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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

Lecture Note

Exotic Superconductivity
Department of Physics, University of Illinois at Urbana-Champaign
Anthony J. Leggett(Received 18:00, 2 March 2012)

These are the lecture notes for a series of lectures “Exotic Superconductivity” done at
the Graduate School of Science of The University of Tokyo, Japan, in May-June, 2011.
These lectures are an introduction to those superconductors, all discovered since the 1970s,
which do not appear to be well described by the traditional BCS theory. While the mainemphasis will be on the most spectacular member of this class, the cuprates, I shall also
discuss briefly the heavy fermion, organic, ruthenate and ferropnictide superconductors
as well as superfluid 3 He for reference. I shall try to provide a general framework forthe analysis of all-electronic superconductivity (i.e. that in which the Cooper pairing is
induced wholly or mainly by the repulsive Coulomb interaction).
These lecture notes were written by the following graduate students at The University of Tokyo: Haruki Watanabe (UC Berkeley from Aug. 2011, Lec. 4, 7, 8), Hakuto
Suzuki (Lec. 5, 6), Yuya Tanizaki (Lec. 1, 4, 6), Masaru Hongo (Lec. 2, 3), Kota Masuda(Lec. 1, 2, 3), and Shimpei Endo (Lec. 1, 5, 7, 8). We would like to thank Profs. Hiroshi
Fukuyama and Masahito Ueda for the organization of the lecture series as well for thecritical reading of these notes. We also thank Oﬃce of Communication and Oﬃce of Internationalization Planning at the Graduate School of Science, and Oﬃce of Student Aﬀairs
at Physics Department of The University of Tokyo for their supports. The lectures were
hosted as the Sir Anthony James Leggett Visit Program 2011-2013, financially supportedby the JSPS Award for Eminent Scientists (FY2011-2013).
Slides and videos of the lectures are available at UT OpenCourseWare:http://ocw.u-tokyo.ac.jp/eng_courselist/828.html

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Contents
Lec. 1 Reminders of the BCS theory
1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 BCS theory at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 BCS wave function . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Alternative form of the BCS wave function . . . . . . . . . . . . . .
1.2.3 Pair wave function . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Quantitative development of the BCS theory . . . . . . . . . . . . .
1.3 BCS theory at finite temperature . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Derivation of the gap equation . . . . . . . . . . . . . . . . . . . . .
1.3.2 Fk at finite temperature . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 hnk i at finite temperature . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Properties of the BCS gap equation . . . . . . . . . . . . . . . . . .
1.3.5 Properties of the Fock term . . . . . . . . . . . . . . . . . . . . . .
1.3.6 Pair wave function . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Generalization of the BCS theory . . . . . . . . . . . . . . . . . . . . . . .

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15
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23
25

Lec. 2 Superfluid 3 He: basic description
28
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Landau Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Eﬀects of (spin) molecular field in 3 He . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Enhanced low-energy spin fluctuations . . . . . . . . . . . . . . . . 31
2.3.2 Coupling of atomic spins through the exchange of virtual paramagnons 32
2.3.3 Pairing interaction in liquid 3 He . . . . . . . . . . . . . . . . . . . . 32
2.4 Anisotropic spin-singlet pairing (for orientation only) . . . . . . . . . . . . 33
2.5 Digression: macroscopic angular momentum problem . . . . . . . . . . . . 34
2.6 Spin-triplet pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Equal spin pairing (ESP) state . . . . . . . . . . . . . . . . . . . . 36
2.6.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.3 d-vector (unitary states) . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Ginzburg–Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.1 Spin-singlet case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.2 Spin-triplet case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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A. J. Leggett

Lec. 3 Superfluid 3 He (continued)
3.1 Experimental phases of liquid 3 He . . . . . . . . . . . . . . . . . . . . . . .
3.2 Nature of the order parameter of diﬀerent phases . . . . . . . . . . . . . .
3.2.1 A phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 A1 phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 B phase: naive identification . . . . . . . . . . . . . . . . . . . . . .
3.3 Why A phase? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Generalized Ginzburg–Landau approach . . . . . . . . . . . . . . .
3.3.2 Spin fluctuation feedback . . . . . . . . . . . . . . . . . . . . . . . .
3.4 NMR in the new phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 What can be inferred from the sum rules? . . . . . . . . . . . . . . . . . .
3.6 Spontaneously broken spin-orbit symmetry . . . . . . . . . . . . . . . . . .
3.7 Microscopic spin dynamics (schematic) . . . . . . . . . . . . . . . . . . . .
3.8 Illustration of NMR behavior: A phase longitudinal resonance . . . . . . .
3.9 Digression: possibility of the “fragmented” state . . . . . . . . . . . . . . .
3.10 Superfluid 3 He: supercurrents, textures, and defects . . . . . . . . . . . . .
3.10.1 Supercurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10.2 Mermin–Ho vortex and topological singularities . . . . . . . . . . .

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Lec. 4 Definition and diagnostics of “exotic” superconductivity
4.1 Diagnostics of the non-phonon mechanism . . . . . . . . . . . . . . . . . .
4.1.1 Absence of isotope eﬀect . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Absence of phonon structure in tunneling I-V characteristics . . . .
4.2 General properties of the order parameter . . . . . . . . . . . . . . . . . .
4.2.1 Definition of the order parameter . . . . . . . . . . . . . . . . . . .
4.2.2 Order parameter in a crystal . . . . . . . . . . . . . . . . . . . . . .
4.3 Diagnostics of the symmetry of the order parameter . . . . . . . . . . . . .
4.3.1 Diagnostics of the spin state . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Diagnostics of the orbital state . . . . . . . . . . . . . . . . . . . .
4.3.3 Eﬀect of impurities . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Addendum: the eﬀect of spin-orbit coupling . . . . . . . . . . . . . . . . .

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69
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74

Lec. 5 Non-cuprate exotic superconductivity
5.1 Alkali fullerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Fullerene crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Alkali fullerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Superconducting state . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Organics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Normal state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Superconducting state . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3

5.4

5.5

Heavy fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Normal-state behavior . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Superconducting phase . . . . . . . . . . . . . . . . . . . . . . . . .
Strontium ruthenate: Sr2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Experimental properties of Sr2 RuO4 . . . . . . . . . . . . . . . . .
Ferropnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Structure (1111 compounds) . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.4 Experimental properties (normal state) . . . . . . . . . . . . . . . .
5.5.5 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.6 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.7 Experimental properties (superconducting state) . . . . . . . . . . .
5.5.8 Pairing state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82
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94
95

Lec. 6 Cuprates: generalities, and normal state properties
98
6.1 Basic chemical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 Construction of phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4 Determinants of Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Other remarks: carrier density and list of cuprate superconductors . . . . . 104
6.6 Experimental properties of the normal state: general discussion . . . . . . 105
6.7 Experimental properties at the optimal doping . . . . . . . . . . . . . . . . 106
6.7.1 Electronic specific heat . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7.2 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7.4 Spectroscopic probes: Fermi surface . . . . . . . . . . . . . . . . . . 108
6.7.5 Results of ARPES experiments at the optimal doping . . . . . . . . 109
6.7.6 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.7.7 Optics (ab-plane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.8 Experimental properties at the underdoped regime . . . . . . . . . . . . . . 111
6.8.1 Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.8.2 ARPES in the pseudogap regime: the puzzle of the Fermi surface . 113
Lec. 7 Cuprates: superconducting state properties
117
7.1 Experimental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1.1 Structural and elastic properties and electron density distribution . 117
7.1.2 Macroscopic electromagnetic properties . . . . . . . . . . . . . . . . 117
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7.2
7.3
7.4
7.5

7.1.3 Specific heat and condensation energy . . . . . . . . . . . . . . . . 118
7.1.4 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.5 Penetration depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.6 AC conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.1.7 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.8 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.9 ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.1.10 Neutron scattering (YBCO, LSCO, and Bi-2212) . . . . . . . . . . 123
7.1.11 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.1.12 Electron energy loss spectroscopy (EELS) . . . . . . . . . . . . . . 125
What do we know for sure about superconductivity in the cuprates? . . . . 125
Symmetry of the order parameter (gap) . . . . . . . . . . . . . . . . . . . . 128
Josephson experiment in cuprate (and other exotic) superconductors . . . . 131
What is a “satisfactory” theory of the high-Tc superconductivity in the
cuprates? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Lec. 8 Exotic superconductivity: discussion
136
8.1 Common properties of all exotic superconductors . . . . . . . . . . . . . . 136
8.2 Phenomenology of superconductivity . . . . . . . . . . . . . . . . . . . . . 137
8.2.1 Meissner eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.2 Persistent current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3 What else do exotic superconductors have in common? . . . . . . . . . . . 140
8.4 Theoretical approaches (mostly for the cuprates) . . . . . . . . . . . . . . . 140
8.5 Which energy is saved in the superconducting phase transition? . . . . . . 143
8.5.1 Virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.5.2 Energy consideration in “all-electronic” superconductors . . . . . . 147
8.5.3 Eliashberg vs. Overscreening . . . . . . . . . . . . . . . . . . . . . . 148
8.5.4 Role of two-dimensionality . . . . . . . . . . . . . . . . . . . . . . . 149
8.5.5 Constraints on the Coulomb saved at small q . . . . . . . . . . . . . 150
8.5.6 Mid-infrared optical and EELS spectra of the cuprates . . . . . . . 152
8.6 How can we realize room temperature superconductors? . . . . . . . . . . . 154

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List of Symbols
Symbol
|↑i
|↓i
|00i
|11i
a, a†
†
αkσ , αkσA(k, ε)
ck
CV
d
dn/dε
dx2 −y2
Ds
e
E0
Ek
EBP
EEP
EGP
EJ
F
F (k)
f (pp0 σσ 0 )F (ω)
F`s , F`a
gD
Hc1
Hc2
ĤD
Hext
k
kB

Meaningspin up statespin down state
state with (k ↑,−k ↓) states unoccupied
state with (k ↑,−k ↓) states occupied
creation/annihilation operators
creation/annihilation operators for quasiparticles (not for bare particles)
spectral function
coeﬃcient of the pair wave function
specific heatd-vector
density of states of both spins at the Fermi
surfacemost popular symmetry of cupurate orderparameterspin diﬀusion constantelectron charge
ground state energy of the interacting system
BCS excitation energy
energy of the broken-pair stateenergy of the excited-pair stateenergy of the ground-pair state
Josephson coupling energy
free energy
relative wave function of the Cooper pair
Landau interaction function
phonon density of states
Landau parameters
nuclear dipole coupling constantlower critical field
upper critical field
nuclear dipole energy
external magnetic field
wave vector
Boltzmann’s constant

6

Page where defined/introduced
11
11
12
12
10
33
108
36
28
38
21
130
28
26
29
16
18
18
18
131
39
23
29
63
30
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81
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Symbol
kF
Ks
`ˆ
`el
`ph
L
L(ω)
m
m∗
n
N (0)
NCooper
Ns
p
pF
qTF
Q
R
R(ω)
S
T∗
T1
Tc
TCurie
TN
uk
vF
vk
vs
αI
α(ω)
β
γ
Γl

MeaningFermi wave number
Knight shift
direction of relative orbital angular momentum of pairs in the A phasemean free paths of electrons
mean free paths of phonons
orbital angular momentum
loss function
mass of atoms/electronseﬀective mass of atoms/electrons
particle density
density of states of one spin at the Fermi
surface (≡ 1/2(dn/dε))
number of the Cooper pairsdensity of states
doping (of cuprates)
Fermi momentum
Thomas-Fermi wave number
pseudo-Bragg vector
center-of-mass coordinate
optical reflectivity
total spincrossover line (in the phase diagram ofcuprates)
nuclear spin relaxation time
critical temperature
Curie temperatureNèel temperaturecoeﬃcient in the BCS wave function
Fermi velocity
coeﬃcient in the BCS wave function
superfluid velocity
isotope exponent
phonon coupling function
1/kB T
coeﬃcient of the linear term in the specificheatrelaxation rate of `-symmetry distortion

7

Page where defined/introduced
10
73
43
121
121
45
110
10
30
13
21
24
72
101
29
26
142
72
110
31
102
73
20
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85
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12
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SymbolΓUδnMeaning
relaxation rate of the T -reversal operator
deviation of (quasi)particle occupation
number from the normal-state value∆k
BCS gap parameterεccutoﬀ energy in the BCS modelεF
Fermi energyεkkinetic energy relative to the Fermi energy
ηviscosity
ΘD
Debye temperatureκ
thermal conductivity
κ0
static bulk modulus of the noninteractiong Fermi gas
λLondon penetration depthµchemical potential∗µ
Coulomb pseudopotentialµB
Bohr magnetonµn
nuclear magnetic momentξhealing lengthξPRpair radiusξabin-plane Ginzburg-Landau healing lengthξkkinetic energy
ξcc-axis Ginzburg-Landau healing lengthρ(T )resistivity
ρ1 , ρ 2
single-particle/two-particle density matrices
ρksingle-particle density
ρssuperfluid densityσ(ω)
AC conductivityτrelaxation timeφ(k)phase of the condensate wave function
φ(r1 σ1 ; r2 σ2 )pseudo-molecular wave function in the
BCS problem
Φ0(superconducting) flux quantum (≡ h/2e)χPauli spin susceptibility
spχspin responseΨ(n̂)
Ginzburg-Landau order parameterΨ(r1 σ1 , r2 σ2 · · · rN σN ) many-body wave function
ωD
Debye frequency

8

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74
29
16
21
10
10
28
63
28
26
79
10
63
52
50
24
14
117
10
117
80
65
18
57
107
107
13
11
131
28
31
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Symbol
ωe
ωp
ωph
ωres
ωSF

Meaningsome characteristic frequency of the orderof the plasma frequency
plasma frequency
frequency of the longitudinal sound waves(phonons)
response frequency
AF fluctuation frequency

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153
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Lec. 1 Reminders of the BCS
theoryFirst we give a brief review of the theory of conventional superconductors, namely the
BCS theory. In this section, we start with the singlet pairing case to describe the basic
physics of the superconductivity.

