. 93, pp. 68–71, Jan 1954.
[4] Roald K. Wangsness. Ferrimagnetic resonance and some related effects. American Journal of Physics, Vol. 24,
No. 2, pp. 60–66, February 1956.
[5] M. H. Seavey and P. E. Tannenwald. Direct observation of spin-wave resonance. Phys. Rev. Lett., Vol. 1, pp.
168–169, Sep 1958.
[6] C. Kittel and Conyers Herring. Effect of exchange interaction on ferromagnetic microwave resonance absorption.
Phys. Rev., Vol. 77, pp. 725–726, Mar 1950.
[7] Sergej O. Demokritov, editor. Spin Wave Conﬁnement: Propagating Waves, Second Edition (English Edition). Jenny
Stanford Publishing, 9 2017.
[8] 中嶋貞雄. 量子の世界 新版 UP 選書. 東京大学出版会, 2 1975.

E11-10[9] Kenneth G. Wilson. The renormalization group: Critical phenomena and the kondo problem. Rev. Mod. Phys.,Vol. 47, pp. 773–840, Oct 1975.
[10] 西森秀稔. 相転移・臨界現象の統計物理学 新物理学シリーズ. 培風館, 11 2005.
[11] Hidetoshi Nishimori and Gerardo Ortiz. Elements of Phase Transitions and Critical Phenomena (Oxford Graduate
Texts) (English Edition). OUP Oxford, 12 2010.
[12] Daniel J. Amit and Victor Martin-mayor. Field Theory, the Renormalization Group, And Critical Phenomena:
Graphs To Computers. World Scientiﬁc Pub Co Inc, 6 2005.
[13] 大谷聡. 統計力学 i –臨界現象と繰り込み群–, 2022. http://aries.phys.cst.nihon-u.ac.jp/˜ohya/
stat-mech/main.pdf.
[14] Andrea Pelissetto and Ettore Vicari. Critical phenomena and renormalization-group theory. Physics Reports,
Vol. 368, No. 6, pp. 549–727, October 2002. available in arXiv: https://doi.org/10.48550/arXiv.
cond-mat/0012164.
[15] Richard P. Feynman, Albert R. Hibbs, and Daniel F. Styer. Quantum Mechanics and Path Integrals: Emended
Edition (Dover Books on Physics). Dover Publications, 7 2010.
[16] Richard P. Feynman. Statistical Mechanics: A Set Of Lectures (Frontiers in Physics). CRC Press, 3 1998.
[17] Tohru Koma and Hal Tasaki. Symmetry breaking and ﬁnite-size effects in quantum many-body systems. Journal of
Statistical Physics, Vol. 76, No. 3-4, pp. 745–803, August 1994.

E11-11

Lecture note
29th June (2022)

Magnetism (12)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 6

Magnetism of Itinerant Electrons

In insulators, the following picture is expected to describe the magnetism: the spins localized at the lattice point cause
magnetism through various spin-to-spin interactions. Common ferromagnetic metals like iron, cobalt, and nickel aresupposed to have largely different mechanism from the above. Namely, electrons migrating in crystals (itinerant electrons)
have their spins aligned partially due to the correlation effect, which phenomenon may generate the ferromagnetism.
When the correlation effect is not very strong and the system is paramagnetic, such itinerant electron systems can betreated within the Landau’s Fermi liquid theory[1] as mentioned in Ch. 3. On the other hand, the correlation is so strongthat a ferromagnetism appears, the difﬁculties in theories increase largely. It was not easy for theoretical models to
explain such types of ferromagnetism to the level in which the theories can be comparable with experiments. Including
such difﬁculties, we would like to review the present understandings and open questions in the last three weeks.

6.1 Hartree-Fock approximation of electron gas
We would have a brief look at the difﬁculty to have a realistic ferromagnetism in the model of electron gas. We use thesimplest Hartree-Fock approximation (not very simple actually) of electron correlation.

6.1.1 Hartree-Fock approximation
Though you are already familiar with Hartee-Fock (HF) approximation, we will shortly review it here (we will notuse such details in analysis of electron gas here). We consider an N -particle system, in which the particles occupy
single-particle wavefunctions
φk1 , φ k 2 , · · · , φ kN .
E12-1

(6.1)

The many body state of this N -body system can be expressed by the Slater determinant:
1
Φ= √
N!

φk1 (x1 )..
.
φk1 (xN )

···
..
.
···

φkN (x1 )
..
,
.

(6.2)

φkN (xN )which satisﬁes the ferminon particle exchange statistics. xi is a general coordinate, which contains all of single particle
freedoms. We write the Hamiltonian in the form
H =

N
X

h(xj ) +Xv(xi , xj ),

(6.3)

⟨i,j⟩

j=1

where h is a single-particle Hamiltonian, v is a two-body interaction. In the HF approximation, we calculate the expectation value
W = ⟨Φ|H |Φ⟩ ,(6.4)

and look for the set {φkj } that minimize W with a variational method.
Here we use the ket representation |kj ⟩ for φkj . We assume the orthonormal basis condition:⟨ki |kj ⟩ = δij .

(6.5)

In this representation, W is written asW=NX⟨kj |h|kj ⟩ +X[⟨ki kj |v|ki kj ⟩ − ⟨ki kj |v|kj ki ⟩].

(6.6)

⟨i,j⟩

j=1

In the rhs, in the term ⟨ki kj |v|ki kj ⟩, the two particles interact in the same order but in ⟨ki kj |v|kj ki ⟩, the two particles
exchange their positions during the interaction. The former is called the direct integral, the latter is called the exchange
integral (exchange interaction). In the following, we adopt the Lagrange multiplier method to minimize the energy of thesystem (6.6) under the constraint of (6.5). That is, we consider the quantityXW−
λij ⟨ki |kj ⟩ ,

(6.7)

⟨i,j⟩

which should be minimized. In order for that, we consider the condition that the variations of the above quantity with{φ∗kj } are zero, which can be expressed as
hφkj +

X[⟨ki |v|ki ⟩ φkj − ⟨ki |v|kj ⟩ φki ] =

i=1

N
X

λij φki .

(6.8)

i=1

Here we deﬁne a single-body density matrix asρ(x, x′ ) =NXφ∗ki (x)φki (x′ ),

(6.9)

i=1

with which we further deﬁne veﬀ and A asZ
veﬀ (x) = dx′ v(x, x′ )ρ(x′ , x′ ),Z
A(x)φ(x) =dx′ v(x, x′ )φ(x′ )ρ(x′ , x).

(6.10)

λij φki (x).

(6.11)

Then eq. (6.8) is written as[h(x) + veﬀ (x) − A(x)]φkj (x) =

N
X
i=1

In eq. (6.11), the Hermite operator given in [· · · ] in the left hand side does not depend on the speciﬁc selection of kj
in the operand. Also the eigenfunctions are orthogonal to each other. Hence by taking φkj as the eigenfunctions, we canwrite[h(x) + veﬀ (x) − A(x)]φkj (x) = ϵkj φkj (x).
E12-2

(6.12)

Then we take N solutions in the ascending order from the lowest {ϵkj }. The Slater determinant of these N solutions
is the ground state in the HF approximation. However, the operator in the [· · · ] part also depends on {ϵkj } (though notdepends on speciﬁc selection of the operand), (6.12) should be solved self-consistently. Equation (6.12) is called HartreeFock equation. This is essentially the same as the self-consistent equation in the molecular ﬁeld approximation of the
Heisenberg model. To see how to proceed the calculation in more speciﬁc physical problems, or how to go beyond the
HF approximation (e.g., taking the effect of higher order term into account), refer to textbooks on many-body problems(e.g., [2, 3, 4]).

6.1.2 Jellium model and ferromagnetism
Here we adopt jellium model, in which the lattice potential is approximated by a uniform background with a plus
charge. And we consider a free electron system in the background. The ground state in the absence of electron mutualinteraction is the state in which the Fermi sphere is ﬁlled up asY|Ψ⟩ =c†kσ |0⟩ .

(6.13)E(k,σ)≤EF

Next we write down the Hamiltonian in the presence of electron-electron interaction asH =X

ϵk c†kσ ckσ +

k,σ

1
2V

X
k,k′ ,σ,σ ′ ,q̸=0

vq c†k+q,σ c†k′ −q,σ′ ck′ σ ckσ ,(6.14)

where V is the system volume, ϵk = ℏ2 k 2 /2m and vq = 4πe2 /q 2 . The Fermi wavenumber kF is determined from theFermi energy EF , and the only parameter that characterizes the system is the averaged electron distance measured by
Bohr radius aB :

1/3

3
1
,
rs ≡
aB 4π(kF3 /3π 2 )(6.15)

in the jellium model.
Because the system has spatially translational symmetry in the jellium model, plane waves are already the solutions ofself-consistent HF equation (6.12), and then the residual procedure of the HF approximation is to minimize the energy of
many-body state. The kinetic energy per single electron isZ
1 X
2V
d 3 k ℏ2 k 2
3 ℏ2 kF2
2.21
ϵke =ϵk nks =n== 2 Ry,k3NN(2π) 2m
10 m
rs(6.16)

ks

where Ry=ℏ2 /2ma2B =13.6 eV is the unit named “Rydberg,” and is the binding energy of hydrogen atom. The Hartree
term that corresponds to the direct integral is vanished by the charge neutral condition of jellium model. The exchange
term, that is the expectation value of interaction between the states of exchanged particles. It is thenϵex = −

1
2N V

X

vq ⟨ψ|ck+q,s ck+q,s c†ks cks |ψ⟩ =k,q̸=0,s

1 X
vq nk+q nk ,2N V

(6.17)ks

per an electron. The summation can be carried out as an integral over q to beϵex = −
Then the total Hartree-Fock energy is

3e2 kF
0.92
=−
Ry.
4 π
rs


ϵhf =

2.21 0.92−
rs2
rs(6.18)

Ry.

