Lecture 7: Consequences of Renormalization Group
Naoki KAWASHIMAISSP, U. Tokyo

June 3, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20191 / 25

In this lecture, we see ...
The free energy (and therefore all the quantities derived from it) canbe expressed as the sum of a singular part and a regular part.
The critical phenomena can be systematically derived from thesingular part of the free energy.
By RG flow diagram, we can understand cross-over phenomena,
which is the scale-dependent critical phenomena.
From RG, we can derive “finite-size scaling (FSS),” which is useful inestimating scaling dimensions through numerical simulation of finitesystems.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20192 / 25

[7-1] Singular part of free energy
The free energy of a finite system can be split into two non-singularparts: the first part is purely extensive, whereas the second is RGTinvariant and becomes singular in the thermodynamic limit. The
latter is called the singular part of the free energy (though it isnon-singular for finite systems).

Naoki KAWASHIMA (ISSP)Statistical Machanics I

June 3, 20193 / 25

Singular part of free energy
As we see later, the RGT invariant function produces a singularitythat explains critical behaviors.
However, the free energy itself cannot be RGT invariant at the criticalpoint due to cotribution from short-range fluctuations.
These observations motivate the following form for the free energy:
F (K, L) = Fs (K, L) + Ld γ(K)(1)

where Fs is RGT-invariant
Fs (Ḱ, Ĺ) = Fs (K, L)(2)and γ(K) is a non-singular function of K

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20194 / 25

Example: 1D Ising model (1)The partition function can be expressed with the transfer matrix as K +hee −K
KS1 S2 +h(S1 +S2 )/2L
=Z = Tr T
TS1 ,S2 ≡ ee −K e K −h
The eigenvalues of T arepλ± ≡ e K cosh h ± sinh2 h + e −4K(3)The correlation length is thenξ −1 = − logpλ−≈ 2 h2 + t 2λ+Z = Tr T L = λL+ + λL− = λL+Naoki KAWASHIMA (ISSP)
1+λ−λ+

Statistical Machanics It ≡ e −2K .

(4)

L !

June 3, 20195 / 25

Example: 1D Ising model (2)F = Lf + ∆Ff ≡ − log λ+ , ∆F ≡ − log 1 + e −L/ξNotice that ∆F is obviously RGT-invariant. However, we cannot takef and ∆F as γ and Fs , respectively, because f is singular.
LNotice also that, for ξ  L, we have ∆F ≈ − log 2 + .
2ξTherefore, by subtracting L/2ξ from ∆F and add it to Lf , we canmake both terms non-singular∗ , while keeping ∆F RGT-invariant:
F = Fs + Lγwith Fs ≡ ∆F −(5)L2ξFsL=−dL→∞ L
2ξ

fs ≡ lim, and γ ≡ f +12ξ(∗) Since F is the free energy of a finite system, it must be regular. Therefore, regularity of γ automatically means
regularity of Fs even if it produces a singularity in the L → ∞ limit. In addition, strictly speaking, γ is singular, but
this singularity is physically uninportant, because it can be removed by adding a constant to the Hamiltonian.

Naoki KAWASHIMA (ISSP)Statistical Machanics I

June 3, 20196 / 25[7-2] Scaling form
It is convenient to introduce the scaling form of the singular part of
the free energy.
From it, we can systematically derive various scaling relations.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20197 / 25

Finite system
The RGT invariance, Fs (K, L) = Fs (Ḱ, Ĺ) , can be rewritten in termsof scaling field, uµ ,Fs (u1 , u2 , · · · , L) = Fs (u1 b y1 , u2 b y2 , · · · , L/b).

(6)

By setting b = L/L0 where L0 is some constant length scale, anddropping the L0 dependence of the function, we may writeFs (u1 , u2 , · · · , L) = F̃s (u1 Ly1 , u2 Ly2 , · · · ),(7)

which is called the “scaling form” of Fs .

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20198 / 25

Infinite system
Another way of rewriting Fs isFs (u1 , u2 , · · · , L) = Ld fs (u1 , u2 , · · · , L)
= (L/b)d fs (u1 b y1 , u2 b y2 , · · · , L/b).
Let us assign a special role to the first scaling field, u1 , which we
assume to be relevant, and denote it as t (t ≡ u1 ).
By taking b so that tb yt = t0 is a constant,dfs (u1 , u2 , · · · , L) = t yt fs (t0 , u2 t −y2 /y1 , · · · , Lt 1/yt )In the thermodynamic limit, the L dependence on the both sideshould vanish. Then, by also droping the t0 dependence,yyd− 2− 3fs (u1 , u2 , u3 , · · · ) = t yt f˜s u2 t y1 , u3 t y1 , · · ·(8)which is called the scaling form of the free energy density.
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 20199 / 25