1.1

Basic model

Just as in the original BCS theory, we consider here the Sommerfeld model for simplicity: we consider N spin-1/2 fermions in a free space. We assume N to be suﬃciently
large and even. For such a system, the kinetic energy for a free particle isεk ≡ ξk − µ(T ),

(1.1)

k2
, and µ(T ) is the chemical potential of the system. We note here that2mµ(T ) can be regarded as a constant1 , and it is equal to the Fermi energy
where ξk =kF2µ(T ) = εF =
.
2m

(1.2)

We assume that the fermions are interacting via an attractive potential, so that theinteraction part can be represented asV̂ =1∑
Vp,p0 ,q a†p+ q ,σ a†p0 − q ,σ0 ap0 + q2 ,σ0 ap− q2 ,σ .
222 p,p0 ,q

(1.3)

σ,σ 0

We do not discuss here the origin of this interaction (we will present the discussion in
Sec. 1.3.4), but rather try to see how the system behaves under such an attractive interaction.

1
In fact, the temperature in question is very small in discussing the BCS theory, and thus the temperature dependence of the chemical potential due to the Fermi statistics is negligible. In addition, theeﬀect of the superconducting phase transition to the chemical potential is very small in the BCS theory.
Therefore, we can regard µ(T ) as a constant.

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LEC. 1. REMINDERS OF THE BCS THEORY

1.2

BCS theory at T = 0

1.2.1

BCS wave function

Under an attractive interaction, the Fermi system forms Cooper pairs and they undergo
Bose-Einstein condensation. When the Bose-Einstein condensation occurs, a macroscopic
number of bosons occupy the same state. Therefore, as a fundamental assumption, we
think that all the pairs of fermions occupy the same pair wave function φ:ΨN = Ψ(r1 σ1 , r2 σ2 ...rN σN ) = A[φ(r1 σ1 ; r2 σ2 )φ(r3 σ3 ; r4 σ4 ) · · · φ(rN −1 σN −1 ; rN σN )],(1.4)
where A is the antisymmetrizer. For now, we restrict our attention to the case where pairs
are formed in the spin-singlet, s-wave orbital angular momentum state, and the center ofmass of the pairs is at rest. Then the pair wave function φ becomes]1 [φ(r1 σ1 ; r2 σ2 ) = √ |↑i1 |↓i2 − |↓i1 |↑i2 φ(r1 − r2 ),(1.5)
2
where φ(r) = φ(−r). If we define the Fourier transform χ(k) by∑
χ(k)eikr ,(1.6)
φ(r) =kthen we find]∑1 [φ(r1 σ1 ; r2 σ2 ) = √ |↑i1 |↓i2 − |↓i1 |↑i2
χ(k)eik(r1 −r2 )2k]∑ χ(k) [√=|k ↑i1 |−k ↓i2 − |k ↓i1 |−k ↑i22k]∑ χ(k) [√|k ↑i1 |−k ↓i2 − |−k ↓i1 |k ↑i2=2k∑
χ(k)a†k↑ a†−k↓ |vaci ,
=

(1.7)

k

where we have used χ(k) = χ(−k) in the second last line. Therefore, if we define∑Ω† ≡χ(k)a†k↑ a†−k↓ ,

(1.8)

k

the N -body wave function defined in Eq. (1.4) is rewritten as
[∑
]N/2
†
†
† N/2
ΨN = (Ω ) |vaci =χ(k)ak↑ a−k↓|vaci .

(1.9)

kNote that this is automatically an eigenstate of N̂ . We also note that the normal groundstate is a special case of this form of the wave function, since we can see
(∑
)N/2
∏ † †
†
†
=
a
a
|vaci
=
a
a
|vaci
(1.10)
ΨnormNk↑ −k↓k↑ −k↓
k<kF

k<kF

from the Fermi statistics, and the final expression corresponds to the BCS wave function
with χ(k) = θ(kF − |k|).
11

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

1.2.2

LEC. 1. REMINDERS OF THE BCS THEORY

Alternative form of the BCS wave function

In the previous subsection, we have obtained the many-body wave function which automatically conserves the number of particles N . In principle, we can minimize the freeenergy with this class of wave functions and study the thermodynamic properties of thesystem, but it is a tough work. Therefore, we replace the wave function in the following
way:
∞
∑
1
† N/2
†
(Ω† )N/2 ,(1.11)(Ω )→ exp Ω ≡(N/2)!
N/2=0

and we try to minimize Ĥ − µN̂ instead of Ĥ. Hence, up to the normalization, the wave
function becomes
(
)
(
)
∑
∏
†
†
†
†
Ψ ∝ exp
χ(k)ak,↑ a−k,↓ |vaci =exp χ(k)ak,↑ a−k,↓ |vaci.
(1.12)
kk

Since (a†k,↑ a†−k,↓ )2 = 0 due to the Fermi statistics, it reads∏(1 + χ(k)a†k,↑ a†−k,↓ )|vaci.
Ψ∝(1.13)k

To make clear the physical meanings of the following calculations, we go over to therepresentation in terms of occupation spaces of k ↑, −k ↓; let |00ik be the corresponding
vacuum, and define|10ik = a†k,↑ |00ik , |01ik = a†−k,↓ |00ik , and |11ik = a†k,↑ a†−k,↓ |00ik .
Then the wave function Ψ can be represented as
∏
Ψ=
Φk ,

(1.14)(1.15)kwhereΦk ∝ |00ik + χk |11ik .

(1.16)√
To satisfy the normalization condition, multiply by the factor 1/ 1 + |χk |2 , and then we
obtainΦk = uk |00ik + vk |11ik ,(1.17)
1χkwith uk = √and vk = √. Thus, we have obtained the general form
1 + |χk |2
1 + |χk |2of the BCS wave function as
)
∏
∏(
†
†
ΨBCS =(uk |00ik + vk |11ik ) =uk + vk ak,↑ a−k,↓ |vaci,(1.18)kkwhich does not conserve the number of particles. The normal ground state corresponds
to a special case of this wave function, which can be obtained by setting uk = 0, vk = 1
for k < kF and uk = 1, vk = 0 for k > kF .
12

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORYWe should make some remarks on the BCS wave function and the above derivation.
At first we should notice that this is the very general result for the spin-singlet paring
systems in the sense that the coeﬃcients uk and vk can depend on the direction of the
momentum k. Since the phase transformation (uk , vk ) → eiφk (uk , vk ) has no physicaleﬀect, we can choose all uk to be real.
As a consequence of the number conservation, we can find that the transformation
vk → eiφ vk , where φ is independent of k, has no physical eﬀects either. To see this, let
us define)∏(ΨBCS (φ) =uk + eiφ vk a†k,↑ a†−k,↓ |vaci.
(1.19)k

From this, we can easily check that∂
ΨBCS (φ) = iN̂ ΨBCS (φ). When we define∂φhÂiφ = Ψ†BCS (φ)ÂΨBCS (φ),(1.20)

where Â is a physical (hence number-conserving) operator, we can see that this expectation
value does not depend on the phase φ:dhÂiφ = iΨ†BCS (φ)[Â, N̂ ]ΨBCS (φ) = 0.
dφ
We can, therefore, construct the number-conserving many body wave function:
∫ 2π
N
1
ΨN =
dφΨBCS (φ)e−iφ 2 .
2π 0

1.2.3

(1.21)(1.22)

Pair wave function

Let us discuss the relative wave function of a Cooper pair. In the BCS theory , the pairwave function at T = 0 is expressed as
Fk = uk vk ,(1.23)∑or as its Fourier transformation F (r) = k Fk eikr .
The physical meaning of the pair wave function becomes clearer if we evaluate theexpectation value of the potential energy hV̂ i:1∑
Vpp0 q ha†p+q/2,σ a†p0 −q/2,σ0 ap0 +q/2,σ0 ap−q/2,σ i.
hV̂ i =
(1.24)
2 pp0 qσσ 0

For the BCS wave function, only three types of terms contribute to the expectation
value: the Hartree term (q = 0), the Fock term (σ = σ 0 , p = p0 ), and the pairing term())(
p + q2 = − p0 − q2 , σ = −σ 0 . The Hartree term can be evaluated as
hV̂ iHartree =1∑
Vpp0 0 hnpσ np0 σ0 i.
2 pp0
σσ 0

13

(1.25)

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

Especially, for the case of the local potential V = V (r), the Hartree term hV̂ iHartree
becomes a constant 21 V (q = 0)hN̂ 2 i.
The Fock term, corresponding to σ = σ 0 , p = p0 , is given by1∑
1∑
hV̂ iFock = −Vppq hnp+q/2,σ np−q/2,σ i = −|vp+q/2 |2 |vp−q/2 |2 .
(1.26)
2 pqσ
2 pq
The last equality is a consequence of the uncorrelated nature of the BCS wave function,and it can be easily checked by a direct calculation.
Finally we evaluate the pairing term. For convenience, we introduce the following
variables: k = p + q/2 and k0 = p − q/2. Then, we have1∑
(1.27)hV̂ ipair =Vkk0 ha†k0 ,σ a†−k0 ,−σ0 a−k,−σ ak,σ i,
2 k,k0where Vkk0 = Vk+q/2,k0 −q/2,k−k0 , which is V (k − k0 ) for a local potential V (r). Again using
the factorizable nature of the BCS wave function except for the O(1/N ) contributions,this reduces to1∑
hV̂ ipair =Vkk0 ha†k0 ,σ a†−k0 ,−σ0 iha−k,−σ ak,σ i
2 kk0 σ(1.28)
1∑††=
Vkk0 hak0 ↑ a−k0 ↓ iha−k↓ ak↑ i.
2 kk0 σAt last, we have used the spin-singlet nature of the BCS wave function. We can find by
an explicit calculation that
ha−k,↓ ak,↑ i = u∗k vk h00|a−k,↓ ak↑ |11i = uk vk = Fk .

(1.29)

Similarly, we can obtain that ha†k,↑ a†−k,↓ i = uk vk∗ = Fk∗ . Hence, the pairing interaction is∑hV̂ ipair =Vkk0 Fk Fk∗0 .
(1.30)
kk0

In the case of a local potential V (r), we can rewrite this in terms of the Fourier componentof F (r):∫hV̂ ipair =d3 rV (r)|F (r)|2 .

(1.31)

The comparison of this result with the interaction between two particles in free space∫hV̂ i = d3 rV (r)|Ψ(r)|2 tells us that F (r) essentially works as the relative wave function
Ψ(r) of the pair in the superfluid Fermi system. It is a much simpler quantity to deal
with than the quantity φ(r), which appears in the N -conserving formalism.
We do not yet know the specific form of u’s and v’s in the ground state, and we cannot
calculate the form of F (r) now. We, however, anticipate that it will be bound in relative
space and that we will be able to define a “pair radius” by the quantity∫ 3
d r|F (r)|2 |r|2
2.
(1.32)
ξPR = ∫ 3
d r|F (r)|2
It cannot be too strongly emphasized that everything above is very general and true
whether or not the state we are considering is actually the ground state.
14《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

1.2.4

LEC. 1. REMINDERS OF THE BCS THEORY

Quantitative development of the BCS theory

We consider a fully condensed BCS state described by the N -nonconserving wave function:∏Ψ=Φk , Φk ≡ uk |00ik + vk |11ik .
(1.33)k

From the normalization condition, uk and vk should satisfy the following relation:|uk |2 + |vk |2 = 1.

(1.34)The values of uk , vk are determined by minimizing the free energy:hĤi = hT̂ − µN̂ + V̂ i.