(6.19)

From eq. (6.16), the total kinetic energy of electrons per unit volume is3 ℏ2 kF2 4π
Eke =10 m 3



kF
2π

E12-3

3
=

ℏ2 kF5.
20π 2 m(6.20)

Let (kF↑ , n↑ ), (kF↓ , n↓ ) be the Fermi wavenumber and the electron number of up and down spin states respectively. Thenℏ23(6π 2 )2/3 ℏ2 5/3
3(6π 2 )2/3 ℏ2 5/3
5/3
5/3
5
5
(kF↑
+ kF↓
)=
(n↑ + n↓ ) =[p − (1 − p)5/3 ]n0 ,
2
20π m
10m
10m
 1/3 1/3
3e2 6
3e2 6
4/3
4/3
4/3
Eex (p) = −(n↑ + n↓ ) = −[p4/3 − (1 − p)4/3 ]n0 .
4π4πEke (p) =(6.21)(6.22)

Here because polarization 0.5 ≤ p ≤ 1 is deﬁned as n↑ = pn0 , n↓ = (1−p)n0 , p = 1 corresponds to perfect polarization,p = 0.5 corresponds to no polarization. Then let ∆E be∆E = [Eke (1) + Eex (1)] − [Eke (0.5) + Eex (0.5)],(6.23)

then if ∆E < 0, the ground state is a spin-polarized ferromagnetic state. This condition can be calculated from the above
and expressed with the average distance parameter deﬁned in eq. (6.15), as
rs > 5.4531.

(6.24)

However, we found several ordinary metals with rs ∼ 5. Hence the above criterion does not ﬁt the reality.

6.1.3 Electron correlation
The above discrepancy should come from either overestimation of lowering in the Coulomb energy with avoiding eachother between parallel spins, or that of the Coulomb repulsion energy between antiparallel spins. The latter seems to be
more plausible because the itinerant electrons with antiparallel spins also should avoid each other to lower the Coulomb
repulsion energy. That means, in the jellium model, the solution of antiparallel spins in HF (mean ﬁeld) approximation
is not good enough for discussion of ground state. We need to go for higher order approximation or to consider somedifferent approximation. We here deﬁne correlation energy as the energy difference between the energy of true groundstate and that of mean ﬁeld approximation.
There exist various methods in the estimation of correlation energy. Figure 6.1 shows a phase diagram of charged
fermion gas in the jellium model calculated by diffusion Monte-Carlo method[5]. The red line indicates spin-polarized gas
and the ferromagnetism appears in the region that the red line lies below the blue broken line which indicates unpolarizedgas. However as can be seen, rs for the appearance of the ferromagnetism is around 70, which is too sparse for real
metals. Around rs ∼ 90 the Wigner crystal state in which the electrons form a crystal and localize. This means if thecorrelation energy is correctly estimated, the ferromagnetism in common metallic ferromagnet like iron or nickel cannot
be explained at all.

Fig. 6.1 Phase diagram of electron gas calculatedby diffusion Monte-Carlo method. The energy origin (0) is taken to that of a bosonic system. The region where the red line lies lowest position should
be ferromagnetic. From [5].

E12-4

6.2 HF approximation in Hubbard model
The jellium model, not only in the mean ﬁeld approximation, even with correct estimation of correlation energy, isfar from the realistic explanation of ferromagnetism in metals. In the case of ferromagnets caused by double-exchangeinteraction, which we have seen in the previous chapter, the electrons hopping between local sites mediate ferromagnetic
interaction between localized spins. Though the situation is different in the case of 3d transition metals, it can be a hintthat the coexistence of localization and hopping can cause ferromagnetism in realistic conditions. Then we try to consider
the Hubbard model introduced in Sec. 4.1.3.

6.2.1 Hubbard model for multiple site
We introduced two-site Hubbard model (sometimes referred to as Kanamori-Hubbard model) in Sec. 4.1.3. It is expected to describe from insulators, metals, ferromagnets and superconductors in spite of its compactness, and has beenlong used in theories. As we saw there, the long range part of the Coulomb interaction is ignored and the Coulomb repulsion only works between electrons in the same site (on-site interaction). On the other hand, the electrons hop betweensite i and site j with probability tij . The Hamiltonian for a general number of sites isHubbard Hamiltonian (1)H =X

tij c†is cjs + U

i,j,s

N
X



n̂i↑ n̂i↓ ,(6.25)iwhere s is the spin freedom. cis satisﬁes the fermion commutation relation{c†is , cis′ } = δij δss′ .

(6.26)In the ﬁrst hopping term, the annihilation operator cis and hopping transition matrix element tij are Fourier-expandedas

1 X iRi ·k
cis = √
e
aks ,
N k

tij =

1 X
ϵk eik·(Ri −Rj ) .
N(6.27)

k

Ri is the spatial coordinate of site i. With substituting these into the ﬁrst term in the right hand side of (6.25), becauseX

tij c†is cjs =⟨i,j⟩,s

X 2
N2
i,j,s

X

ϵk1 eik1 ·(Ri −Rj ) e−ik2 ·Ri a†k2 s eik3 ·Rj ak3 s =k1 ,k2 ,k3Xϵk a†ks aks ,(6.28)

k,s

we can view the hopping (1st) term as kinetic energy term of wide-spreading electrons. From this, (6.25) can be expressed
as follows.
H =X

ϵk a†ks aks + UNXn̂i↑ n̂i↓ .

(6.29)ik,s

The Hubbard model is widely used in the study of many-body problem. Particularly it is practical to be applied to
the ferromagnetism of 3d transition metals. In such systems, 4s and 3d electrons coexist in single band. 3d electrons
are the origin of ferromagnetism and though the wavefunctions spread over the whole crystal (itinerant) they have highprobabilities of existence at atomic positions, i.e. tendency to localize. On the other hand, 4s electrons tend to delocalize
and screen the Coulomb interaction between 3d electrons. This property justiﬁes the approximation of on-site short range
Coulomb interaction.

E12-5

6.2.2 HF approximation
We apply HF (mean ﬁeld, molecular ﬁeld) approximation to the Hubbard model. There is the same problem of overestimation as that in electron gas, but we will see what is different in the case of Hubbard model.
To estimate the ferromagnetic transition, we consider magnetization and electron number per site:m = ⟨n↑ ⟩ − ⟨n↓ ⟩ ,n = ⟨n↑ ⟩ + ⟨n↓ ⟩ ,(6.30)and compare the expectation values of energy for the states of m = 0 and m ̸= 0. The magnetization is expressed in theunit of µB and g-factor is set to 2.
In the HF approximation of Hubbard model (6.25), the second term (interaction term) is simpliﬁed asUXn̂i↑ n̂i↓ = UiX[⟨n̂↑ ⟩ n̂i↓ + ⟨n̂↓ ⟩ n̂i↑ − ⟨n̂↑ ⟩ ⟨n̂↓ ⟩ + (n̂i↑ − ⟨n↑ ⟩)(n̂i↓ − ⟨n↓ ⟩)]

i

≃U

X

(⟨n̂↑ ⟩ n̂i↓ + ⟨n̂↓ ⟩ n̂i↑ ) − N U ⟨n↑ ⟩ ⟨n↓ ⟩(6.31)i

Take average → =NU 2(n − m2 ).
4(6.32)

Namely, the second order term of ﬂuctuation (n̂i↑ − ⟨n↑ ⟩)(n̂i↓ − ⟨n↓ ⟩) is ignored. In this approximation, ↓-electrons
work as an average on an ↑-electron, conversely, ↑-electrons work as an average on a ↓-electron. The last equation showsthe expectation value for eigenstates.
We can rewrite eq. (6.31) with the Fourier expansions in (6.27), (6.28) to a Hamiltonian with the operators n̂ks as
HHF =

X

(ϵk + U ⟨n−s ⟩)nks − N U ⟨n↑ ⟩ ⟨n↓ ⟩ .

(6.33)

k,sHere we assign s = ±1 to spin ↑, ↓, and use the relation⟨ns ⟩ =

1(n + sm),2(6.34)

to obtain
HHF =

X

ϵk −

k,s

≡

X

ϵ̃ks n̂ks +k,swhere we take averages as

P

sU m
2


n̂ks +NU 2(n + m2 )4

NU 2(n + m2 ),4(6.35)k,s n̂ks → N (⟨n↑ ⟩ + ⟨n↓ ⟩). That is, the appearance of the magnetization m shifts the single

electron energy by ∆µ = (−s)U m/2. The directions of the shifts are opposite depending on spins. With packing theelectrons into this band up to a common chemical potential µ, the total energy is given by

X 
sU m
NU 2
E=
ϵk −
+
(n + m2 )24ϵ̃ks ≤µ=Xϵ̃ks ≤µϵk +NU 2(n − m2 ).
4(6.36)On the other hand, the spin-dependence of ∆µ brings about the difference in the electron numbers in ↑ and ↓ states.
And the difference should be equal to m. This is the self-consistent condition, which commonly appears in mean-ﬁeld
approximation. For simplicity, we assume that the density of states D(EF ) around EF is constant for energy. Then fromm = 2D(EF )∆µ = D(EF )U m,

E12-6(6.37)

the condition m ̸= 0 gives the criterion of appearance of nonzero m as
U D(EF ) = 1.

(6.38)

We estimate the enhancement of kinetic energy by magnetization as we did in the electron gas. From the above
condition we obtain
D(EF )(∆µ)2 =m2.
4D(EF )

(6.39)As in eq. (6.36), the decrease in the Coulomb repulsion energy by appearance of m is −N U m2 /4. Summing up of these
gives
∆E =

N
4




m2
− U m2 .
D(EF )(6.40)

The condition ∆E < 0 is
U D(EF ) ≥ 1,

(6.41)

which agrees with the criterion (6.38). This is called Stoner condition.
−1
Roughly speaking, let Ew be the bandwidth and from D(EF ) ∼ Ew
, we can say that the Stoner condition means

the ferromagnetism appears when the Coulomb interaction width U exceeds the bandwidth Ew . Because it is still a HF
approximation, as in the case of electron gas, we have the problem of overestimation the stability of ferromagnetic state.