Singularity of various quantities (1)We can derive various scaling properties from (7) (or (8)).
Below, we consider only the vicinity of the critical point which allows
us to set all irrelevant fields zero.
As an example, we consider the case where we have only two relevant
fields, t ≡ u1 and h ≡ u2 (like t ∝ T − Tc and h ∝ H in the Isingmodel). So, our singular part of the free energy becomesFs (t, h, L) = F̃s (tLyt , hLyh )Naoki KAWASHIMA (ISSP)

Statistical Machanics I(9)June 3, 201910 / 25

Singularity of various quantities (2)Fs (t, h, L) = F̃s (tLyt , hLyh )For the “specific heat”, we have1 ∂ 2 Fsc(t, L) ∝ − d∼ −L−d+2yt F̃s(2,0) (tLyt , 0)
∂t 2 h→0L
(tL )− 2yty−dc̃(tLyt )∼L=t2y −dyt − tyt2yt −dtyt

2yt −dyt(m,n)

F̃smn∂∂≡ ∂tm ∂hn F̃sF̃s(2,0) (tLyt , 0)× − (tL )
2yt −d(2,0)
yc̃(x) ≡ −x t F̃s (x, 0)

Since limL→∞ c(t, L) is independent of L, c(t, ∞) ∝ t −α where
2yt − d1
α== 2 − dν
ν≡
(10)
ytyt
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201911 / 25

Singularity of various quantities (3)For “magnetization”, we have


1 ∂Fs
= −L−d+yh Fs(0,1) (tLyt , 0)m∝− d∂h h→0Ld−yh= t yt m̃(tLyt ) ∝ t βd − yhyt
For “magnetic susceptibility”, we have1 ∂ 2 Fsχ∝− d= −L−d+2yh Fs(0,2) (tLyt , 0)∂h2 h→0
Lwith β ≡=t−2yh −dytwith γ ≡Naoki KAWASHIMA (ISSP)(11)

χ̃(tLyt ) ∝ t −γ

2yh − dyt

(12)

Statistical Machanics I

June 3, 201912 / 25

Scaling relations
From (10), (11) and (12),α + 2β + γ = 2.

(Rushbrooke)(13)

Similarly, we can also derive thatγ = β(δ − 1)(Griffiths)(14)

where δ is the exponent that characterize the magnetic-fielddependence of the magnetization at the critical temperature,m(t = 0, h) ∝ h1/δNaoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201913 / 25

[7-3] Cross-over phenomena
A cross-over phenomenon is the behavior of the system in which aweak but relevant scaling field manifests itself.
We can understand it from the scaling form.
We can also derive the form of the phase boundary near the criticalpoint.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201914 / 25

Example: Heisenberg model with anisotropy (1)3D classical Heisenberg modelXX
H = −J
Si · Sj − DSiz Sjz
(ij)

(ij)

where S ≡ (Six , Siy , Siz )T (|S| = 1).
If D = 0, the system has a critical point,corresponding to the fixed point “H”.
The anisotropic operator is relevant at H.
If the anisotoropy is strong enough, we canregard the system as an Ising model, whosecritical point is represented by “I”.
Accordingly, there is a RG trajectory (criticalline) starting from H and ending at I.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201915 / 25

Example: Heisenberg model with anisotropy (2)By (8), the free energy around “H” isdfs (t, D) = t yt f˜s Dt −φ
where φ is cross-over exponent φ ≡(15)yD.
yt

Eq.(15) can be re-written asd
fs = t yt f˜s (t/t ∗ (D)) with a “cross-overtemperature” t ∗ (D) ∝ D 1/φ . Then, fs behaves
like an isotropic Heisenberg model (D = 0)
when t  t ∗ , whereas it qualitatively deviatesfrom the Heisenberg-like behavior when t  t ∗ .

γ3DI = 1.237075(10)γ3DH ≈ 1.35(∗)
γ3DXY = 1.3177(5)(∗) Kaupuzs, cond-mat/0101156

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201916 / 25

Example: Heisenberg model with anisotropy (3)Now, we consider the shape of the phaseboundary in the D − T phase diagram.
HWe again use fs (t, D) = t d/yt f˜s (Dt −φ ) ,where φ ≡ yDH /ytH . where we put superscript“H” to make it clear that ytH is the value at H.
Wnen D > 0, the system should show theIsing-like critical behavior. Then, we obtainφ ≈ 1.2 for 3D Heisenberg model.
Ifs (t, D) ∼ (t − tc (D))d/yt .
Now, to satisfy both of these forms at the same time, fs must havethe following form near the criticality.
fs ∝ tdytH

tD− φ1− x0

 dIyt∝DHdφ

1− 1IytHyt

1t − x0 D φ

 dIytITherefore, tc (D) ∝ D 1/φ = D yt /yt
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201917 / 25