(1.35)Let us neglect2 the Fock term in hV̂ i unless mentioned otherwise (we have already seen
below Eq. (1.25) that the Hartree term contributes only a constant for the local potentialcase). Then, the contribution of hV̂ i comes only from the pairing terms∑
hV̂ i =
Vkk0 Fk Fk∗0 , Fk ≡ uk vk .
(1.36)k,k0

Here, Vkk0 is a matrix element for the process where fermions change the state from(k ↓, −k ↑) to (k0 ↑, k0 ↓). Let us consider the term∑∑n̂kσ εk .
(1.37)n̂kσ (ξk − µ) ≡T̂ − µN̂ =
k,σ

k,σ

It is clear that |00ik and |11ik are eigenstates of n̂kσ with their eigenvalues 0 and 2,
respectively. Taking the sum of the spins, we find∑εk |vk |2 ,(1.38)
hT̂ − µN̂ i = 2kand therefore we obtainhĤi = 2∑

εk |vk |2 +k∑

Vkk0 (uk vk )(uk0 vk∗ 0 ).

(1.39)

k,k0This hĤi must be minimized under the constraint |uk |2 + |vk |2 = 1.
We introduce a pretty way of visualizing the problem. Let us put
uk (= real) = cosθkθk, vk = sin expiφk ,22

and rewrite the Hamiltonian as∑∑1∑
hĤi =(−εk cosθk ) +εk .
Vkk0 sinθk sinθk0 cos(φk − φk0 ) +4 k,k0kk
2

In fact, we can shot that the Fock term have little eﬀects. We will consider this eﬀect later.

15(1.40)(1.41)

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

The last term is a mere constant, so that we can neglect it. Next, we introduce theAnderson pseudospin representation of the BCS Hamiltonian. We introduce a unit vector
σk with its polar angle given by (θk , φk ):sinθk cosφk = σxk ,sinθk sinφk = σyk ,(1.42)

cosθk = σzk .
With this representation, the expectation value is rewritten as
hĤi = −∑kεk σzk +∑1∑
Vkk0 σk⊥ · σk0 ⊥ = −σk · Hk ,4 k,k0k(1.43)

where σk⊥ is the xy-component of σk , and the pseudo-magnetic field Hk is defined as
Hk ≡ −εk ẑ − ∆k ,∆k ≡ −

1∑
Vkk0 σk0 ⊥ .
2 k0(1.44)(1.45)

Thus, the z-component of Hk gives the kinetic energy, while the xy-component is thepotential energy (see Fig. 1.1).
It is actually very convenient to represent ∆k and σk⊥ as complex numbers ∆k ≡∆kx + i∆ky , σk⊥ ≡ σkx + iσky rather than representing them as vectors. Evidently, themagnitude of the field Hk is|Hk | = (ε2k + |∆k |2 )1/2 ≡ Ek ,(1.46)In the ground state the spin σk lies along the field Hk , giving an energy −Ek . If the spin
is reversed, this costs 2Ek (not Ek !). This reversal corresponds toθk → π − θk , φk → φk + π,

(1.47)

Fig. 1.1. Schematic illustration of the vectors Hk and σk . At equilibrium, Hk and σkshould point the same direction.

16

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORYandθkexp(−iφk ) = vk∗ ,2θkvk → −cos = −uk .
2uk → sin(1.48)

Therefore, the wave function of the excited state generated in this way is∗ΨEP
k = vk |00i − uk |11i ,(1.49)

We can easily verify that this excited state is orthogonal to the ground state Φk ≡uk |00i + vk |11i (remember we take uk to be real).
Let us derive the BCS gap equation. Since the vector σk must point along the field
Hk in the ground state, this gives a set of self-consistent conditions for ∆k ; since σk0 ⊥ =−∆k0 /Ek0 , we have∑∆k0.
(1.50)∆k = −
Vkk0
2Ek0
k0
This is the BCS gap equation. Note that the above derivation is quite general. In
particular, we have never assumed the s-wave state (though we did assume the spinsinglet pairing).
Let us also introduce an alternative derivation of the BCS gap equation. We simply
parametrize uk and vk by ∆k and Ek as
uk ≡ √

E k + εk,(1.51)∆kvk ≡ √.
|∆k |2 + (Ek + εk )2(1.52)|∆k |2 + (Ek + εk )2This clearly satisfies the normalization condition |uk |2 + |vk |2 = 1, and gives
[
]
[
]
1
εk
1
εk
∆k
2
2
|uk | =
1+
, |vk | =1−
, u k vk =
.
2
Ek
2
Ek
2Ek
The BCS ground state energy can therefore be written in the form) ∑∑ (∆k ∆∗k0εk+Vkk0.
hĤi =
εk 1 −
Ek
2Ek 2Ek0k
kk0(1.53)(1.54)

Here, ∆k for each k are independent variational parameters. By using ∂Ek /∂∆k =∆∗k /Ek , we find
[
]
∗
∑
∆
ε2k
0
∆∗k −
Vkk0 k = 0,(1.55)
Ek3
2E
k0
0
k
so that we again obtain the standard gap equation.

17

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

(a)

(b)

Fig. 1.2. (a) The number distribution and (b) the pair wave function for the BCS groundstate.

For the s-wave state, ∆k is independent of the direction of k and depends only on its
magnitude |k|. Let us expect that, as in most cases of interests, ∆k is approximately aconstant ∆ over a wide range of energy ε  ∆. Then, we obtain
)
(
1
ε
k
hnk i = |vk |2 =,(1.56)1− √ 22
εk + |∆|2andFk = uk vk =

∆
.
2Ek

(1.57)The behavior of hnk i and Fk are illustrated in Fig. 1.2: hnk i behaves qualitatively
similarly to the normal-state at T = Tc , but falls oﬀ very slowly ≈ ε−2 , rather than
exponentially. On the other hand, Fk falls oﬀ as |ε|−1 for large ε.

1.3

BCS theory at finite temperature

1.3.1

Derivation of the gap equation

To generalize the BCS theory to a finite temperature, we have to use the density
matrix formalism. An obvious way is to assume that the many-body density matrix canbe written in a product form just like the ground state wave function:
∏
ρ̂ =
ρk .
(1.58)k

Here, each Hilbert space labeled by k is spanned by the following four states:ΨGP = uk |00i + vk |11i : “Ground pair states” ;(1.59)

ΨEP = vk∗ |00i − uk |11i : “Excited pair states” ;(1.60)(1)(2)

ΨBP = |10i ΨBP = |01i : “Broken pair states” .
18(1.61)

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORYAs regards the first two states, they can be parametrized by the Anderson variables θk(k)and φk . The diﬀerence from T = 0 is that there is a finite probability PEP for a given“spin” σk to be reversed, i.e., the pair is in the ΨEP state rather than the ΨGP state.
(k,1)
(k,2)
There is also finite probability, PBP and PBP , that the pair is a broken-pair state. As
to the broken-pair states, they clearly do not contribute to hV̂ i and thus do not to theeﬀective field. Thus, we can go through the argument as above and obtain the result1∑(1.62)∆k = −
Vk,k0 hσ⊥k0 i,
2 k0
where hσ⊥k0 i is now given as
( 0
)
(k )
(k0 ) ∆k0.
hσ⊥k0 i = PGP − PEP
Ek0(1.63)

Therefore, the gap equation becomes∆k = −( 0)(k )(k0 ) ∆k0
Vk,k0 PGP − PEP
.
2E
k0
0
k

∑

(k)

(1.64)(k)

We therefore need to calculate the quantities PGP and PEP 3 . This is simply given by
the canonical distribution4
(k)

(k)

(k)

PGP : PBP : PEP = exp(−βEGP ) : exp(−βEBP ) : exp(−βEEP ).

(1.65)

As we already noted in the discussion below Eq. (1.46), the energy diﬀerence between the
ground pair and excited pair states is EEP − EGP = 2Ek 5 . What is EBP − EGP ? Here, aspecial care should be paid. If all energies are taken relative to the normal-state Fermisea, then evidently the energy of the “broken pair” states |01i and |10i is εk (which canbe negative!). In writing down the Anderson pseudospin Hamiltonian, we omitted the∑constant term k εk . Hence, the energy of the GP state relative to the normal Fermi sea
is not −Ek , but εk − Ek . Thus, we have
EBP − EGP = Ek ,(1.66)EEP − EGP = 2Ek .

(1.67)The broken-pair states can be regarded as states with one quasi-particle, the excited pairstate as one with two quasi-particles.
From Eqs. (1.66) and (1.67), we obtain( βE )
1 + exp(−βEk )
k
(k)
(k)
= tanhPEP − PGP =,(1.68)
1 + 2 exp(−βEk ) + exp(−2βEk )2(k)Since the states |10i and |01i are degenerate, we can calculate PBP immediately from these quantities
(k)
(k)
(k)
and from PGP + PEP + 2PBP = 1.
4
Since we are talking about diﬀerent occupation states, Fermi/Bose statistics are irrelevant, and theprobability of a given state with its energy E is simply proportional to exp(−βE).
5Note that Ek here is temperature dependent!
3

19

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

and can derive the gap equation:∆k = −

∑

Vk,k0

k0

( βE 0 )
∆k0
k
tanh
.
2Ek0
2

(1.69)

Note that this gap equation can also be derived in a brute force manner: by minimizing
the the free energy F (∆k ) with respect to ∆k (see appendix 5D of Ref. [1]).

1.3.2

Fk at finite temperature

As for hσ⊥k i, if we recall the definition hσ⊥k i ≡ ha−k↓ ak↑ i, we can easily see that thebroken pair states do not contribute to this quantity. Recalling that the EP and GP
states have the opposite direction in the Anderson pseudospin representation, and thatthe magnitude of its xy-component is ∆k /2Ek , we obtain()( βE )∆kk(k)(k) ∆khσ⊥k i = Fk = PGP − PEP=
tanh.
(1.70)
2Ek
2Ek
2
(
)
Thus, the pair wave function is also reduced by a factor of tanh βE2 k from the groundstate6 .

1.3.3

hnk i at finite temperature

By using Eqs. (1.59) to (1.61) and recalling that nk − 12 is zero for the BP states, we
obtain( βE )
1
εk
k
(k)
(k)
.
(1.71)
hnk i − = (|vk |2 − |uk |2 )PGP + (|uk |2 − |vk |2 )PEP = −
tanh
2
Ek
2
By introducing the occupation number of the ideal Fermi gas hnk i0 = θ(kF − |k|), we findhnk i − hnk i0 =

[1
2

−

( βE )]|εk |ksgn(εk ).
tanh
Ek
2

(1.72)

From this, we can see that the occupation number will reduce to that for the normal
Fermi gas as T → Tc and ∆ → 0.

1.3.4

Properties of the BCS gap equation

We can immediately see that the BCS gap equation always has a trivial solution ∆k =0 regardless of the form of the potential Vkk0 , which corresponds to the normal state.
Thus, we concentrate only on nontrivial solutions, which, as we shall see soon, depend
significantly on the form of the potential Vkk0 and the temperature T .
6Note that the value itself is far more reduced than this factor, since the value of gap ∆k decreases
from its ground state value as the temperature is raised.

20

《講義ノート》

物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

We can find two rather simple cases where no nontrivial solution exists. One is when all
Legendre components V` of Vkk0 are non-negative; see Sec. 2.4. The other case is the hightemperature limit T → ∞. In this limit, the right-hand side of the BCS gap equation(1.50) reduces to∑∆k0 βEk0
1 ∑Vkk0=
Vkk0 ∆k0 .
(1.73)
0
2E
2
4k
T
k
B
k0
k0
For the existence of the nontrivial solution, the potential −Vkk0 should have the eigenvalue
4kB T , which is impossible in the high-temperature limit. Thus, there is no nontrivial
solution in the limit T → ∞. We can also conclude that if there is a nontrivial solution
at T = 0, there must exist a critical temperature Tc at which this solution vanishes.
So far, we have considered the general features of the BCS theory, but in order toobtain further insights, we confine ourselves to the case of the original BCS form, where
we approximate the potential as Vkk0 ' Vo with an energy cutoﬀ εc around the Fermi
1 dn.
surface. Let us introduce the density of states at the Fermi surface by N (0) =2 dε ε=εF∫∑
With the replacement k → N (0) dε, we have
∫ εc
∫ εc
tanh βE/2
tanh βE/2
−1
λ =
dε =
dε
(1.74)
2EE−εc0with λ = −N (0)Vo . It is obvious that nontrivial solutions do not exist for Vo > 0, as
already remarked. We therefore consider the case Vo < 0 below.
At first, we calculate the critical temperature Tc . Put β = βc , then ∆ goes to zero andE → |ε|:∫ εctanh(βc ε/2)
−1
λ =
dε
= ln(1.14βc ε).
(1.75)
ε0Thus the critical temperature Tc is given by(kB Tc = 1.14εc exp(−λ ) = 1.14εc exp −−1

1
N (0)|Vo |).

(1.76)

This expression does not depend on the choice of the cutoﬀ εc because the renormalized
potential |Vo | ∼ const. + ln εc 7 cancels its dependence. Therefore, it is plausible to set thevalue of the energy cutoﬀ to be εc ∼ ωD as in the original BCS paper. By recalling thatthe Debye frequency depends on the mass of the ions M as ωD ∼ M −1/2 , this predicts
Tc ∼ M −1/2 , which explains the isotope eﬀect. It also ensures the self-consistency of theabove calculations: we focus on the energy region close to the Fermi surface, which canbe seen to be true since it is known experimentally that the transition temperature scales
as Tc  ωc .