6.2.3 Susceptibility
Before going into the problem, we see the magnetic susceptibility given in the HF approximation. The magnetization
M is expressed as
M=

gµB X
gµB X
[⟨ni↑ ⟩ − ⟨ni↓ ⟩]
ni− .
2 i
2 i

(6.42)

gµB n−
M
=
.
NB
2 B

(6.43)The susceptibility χ per an atom isχ=Since the interaction energy of magnetic ﬁeld and magnetic moment (Zeeman energy) is −M B, the energy of electrons
in magnetic ﬁeld is written as
EB = E(0) + E2 n2− − N
where n− is small. And E2 is
E2 =

gµB
Bn− ,
2

1 d2 (∆E),
2 dn2−(6.44)(6.45)

where ∆E corresponds to ∆E in eq. (6.40).
In paramagnetic state, the coefﬁcient of the term with the second order in M is positive in the GL theory. Then E2 isalso positive and n− that minimizes EB , gives the susceptibility. Namely,χ=(gµB )2 N.
4E2(6.46)

If we calculate E2 from the HF approximation in eq. (6.40), we obtainχ= gµ 2
B

2

χPauli(a)D(EF )=,
1 − U D(EF )1 − U D(EF )(6.47)

where χPauli(a) is the Pauli paramagnetic susceptibility in eq. (3.8) per an atom.
In the HF approximation, when the system does not fulﬁll the Stoner condition, it keeps paramagnetism though as ineq. (6.47), the susceptibility is enhanced from the Pauli paramagnetic susceptibility by the factor [1 − U D(EF )]−1 . Thisis called Stoner factor.
E12-7

Here from the identity of temperature expansion of the chemical potential when the system is strongly Fermi degener"ated:µ = µ0π 2 d log D(µ0 )1−6 d log µ0



kB T
µ0

#

2

+ ··· ,

(6.48)

the temperature variation of chemical potential δµ is written asδµ = −

π 2 DF′(kB T )2 .
6DF

(6.49)For simpler expression, dD(E)/dE|E=EF is simply written as DF′ . To add this to ∆µ in eq. (6.37) as correction, taking
the second derivative by the energy is required. Hence we write
π2
A=
6




′
(DF )2′′− DF ,
DF(6.50)and the temperature dependence of susceptibility is written asχ= gµ 2
B

2

D(EF ).
1 − U D(EF ) + U A(kB T )2

(6.51)The temperature that gives zero for the denominator is TC , and by using this, we can writeχ=C.
T 2 − TC2

(6.52)

This does not agree with the Curie-Weiss law, which is also observed in 3d transition metals with ferromagnetism. This
indicates that the HF approximation has problems other than the quantitative problem we will see in the second next
section.

6.3 Ferromagnetism in 3d transition metals
Before going into more realistic theories, we would like to have a look on experimental facts on the ferromagnetism in
3d metals. Those should be also the target of more realistic theory. We also see “tuning” of the exchange parameter gives
a qualitative understanding of the experiments even within the HF approximation.
In the table below, the bulk parameters of the three elemental ferromagnetic metals are listed[6]．
structure/density

parameters

TC

MS

K1
−1

)

P

287

1044

1.71

48

−7

1.6

45

hcp

251

1388

1.45

530

−62

8.0

42

8836

407 (fcc)

fcc

352

628

0.49

−5

−34

bcc

)

(kJm

α
−6

(MAm

(kgm

)

λS
−3

(K)

−3

Fe

lattice

(10

)

(%)

(pm)

7874
Co
Ni

44

8902
Tab. 6.1 Bulk properties related to the ferromagnetism in Fe, Co, Ni. K1 is density of anisotropic energy; λS spin
diffusion length; α damping factor of spin resonance; P spin polarization. P is measured by the Andreev reﬂection
at 4.2 K. The others are measured at room temperature. From [6].

E12-8

Fig. 6.2 Averaged spontaneous magnetic
moment (in unit µB ) is plotted as a functionof averaged valence electron number (horizontal axis) for various 3d transition metals(elemental metals, and alloys). From [7].
Valence electron number 6 corresponds to
Cr, 11 corresponds to Cu.

6.3.1 Slater-Pauling’s curve
As shown above, among 3d transition metals, elemental metals that show ferromagnetism are Fe, Co, and Ni. The
averaged valence electron numbers are 8, 9, 10 respectively. However, with making alloys with other 3d metals or
Heusler alloys, which contains group III, or IV elements, we can synthesis metallic ferromagnets with the valence electron
number from around 6 to about 10.5. Then we plot the spontaneous magnetization per an atom obtained from the saturated
magnetization as a function of number of valence electrons. The data points align regularly as in Fig. 6.2. They are on
the lines forming a triangle with Fe around the peak. And the edges of the triangle have gradients of ±1. This curve iscalled Slater-Pauling’ curve.
In Fig. 6.2 the alloys plotted in the left region than Fe are mainly Heusler alloys. For Heusler alloys, the following
relation is reported to hold[8]:mmatnetization per atom in unit of µB = Z − 24.

(6.53)

Relation like eq. (6.53) is called Slater-Pauling law.

6.3.2 Spin-band structure in Ni
This relation should come from the number of 3d electron spins in the open shell. Here, however, the “open shells”
form an energy band and the electrons in it are itinerant. Then numerically calculated band structures are often used forthe explanation of Slater-Pauling’s curve. The band structure must have spin-dependence, that means the electron mutualinteraction should be taken into account in some way. Many of them are the HF approximation, which is known tooverestimate the exchange energy gain. But still, qualitative explanation of Slater-Pauling’s curve is possible as follows.
Nickel (Ni) has fcc structure and TC is comparatively low among the three elemental ferromagnets. Figure 6.3(a)shows the spin-dependent density of states calculated by the APW method (Appendix 12A)[9]. The 4s electron bandhas a widely spread density of states with low amplitudes. Conversely, the 3d electron band has a comparatively narrow
distribution and has several high sharp peaks. A single spin subband of 4s electrons can accommodate a single electron
per atom while that of 3d electrons can accommodate 5 electrons per atom. The bottom of 4s band is lower than that of
3d band. The position of EF shown in Fig. 6.3(a) indicates that the 4s band as 0.6 electron, the 3d ↑ band (the upperside in the ﬁgure. the major spin subband) has 5 electrons, the 3d ↓ band has 5.4 electrons. 10 valence electrons are

E12-9

(a)

(b)
Fig. 6.3 Spin-dependent density of states in (a) Ni, (b) Fe, calculated by the APW method. (a) is from [9]. (b) is from [10].

in an atom in total. In total, the total spin for 0.6 electron remains, which fact explains the appearance of spontaneous
magnetic moment with 0.6µB . As above, the non-integer spontaneous magnetic moment is explained by the coexistenceof 4s band and 3d band and partial occupation of them by valence electrons.
In Ni-Cu alloys, the spontaneous magnetizations vanishes around 60% of Cu content. A Cu atom has 11 valence
electrons, which is one more than Ni. Hence with increasing the Cu content, the extra one electron ﬁlls the vacant spaceof 0.6 electron in 3d ↓ band. Just at 60%, the space is ﬁlled up and the spontaneous magnetization vanishes. Also, it is
now clear that the gradient is −1 in Slater-Pauling’s curve from Ni to Cu.

6.3.3 Spin band structure in Fe
Figure 6.3(b) shows the spin dependent density of states in bee Fe calculated by the APW method. Though the shapeof density of states resembles to that of Ni, there is a large difference in the position of EF . The total electron number is
8 and the 4s band has 0.8 electron. Of the remaining 7.2 electrons, 4.7 electrons are in the major spin subband (the lower
in the ﬁgure), 2.5 electrons are in the minor spin subband. The conﬁguration brings about the spontaneous magnetizationof 2.2µB .
The reason why the major spin subband does not have full 5 electrons can be explained in Fig. 6.3(b). In the case of Ni,
EF places above the 3d major spin subband while in the case of Fe, EF hits a valley of the minor spin subband. In sucha situation, a variation of electron number in the minor spin subband causes a large shift in the relative EF position. The
position of EF in ﬁxed energy space does not move largely. That means actually the whole band should move largely.
Summing up the discussion, once EF hits a valley of density of states, change of electron number is mostly absorbed byother subbands and EF is locked to the valley. This makes it hard to increase the space (hole) in the minor spin subband.
And the major spin subband still has some hole in it.
Alloying with Co increases the number of electrons, with which we can test the above hypothesis. When the Coconcentration is low, the increasing electrons ﬁll the major spin band, and the spontaneous magnetic moment increases
with the Co concentration. At 30% where the major spin band is completely ﬁlled (5−4.7 = 0.3), the increasing electrons
work in the direction of ﬁlling the holes in the minor spin band, and the number starts to decrease. For the same reason,

E12-10the moment decreases due to alloying with Cr, which reduces the number of electrons.

Appendix 12A: An example of band calculation (APW)
Numerical calculations that exceed the HF approximation are still not vey common, and many theories used to explain
the Slater-Pauling law are by (a bit-modiﬁed) HF approximation. Band calculation is often the basis for considering
the electron correlation effect, so let us take a brief look at a type of calculation method here. Here we introduce APW(augmented plane wave) method, which is one of the techniques to ﬁnd solutions of Kohn-Sham equation *1 , for thedetails, refer to [11, 12] and the references therein.
Let us consider the Schrödinger equation of the state ϕ(r) in potential V (r):



ℏ2 2
H ϕ(r) = −∇ + V (r) ϕ(r) = Eϕ(r).
2m(12A.1)As the potential, we consider one called Mufﬁn-tin potential. Let rc be the radius of Mufﬁn-tins, which must be shorter
than the half of the distance between neighboring atoms. Then the potential is described as(Fig. 12A.1)(
Va (r) (spherical)(r < rc )V (r) =Vo (= Va (rc ): const.) (r ≥ rc ).

(12A.2)As V (r), Hartree potential, which corresponds to the direct integral in eq. (6.6), is adopted. That is
Vd (r) =X⟨ϕi (r ′ )|ie2|ϕi (r ′ )⟩ .
|r − r ′ |(12A.3)And exchange potential which corresponds to the exchange integral isVex↑ = −3e2

3
4π

1/3
ρ↑ (r)1/3 .

(12A.4)

The above is from spin density function approximation. The above Hartree and exchange potentials are obtained fromthe eigen functions, which are the solutions of eq. (12A.1). Hence the equation constitutes the self-consistent equation.
Actual calculation is on the variational method. As the variation functions, we adopt(Pmr < rc ,l,m Alm Rl (r)Yl (θ, φ)Φvr (r) = PN
n=0 Bn exp[i(k + Kn ) · r] r > rc .