Dimensional crossover
Some systems have phase transitions even when
the size is finite in one direction. However, thecritical properties are different from the casewhere the system is infinite in all directions.
Though the system size is not a “field” in theconventional terminology, we can treat (size)−1
as if it were a relevant field.
In doing so, the scaling eigenvalue of D ≡ L−1
k ,where Lk is the size that is finite, is obviously 1.
Therefore, we have φ = 1/yt for the crossover exponent, which leads
ttto t ∗ ∝ L−yfor the crossover temperature, and tc (Lk ) ∝ L−yfor thekk
transition temperature near the Lk = ∞ critical point.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201918 / 25

Quantum critical point
By Feynman’s path integral formulation,
d-dimensional quantum system can berepresented as (d + 1)-dimensional classicalsystem with size 1/T in the new direction.
In some special cases, the extra dimension, called the “imaginarytime”, is essentially equivalent to one of the spatial directions.
For example,transverse field Ising model
X the 2-dimensionalXz zx
H = −J
Si Si − ΓSi has a quantum phase transition at(ij)i

T = 0 and Γ = Γc , and it can be mapped to 3-dimensional classical
Ising model with the size 1/T in the 3rd dimension.
Then, we can apply the dimensional cross-over to this system:−y 3DIt ∗ (T ) ∝ tc (T ) ∝ Lk t

∝ T yt(16)

where t ≡ Γ − Γc and yt is for the 3D Ising model.
Naoki KAWASHIMA (ISSP)

Statistical Machanics IJune 3, 201919 / 25

[7-4] Finite-size scaling
As we have seen, the RGT-invariant quantity can be used tocharacterize a critical phenomena.
We’ll see a practical way for obtaining the critical indices (scalingdimensions).
We can define such a computable RGT-invariant quantity as adifference in the free energy of two system-sizes.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201920 / 25

Specific heat (1)Suppose we have obtained Fs (K, L) as afunction of K and L, and it has the formFs (K, L) = F̃s (tLyt , hLyh , · · · ).
From F̃s , the scaling form of the specific heat isc≈

−T ∂ 2 Fs∼ L2yt −d c̃(tLyt ).
Ld ∂T 2(17)If c(T , L) diverges at the critical point for
L → ∞, we expect that c has a peak aroundT ≈ Tc even if L is finite.
This is compatible with (17) only if c̃ has apeak itself.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201921 / 25

Specific heat (2)c(T , L) ∼ L2yt −d c̃(tLyt ).
Suppose c̃(x) has a peak at x = xp . It meansthat c(T , L) has a peak when tLyt = xp .
Let Tc (L) be the temperature at which c(T , L)
has the peak. Then,Tc (L) − Tc ∝ tc (L) ∝ L−yt .

(18)

The height of the peak also carries some
information on the critical behavior, i.e., it is
proportional toc(Tc (L), L) ∝ L2yt −dNaoki KAWASHIMA (ISSP)(19)

Statistical Machanics I

June 3, 201922 / 25

Specific heat (3)More directly, by plotting c/L2yt −d against(T − Tc )Lyt , we expect that curves
corresponding to varying system sizes fall on
top of each other.
Of course, to do this, we need to choose theright values for Tc and yt , which we do notknow initially.

⇓We can fix these values by sometrials-and-errors, like using an analog camera
and adjusting the focus.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201923 / 25

Remark: Practical substitute of Fs
So far, we have been implicitly assuming that we can compute Fs .
But it is not usually true even for finite systems. That’s why people
often simply use F itself in the place of Fs in numerical calculation.
(This is equivalent to using the specific heat itself instead of its singular part.)

This “approximation” is bad when the divergence is “weak.”
We had better use the following quantity, not F , in the place of Fs :∆b F (K, L) ≡ (b d F (K, L/b) − F (K, L))/(b d − 1)
= (b d Fs (K, L/b) − Fs (K, L))/(b d − 1).
The last expression tells us that ∆b F is RGT-invariant, i.e., free fromthe regular part, while the second expression is computable.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201924 / 25

Exercise
Consider the 1-dimensional q-state Potts model. Following the similarargument as in the lecture, obtain the singular part of the free energy.
Consider S = 1 Ising model which is described by the same form of
the Hamiltonian as the conventional Ising model, whereas each spin
variable takes one of three values, −1, 0, 1 instead of two. Confirm
that ξfs = −1/2 for this model, the same as the S = 1/2 Ising model.
Derive Griffiths’ scaling relation (14).

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 3, 201925 / 25