7

In the renormalization group analysis, the four-Fermi coupling turns out to obey the flow V (ε) =V
1+N (0)V ln(εc /ε) , where ε is the characteristic energy scale of the system, and V is the bare coupling
constant.

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORYAt zero temperature, the gap equation reads
∫ εc
dε
2εc
−1
√
λ =
' ln
.
∆(0)
ε2 + |∆(0)|20(1.77)

Then, we can find the ratio between the energy gap and the transition temperature as
follows∆(0)= 1.76.
(1.78)
Tc
Note that this ratio is a universal constant independent of the detail of the materials. Since∆(0) can be measured by tunneling experiments, we can confirm this relation experimentally. This relation usually works quite well for weak coupling superconductors, while theratio becomes usually somewhat larger than 1.76 for “strong-coupling” superconductors,
where Tc /ωc is not very small.
At finite temperature T < Tc , the gap equation can be written as∫ ∞ [tanh(βE(T )) tanh(βc ε) ]= 0.
(1.79)
dε
−
E(T )
ε
0
Since this integral converges, we can extend εc to εc → ∞. Then, we can easily see thatthe energy gap should be written as∆(T ) = Tc f (T /Tc ),(1.80)∆(T )= f˜(T /Tc ).
∆(0)(1.81)

or equivalently,

The temperature dependence of the energy gap is pretty close to the following form8( )4 ]
1/2∆(T ) [
T
= 1−.
(1.82)∆(0)Tc
On the other hand, near Tc , we can obtain the following result from the gap equation

or equivalently,

(
)1/2
∆(T )
T
∼ 1.74 1 −
,
∆(0)
Tc

(1.83)

(
)1/2
∆(T )
T
∼ 3.06 1 −.
Tc
Tc(1.84)

8

Before the BCS theory, this temperature dependence of the energy gap was presented theoreticallyby Casimir and Gorter with a phenomenological two-fluid model, which agrees well with experiments.

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A. J. Leggett

1.3.5

LEC. 1. REMINDERS OF THE BCS THEORY

Properties of the Fock term

In the calculation above, we have neglected the Fock termhH − µN iFock = −1 ∑Vk,k0 hnkσ ihnk0 σ i.
2 k,k0 σ(1.85)For a weak coupling s-wave superconductor, this is indeed a valid approximation. In fact,
if we look at the Fock term we can regard∑−
Vk,k0 hnk0 σ i(1.86)k0

as a molecular field acting on nk , changing the single particle energy as∑
εk → ε̃k,σ ≡ εk −Vk,k0 hnk0 σ i.

(1.87)

k0

As long as Vk,k0 can be regarded as a constant around the Fermi surface, this molecularfield is simplified as∑V
hnk0 σ i.
(1.88)
k0∑
Since the integration range of k is far larger than the energy gap,
k0 hnk0 i can be
regarded also as a constant, so that the eﬀect of the Fock term is only to shift thechemical potential.
For an anisotropic case, this molecular field term depends on the angle, so that theeﬀect of the Fock term cannot be absorbed into the chemical potential. However, theeﬀect of the Fock term can be taken into account in a similar way as in Landau’s Fermi
liquid theory (see Sec. 2.2) as far as Vk,k0 is a constant with respect to |k|.
0

1.3.6

Pair wave function

The most important quantity characterizing the superconducting phase is the “pair∫wave function” F (r) = hψ↓ (r)ψ↑ (0)i, or its spatial Fourier transform Fk = drF (r)e−ik·r =a−k,↓ ak,↑ . We already saw the physical significance of this quantity in evaluating the expectation value of the interaction term in Eq. (1.30): F (r) behaves as the two-particle
wave function. As we will see later, F (r) still behaves as the pair wave function of the
Cooper pairs even when we go beyond the BCS theory, and it is essential quantity in the
superconducting phase.
At finite temperature temperature, the expression of Fk is modified into
)
(
)
(
∆k
βEk
βEk
=
tanh
,
(1.89)
Fk = uk vk tanh
2
2Ek
2

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

so that its spacial dependence is given as
(
)
∑ ∆k
βEk
tanh
F (r) =
exp(ik · r).
2E
2
k
k

(1.90)In the case of the s-wave pairing, ∆k and Ek are independent of the direction k̂. Therefore,
we can perform the integration over the angle:∫∫∫∑
sin krdΩkexp(ik · r) = N (0) dεkexp(ik · r) = N (0) dεk.
(1.91)
4πkrkTherefore, we find Fk for the s-wave pairing as
(
)
∫
βEk
sin kr ∆k
F (r) = N (0) dεktanh.
kr 2Ek2(1.92)

To go further, therefore, let us assume as always the weak coupling limit. Then, we obtain
Tc  εF and we find kF ξ  1, where ξ = ~vF /∆(0) is the healing length. This healing
length is of the order of the “pair radius” defined in Eq. (1.32).
One important remark on F (r) is that it is not normalized to unity, but rather one canregard the integral of its squared as the number of Cooper pairs9 :
)
(
∫
∑ ∆2
βEk
2
k
3
2
NCooper ≡ d r|F (r)| =tanh.
(1.93)
24E
2kk
It is clear that the main contribution to this integral comes from a small energy region|ε| ∼ ∆(T ) ∼ kB T . In this region, we can approximate ∆k (T ) by its value at the Fermi
surface, simply denoted by ∆(T ). In this approximation, the total number of Cooper
pairs is given by
)
(
∫
dεk
βEk
2
2
NCooper = |∆(T )| N (0)tanh.
(1.94)
4Ek2
2
In the limit T → 0, this must be on the order of N ∆(0)/εF , where N is the total numberof the fermions. One can obtain an important insight from this equation: for the oldfashioned BCS superconductors, the number of Cooper pairs is much less than that of
the fermions. We can see this point easily by using ∆(0)/εF ∼ 10−4 . As the temperature
is increased, the number of Cooper pairs decreases, and in the limit T → Tc , we findN |∆(T )|2 /Tc εF .
Let us discuss general behaviors of the pair wave function F (r). What we can expect
is that
1. At short distance r  kF−1 , some of the above approximations break down, andequations given above are not valid. Since the Coulomb repulsion between thetwo electrons becomes dominant when the two electrons come close, the pair wave
function at short distance behaves as F (r) ∝ ϕ(r), where ϕ(r) is the relative wave
function of the two colliding electrons in the free space10 with E ∼ εF .
9Again, this physical meaning is a much more general property that goes beyond the BCS theory,
although the above equation does no longer hold (we will come back this point later).
10
The modification to the Coulomb interaction could be important in some applications.

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A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

2. In the intermediate region kF−1  r  ~vF /∆(0), we can find F (r) ∝ ϕfree (r), where
ϕfree (r) is the wave function of two freely moving particles with zero center of mass
momentum at the Fermi energy.
3. At large distance r >> ~vF /∆(0), F (r) falls oﬀ exponentially F (r) ∝ exp (−r/ξ)with ξ ∼ ~vF /∆(0). Here, the spatial extent of the pair wave function ξ can beshown to be only weakly T -dependent [2].
The bottom lines are that
1. The radius of the Cooper pairs is always of the order of ~vF /∆(0), and is very hugecompared to the microscopic length scale. Even when we vary the temperature, thesize of the pairs does not change significantly, and this point remains to be true.
2. Even at T = 0, the number of Cooper pairs NCooper is far smaller than that of thefermions N . As the temperature increase, NCooper decreases and finally it vanishes
at T = Tc .

1.4

Generalization of the BCS theory

Here let us consider the generalization of the previous discussion.
1. From the beginning, we have assumed the Sommerfeld model; thus we have ignored
the existence of the crystalline and the Coulomb interaction between the electrons.
The periodic potential can be taken into account by replacing the free wave functionsin the previous discussions with the Bloch waves
ψk,n (r) = uk,n (r)eikr .

(1.95)

2. Next, let us take the Coulomb interaction into consideration. Here, we apply Landau’s Fermi liquid theory, and assume that the states of the interacting system canbe labeled with those of the non-interacting system under the adiabatic switchingof the interaction. Then, the net “polarization” of the states is given by∑
hnk,σ i.
(1.96)|k|

As long as the net polarization remains unchanged across the normal-superfluid
phase transition, the molecular field terms do not play any role. Therefore, theonly eﬀect of the interaction is to replace the bare mass with the eﬀective mass
m → m∗ , leaving the gap equation intact. They do aﬀect, however, the responses
to the external fields, just as in the normal state.

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A. J. Leggett

LEC. 1. REMINDERS OF THE BCS THEORY

3. The Coulomb interaction
VCoulomb (q) =e2εo q 2(1.97)

is long ranged, so that it is diﬃcult to treat it straightforwardly. However, if we takethe screening eﬀect into account, it becomes short ranged, and we can show that it
does not have significant eﬀect. In fact, if we use the random-phase approximation(RPA), the eﬀective potential is modified due to the screening as
Veﬀ (q) =κo,2
1 + q 2 /qTF

(1.98)

where κo is the static bulk modulus of the non-interacting Fermi gas and qTF is theThomas-Fermi wave number. Since this is valid only in the static limit, ω should
be much smaller that vF q, where vF is the Fermi velocity. As long as we restrict
ourselves to the classical superconductors, this condition is usually satisfied and theabove expression can be used safely. If we assume that only the interaction for small
q’s is important, the long-range part of the Coulomb interaction merely shifts thestrength of the potential, and it has no eﬀect on the gap equation. However, it does
aﬀect the responses and the value of Tc .
4. Finally, let us consider the strong coupling case. Generally speaking, this kind ofinteraction requires much more complicated treatments as Eliashberg has pointedout (see Sec. 4.1.1). However, it provides only fairly small corrections to the naive
BCS theory. In fact, the ratio ∆(0)/kB Tc can be 2.4 in Hg, and Pb at the largest,
while it is about 1.76 in the BCS theory.

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

References[1] A. J. Leggett, Quantum Liquids (Oxford University Press, New York, NY, USA, 2006).
[2] Question from a student: We have defined the pair radius by the eﬀective radius of thepair wave function F (r). Also we know that there is another length scale called theGinzburg–Landau healing length, which diverges as T → Tc . What is the diﬀerence
between these two?
Answer: Thank you. A good question. What we talk about here is eﬀectively theradius of the Cooper pair. So you may think it is the radius of the eﬀective moleculeof the Cooper pair, described in their relative coordinate. On the other hand, theGinzburg–Landau healing length is, crudely speaking, the length which characterizes
the behavior of the pair wave function in the bulk. If I consider the pair wave function
around the bulk boundary, the pair wave function goes to zero at the wall. When
we discuss the Ginzburg–Landau healing length, we are talking about the center ofmass coordinate. Suppose that the pair wave functions go to zero at the wall, thenit must have an exponential behavior. How long does it take to? The answer is theGinzburg–Landau healing length.
Another possible interpretation of the Ginzburg–Landau healing length is that itis the length over which the order parameter has to distort such that the bending
energy is equal to the bulk condensation energy. According to this criterion, it is not
surprising, although not obvious, that the Ginzburg–Landau healing length tends toinfinity in the limit T → Tc .

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

Lec. 2 Superfluid 3He: basic
description
In this and next sections, we briefly review the 3 He system. First, we deal with a normal
phase of 3 He by the famous Landau Fermi liquid theory. Next, we describe the theoryof superfluid 3 He, where, unlike the simple BCS theory presented in the previous section,
the anisotropy becomes important. Finally, the Ginzburg–Landau theory is formulated
for both the singlet and triplet superfluids.

2.1

Introduction

The liquid 3 He has become available since the 1950s. Since it does not exist in nature,most of 3 He people actually use is produced from tritium through the reaction1 (3 H→
3He + e + ν̄). 3 He is an inert atom having the stable electronic state (1s)2 S0 with a huge
excitation energy. Therefore, we can regard it as a point particle with a (nuclear) spin
1/2, obeying the Fermi statistics just as an electron in metals.
The interaction potential between 3 He atoms is showed in Fig. 2.1. At short distance,
it has a “hard-core” repulsive region, originating from the Pauli principle between theelectrons. At large distance, on the other hand, the van der Waals interaction makes thepotential attractive.
For T . 100 mK, the liquid 3 He behaves much like a textbook normal metal. For
example, the specific heat CV , Pauli spin susceptibility χ, viscosity η, spin diﬀusion
constant DS , and thermal conductivity κ behave as
CV ∝ T,χ = const.,

η, DS ∝ T −2 ,κ ∝ T −1 .

(2.1)

It turns out, however, that the inter-atomic interaction is rather strong. For example,
the spin susceptibility χ is 20 times larger than that for the ideal Fermi gas. How can
we justify the above seemingly non-interacting behavior in the presence of such a strong
interaction?

1

Recently, there is a shortage of 3 He in order to use it in neutron detectors, and its price is growing.