(12A.5)That is, inside the mufﬁn-cup, wavefunctions have the same form for an isolated atom potential, and outside, the superpositions of plane waves. Kn are inverse-lattice vectors.

Fig. 12A.1 Schematic diagram of Mufﬁntin potential. From [13].

*1 Though APW method was invented long before the Kohn-Sham equation, now it can be placed at such a position in the present understandings.

E12-11

The wavenumber k is ﬁxed and the variation is taken under the boundary condition of wavefunction Φ connection atr = rc . Actually, the coefﬁcients {Bn } is determined for ⟨Φ|H |Φ⟩ to take the extremals. N cannot be taken to inﬁnity
and the calculation is done within a ﬁnite number. Thus obtained Φ is used to calculate the potential and the procedure
is continued to reach conversion. Then k is varied and the same procedure is repeated to obtain the band structure as wehave seen in Sec. 6.3.

Appendix 12B: MateriApps
MateriApps (https://ma.issp.u-tokyo.ac.jp/) is a portal site for material science simulations operated in cooperation
with the Institute for Solid State Physics CMS and others, and has information and download links for many related
applications.

Among them, Quantum Espresso (https://www.quantum-espresso.org/) is an application that performs a wide range of
calculations such as ground state calculation, DFT calculation, and quantum transport, and is characterized by being able
to run on a PC.

References[1] A. A. Abrikosov. Fundamentals of the Theory of Metals. Dover Publications, 10 2017.
[2] Alexander L. Fetter and John Dirk Walecka. Quantum Theory of Many-Particle Systems (Dover Books on Physics).
Dover Publications, 6 2003.
[3] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski. Methods of Quantum Field Theory in Statistical Physics(Dover Books on Physics). Dover Publications, 10 1975.
[4] Henrik Bruus and Karsten Flensberg. Many-body Quantum Theory In Condensed Matter Physics: An Introduction(Oxford Graduate Texts). Oxford Univ Pr, 11 2004.
[5] D. M. Ceperley and B. J. Alder. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett., Vol. 45,
pp. 566–569, Aug 1980.
[6] J. M. D. Coey. Materials for spin electronics. In Lecture Notes in Physics, pp. 277–297. Springer Berlin Heidelberg.
[7] Benjamin Balke, Sabine Wurmehl, Gerhard H Fecher, Claudia Felser, and Jürgen Kübler. Rational design of newmaterials for spintronics: Co2 fez (z=al, ga, si, ge). Science and Technology of Advanced Materials, Vol. 9, No. 1,
p. 014102, January 2008.
E12-12

[8] I. Galanakis, P. H. Dederichs, and N. Papanikolaou. Slater-pauling behavior and origin of the half-metallicity of the
full-heusler alloys. Phys. Rev. B, Vol. 66, p. 174429, Nov 2002.
[9] John W. D. Connolly. Energy bands in ferromagnetic nickel. Phys. Rev., Vol. 159, pp. 415–426, Jul 1967.
[10] Rastko Maglic. Van hove singularity in the iron density of states. Phys. Rev. Lett., Vol. 31, pp. 546–548, Aug 1973.
[11] 和子望月, 直鈴木. 固体の電子状態と磁性. 大学教育出版, 7 2003.
[12] Jorge Kohanoff. Electronic Structure Calculations for Solids and Molecules. Cambridge University Press, June
2006.
[13] Walter A. Harrison. Solid State Theory (Dover Books on Physics) (English Edition). Dover Publications, 4 2012.

E12-13

Lecture note
6th July (2022)

Magnetism (13)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Last week, we saw that the HF approximation on the Hubbard model leads to the Stoner condition, which brings about
an energy difference between the ↑-band and the ↓-band. This modeling and the approximation enable us to explain someexperimental facts like Slater-Paulings’ curve qualitatively. On the other hand, the approximation still has qualitativeand quantitative problems naturally for such a simple approximation. In such situations, self-consistent renormalization(SCR) spin-ﬂuctuation theory gave satisfactory results at least for the ground states. This week we would like to reachthe entrance of SCR theory but before that we need a bit heavy preparation, to which most of the lecture time is devoted.

6.4 Dynamic susceptibility
So far we have seen responses of materials to static magnetic ﬁelds. Here we turn our attention to responses to vibratingexternal ﬁelds. In such a case, we need to use the linear response theory.

6.4.1 Linear response
Let Hext (t) be the Hamiltonian of time-dependent external ﬁeld. The total Hamiltonian is expressed as H0 + Hext (t).
From the time dependence of density matrix ρ deﬁned in the previous section in eq. (6.9) is,iℏ∂ρ= [H0 + Hext (t), ρ(t)],∂t(6.54)

where only time t is shown explicitly as a variable. The initial state, for time t = −∞, is set to the thermal equilibrium
state of H0 , that is



1
H0
ρ(−∞) = ρeq =
,
exp −
Z0
kB T

(6.55)

where Z0 = Tr[exp(−H0 /kB T )] is the partition function of unperturbed state.
As shown in Appendix 13A, the density matrix satisﬁes the following integral equation.
ρ(t) = ρeq +1iℏ= ρeq +

1
iℏ

Z t
−∞
Z t
−∞

dt′ [U0 (t − t′ )Hext (t′ )U0−1 (t − t′ ), U0 (t − t′ )ρ(t′ )U0−1 (t − t′ )](6.56)

dt′ U0 (t − t′ )[Hext (t′ ), ρ(t′ )]U0−1 (t − t′ ),(6.57)

where also as in (13A.1),


U0 (t) ≡ exp


H0
t .
iℏ

(6.58)

Since the commutation relation in the right hand side of eq. (6.57) is the response to an external ﬁeld in addition to ρeq ,the lowest order in it is the ﬁrst order of t. In the same way, Hext is a time-dependent part with no constant. As longas we consider the liner response, ρ(t′ ) in the commutation relation in eq. (6.57) can be replaced with time-independentρeq . ρeq commutes with Hamiltonian H0 , being made from the eigenstates of it. Then we can write
ρ(t) ≃ ρeq +

1
iℏ

Z t
−∞

dt′ [U0 (t − t′ )Hext (t′ )U0−1 (t − t′ ), ρeq ].

E13-1(6.59)

Now we assume the Hamiltonian of external ﬁeld can be written in the form of
Hext (t) = −P F (t),(6.60)

where F (t) is a quantity to represent the strength of the ﬁeld and P is the operator corresponding to the ﬁeld. With thisdensity matrix, an expectation value of a general physical quantity Q can be obtained as Tr{ρ(t)Q}[1, 2] in the following.
⟨Q(t)⟩ = Tr{ρ(t)Q} = ⟨Qeq ⟩ +

1
iℏ

Z t
−∞

dt′ ⟨[P, Q(t − t′ )]⟩ F (t′ ).

(6.61)

Here ⟨Qeq ⟩ and Q(t) are deﬁned as
Q(t) = U0 (t)−1 QU0 (t).

⟨Qeq ⟩ = Tr{ρeq Q},(6.62)

From eq. (6.61), we know that the expectation value ⟨[P, Q(t − t′ )]⟩ is a pure imaginary.
Now we consider the external ﬁeld of frequency ω and write
F (t) = F0 cos(ωt) = Re[F0 e−iωt ].

(6.63)∆Q(t) = ⟨Q(t)⟩ − ⟨Qeq ⟩ = Re[χ(ω)F0 e−iωt ].

(6.64)We here deﬁne the susceptibility χ(ω) as

On the other hand, from eq. (6.61)∆Q(t) =

1
iℏ

Z t
−∞

′

dt′ ⟨[P, Q(t − t′ )]⟩ Re[F0 e−iωt ].

(6.65)

Now we know that the right-hand sides of eq. (6.64) equates eq. (6.65). The right hand side of (6.64) can be written as
Re[χ(ω)F0 e−iωt ] =F0 ∗[χ (ω)eiωt + χ(ω)e−iωt ].
2

Similarly (6.65) is developed to
Z ∞

Z ∞


F0
dτ ⟨[P, Q(τ )]⟩ e−iωτ eiωt +dτ ⟨[P, Q(τ )]⟩ eiωτ e−iωt ,2iℏ
00(6.66)(6.67)

where τ = t − t′ ). By remembering ⟨[P, Q(τ )]⟩ is a pure imaginary, we compare the above two to obtain the following.

Kubo formulaχQP (ω) =


i
ℏ

Z ∞

⟨[Q(τ ), P ]⟩ eiωτ dτ.

(6.68)

0

Here we add a subscript to χ, which shows the relation of response P → Q. Equation (6.68) is one of the formulas
called Kubo formula, which give linear response functions[3]. This can be viewed as a terminus ad quem of the linear
response theory initiated by Nyquist and by others. The Kubo formula has been applied to a vast ﬁeld of science withfruitful results. It should be used probably forever in science. On the other hand, there are various different formalisms
in linear response. We need to select one of them according tot the character of the problem[2].

6.4.2 Fluctuation-dissipation theorem
Equation (6.68) is a Fourier transformation from (time) to (frequency). At the same time it is a response of a physicalquantity Q to the external ﬁeld in eq. (6.60). The correlation function ⟨[Q(τ ), P ]⟩ represents transfer on time (τ ) axis toE13-2

the susceptibility *1 . Then we deﬁne a Green’s function for physical quantity P , Q asi
G±
QP (t) = ∓ θ(±t) ⟨[Q(t), P ]⟩ ,ℏ
in which we restore the symbol for time as τ → t. Here θ(t) is the Heaviside function deﬁned as(
1 (t ≥ 0),
θ(t) =
0 (t < 0).