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett“hard-core” repulsionr

van der Waals attraction

Fig. 2.1. The interaction potential between 3 He atoms.

2.2

Landau Fermi liquid theory

A very nice explanation of the normal liquid 3 He for T . 100 mK was given by theLandau Fermi Liquid theory. This theory is based on the following qualitative assumption
about the behavior of the system: we turn on the interaction adiabatically to the freeFermi gas, and assume that the ground state and all low-energy excited states of the noninteracting system evolve continuously into those of the interacting system. Obviously,
we exclude the possibility of any phase transitions in the above adiabatic process, suchas the normal liquid-superconductivity phase transition, the disorder-ferromagnetic phasetransition and the liquid-crystal phase transition.
The low-energy excited states are labeled by specifying the diﬀerence in the occupation
number δn(pσ) of the state with the momentum p and the spin σ measured from theground state. As long as the above assumption holds, we are able to do this even if theinteraction is pretty strong. The diﬀerence δn(pσ) can only take the following values (pF :the Fermi momentum):{δn(pσ) = 0 or − 1 (|p| < pF ),δn(pσ) = 0 or 1

(2.2)(|p| > pF ).

The energy E of the whole system can be expanded as
E = E0 +∑pσ

ε(pσ)δn(pσ) +1 ∑f (pp0 σσ 0 )δn(pσ)δn(p0 σ 0 ),2 pp0 σσ0(2.3)

where E0 is the ground state energy of the interacting system. We defined ε(pσ) andf (pp0 σσ 0 ) as the coeﬃcients in this expansion, and f is called the Landau interaction
function.
Now, we make use of the symmetry of the system to restrict the general form of thecoeﬃcients. First of all, ε(pσ) must be spin-independent and isotropic; i.e., ε(pσ) = ε(|p|).
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LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett

Since we are interested in the low-energy excitation, we expand it asε(pσ) = ε(|p|) ' ε(pF ) + vF (|p| − pF ).
The eﬀective mass m∗ and the Fermi velocity vF are defined as
( )
pF
dε
∗
m ≡
, vF ≡.
vF
dp p=pF

(2.4)

(2.5)

From the symmetry argument, we can also see that f (pp0 σσ 0 ) is a function of |p|, |p0 |,p · p0 and σ · σ 0 . Hence, it can be expanded in terms of the Legendre polynomials P` as∑f (pp0 σσ 0 ) '(2.6)(f`s + f`a σ · σ 0 ) P` (p̂ · p̂0 ) .
`Since the coeﬃcients f`
have the dimension of (energy) × (volume)−1 , it is convenient
to define dimensionless quantities(s,a)

F`s ≡ Ω

dn s
f ,
dε `

F`a ≡ Ω

dn a
f ,
dε `

(2.7)

where Ω is the total volume of the system. For the liquid 3 He, the values of these parameters arem∗ /m ∼ 3-6,
 Fs
∼ 10-100,
0
(2.8)
s

F`
∼ 1 (` 6= 0),


 a
F`
∼ 1.
The Landau Fermi liquid theory may be very informally summarized as follows:• Instead of real particles with the bare mass m, we deal with “quasi-particles” with
their eﬀective mass m? .
• The system is subject to the molecular fields which are proportional to F`s andF`a and generated by the polarizations of the system (see below).
Molecular fields
Now, we review the molecular field theory in order to examine the spin response. Using
Eq. (2.7), we rewrite Eq. (2.6) as
( )−1
∑
dn
0
0
f (pp σσ ) =
(2.9)
Ω−1
(Fls + Fla σ · σ 0 ) P` (p̂ · p̂0 ).
dεl
If F0a is much larger than the other terms, we keep only this term in Eq. (2.3):( )−1∑
1 −1 dnE= ΩF0aσ · σ 0 δn(pσ)δn(p0 σ 0 ).
2dε
pp0 σσ 030(2.10)

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett

Since the total spin S =∑pσ σδn(pσ) is conserved, Eq. (2.10) is reduced to1
E = Ω−1
2

(

dn
dε

)−1

F0a S · S.

(2.11)

This expression is the same as the energy of the free Fermi gas with total spin S in themolecular field( )−1dn(2.12)
Hmol = −F0a S.
dε
Since we know the spin response of the ideal Fermi gas to an external field Hext (kω),
we obtain



S(kω)
= χsp
0 (kω)Htot (kω),

Htot (kω) = Hext (kω) + Hmol (kω),(2.13)
( )−1


dn


F0a S(kω),
Hmol (kω) = −dε
where χsp
0 (kω) is the spin response function of the noninteracting Fermi gas with theeﬀective mass m∗ . These relations are generalizations of the very familiar mean field
theory of ferromagnetism.
We can easily derive the true spin response function
χtrue (qω) =χsp
0 (qω).
1 + (dn/dε)−1 F0a χsp(qω)
0(2.14)dn
By substituting χsp
into Eq. (2.14), we immediately obtain the static spin suscep0 =dεtibilitydn/dεχ=.
(2.15)
1 + F0aThis formula is exactly the same as the one in the Landau Fermi liquid theory. Here, wehave derived it based on the molecular field theory and our knowledge on the ideal Fermi
gas.

2.3

Eﬀects of (spin) molecular field in 3He

2.3.1

Enhanced low-energy spin fluctuations

The left figure of Fig. 2.2 shows the frequency-dependence of the χsp
0 of the free Fermi
spgas, while the right figure is χtrue corresponding to the Fermi liquid. Their relation is
described in Eq. (2.14) with dimensionless parameter F0a ∼ −0.7. We notice that there isa peak for the Fermi liquid. Although this peak does not resemble the delta-function andthus does not represent a real propagating excitation, we can think of this peak as a sortof an elementary excitation, so-called “paramagnon”. The strong peak at low frequency
suggests that the elementary excitation is long-lived. Thus, the excitation is also referred
to as the “persistent spin fluctuation”.
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A. J. Leggett

LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION“paramagnon”

Fig. 2.2. The imaginary part of the spin susceptibility χsp
0 .

2.3.2

Coupling of atomic spins through the exchange of virtualparamagnons

In metals, the eﬀective electron-electron interaction arises from the exchange of virtual
phonons. This is illustrated schematically in the left figure of Fig. 2.3. An electron
attracts positive ions on the way and other electrons feel these positive charges. Hence,the eﬀective electron-electron interaction is attractive.
The eﬀective interaction between 3 He atoms due to spin fluctuations is illustrated
schematically in the right figure of Fig. 2.3. In this case, virtual paramagnons mediate the attractive interaction between 3 He atoms, just as phonons do in metals. There
are, however, several important diﬀerences between paramagnons and phonons. For example, the interaction due to the exchange of paramagnons is spin-dependent. In thelimit q, ω → 0, the interaction induced by the virtual paramagnon is always attractive inthe spin-triplet state, while it is repulsive in the spin-singlet state.

2.3.3

Pairing interaction in liquid 3 He

Let us examine the possibility of forming Cooper pairs in the 3 He system. To thisend, let us consider the interactions between 3 He atoms. The bare atom-atom potential
shown in Fig. 2.12 has a strong hard core repulsion at short distance, much stronger
2
The attractive part of the potential has the maximum around r ∼ r0 , which we can assume to be of
the order of the radius of the Cooper pairs. On the other hand, the Cooper pairs must be formed fromstates near the Fermi surface, k ∼ kF . Therefore, we infer the following relation` ∼ kF r0 (` = 1, 2, or 3),
where ` is the angular momentum.

32(2.16)

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett

3He

Metals
−

＋

−
＋

−

↑ ↑ ↑

− ＋
−

＋
＋
−

＋

＋
−

↓

↓
↓

↑
↑
↓
↑
↓
↑
↓
↓
↑
↑
↓
↑
↓
↑
↑
↓
↑
↓
↓

−

＋

↑

↓

−

＋
−

Fig. 2.3. The mechanism to induce the interaction in the liquid 3 He is analogous to thatof metals.

than the Coulomb repulsion for electrons. Due to this strong repulsion, the Cooper
pairs with zero angular momentum are disfavored in 3 He. Furthermore, the eﬀective
interaction originated from the spin-fluctuation exchange, discussed above, is attractive
for the spin-triplet case and repulsive for the spin-singlet case. Recalling that the Pauliprinciple constrains that states with even (odd) angular momentum ` must be in spinsinglet (triplet) state, we can expect, all in all, that the ` = 1 or possibly ` = 3 pairingwith S = 1 may be favored3 . Even before the experimental discovery of 3 He, people
discussed the possibility of ` = 1: p-wave state. Now it is clear that we have to generalize
the BCS theory to the ` 6= 0 pairing.

2.4

Anisotropic spin-singlet pairing (for orientation
only)

We begin with the easiest anisotropic pairing; that is, the ` = 2 spin-singlet pairing. Our
strategy here is basically using the usual BCS theory and making necessary modifications
to describe the anisotropic pairing. Let us assume the BCS ansatz similar to Eq. (1.9),
(
)N/2
∑
†
†
ΨN =
ck αk↑
α−k↓|vaci .
(2.17)k†
Note that αkσhere creates quasi-particles, not bare particles, since there is a strong
inter-atomic interaction. We have relations similar to Eqs. (1.57) and (1.69),

Fk = ∆k /2Ek ,∆k0∆k = −
Vkk0
tanh(Ek0 /2kB T ).
2E
k0
k
∑

(2.18)(2.19)The pair wave function Fk and the gap function ∆k now depend on both the direction
and the magnitude of the momentum k. The interaction Vkk0 is a nontrivial function of3

The ` = 2 pairing was considered in the original theory [1].

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LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggettk − k0 , and it can have a complicated form under the constraint that it must be invariant
under the spatial rotation. Since we are only interested in behaviors of the system close
to the Fermi surface and we can set |k|, |k0 | ∼ kF , Vkk0 can always be expanded as∑
Vkk0 =
V` P` (k̂ · k̂0 ),(2.20)`

where P` (k̂ · k̂0 ) are the Legendre polynomials. If V` is negative for some `0 and if |V`0 |is appreciably larger than other V` ’s, we keep only the `0 component, ignoring all othercomponents. Now let us assume this is the case.
We can also decompose ∆k into spherical harmonics Y`m (θk , φk ) as∑∆k =∆`0 m Y`0 m (θk , φk ).
(2.21)m

To find coeﬃcient ∆`0 m , we consider the free energy and minimize it. Note that the
optimal solution ∆`0 m can be a nontrivial complex number, which in turn means a nonzero angular momentum of the paired state. This is because, as we will see later, if thegap function ∆k is complex, the pair wave function Fk is also complex. More generally,∆`0 m 6= 0 for `0 6= 0 implies that some physical quantities, such as the density of states,
would be anisotropic.

2.5

Digression: macroscopic angular momentum problem

In this section, let us consider one of the long-standing questions about the anisotropic
pairing: can a superfluid state with an anisotropic coupling have a macroscopic angularmomentum?
We take the BCS wave function Eq. (2.17) with the following coeﬃcients:
ck = f (|k|, θk )exp(2iφk ) (d-wave).

(2.22)(b)(a)

Fig. 2.4. (a) The definitions of θk and φk . (b) The BCS ground state constructed fromthe Fermi sea, not from the Fock vacuum.

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LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett

See Fig. 2.4 (a) for the definition of θk and φk . With this wavefunction, we can calculate
the commutator of the operator Ω̂† defined in Eq. (1.8) and the generator of rotations
around the `-axis L̂z as[L̂z , Ω̂† ] = −i~∂ck †α · α† = 2~Ω̂† .
∂φk k↑ −k↓(2.23)

Therefore, it turns out thatL̂z ΨN = N ~ΨN .

(2.24)

This result is somewhat counterintuitive because it implies that this d-wave superfluid
state has a macroscopic angular momentum at any temperature below Tc ! Why did we
get this seemingly unphysical result? Obviously, this is because we started from the Fockvacuum |0i. All pairs of electrons below the Fermi sea gave finite contributions to thetotal angular momentum.
For comparison, let us change our starting point from the Fock vacuum |0i to the Fermi
sea |FSi, the ground state of the non-interacting system. The BCS ground state can beconstructed by moving electron pairs from inside of the Fermi surface to outside of it, asshown in Fig. 2.4 (b). The corresponding formula would beΨN ≡ (Ω̂+ )N+ (Ω̂− )N− |FSi (N+ − N− = 2Nm ),(m)(2.25)

where Ω̂± is defined as
Ω̂+ =

∑

†
ck αk† α−k
,

Ω̂− =

∑

c−1
k α−k αk .

(2.26)

k<kF

k>kF

(m)

We can easily convince ourselves that the state ΨN is an eigenstate of L̂z with the
eigenvalue Nm ~. If we use the result Nm /N = ∆/εF obtained in Lec. 1, we see that the
eigenvalue is Lz ∼ N ~(∆/εF )  N ~.
We therefore have these two diﬀerent conclusions depending on the starting points. It
turns out that these two ground states give the same prediction for almost all physicalproperties, except for the angular momentum as we have seen above. After all, which is thetrue value of the angular momentum of the superfluid 3 He under a specific geometry anda boundary condition? This problem is not fully resolved yet, and remains controversial
even today.

2.6

Spin-triplet pairing

In this section, we will discuss the spin-triplet pairing in detail.

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A. J. Leggett

Fig. 2.5. The response to the external field.

2.6.1

Equal spin pairing (ESP) state

Let us start oﬀ with the simplest case, the equal spin pairing (ESP). With a suitablechoice of the spin axes, the ESP state is characterized by the wavefunction
)N/2
∑(
†
†
†
†
|vaci .
(2.27)
ck↑ αk↑α−k↑+ c−k↓ αk↓
α−k↓
ΨN =
k

Although the Pauli principle implies ckσ = −c−kσ , there is no particular relation between
ck↑ and ck↓ in general. Note that the above state is not equivalent to)N/4)N/4 (
(
∑
∑
(F )
†
†
†
†
c−k↓ αk↓α−k↓|vaci .
(2.28)
ck↑ αk↑
α−k↑
ΨN =
k

k

Equation (2.27) is a coherent superposition of ↑↑ and ↓↓ pairs, while Eq. (2.28) represents
a Fock state. For a spin-conserving potential, the gap equation for σ =↑↑, ↓↓ decouples:∑∆k 0 σ1∆kσ = −
Vkk0
tanh βεk0 σ .
(2.29)
2E
2
k0 σ
0
k
One important remark for the ESP state is the spin susceptibility. Let us imagine that
we prepare an ESP state with a certain pairing axis and apply a small magnetic fieldalong the axis. The reactions of ↑↑ pairs and ↓↓ pairs to the magnetic field are completely
independent, and thus the magnetic field does not aﬀect the Cooper-pair formation. The
spin susceptibility for the ESP state χESP is therefore approximately equal to that of thenormal state χn (Fig. 2.5).

2.6.2

General case

We consider the most general spin-triplet pairing state,
(
)N/2
∑
†
†
ΨN =
ckαβ αkα
α−kβ|vaci .
kαβ36(2.30)

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LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

A. J. Leggett

The coeﬃcients ckαβ must be an odd function of k and symmetric with respect to α andβ:
ckαβ = ckβα = −c−kαβ .
(2.31)For fixed k, we can always choose the spin axis which makes ckαβ diagonal; i.e., ck↑↓ =ck↓↑ = 0. This spin axis may not be unique, and also can depend on k. What is worse,
even if we use this axis, we cannot proceed much further. The gap equation and otherformulas take too complicated forms in general.
These formulas are, however, enormously simplified if we restrict ourselves to the unitarycase, where |ckσ |2 is independent of σ. In this case, the pair wave function would be givenby∆k,αβFk,αβ =,
(2.32)
2Ek
where we have defined∑()1/2|∆k,αβ |2 .
(2.33)
Ek ≡ ε2k + |∆k |2, |∆k |2 ≡β

We can check that |∆k |2 and hence Ek are independent of α.
If the potential is spin-independent, then the expectation value of the interactions termis reduced to∑hV̂ i =
Vkk0 Fkαβ Fk0 βα .
(2.34)
kk0 αβAs a consequence, the gap equation is decoupled and does not mix the spins:∆kαβ = −

∑

Vkk0

k0

2.6.3

∆k0 αβ.
2Ek0(2.35)

d-vector (unitary states)

Let us introduce the d-vector, which is very useful for describing the unitary state. In
an arbitrary reference frame, the d-vector is defined by∑di (k) ≡ −i(σ2 σi )βα Fαβ (k),(2.36)
αβwhere σi are Pauli matrices. For any given k, we can choose the spin axis in such a
way that F↑↓ (k) = F↓↑ (k) = 0. The definition of the unitary state further imposes the
restriction,|F↑↑ (k)| = |F↓↓ (k)| ≡ |Fk |.
(2.37)In this case, Fk can be written as
Fk = (d1 (k) + id2 (k))/2,37

d3 (k) = 0.

(2.38)

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LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

Therefore, d is a real vector up to an overall phase (i.e., d × d∗ = 0). In the xy-plane,1
the angle of the d-vector with y-axis is arg(F↑↑ /F↓↓ ), while the magnitude |d| is equal2
to |Fk |.
1
In other words, the two-particle state of spin- particles of this form is given by2
S = 1, S · d = 0.

(2.39)

Even in a more general reference frame, the unitary phase has d(k) such that d(k) ×d (k) = 0 for each k. In the BCS case d(k) depends only on the direction n̂ = k/|k| ≈k/kF , not on the magnitude |k|. If the direction of d(k) is independent of k, it represents
the ESP state.
∗

2.7

Ginzburg–Landau theory

2.7.1

Spin-singlet case

The Ginzburg–Landau theory was developed in the context of the old-fashioned superconductors. Actually, this theory was developed before the microscopic works of the BCS
theory were done. We consider a general BCS state (not necessarily the ground state) ina uniform space. Let us define the order parameter of the system based on the pair wave
function Fk ≡ ha†k↑ a†−k↓ i = uk vk :Ψ(n̂) ≡∑

Fk .

(2.40)|k|

The pairing potential energy hV̂ i is given by
∫
∫
∑
dΩ
dΩ0
∗
hV̂ i =
Vkk0 Fk Fk0 =V (n̂, n̂0 )Ψ(n̂)Ψ∗ (n̂0 ).
4π
4π
kk0

(2.41)

We confine ourselves to the case where Ψ(n̂) contains only the ` = `0 component of thespherical harmonics Y`m (n̂) which corresponds to the most negative component V`0 ofV (n̂, n̂0 ) (see Eq. (2.20) and the discussion below). Then hV̂ i can be rewritten as∫dΩ|Ψ(n̂)|2 .
(2.42)hV̂ i = V`0
4π∑
Next we consider the kinetic energy hK̂ − µN̂ i = kσ εk hnkσ i, and the entropy −T S.
It is clear that hK̂ − µN̂ i − T S is a sum of contributions f {Ψ(n̂)} from each point on the
Fermi surface:∫dΩ
f {Ψ(n̂)}.
(2.43)hK̂ − µN̂ i − T S =4π

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A. J. Leggett

From symmetry considerations4 , the function f {Ψ(n̂)} can be expanded as
1
f {Ψ(n̂)} = f {|Ψ(n̂)|2 } = const. + α(T )|Ψ(n̂)|2 + β(T )|Ψ(n̂)|4 + O(|Ψ(n̂)|6 ). (2.44)
2Hence the total free energy F ≡ hK̂ − µN̂ + V̂ i − T S is given by
∫
∫
1
dΩ
dΩ
2
F = const. + (V` + α(T ))|Ψ(n̂)| + β(T )|Ψ(n̂)|4 + O(|Ψ(n̂)|6 ).
4π
24π(2.45)The explicit forms of α(T ) and β(T ) are not necessary for our purpose. We will drop theconstant term below.
If the coeﬃcient V` + α(T ) is positive, the normal state Ψ(n̂) = 0 minimizes the freeenergy. In contrast, if the coeﬃcient is negative, the order parameter Ψ(n̂) has a finite
value. Therefore, the critical point Tc can be determined from the equationV` + α(T ) = 0.

(2.46)

By expanding V` + α(T ) around T = Tc as α0 (Tc )(Tc − T ) (α0 (Tc ) < 0) and β(T ) asβ ≡ β(Tc ), we have
∫
∫
dΩ
1
dΩ
0
2
F = α (Tc )(Tc − T )|Ψ(n̂)| + β|Ψ(n̂)|4 + O(|Ψ(n̂)|6 ).
(2.47)4π2
4π
Note that V` has been dropped out of the problem!
For the s-wave pairing, since Ψ(n̂) becomes a constant Ψ, we get the standard Ginzburg–
Landau expression,1
F = α0 (Tc )(Tc − T )|Ψ|2 + β|Ψ|4 + O(|Ψ|6 ).
2(2.48)

Minimizing this with respect to Ψ gives F = −(α0 (Tc )2 /2β)(Tc − T )2 , which agrees with
the BCS theory.
For the ` 6= 0 pairing, the free energy F depends on the specific form of Ψ(n̂). By
minimizing the free energy with respect to the overall magnitude of Ψ, we find
)2
(
02
2
(2.49)
F = −(α /2β) |Ψ|/|Ψ|4 ,∫
where the average X of a function X(n̂) is defined as X ≡ dΩX(n̂). Hence, the lowest
4π
2free energy is achieved by minimizing the anisotropy of |Ψ| over the Fermi surface.

2.7.2

Spin-triplet case

To examine the spin-triplet case, it is convenient to define∑d(n̂) ≡
dk .

(2.50)|k|

The free energy should be invariant under the global gauge transformation Ψ → exp(iα)Ψ, which isa natural consequence of the assumption that Ψ has the meaning of a Schrödinger-like wave function.
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A. J. Leggett

LEC. 2. SUPERFLUID 3 HE: BASIC DESCRIPTION

The potential energy and the kinetic energy become
∫
∫
∫
dΩ
dΩ0
dΩ
0
∗
0
hV̂ i =
V (n̂, n̂ )d(n̂) · d (n̂ ) → V`0|d(n̂)|2 ,
4π
4π
4π
∫
dΩ
hK̂ − µN̂ i − T S =f {|d(n̂)|2 }.
4π
Therefore the free energy F is
∫
∫
dΩ
1
dΩ
0
2
F = α (Tc )(Tc − T )|d(n̂)| + β|d(n̂)|4 + O(|d(n̂)|6 ).
4π
24π(2.51)(2.52)(2.53)

As we can see, |d(n̂)|2 in the spin-triplet case plays the role of |Ψ(n̂)|2 in the spin-singlet
case. Hence, the lowest free energy is obtained by minimizing the anisotropy of |d(n̂)|2
over the Fermi surface.

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References[1] P. W. Anderson and P. Morel, Phys. Rev. 123, 1911 (1961).

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Lec. 3 Superfluid 3He (continued)
In this section, we focus specifically on superfluid 3 He and some interesting topics.

3.1

Experimental phases of liquid 3He

Figure 3.1 is the phase diagram of 3 He at the temperature below about 3 mK. Solid3
He is also interesting, but in this lecture, we focus on the liquid phases of 3 He. As wesaw in the previous section, at higher temperature (but below 100 mK), this liquid 3 He
behaves very much like a pure textbook Landau Fermi liquid. The Landau Fermi liquid
is basically very much like a free degenerated Fermi gas with two diﬀerences: (a) The real
particle excitation in the free Fermi gas becomes the quasiparticle excitation in the Fermi
liquid, and the eﬀective mass m∗ of the quasiparticle is diﬀerent from an atomic mass mdue to the inter-atomic interaction. (b) The system is subject to a set of molecular fields
which generate various kinds of macroscopic polarizations.
So far, the liquid 3 He is known to have at least three new phases. Most of the phase
diagram is occupied by the B phase, and the A phase exists between the B and the normal
phases. The last one is the A1 phase, which only appears in a strong magnetic field. The
phase transitions between the normal phase and the A phase, or the normal phase and theB phase are second-order phase transitions. On the other hand, the transition between

SOLID 3HeA
A1 (in ࠉࠉ)red curve: 1st-order transition
green curve: 2nd-order transition
broken curves: curves for

B
N

1

2

3

Fig. 3.1. Phase diagram of 3 He (See [1, 2]).

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A. J. Leggett

the A and B phases is a first-order phase transition, and this is typically accompanied bya certain amount of supercooling1 .
In the above paragraph, I avoid referring to the superfluid, because the concept of superfluid 3 He was originally based more on theoretical presumptions than on experimental
phenomenologies. In fact, it is not easy to show that these phases are really superfluid inexperiments2 , but here, we use a usual convention in referring to the phases of superfluid
3
He.

3.2

Nature of the order parameter of diﬀerent phases

It is believed that we are fairly confident in identifying the order parameters of pair wave
functions of diﬀerent phases of 3 He. We can identify the order parameters by susceptibility,
nuclear magnetic resonance (NMR) and also indirect arguments based on stability.

3.2.1

A phaseFirst of all, the A phase (spin-triplet) is believed to be characterized by the d-vector.
The d-vector depends on the position on the Fermi surface, which we specify by k̂, andthe d-vector can written down in the form
ˆ (k̂), f (k̂) = sin θ eiφk̂ ,d(k̂) = df
k̂

(3.1)

where dˆ is a characteristic spin vector, which is independent of the position on the Fermi
surface, `ˆ is a characteristic vector describing the orbital space, and θk̂ and φk̂ are thepolar angles of k̂ with respect to `ˆ (see Fig. 3.2). This phase is usually known as theAnderson–Brinkman–Morel (ABM) state.
In the A phase, the pair wave function can be broken
up into a spin wave function χ(σ) and a spatial wave
function F (r):F (r, σ) = F (r)χ(σ).