(6.69)(6.70)−We also call G+
QP (t) as a retarded Green’s function, GQP (t) an advanced Green’s function. Then eq. (6.68) is a Fouriertransformation of G+
QP (t), and can be written in the form+
χQP (ω) = −GQP(ω) = −Fω {G+QP (t)},(6.71)

where Fω {· · · } expresses the Fourier transform of · · · to ω-space.
A Fourier transform of a correlation function of a perturbation and a response:Z ∞SQP (ω) =dt ⟨Q(t), P ⟩ eiωt

(6.72)−∞

is called a dynamical form factor. Here we can show the following as in Appendix 13B.
SQP (ω) =i−[G + (ω) − GQP(ω)],
1 − e−βℏω QP(6.73)

where β ≡ (kB T )−1 . The left-hand side of eq. (6.73) is a Fourier transform of a correlation function and the righthand side is a susceptibility of linear response. Such formulas that show linear relations between correlation functions
and coefﬁcients of linear responses are called ﬂuctuation dissipation theorem. Of course ﬂuctuations and energydissipations are different physical quantities. The theorems are not saying that they are the same but that ﬂuctuations canbe expressed by coefﬁcients of linear response, which are parameters of energy dissipation.
Let {|n⟩} be a complete set of eigenfunctions of H , then
+
GQP
(ω) =

X

⟨n|Q|m⟩ ⟨m|P |n⟩n,me−βEn − e−βEm.
En − Em + ℏω + iη

(6.74)

6.4.3 Random Phase Approximation (RPA)
We consider an external magnetic ﬁeld
B(r, t) = B(q, ω)ei(q·r−ωt) ,(6.75)

applied on a Hubbard modelH =X

tij c†is cjs + U

i,j,s

N
X

n̂i↑ n̂i↓ .

(6.25)i

We write the local magnetization density in unit of −gµB as
S(r) =1 XX
δ(r − ri )c†iα σαβ ciβ ,2 i(6.76)

α,β*1 Green’s function was invented by George Green (1973-1841). As you probably used in the electromagnetism, it is frequently used for ﬁnding

solutions of differential equations. A Green’s function generally expresses an effect of some local cause to an away point. It appears in various
formalisms and the name “Green’s function” is now also applied to general correlation (transfer) functions. I should note that according toref. [4], the naming “Green’s function” is strange and we should call it “Green function.”E13-3

where σ = (σx , σy , σz ) is a vector with Pauli matrices as elements. Accordingly, Hamiltonian(6.60) in the present caseZisB(r, t) · S(r)d3 r = gµB S−q · B(q, ω)e−iωt .

Hext (t) = gµB(6.77)

Here Fourier-q components of magnetization Sq are deﬁned as follows.
Sq+ = Sqx + iSqy =











X †
ak↑ ak+q↓ ,k

X †Sq− = Sqx − iSqy =ak↓ ak+q↑ ,Sqz = (1/2)

Xk(6.78)






†
†
(ak↑ ak+q↑ − ak↓ ak+q↓ ). kIn comparison of eq. (6.77) and eq. (6.60), the quantity which corresponds to P is gµB S−q . On the other hand, theresponse is also magnetization and in linear response, that is gµB Sq . Hence the z-component of dynamic susceptibilityisχzz (q, ω) = (gµB )

2i

Z ∞

ℏ

dt ⟨[Sqz (t), S−qz ]⟩ eiωt .

(6.79)

0

Similarly, considering non-zero part after taking correlation function, the transverse component is written asZi ∞dt ⟨[Sq+ , S−q− ]⟩ eiωt .
χ+− (q, ω) = (gµB )2ℏ 0(6.80)Let us calculate χ+− (q, ω) in the following way. We take a k term in the expression of Sq+ (t) in eq. (6.78). The
corresponding Green’s function is†G+
kq (t) = −iθ(t) ⟨[ak↑ (t)ak+q↓ (t), S−q− ]⟩ .

(6.81)

Henceforth we omit + to specify “retarded.” The time derivative of this Green’s function (equation of motion) isiℏ∂Gkq= −iθ(t) ⟨[eiH t/ℏ [a†k↑ ak+q↓ , H ]e−iH t/ℏ , S−q− ]⟩ + δ(t)ℏ ⟨[a†k↑ (t)ak+q↓ (t), S−q− ]⟩ .
∂t

(6.82)We divide the Hubbard Hamiltonian into the kinetic energy term Hk and the on-site interaction term Hint , and calculate
the commutation relation in the right-hand side as follows.
[a†k↑ ak+q↓ , S−q− ] =X[a†k↑ ak+q↓ , a†k′ +q↓ ak′ ↑ ]k′= a†k↑ ak↑ − a†k+q↓ ak+q↓ ,[a†k↑ ak+q↓ , Hk ] = (ϵk+q − ϵk )a†k↑ ak+q↓ ,[a†k↑ ak+q↓ , Hint ] = (U/N )X(6.83a)(6.83b)[a†k↑ ak+q↓ , a†k1 +p↑ a†k2 −p↓ ak2 ↓ ak1 ↑ ]k1 ,k2 ,p= −(U/N ) Xa†k↑ a†k1 +p↑ ak+q+p↓ ak1 ↑ +

k1 ,p

X


a†k+p↑ a†k2 −q↓ ak2 ↓ ak+q↓  .

(6.83c)k2 ,pThere are terms with four (2+2) operators products term of annihilation-creation operators in eq. (6.83c) representing
the interaction, to which we apply mean ﬁeld approximation. That is, we replace two of the four operators with theaverage of them in [· · · ] as
−

X †
X †
ak↑ ak+q↓ ⟨a†k1 ↑ ak1 ↑ ⟩ak+p↑ ak+q+p↓ ⟨a†k↑ ak↑ ⟩ +
p

k1

−

X †
X †
ak↑ ak+q↓ ⟨a†k2 ↓ ak2 ↓ ⟩ +ak+p↑ ak+q+p↓ ⟨a†k+q↓ ak+q↓ ⟩ .
p

k2

E13-4

(6.84)Mean ﬁeld approximation to such dynamic quantity is generally called Random Phase Approximation(RPA). The
naming means a quantity with a phase factor (exp(iθ)) of randomized phase (θ) should vanish.
In eq. (6.84), a difference between an average on ↑ and that on ↓ is taken in the second and the thrid terms. In theparamagnetic states, they cancel each other, and the time derivative of Green’s function in RPA (6.81)isiℏ∂Gkq= (ϵk+q − ϵk )Gkq (t)∂tX− (U/N )(⟨a†k↑ ak↑ ⟩ − ⟨a†k+q↓ ak+q↓ ⟩)G(k+p)q (t)p+ (⟨a†k↑ ak↑ ⟩ − ⟨a†k+q↓ ak+q↓ ⟩)δ(t).

(6.85)Taking Fourier transformation of both side we get"#
U X
fk↑ − fk+q↓
1−
Gpq (ω) ,Gkq (ω) =ℏω + ϵk − ϵk+q
N p(6.86)

where fks = ⟨a†ks aks ⟩ is the Fermi distribution function. Summation over k givesχ+− (q, ω) = N (gµB )2
where
χ(0) (q, ω) =2χ(0) (q, ω),
1 − 2U χ(0) (q, ω)

1 X fk+q↓ − fk↑
2Nℏω + ϵk − ϵk+q(6.87)(6.88)

kis the susceptibility of non-interacting system per site normalized by (gµB )2 .
For the calculation of above χ(0) (q, ω) we calculate the following. Here for clearness of expression, we adopt ℏ → 1,
the unit of wavenumber is taken to kF , the unit of energy is taken to ϵF . With 3D Jacobian, the integral is written as
1 X
fk
1= ρ(ϵF )2N
ω + ϵk−q − ϵk
2
k

=

1
ρ(ϵF )
2

Z 1

Z 1
k 2 dkd(cos θ)2 − 2kq cos θ
ω
+
q
−1

k 2 dk1
ω + q 2 + 2kq
log
.
2kq
ω + q 2 − 2kq

0

Z 1
0

From a mathematical identityZ" 2 #
1 2bbx2
x log(ax + b)dx =x −+ x,log(ax + b) −
2
a
4
2a

Fig. 6.4 Boundary lines of Kohn anomaly expressed ineq. (6.91), and four regions separated by them in the upper halfof q-ω plane.

E13-5(6.89)

Fig. 6.5 Plot of the real part of χ(0) (q, 0) in
eq. (6.93). The derivative diverges at q = 2.

the integration can be performed as
χ

(0)

( "

2 #
1
ω + q2
ω + q 2 + 2q ω + q 2
1−
log
+
2
2q
ω + q 2 − 2q
2q
"
)
#


2
1
−ω + q 2−ω + q 2 − 2q −ω + q 2
− 1−
+
.
log
2
2q
ω + q 2 + 2q
2q

ρ(ϵF ) 1(q, ω) =2 2q(6.90)In (6.90), when the arguments of log is negative, the susceptibility has a ﬁnite imaginary part that leads to damping.
The boundary (Kohn anomaly boundary) is given byω = ±(q 2 ± 2q).

(6.91)

Figure 6.4 shows these boundaries on q − ω plane. They divide the upper half plane to four regions I∼IV. In regions I
and IV, the imaginary part is zero. In region III, the imaginary part isIm[χ(0) (q, ω)] =The real part is for ω = 0
Re[χ(0) (q, 0)] =ρ(ϵF ) 12 2q

ρ(ϵF ) π ω
.
2 4q


1−

q2
4


log

(6.92)

2+q
+q .
2−q

(6.93)

This is plotted as a function of q in Fig. 6.5. At the Kohn anomaly boundary q = 2(kF ), the curve shows a divergence of
derivative by q.
From eq. (6.87) the RPA on dynamical susceptibility of Hubbard model predicts appearance of magnetic order forU χ(0) (qmax , 0) ≥1.
2

(6.94)

Here qmax is the wavenumber that gives the maximum value of χ(0) . In the case of qmax = 0, as can be seen in Fig. 6.5,chi(0) (qmax ) → ρ(ϵF )/2, then naturally this agrees with the Stoner condition. On the other hand, when qmax ̸= 0, amagnetic order with ﬁnite wavenumber exists. This corresponds to spin density wave (SDW).