(3.2)

Here, in the ABM state, χ(σ) is the state which satisfies
ˆ and F (r) is the state with theS = 1 and Sz = 0 along d,
ˆ The gap function
apparent angular momentum ~ along `.
in the ABM state is explicitly given by the formula here:

Fig. 3.2. Angles θk̂ and φk̂ˆare relative to `.

∆k̂ = ∆0 sinθk̂ exp(iφk̂ ).

(3.3)

1
There are some very interesting problems associated with the supercooling, but we do not discuss indetails here. See Ref. [2] for more details.
2
The healing length of 3 He is at least of the order of a pair radius and much larger than that of 4 He.
Therefore, it is much easier to destroy the superfluidity in 3 He than in 4 He. This is the main reason why
3
He experiments is diﬃcult.

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A. J. Leggett

It is important that ∆k̂ is complex, and hence it has an angular momentum. Furthermore,
the gap function has nodes at k̂ = ±`ˆ (i.e., θk = 0, π), which suggests the low-energy
quasiparticles are more easily excited at low temperature compared with the fully gappedstate.
The fact that the vector dˆ is independent of k̂ indicates the ABM state being the EqualSpin-Pairing (ESP) state, which has a susceptibility equal to the normal state value. Thus,ˆχ is not reduced so long as the magnetic field is applied perpendicular to d.

Alternative representation of the ABM state
We can use an alternative notation of the ABM state, in which the many-body wave
function has the following form
ΨN = const.

[
∑

(

†
†
†
†
ck αk↑
α−k↑+ exp(iχd )αk↓
α−k↓

]
) N/2

|vaci ,

(3.4)

kwhere χd = −2ϕd , and ϕd is the angle of d in the xy-plane measured clockwise fromthe positive y-axis. This wave function brings out its ESP nature explicitly. There isa definite phase between the two up pairs and the two down pairs: the ESP state is acoherent superposition of them. Here, ck is the orbital part of the wave function
ck = f (|k|)sinθk exp(iφk ).

3.2.2

(3.5)

A1 phase

Let us consider another phase, the A1 phase, which only appears under a strong magnetic field. If we take the coordinate axis such that the magnetic field is applied in theup (↑) direction, the many-body wave function of the A1 phase is expressed as(
ΨN = const.

∑

)N/4
†
†
ck αk↑
α−k↑× (↓ Fermi Sea).

(3.6)

k

Crudely speaking, in the A1 phase, a pair of up spins form a Cooper pair, while down
spins just behave like in the normal phase.
It is tempting to think that in the phase of this type, the spins are automaticallypolarized, but that is not true. Because this expression does not mean creating moreparticles in one spin direction ↑ than the normal phase, they are just reorganized to makepairs in the real space. Therefore in this case, up spins have a gap on the Fermi surface
which is the same as in the A phase, but down spins have no gap.

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LEC. 3. SUPERFLUID 3 HE (CONTINUED)

A. J. Leggett

3.2.3

B phase: naive identification

The naive identification of the B phase is first proposed by R. Balian and N. R.
Werthamer3 [4] in 1963, and this state is called the BW state. In the d-vector notation, the state is described by k̂:d(k̂) ∝ k̂.
(3.7)
In other words, the spin state is S = 1, Sz = 0 along k̂. It turns out that the orbital
angular momentum satisfies L̂z = 0. Thus, in the BW state, we can show that the spin
vector S of the Cooper pairs is always directed oppositely to L:
L + S ≡ J = 0.

(3.8)

In the language of molecular physics, the BW state is 3 P0 state. In particular, by the
Wigner–Eckart theorem, any anisotropy in physical properties must be determined by thevalue of J . In this case J = 0, which means all properties have to be isotropic, and thegap function ∆ satisfies|∆(n)| ∝ |d(n)| = const.
(3.9)
This means that ∆ has no nodes (as it does not in the ABM state), and the numberof quasiparticles is exponentially small (∝ exp(−∆/T )) at low temperature. As to spin
susceptibility χ, for any particular direction of the field, some pairs are formed in the
Sz = 0 state and these pairs will not respond to the field. Therefore, χ is reduced fromthe normal-state value.
Combined with the result on the A phase, this has an important consequence that amagnetic field always advantages the A phase over the B phase.

3.3

Why A phase?

The very important question which arose soon after the phase diagram was discovered
experimentally, is “Why should the A phase be there?”. To see this problem, remember
the Ginzburg–Landau theory discussed in the previous section. For the pairing withgiven `, it was shown that the free energy is minimized by the choice which minimizes theanisotropy of |d(n̂)|2 over the Fermi surface. We can define this anisotropy by
K≡(

|d|4
)2 ,
|d|2

(3.10)

and all we have to do is to minimize K.
Now we can estimate K by using the results of the gap function of each state, and see:3

Actually, there was an earlier discussion of this phase by Yu. Vdovin [3], but this work was notwidely known outside the former Soviet Union.

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A. J. Leggett

For the ABM state, |d(n̂)|2 ∝ sin2 θ

8/15⇒ KABM = (2/3)
2 = 6/5.

For the BW state, |d(n̂)|2 = const.

⇒ KBW = 1.

Thus, in the generalized BCS theory, we always have FBW < FABM !
This was the major problem when superfluid phases of 3 He were discovered experimentally, and some approaches were developed to examine this problem. The first approachis to generalize Ginzburg–Landau approach, and the second one is to consider spin fluctuation feedback. In the next two subsections, I will introduce these two approaches.

3.3.1

Generalized Ginzburg–Landau approach

In this approach, our discussion is only based on the symmetry of the states, and there
is no particular assumption on energetics. Since d(n̂) is a vector in spin space, and theorbital dependence is assumed to be p-wave, we can always write the quantities dα as∑
dα (n̂) ≡diα n̂i ,(3.11)iwhere i is a spin subscript, and α is an orbital one. This means that all p-wave triplet
states are completely parameterized by 9 complex quantities diα , and the order parameteris defined in the 18-dimensional space.
Here, let us follow the original argument by Ginzburg and Landau, and assume theexpansion:
F (T : {diα }) = α0 (T − Tc ) · O(|d|2 ) + β(T ) · O(|d|4 ) + ...,
(3.12)
where α0 > 0 is a constant. Then, we consider some basic symmetries: invariance underglobal gauge transformations, and under rotations of the spin and orbital coordinate
system separately. These symmetry conditions constrain the term of O(|d|2 ) to have theunique form∑
O(|d|2 ) ∝|diα |2 ,(3.13)iα∫ dΩwhich is 4π |d(n̂)|2 , just as in the BCS theory.
For the O(|d|4 ) term, on the other hand, there are five terms which are invariant under
the above symmetries. For example,
I1 =

∑

2

d2iα

,

(3.14)

iα

I2 =

∑

d∗βi d∗βj dαi dαj , etc.

(3.15)αβij∑
2Here, note that I1 is diﬀerent from I4 ≡ ( iα |diα |2 ) = 1. We can write down any fourthorder invariants by the linear combinations of these five invariants4 . Therefore in general,
4

Readers interested in the other two invariants I3 and I5 may check them in Ref. [5].

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setting some coeﬃcients βs , we obtain
O(|d|4 ) =5∑βs Is ≡ K4 .

(3.16)

s=1

Is ’s are fourth-order invariants characteristic of particular kinds of states (e.g., the ABM
state, the BW state ...), but βs ’s are parameters which depend on energetic assumptions
in general. Therefore we cannot determine βs a priori unlike the simple BCS case.
Next, we should tackle the problem of finding diα , which minimize the fourth-order freeenergy K4 for given parameters βs and normalization conditions:∑(3.17)|diα |2 ≡ |d|2 = 1.
iαThe problem is to find all possible states (all possible forms of diα ) which can be minimaof free energy (i.e., minima of K4 ) for some choice of the βs under the above constraints.
This problem is solvable in principle, but is quite messy and nearly unsolvable in reality
without constraints. Therefore we restrict possible states to unitary states. Here, unitarystates mean that for every n̂, Sz = 0 is satisfied in some set of spin axes.
If we assume this, the problem becomes simple, and it is found that only four states
can be the extrema of free energy (for more details, see Ref. [5]):1
1. The BW (“isotropic”) state: dαi = √ δαi3This state is already introduced. In the BW state, we have pairing in all possible
directions.
1
2. The 2D (“planar”) state: dαi = √ δαi (1 − δiz )2This state is essentially very similar to the BW state because if the z-component
is removed from the BW state, it becomes the planar state. This state is an ESP
state, but is diﬀerent from the ABM state.
1
3. The ABM (“axial”) state: dyx = −idzx = √ , all other dαi = 02This state is already introduced.
4. The 1D (“polar”) state: dzz = 1, all other dαi = 0
This state is also an ESP state.

Remarkable Theorem
For the BCS values of β 0 s, the BW state turns out to be most stable. If the non-BCS
contributions to β are taken into account, however, other states may be more stable. In
fact, the polar state can be more stable than the BW state only if the non-BCS contributions to β 0 s are comparable to BCS ones. For the ABM state, its energy become smaller
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than that of the BW state for relatively small non-BCS contributions. Furthermore, we
can show that the 2D (planar) state can never be the absolute minimum of free energy
for any choice of β 0 s. Therefore, in general, the BW or the ABM states are most likely toappear.
It is important to note that in the above analysis, both orbital subscripts α’s and spin
subscripts i’s always occur in pairs. Thus, Is0 s are invariant under spin and orbital rotation
separately, and the BW state, the ABM state, etc. represent classes of states transforming
into one another under these rotations.

3.3.2

Spin fluctuation feedback

In this subsection, we view the second approach which is much more based on physicalmechanism. This approach is proposed by Anderson and Brinkman in 1973 [6].
The basic physical idea of Anderson–Brinkman theory is shown in Fig. 3.3. In a standard superconductor, which is described by the BCS theory, the mechanism of the formation of Cooper pairs is an exchange of virtual bosons, or phonons between the electrons. In
the case of 3 He, quasiparticles also exchange bosonic degree of freedom called “spin fluctuation”, where additional quasiparticle and quasihole pairs are formed and the medium
becomes virtually polarized by the strong inter-atomic interaction. Thus, the mechanism
for the superfluidity sounds rather similar to the BCS one, but there is a critical diﬀerence.
Unlike the superconductor, in the case of 3 He, the medium being polarized is precisely themedium in which Cooper pairs are formed, and the superfluid phase transition modifies
the pairing interaction between quasiparticles. Therefore, the original force of the superfluidity, the spin fluctuation, is in turn modified in the presence of the superfluidity, andin general, we must include this “spin-fluctuation feedback”. The amount of this feedback
eﬀect depends on a particular kind of superfluid (the ABM state, the BW state, ...), and
it is, in fact, significant in discussing the ABM and the BW states.
The spin-fluctuation-induced interaction (See Sec. 2.3) is given by
V̂eﬀ (qω) ≈ −(F0a )2 χsp (qω)σ (1) · σ (2) ,(3.18)

where σ (1) and σ (2) are spin operators for each atom of the Cooper pairs. This interaction
is attractive in the spin-triplet state, and repulsive in the singlet state. The point is thatthe spin susceptibility χsp (qω) is modified by pairing. To obtain quantitative results,
we need a microscopic complicated calculation [6], but we can easily obtain qualitative
understanding by assumingδχsp (qω) ∝ δχ,(3.19)
where χ is a static spin susceptibility. Crudely speaking, we assume that the modificationof the spin susceptibility is similar to the static part. With this assumption, we canimmediately see that the BW state is disfavored. In fact, in the BW phase, the spinfluctuation-induced attraction V̂eﬀ gets smaller since the susceptibility is reduced χ < χn .
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Superconductor:

Lattice vibration,
insensitive to onset of electron pairing

Liquid 3He:=

Spin fluctuation of 3He system,
sensitive to onset of pairingFig. 3.3. Schematic illustration of the Anderson–Brinkman theory.

In the ABM phase, for fixed d, δχ is actually anisotropic. In this case, we can write down
the modification δχ and potential asδχij ≈ −f (T )di dj ,(3.20)(1) (2)(1) (2)∆hV̂eﬀ i ≈ −δχij hσi σj i ≈ +f (T )di dj hσi σj i.

(3.21)(1) (2)

If we take the axis of d in the z-direction, we find ∆hVeﬀ i ∝ d2z hσz σz i. In the ABM
state, all Cooper pairs are in a state with S = 1, Sz = 0, so that the spins of the pairs are(1) (2)in the z-direction are antiparallel and hσz σz i is strongly negative. Hence, in the ABM
state, the spin-fluctuation attraction is increased in the ABM state over the normal-statevalue. If the spin-fluctuation feedback eﬀect is strong enough, the ABM state may become
even more stable than the BW state.