6.5 Self-consistent renormalization spin ﬂuctuation theory
As we have seen above, the mean ﬁeld (HF approximation) theory based on the Hubbard model has various problems
both in principle and in comparison with experiments. On the other hand, although parameter tuning may be included, it
E13-6

explains some aspects of experiments such as the Slater-Pauling’s curve. It is impossible to discuss trends of researches
from a single point of view. However a way to view the ﬂows of research on magnetism is that there were two ways
to go after the HF approximation. One is to look for different ways from HFA by simplifying models, by considering
extreme cases and the strong correlation is taken into account more seriously. The other is improvement of HFA to solve
the difﬁculties. The former has produced many interesting results on mathematical physics and conversely experiments
appeared aiming at realizing such mathematical models. A big success of the latter is self-consistent renormalization(SCR) spin ﬂuctuation theory[5]．
Since it is difﬁcult to see the mathematical scientiﬁc direction in the remaining one lecture, I would like to brieﬂy
explain the SCR theory and ﬁnish it. Many textbooks on mathematical science directions and strongly correlated systems
have been published during the last quarter century[6, 7, 8, 9]. If you are interested, please refer to them.
(To be continued)

Appendix 13A: Derivation of integral equation
We deﬁne the interaction representation of ρ (ρ(I) ) as
ρ(t) = e

H0 t/(iℏ)ρI (t)e−H0 t/(iℏ)= U0 (t)ρI (t)U0−1 (t),
U0 (t) ≡ exp


H0
t .
iℏ

(13A.1)

Here, ρI = U0−1 ρU0 , [H0 , U0 ] = 0. Also
∂U0
H0
1
=
U 0 = U 0 H0 ,
∂t
iℏ
iℏ

∂U0−1
H0 −1
1
=−
U = − U0−1 H0 ,
∂t
iℏ 0
iℏ

U0−1 (t) = U0 (−t).

Then from eq. (6.55), the equation of motion for ρI (t) is

∂U0−1
−1 ∂ρ
−1 ∂U0ρU0 + U0
U0 + U0 ρ
∂t
∂t
∂t


H0 −1
1 −1
−1 H0
= iℏ −U ρU0 + U0 [H0 + Hext , ρ]U0 + U0 ρ
U0
iℏ 0
iℏ
iℏ

∂ρI
iℏ
= iℏ
∂t



= U0−1 (Hext ρ − ρHext )U0= [U0−1 Hext U0 , ρI ].

(13A.2)

From the condition ρ = ρeq and Hext = 0 for t = −∞, by integrating both sides of eq. (13A.2) with (−∞, t],
ρI (t) − ρI (−∞) = −

1
iℏ

Z t
−∞

dt′ [U0−1 (t′ )Hext U0 (t′ ), ρI (t′ )].

(13A.3)

The we obtain
Z t

1
−1 ′
−1 ′
′
′
′
′
ρ(t) = ρ(−∞) + U0 (t)dt [U0 (t )Hext U0 (t ), U0 (t )ρ(t )U0 (t )] U0−1 (t)
iℏ
−∞
Z
1 t
dt′ U0 (t − t′ )[Hext (t′ ), ρ(t′ )]U0−1 (t − t′ ),= ρeq +iℏ −∞which is eq. (6.57).

E13-7(13A.4)

Appendix 13B: Fluctuation-dissipation theorem
We change the order of Q and P in
SQP (ω) =Z ∞to write−∞dt ⟨P Q(t)⟩ eiωt =Z ∞−∞dt ⟨Q(t), P ⟩ eiωt ,Z ∞1
dt Tr{e−βH P eiH t/ℏ Qe−iH t/ℏ }eiωt .
Z−∞(13B.1)

Now we use mathematical identity that for operators A, B, C, Tr{ABC} satisﬁes
Tr{ABC} = Tr{CAB} = Tr{BCA}.

(13B.2)

Then
Z ∞
−∞

dt ⟨P Q(t)⟩ e

iωt

Z ∞

1
dt Tr{eiH t/ℏ Qe−iH t/ℏ e−βH P }eiωt
Z
−∞
Z ∞
1
=
dt Tr{e−βH e(i/ℏ)H (t−iβℏ) Qe−(i/ℏ)H (t−iβℏ) P }eiω(t−iβℏ) e−βℏωZ−∞== e−βℏω SQP (ω).

(13B.3)

Namely we reach
1
SQP (ω) =1 − e−βℏωZ ∞−∞dt ⟨[Q(t), P ]⟩ eiωt =i[G+ (ω) − G−
QP (ω)].
1 − e−βℏω QP(13B.4)

References[1] 川畑有郷. 電子相関 (パリティ物理学コース クローズアップ). 丸善, 3 1992.
[2] Giuseppe De Nittis. Linear Response Theory: An Analytic-Algebraic Approach (SpringerBriefs in Mathematical
Physics). Springer, 6 2017.
[3] Ryogo Kubo. Statistical-mechanical theory of irreversible processes. i. general theory and simple applications tomagnetic and conduction problems. Journal of the Physical Society of Japan, Vol. 12, No. 6, pp. 570–586, June
1957.
[4] Philip Phillips. Advanced Solid State Physics (English Edition). Cambridge University Press, 3 2012.
[5] Tôru Moriya. Theory of itinerant electron magnetism. Journal of Magnetism and Magnetic Materials, Vol. 100, No.
1-3, pp. 261–271, November 1991.
[6] 青木秀夫, 草部浩一. 強磁性 (多体電子論). 東京大学出版会, 12 1998.
[7] Adolfo Avella and Ferdinando Mancini, editors. Strongly Correlated Systems: Theoretical Methods (Springer Series
in Solid-State Sciences). Springer, 1 2014.
[8] 勝藤拓郎. 基礎から学ぶ強相関電子系 – 量子力学から固体物理、場の量子論まで (物質・材料テキストシリーズ).
内田老鶴圃, 6 2017.

[9] Daniel P. Arovas, Erez Berg, Steven A. Kivelson, and Srinivas Raghu. The hubbard model. Annual Review ofCondensed Matter Physics, Vol. 13, No. 1, pp. 239–274, March 2022.

E13-8

Lecture note
13th July (2022)

Magnetism (14)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Let us go into self-consistent renormalization (SCR) spin ﬂuctuation (SF) theory, the entrance of which we ﬁnallyreached last week. As mentioned in the last lecture, SCR-SF theory approaches the problem with improving the approximation not taking completely different ways. The theory then gained great successes[1]. We have used two weeks toreach the entrance. It is impossible to introduce the details of the ﬁrst paper[1] in a single lecture. Here I would liketo introduce only the framework of the theory along with resumes like [2, 3].e Those who wish to know the details arerequested to refer to monographs of this topic[4, 5, 6], and to original papers *1 .

6.5.1 Weak metallic ferromagnetism, paramagnonr
Kawabata, one of the founders of SCR-SP theory, says a strong motivation of the theory is the ﬁnding of weak metallic
ferromagnetism like ZrZn2 , Sc3 In, Ni3 Al[2], Also in the textbook[4] by Moriya, who is synonymous with SCR-SF
theory, paramagnon theory by Izuyama and Kubo[7] is introduced as a preliminary form of SCR-SF theory. Let us
then consider paramagnons, which are spin waves in non-magnetically-ordered metals with strong effective interactions
between spins. Such metals are considered to be marginal to ferromagnetism (anti-ferromagnetism). Palladium (Pd) tends
to form ferromagnetic alloys with non-magnetic elements and is considered to be close to ferromagnetism[8]. Helium-3,
which is not a metal but has a nuclear spin, can also be mentioned. It is long since the magnon (paramagnon) systemof superﬂuid helium-3 was found to cause BEC[9, 10]. In high-TC cuprates close to anti-ferromagnetic order[11], also
in iron-based high-TC superconductors[12], existences of paramagnons have been conﬁrmed and the dispersion relations
etc. have been determined.
We saw in the last lecture that in RPA, which is the mean ﬁeld approximation on dynamic susceptibility, magnon states
may be magnetically ordered ground states. This suggests that spin-ﬂuctuations may be considered as excited states in themagnon-system picture even when the ground state is paramagnetic. It also suggests that the existence of such collective
excitations may greatly lower the energy of excited states compared with those in static HF approximations. In spite ofsuch signs of improvement, the simple RPA does not solve the problems of mean ﬁeld theory in temperature dependenceof susceptibility etc. This is probably due to ignoring spin ﬂuctuation in ﬁnite temperature thermal equilibrium state.
Then in the paramagnon theory we consider the spin ﬂuctuation in paramagnetic state.
Hellmann-Feynman theorem
First we prove the following general theorem. We consider a Hamiltonian with parameter p
H (p) = H0 + H1 (p)(6.95)

with normalized eigenstate |p, n⟩ and the eigen energy En (p). A change in an eigenstate due to an inﬁnitesimally smallchange δp of p H (p + δp) is expressed as a linear sum of the original eigenstates as|p + δp, n⟩ = |p, n⟩ +X

Cm |p, m⟩ .

(6.96)

m*1 As far as I checked, most of textbooks written in English, do not even refer to the SCR-SF theory, whose logic is even difﬁcult to be followed.

They usually introduce the simple HF approximation, then shift to DFT or GGA, in which the approximation is improved. These ab initio
calculation may doing similar improvement for the HF as SCR-SF though the details cannot be known.

E14-1

If we assume linear approximation Cm = cm δp, from⟨p + δp, n|p + δp, n⟩ = |1 + cn δp|2 ⟨p, n|p, n⟩ +X

|cm |2 |δp|2 ⟨p, m|p, m⟩ ,m̸=nthe condition of normalization leads to cn = 0, that is, within linear approximation in δp, Cn = 0. Namely, within linear
approximation of δp,⟨p + δp|H (p)|p + δp⟩ = ⟨p|H (p)|p⟩ = En (p).

(6.97)

Then we can write the shift in the eigenenergy as
En (p + δp) = ⟨p + δp, n|H (p + δp)|p + δp, n⟩∂H (p)= p + δp, n H (p) + δp
p + δp, n∂p∂H (p)= En (p) + δp p, n
p, n .
∂p
We can thus conclude

dEn (p)=dp
∂H1 (p)p, np, n .
∂p(6.98)(6.99)

This is called Hellmann-Feynman theorem.
From this theorem, e.g., the free energy F (p) of the system of Hamiltonian H (p) can be written as∂F (p)−En (p) ∂En (p)
1 X
=
exp
.
∂p
Z n
kB T
∂p

(6.100)In eq. (6.95), H1 (p) can be an interaction Hamiltonian HI of interaction constant I. From Hellmann-Feynman theorem, we can introduce the interaction with varying the interaction constant as I : 0 → I. The correction term for freeenergy is given as a function of I by

Z IF (I) = F (0) +
0

∂HI ′
∂I ′



dI ′ .

(6.101)

From the above, we consider the contribution of spin ﬂuctuation to the speciﬁc heat. Here Hubbard model in eq. (6.29)
can be written as
H =X

ϵk a†ks aks + UNXn̂i↑ n̂i↓ = H0 + HI .

(6.102)ik,sAs in the transform of ﬁrst term, an operator at lattice point Ri is Fourier-expanded as
1 X iRi ·k
cis = √
e
aks .
N k

(6.103)U X †ak+q↑ ak↑ a†k′ −q↓ ak′ ↓ ,N′(6.104)

And as in eq. (6.83c), we write HI asHI =k,k ,qwhere the interaction parameter is I = U/N . We also change the notation in eq. (6.78) a bit toX †ak↑ ak+q↓ , 


k
X †

S− (q) =ak↓ ak+q↑ . S+ (q) =k

E14-2(6.78)

From fermionic commutation relation, with exchanging the last two operators in the sum in right-hand side, eq. (6.104)
is developed to
HI = IX
k,k′ ,q=IXa†k+q↑ ak↑ (δk′ ,k′ −q − ak′ ↓ a†k′ −q↓ )Xa†k↑ ak↑ −k,k′a†k+q↑ ak′ ↓ a†k−q↓ ak↑  .

(6.105)k,k′ ,q

Wavenumber q, over which the sum is taken, transform q → −q + k′ − k results in k + q → k − q + k′ − k = k′ − q,and similarly k′ − q → k + q. Then from eq. (6.78), we can write


X †
X †
HI = I a ak↑ −a ′ak ′ ↓ a†
ak↑ 
k −q↑

k↑


=I

k,k′

k,k′ ,qX

X

a†k↑ ak↑ −

k,k′

k+q↓





S+ (−q)S− (q) = I Xa†k↑ ak↑ −

k,k′

q

X


S+ (q)S− (−q) .

(6.106)q

Next, in paramagnetic states, HI does not change with spin inversion, thus, we add the expression of ↑↔↓ and divide by
two to obtain
HI =

Ne U
IX
−
{S+ (q), S− (−q)}+ ,2
2 q(6.107)

where Ne is the number of electrons, {A, B}+ = AB + BA represents anti-commutation relation. In eq. (6.106), sums
over k, k′ produce Ne and N respectively. The latter is because the sum comes from that over Ri . Accordingly, thevariation in the free energy due to introduction of interaction is given by
∆F =

Ne U
1X
−
2
2 q

Z I

dI ′ ⟨{S+ (q), S− (−q)}+ ⟩ .

(6.108)0

For usage of ﬂuctuation-dissipation theorem, remember eq. (6.74):
+
GQP
(ω) =

X

⟨n|Q|m⟩ ⟨m|P |n⟩n,me−βEn − e−βEm.
En − Em + ℏω + iη

(6.74)−
If we use the lower half of complex plane by replacing η → −η, we obtain a parallel representation for GQP(ω). Fromthese two, we can write+−
GQP(ω) − GQP(ω) = −2iIm[χQP (ω)],

to further obtain
SQP (ω) =2
Im[χQP (ω)].
1 − e−βℏω(6.109)We also rewrite eq. (6.80) as
χ+− (q, ω) = −(gµB )2

i
ℏ

Z ∞

dt ⟨[S− (−q), S+ (q, t)]⟩ eiωt .

(6.110)0

Let |n⟩ and En be a many-body eigenstate and its eigenenergy of Hamiltonian in eq. (6.102), then the imaginary partof χ+− (q, ω) is given by
Im[χ+− (q, ω)] =whereπ(gµB )2 X
(ρm − ρn )δ(ω − ∆Emn /ℏ) ⟨n|S− (−q)|m⟩ ⟨m|S+ (q)|n⟩ ,
ℏ
m,n



1
En
Boltzmann factor: ρn = exp −
,
Z
kB T
E14-3

∆Emn = Em − En .

(6.111)Multiply both sides of eq. (6.111) by coth(βωℏ/2), and integrate over ω to obtain


Z ∞
ℏω
dωImχ+− (q, ω) coth
2kB T
−∞


π(gµB )2 X
∆Enm=⟨n|S− (−q)|m⟩ ⟨m|S+ (q)|n⟩(ρm − ρn ) coth
ℏ
kB T
m,n
=

π(gµB )2⟨{S− (−q), S+ (q)}+ ⟩ .
ℏ(6.112)

Then we can write the variation of free energy in eq. (6.108) as
Ne U X
∆F =
−
2
q

Z I
dI
0

′ 1

2π



Z ∞
dω coth
−∞

ℏω
2kB T


Im[χ+− (q, ω)].

By applying the RPA formula in eq. (6.87), the above is approximated asZ
Ne U X 1 ∞
ℏω
∆F =
+
dω cothIm{log[1 − 2U χ(0) (q, ω)]}.
2
π
k
T
B
0
q

(6.113)(6.114)For calculation of speciﬁc heat, only temperature-dependent parts in ∆F are necessary.
coth

ℏω
2
= 1 + ℏω/k T
.
B
kB T
e
−1

(6.115)In the above, the ﬁrst term “1” in the right hand side is the zero-point motion of paramagnons and ignored in the calculation of speciﬁc heat. In eq. (6.114), we need to consider the contribution of χ(0) (q, ω), susceptibility of non-interacting
system. In (q, ω)-plane shown in Fig. 6.4, the largest contribution is given in region-III, in which we can expand χ(0) (q, ω)
as
χ

(0)

"
#
 2
1
q
ℏω kF
(q, ω) = ρ(ϵF ) 1 − A0
.
+ iC0
2
kF
ϵF q

(6.116)

Here the expansion coefﬁcients are
A0 =

1
,
12

C0 =

π
.
4

(6.117)For the simplest approximation we use this expression of χ(0) (q, ω). Let α ≡ U ρ(ϵF ) be the interaction constant, we
reach
 

Z qc
Z ∞
N
2
ω
ρ(ϵF )ϵ2F
q 2 dqdω βω
Im log 1 − α + αA0 q 2 − iαC0
2
e −1
q
0
0


Z qc
Z ∞
N
ω
C0
2
= − ρ(ϵF )ϵ2F
arctanq 2 dqdω βω
2 + A q2 .
2
e
−
1
q
K
0
0
0
0

∆F (T ) =(6.118)As before, for simplicity of representation, unit of wavenumber q is taken as kF , ℏ = 1, unit of energy ω is taken as ϵF .
qc is a cutoff of wavenumber. Here expansion in region-III is effective only for ﬁnite q and the cutoff is indispensable andis around 1 (kF ). And K0 is
K02 =

1−α
.
α

(6.119)We further apply low temperature approximation ω ≪ 1 and arctan x ∼ x to obtain∆F (T )
2π 2
C0
K 2 + A0 q 2=−
ρ(ϵF )(kB T )2
log 0 2 c ,
N
3
2πA0
K0

(6.120)from which we know the spin-ﬂuctuation contribution to low temperature speciﬁc heat is proportional to T . Then if wewrite
C = γT,γ0 ≡2π 2 2k ρ(ϵF ),
3 B

E14-4

(6.121)then the coefﬁcient γ isC0
K 2 + A0 q 2
γ = γ0 1 +log 0 2 c ,πA0
K0(6.122)where γ0 is the temperature coefﬁcient of speciﬁc heat of electron systems without contribution from spin ﬂuctuation.
Equation (6.122) tells that the temperature coefﬁcient of speciﬁc heat logarithmically diverges as the condition approaches
Stoner ferromagnetic criterion α → 1, K0 → 0.

6.5.2 SCR spin-ﬂuctuation theory
So far, Hellmann-Feynman theorem and ﬂuctuation-dissipation theorem are used to obtain the expression for freeenergy of thermal equilibrium with dynamic susceptibility. By applying RPA to dynamic susceptibility, the effect of spinﬂuctuation is included. However, at this stage the theory just describes low-temperature approximation of paramagnetic
states. A problem here is, as is always the case of many-body problem, the effect of spin-ﬂuctuation should be reﬂectedon the spin-ﬂuctuation. Hence this simply cannot be applied to ferromagnetic state with spontaneous magnetization. For
that we need to consider self-consistent equation as in the case of Heisenberg model.
Then, here, we consider the free energy in the presence of magnetization M . The free energy is given as the sum of
free energy F0 (M, T ) of non-interacting system and the quantity in eq. (6.113) by
Z
XZ I
Ne U
ℏω
1
F (M, T ) = F0 (M, T ) +− bM −dI ′dω cothIm[χ+− (M, I ′ ; q, ω)].
2
2π
2k
T
B
0
q

(6.123)−bM is the Zeeman term. Here, as a notation, the magnetization M and the interaction parameter I are speciﬁed in the
dynamic susceptibility χ+− .
As we saw in the GL theory static spontaneous magnetization M is given by extremum condition of F as
∂F (M, T )= 0.
∂M(6.124)This can be viewed as magnetic equation of state.
We restart with eq. (6.113), which does not contain any approximation. In HF mean-ﬁeld approximation, integrand of
the second term in right-hand side is replaced with the one without interaction (I = 0), thus the integration with dI ′ overregion [0, I] is replaced with simple product of I.
X 1
Ne U∆FHF =
−I
2
2π
q



Z ∞
dω coth
−∞

ℏω
2kB T


Im[χ+− (M, 0; q, ω)].

(6.125)

From the Hubbard model in eq. (6.102), we consider the quantity:


∂H
∂I


=N
I=0

N
X

⟨ni↑ ni↓ ⟩I=0 = N

i

=

N
X

⟨ni↑ ⟩ ⟨ni↓ ⟩

i

N2 2
N2 2
N2
(n+ − n2− ) =[n − (2m)2 ] = e − M 2 .
4
4
4(6.126)In the above, small characters n, m are the numbers per sites, andn+ = n↑ + n↓ ,n − = n↑ − n↓ ,

m=

n−
,
2

where the unit of magnetization is taken as gµB , the spin is 1/2. We take eq. (6.126) multiplied by I as the I-proportionalterm in (6.125). Then we can write formally
 2

Ne
2
F (M, T ) = F0 (M, T ) + I
−M
− bM
4
Z ∞
XZ I
1
ℏω
dI ′
dω coth−Im[χ+− (M, I ′ ; q, ω) − χ+− (M, 0; q, ω)]
2π
k
T
B
0
−∞
q
E14-5

: HF Approximation: Correction.

(6.127)

As is in paramagnon theory, we apply RPA(eq. (6.87)) to χ+− (M, I ′ ; q, ω). Then the integration with I ′ can be
performed and the result is



Ne2
2
F (M, T ) = F0 (M, T ) + I
−M
− bM
4
Z
∞
X 1
ℏω
−
Im[log{1 − 2U χ(0) (M ; q, ω)} + 2U χ(0) (M ; q, ω)],dω coth
2π
k
BT
−∞
q
where we write
χ(0) (M ; q, ω) =1
χ+− (M, 0; q, ω).
2N(6.128)(6.129)

We obtain the following magnetic equation of state by calculating magnetization-derivative of eq. (6.124) in eq. (6.128).
1 X 1
∂F0
− 2U m − b −N ∂m
N q 2πZ ∞dω coth−∞ℏω
2U χ(0) (M ; q, ω)∂χ(0) (M ; q, ω)
Im
2U
= 0,
2kB T
∂m
1 − 2U χ(0) (M ; q, ω)(6.130)which is given as an equation for magnetization per site m = M/N .
Here we formally deﬁne magnetization per site in paramagnetic states as
∂m
,
∂b

χ=

1
∂b
=
.
χ
∂m

(6.131)

We also write susceptibility per site for non-interacting system as χ0 . Free energy of the system with Zeeman term
F0 − bm gives a magnetic equation of state as∂F0− b = 0.
N ∂m

(6.132)By applying eq. (6.124) to the above with transposition of b, we obtain∂ 2 F0
1
=
.
N ∂m2
χ0

(6.133)

Then in eq. (6.130) for the case of paramagnetic state, substituting the above gives
1
1
=
− 2U
χ
χ0
Z ∞
1 X 1
ℏω
−
dω coth(2U )2
N q 2π −∞
2kB T
"

2 #
∂ 2 χ(0) (q, ω)∂χ(0) (q, ω)
1
2
× Im χ(q, ω)(q, ω)+ χ (q, ω)
,
∂m2
∂m
χ(0)
m=0
m=0

(6.134)the last part of which is calculated as follows. Equation (6.87) of RPA and eq. (6.129) lead toχ( q, ω) =

χ(0) (q, ω).
1 − 2U χ(0) (q, ω)(6.135)We then apply this to obtain(0)∂χ∂χ ∂χ(0)1∂χ(0)
2 1 ∂χ
===χ.
∂m∂χ(0) ∂m
(1 − 2U χ(0) )2 ∂m
χ(0)2 ∂m

Equation (6.134) tells the existence of correction, which is expressed as an integral with ω, to the susceptibility fromthe term indicated as “Correction” in eq. (6.127). In the correction, inside Im[· · · ], within two terms in · · · , the ﬁrst
term is linear in spin-ﬂuctuation, the second is in the second order. Here we only consider the linear term. We further
approximate the ﬁrst term by replacing ∂ 2 χ(0) /∂m2 |m=0 with the value for q = 0, ω = 0.h Then by writing thatg = −(2U )2 χ0∂ 2 χ(0) (q, ω),∂m2
m=0,q=0,ω=0
E14-6(6.136)

we obtain the following.
χ0
g X 1= 1 − 2U χ0 +
χN q 2πZ ∞dω coth−∞ℏωIm[χ(q, ω)].
2kB T

(6.137)

We further ignore the temperature dependence of susceptibility of non-interacting system χ0 . This should be taken into
account, e.g., in comparison with experiments etc., though here our purpose is to see the essential structure of SCR-SF
theory. However, we still have the problem of inconsistency in that application of RPA (eq. (6.135)) to the result of(6.137) results in a disagreement of the divergent point of dynamic susceptibility ω ̸= 0 and that of static one (ω = 0). In
spite of the progress of approximation, a simple RPA still cannot satisfy the self-consistent condition.
The difﬁculty is avoided in the SCR-SF theory in the following way. We write the susceptibility at absolute zero asZ
χ0g X1 ∞= 1 − 2U χ0 +dωIm[χ(q, ω)]T =0 .
(6.138)
χ(T = 0)N q π 0
The third term in the rhs represents the contribution of spin-ﬂuctuation with q ̸= 0 at absolute zero, namely the zero-point
motion. As can be seen here, the effect of spin-ﬂuctuation enters into the denominator of RPA representation making thecondition of ferromagnetism difﬁcult to be satisﬁed.
We take this as an RPA modiﬁed in the simplest way by the effect of spin-ﬂuctuation. As is in the case of eq. (6.115),
we again ignore the temperature dependence of zero-point motion in magnon spectrum and take the difference between
eq. (6.137) and eq. (6.138). The approximation gives
χ0
χ0
g X1
=
+
χ
χ(T = 0) N q π

Z ∞
0

2
dω ℏωβIm[χ(q, ω)].
e
−1Here an expansion of χ−1 (q, ω) around (q, ω) = (0, 0) is possible as in eq. (6.116). The functional form is
 2
χ0
χ0
q
ω kF
=
+A
.
− iC
χ(q, ω)χ(+0, +0)
kF
ϵF q

(a)

(b)
Fig. 6.6 (a) Temperature dependence of susceptibility in a ferromagnet calculated in the SCR-SF theory. The parameter here is the interaction parameter, which determines TC . The broken lines are by the simple molecular ﬁeld
theory. (b) Critical temperature as a function of Stoner parameter α = ID(ϵF ) calculated by the SCR-SF theory. The
broken line is 1/5 of the result in the simple molecular ﬁeld theory. From [1].

E14-7

(6.139)(6.140)(a)(b)
Fig. 6.7 (a) Speciﬁc heat in the vicinity of ferromagnetic transition point calculated with the SCR-SF theory. From[13]. (b) Speciﬁc heat of itinerant electron weak ferromagnet Sc3 In under magnetic ﬁeld around TC . The measured
values in zero-magnetic ﬁeld is subtracted. Solid and broken lines are results of numerical calculation. From [14].

This form requires agreement of χ(q, ω) with the static susceptibility for q → +0, ω → +0.
In summary, calculation of χ(q, ω) from (6.139) and (??) results in a self-consistent solution. The above is the essential
framework of the SCR-SF theory though realistic calculations are far more complicated even for simpliﬁed band structurewith many expansion coefﬁcients[1].
In Fig. 6.6, we show the temperature dependence of susceptibility calculated with the SCR-SF theory for a simple
parabolic band structure and the critical temperature TC as a function of interaction strength α = ID(ϵF ) (Stoner parameter) [1]. In Fig. 6.6(a), compared with the simplest molecular ﬁeld theory (broken lines), the SCR-SF theory gives muchbetter linearity in a wide temperature range, which indicates the Curie-Weiss law. Also as in Fig. 6.6(b), the energy ofparamagnetic state largely lowers due to the contribution of spin-ﬂuctuation in thermal equilibrium, leading to correctionof overestimation in stability of ferromagnetic state. As a result, TC largely lowers from the results of the molecular ﬁeld
theory.
On the other hand, there still exist many problems and open questions. As shown in Fig. 6.7, there is a large difference
between the calculated behavior of speciﬁc heat and those in experiments. Studies on these points have long history andthey are summarized e.g., in [5].

Appendix 14A: Calculation in (6.111)
The calculation shown in the following, very often appears, e.g, in usage of Green’s function (calculation of (6.74) isalmost the same). For your convenience, I would like to show a little of calculation in (6.111). It is probably enough to
see

i
ℏ

Z ∞

dt ⟨S+ (q, t)S− (q)⟩ eiωt = (∗∗).

0This is calculated as follows.
Z ∞
h
i
i
(∗∗) =
dtTr e−βH e(i/ℏ)H t S+ (q)e−(i/ℏ)H t S− (q) eiωt
ℏZ 0
Z ∞ X
i
=
dt
⟨n| e−βH +(i/ℏ)H t S+ (q) |m⟩ ⟨m| e−(i/ℏ)H t S− (q) |n⟩ eiωt
ℏZ 0
n,m
Z ∞ X
i
=
dt
e(−β+(i/ℏ)t)En ⟨n| S+ (q) |m⟩ ⟨m| S− (−q) |n⟩ e−(i/ℏ)Em t eiωt .
ℏZ 0
n,m
E14-8

(14A.1)

Here we add a inﬁnitesimal real part to the pure imaginary argument of exponential function, which is a trick often
appears in Fourier integration over half-inﬁnite region. Rigorous proof is of course possible with ϵ-δ logics. Below, weomit to write but η implicitly means taking the limit η → +0.
Z
ie−βEn ∞(∗∗) =⟨n| S+ (q) |m⟩ ⟨m| S− (−q) |n⟩dte(i/ℏ)(En −Em +ℏω+iη)tℏZ0n,m (i/ℏ)(ℏω−∆Emn +iη)t ∞Xe=⟨n| S+ (q) |m⟩ ⟨m| S− (−q) |n⟩ ρnℏω − ∆Emn + iη 0
n,m
X
1
−ρn
=
⟨n| S+ (q) |m⟩ ⟨m| S− (−q) |n⟩ℏ ω − ∆Emn /ℏ + iη
n,m


X
−ρn
1
=
⟨n| S+ (q) |m⟩ ⟨m| S− (−q) |n⟩− iπδ(ω − ∆Emn /ℏ) .
ℏω − ∆Emn /ℏn,mX(14A.2)

Then eq. (6.111) is obtained. Here in the last part we have usedlim

1

η→+0 x ± iη=1
∓ iπδ(x).
x(14A.3)This equation can be shown from an expression of δ-function as a limit of function:
lim

1

η

η→+0 π x2 − η 2= δ(x).

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[13] Kenji Makoshi and Tôru Moriya. Effect of spin ﬂuctuations on the speciﬁc heat of weakly and nearly ferromagnetic
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E14-10