3.4

NMR in the new phase

The first experiment of the superfluid 3 He was the NMR experiments, which measure thesusceptibility and the resonance frequency. First, we consider the susceptibility (Fig. 3.4.).
In the normal phase and the A phase, χ is constant (we actually know now by experimental results that there is a very small change for χ when we go into the A phase from49

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B

N

A

Fig. 3.4. The behavior of the susceptibility.
the normal phase), but in the B phase χ discontinuously drops. These phenomena are
not particularly mysterious. The A phase could be an ESP state, which is composed ofonly ↑↑, ↓↓ pairs. Thus, there is no reduction in χ. The B phase could be the singlet or
BW state, where ↑↓ pairs are formed for both cases, and χ is expected to be reduced.
Next we consider the resonance frequency (Fig. 3.5 (a)), which is much more puzzling
than the susceptibility. In the normal phase the resonance frequency ωres isωres = γHext ,(3.22)

where γ is µn /~ (µn : the nuclear moment) and Hext is the external magnetic field. This isa standard Larmor frequency expression. When we go into the A phase, the early NMR
experiments show that the resonance frequency shifts upward from the normal value22
ωres= γ 2 Hext
+ ω02 ,

(3.23)

which seems Pythagorean. This result is very well verified in other experiments. Throughthe phase transition from the A phase to the B phase, the resonance frequency appears
to go back to the exactly same form as the normal state. By experiments
ω02 (T ) ≈ A(1 − T /TA ),A≈ 5 × 1010 Hz2 .
(2π)2(3.24)

Thus we need H0 (≡ ω0 (T )/γ) ∼ 30 G (Fig. 3.5 (b)). However, the only spin-nonconserving force in problem is the nuclear dipole-dipole interaction, and the maximum
associated field is less than 1 G even if the atoms approach very close! What develops theextra field H0 which is perpendicular to the external field? Where does the eﬀect comefrom? Is this the first indication of a radical breakdown of quantum mechanics?

3.5

What can be inferred from the sum rules?

From a simple sum-rule argument, if a single sharp resonance is observed (as in theexperiments), then22
ωres
= γ 2 Hext
+ ω02 ,
(3.25)
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(a)

(b)

( “Pythagorean” )Fig. 3.5. (a) The behavior of the resonance frequency. (b) The extra field H0 .

ω02 = χ−1

∂ 2 hĤD i,∂θ2(3.26)

where ĤD is the nuclear dipole energy and θ is the angle of simultaneous rotation of all
spins (Fig. 3.6).
The second derivative of hĤD i is expected to be∂ 2 hĤD i∼ hĤD i,∂θ2(3.27)∑32
2 2
which is shown from the tensorial structure of ĤD thati=1 ∂ hĤD i/∂ θi = −hĤD i
where θi is the i-th component of θ. This implies that the experimental value of ω02 (T ) is)(T
, K ∼ 10−3 ergs/cm3 .
(3.28)
hĤD i(T ) ∼ K 1 −TAThis is a very large value. Since the end-to-end configuration is favored over the side-byside one only by an energy∆E ≤µ0 µ2n∼ 10−7 K  kB T,
r03(3.29)

the “good” orientation is preferred to the “bad” one by (we consider kB T ∼ 10−3 K)∆E/kB T ∼ 10−4 .

(3.30)The real situation is actually worse than this, since because of the Fermi degeneracy thethermal energy kB T should be replaced by kB TF ∼ 1 K, or ∆E/kB TF ∼ 10−7 . Thus thevacuum expectation value of the dipole energy is much too small.

3.6
3

Spontaneously broken spin-orbit symmetry

We consider here the spontaneously broken spin-orbit symmetry (SBSOS) in the liquid
He by an analogy with the behavior of ferromagnets (Fig. 3.7). For the ferromagnetic
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Fig. 3.6. θ is the angle of simultaneous rotation of all spins.
case, in the normal phase, the direction of spin vector is completely random. However,when we go into the ordered phase, all of the spin vectors become parallel,hSi 6= 0.

(3.31)

This phase transition aﬀects the response to an external field significantly. To see this,let us consider the Hamiltonian expressed with the two terms
Ĥ = Ĥ0 + Ĥz ,(3.32)

where Ĥ0 is the exchange interaction between spins, and Ĥz is the Zeeman interactionterm∑Ĥz = −µB Hext
Szi .
(3.33)i

While Ĥ0 is invariant under simultaneous rotation of all spins, the Zeeman term breaks
the spin-rotation symmetry. In the paramagnetic phase (T > Tc ), the spins behave
independently. Thus the polarization value is approximately µB Hext /kB T  1, and the
expectation value of the second term in Eq. (3.32) becomeshĤz i ∼ N (µB Hext )2 /kB T,(3.34)

which is usually a very small value. In contrast, in the ferromagnetic phase (T < Tc ), Ĥ0forces all spins to lie in parallel. Thus the polarization value is approximately 1, and the
expectation value of the second term in Eq. (3.32) becomeshĤz i ∼ N µB Hext .

(3.35)

Next, we consider the liquid 3 He. The Hamiltonian can be expressed asĤ = Ĥ0 + ĤD ,(3.36)

where Ĥ0 is invariant under relative rotation of spin and orbital coordinate systems, andthe dipolar-interaction ĤD is given by()∑ σi · σj − 3σi · r̂ij σj · r̂ij
( 3 3)
ĤD = gD.
(3.37)
rij /r0
ij
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Liquid 3He

Ferromagnet

total spin of pairspin

relative orbital angular momentum

Normalphase

Orderedphase

Fig. 3.7. SBSOS with an analogy with ferromagnets.
This term breaks the relative spin-orbit rotation symmetry.
In the normal phase (T > TA ), the spins of the pairs behave independently. Therefore,
the polarization value is approximately gD /kB T  1, and2/kB T.
hĤD i ∼ N gD

(3.38)In the normal phase of the liquid 3 He, the direction of spin vector and the orbital angularmomentum are completely random. There is no particular relationship between these
vectors.
In the ordered phase (T < TA ), the directions of these vectors are still completely
random, but all pairs have the same relative angles
hSi = 0, hLi = 0,(3.39)

hL × Si 6= 0.

(3.40)

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Furthermore, all the Cooper pairs are in the same spin and orbital state. Therefore, Ĥ0forces all pairs to behave similarly, and thus we find (as was found in Ref. [7])hĤD i ∼ N gD ∼ 10−3 ergs/cm3 .

3.7(3.41)

Microscopic spin dynamics (schematic)

We consider the two important basic variables, the total spin S and the orientation θof the spins of the Cooper pairs. By using hydrodynamic (Born-Oppenheimer) approximation, the Hamiltonian can be expressed asĤ = Ĥ0 (S) + HˆD (θ).

(3.42)The set of semiclassical equations of motion can then be written as follows:dθ∂hĤ0 i== Hext − χ−1 S,
dt∂S(3.43)∂hHˆD idS
= S × Hext −.
(3.44)dt∂θThe last term of Eq. (3.44) expresses the dipole torque acting on the spin. From theseequations, we can show that the linear NMR behavior is completely determined by eigenvalues of∂ 2 hHD iΩ2ij ≡.
(3.45)∂θi ∂θj
We can “fingerprint” A and B phases by NMR [2, 5]! In the ABM phase, we get asingle resonance line. In the planer phase, alternatively, we get split resonances. Thus,
these two phases can be easily resolved by the NMR experiments. The BW is much moreinteresting. The original BW state is L = −S, i.e., J = 0. However, to minimize thedipole energy, the dipole torque rotates d(k̂) by the angle of cos−1 (−1/4) = 104◦ aroundthe axis ω̂. This means that the relative orientation of L and S is changed and J nolonger has a unique value. Under an external magnetic field, the “best” choice of this axisω̂ is parallel to Hext . Thus, once θ is settled on its equilibrium value 104◦ , any further
twisting perpendicular to ω̂ is, to linear order, simply equivalent to a change in ω̂ with θ
fixed. Therefore, this twisting cannot give rise to a dipole torque, and there is no shift intransverse resonance. It only shifts the finite-frequency longitudinal resonance 5 .

5This shift in the longitudinal resonance also appears in the ABM phase.

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3.8

Illustration of NMR behavior: A phase longitudinal resonanceThroughout the experiment, d lies in the xy-plane (but may rotate), and we keep theorbital part of the pair wave function fixed. The spin part of the pair wave function isΨN (t) ≈ (exp (i∆φ(t)/2) |↑↑i + exp (−i∆φ(t)/2) |↓↓i)N/2 ,(3.46)

where ∆φ is the relative phase between |↑↑i and |↓↓i. Note that this state is a coherent
superposition of |↑↑i and |↓↓i, not the mixed state. The variable canonically conjugate to∆φ̂ is the z-component of total (not Cooper-pair) spin Ŝz , and the canonical commutation
relation is defined as[Ŝz , ∆φ̂] = 2i.
(3.47)The dipole interaction does not conserve spin and scatters from |↑↑i to |↓↓i (and viceversa). Therefore, this interaction depends on the relative phase ∆φ̂ as
ĤD = −gD
cos∆φ̂.
4(3.48)In the adiabatic approximation, the polarization energy isĤ0 = Ŝz2 /2χ − Ŝz Hext (t).

(3.49)

Equations (3.47)-(3.49) show the analogy with a simple quantum pendulum (Fig. 3.8),
with the following correspondence:∆φ̂ → θ (the angle of pendulum with vertical),Ŝz → L (the angular momentum).
Equations (3.47)-(3.49) define the problem of a (driven) simple quantum pendulum. Generally, in the real-life experiments and in the semiclassical limit, we can treat Sz (t) and∆φ(t) as classical variables. The canonical equations for these classical variables can be
written as
dSz
∂H
gD
=−
= − sin∆φ,(3.50)
dt
∂(∆φ)
2
(
)
d
∂H
Sz
(∆φ) ==2− Hext (t) .
(3.51)
dt
∂Sz
χ

3.9

Digression: possibility of the “fragmented” state

The standard statement in a superconducting (superfluid) state is that the global U(1)symmetry is spontaneously broken, which is experimentally untestable! What is really

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A. J. Leggett“internal Josephson effect”

small oscillations

rotationFig. 3.8. The analogy with a simple quantum pendulum.
important is that the relative symmetry: relative phase of |↑↑i and |↓↓i. We assume (cf.
Eq. (3.46))
ΨN ∼ (exp(i∆φ/2) |↑↑i + exp(−i∆φ/2) |↓↓i)N/2 ,(3.52)
where ∆φ the definite relative phase, and essentially assume to be a classical value andwe ignore its quantum fluctuations. Why can we neglect the fluctuation? We can ask thesame question in a diﬀerent manner. We can also think about the possibility forΨN ∼ (|↑↑i)N↑ (|↓↓i)N↓ .

(3.53)

This state has no definite phase relation between |↑↑i and |↓↓i, while it has a definite
relative number
Sz = N↑ − N↓ .
(3.54)
Why we do not consider this possibility?
The answer is as follows. We consider the HamiltonianŜz2 gDĤ =−cos∆φ̂,2χ
4(3.55)

in two cases: (a) gD  χ−1 , and (b) gD  χ−1 . In the case (a), the dominant term in
Eq. (3.55) is the second term. Therefore ∆φ is fixed, and Sz fluctuates. In contrast, inthe case (b), the first term is dominant. Thus, Sz is fixed, and ∆φ fluctuates. However
in general gD ≈ N and χ−1 ≈ N −1 . In the thermodynamic limit, the case (a) is always
satisfied, and we need N < 109 to realize the latter6 . While the case (a) corresponds toan ordinary BEC, the case (b) corresponds to the “fragmented” BEC, which we discussin details in Sec. 4.2.

6

In the ordinary bulk liquid, this condition is impossible. However, we can prepare a rather small
system, e.g. in small inclusions of a liquid 3 He in a solid 4 He, so that the case (b) may be observed insuch a case.

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3.10

Superfluid 3He: supercurrents, textures, anddefects

It is important that the pair wave function is expressed by the spin orientation vector
and orbital orientation vector. Before starting our discussions on the supercurrent, let us
briefly summarize the behaviors of the spin and orbital orientation vectors in the uniform
system. In the absence of any perturbation, the A phase is characterized by the spinorientation vector d and orbital orientation vector `, which in a uniform situation are in
arbitrary directions. Similarly, the B phase is characterized by the rotation R̂ away fromthe 3 P0 state; R̂ again is arbitrary.
The dipole force (dominant perturbation in bulk) tends, in the A phase, to orientd parallel (or anti-parallel) to ` because the end-over-end configuration is formed by `lying perpendicular to the spins involved, which means ` lies in parallel to d, but thecommon orientation is still arbitrary. In the B phase, it fixes the angle of rotation R
to be cos−1 (−1/4) = 104◦ , but the rotation axis ω itself is still arbitrary. The external
magnetic field Hext tends to orient d perpendicular to Hext in the A phase, while it tends
to orient ω̂ parallel to Hext in the B phase.
What happens if the orientation vectors vary in space? It turns out that superfluid
currents appear. In this section, we briefly review the properties of the supercurrents inthe superfluid 3 He and phenomena related to them.

3.10.1Supercurrents

When the order parameter changes in space, there appears a current, in general. Since
there are more than one degree of freedom in the triplet superfluid, there can be some
non-trivial currents. We consider three situations where the order parameter varies inspace. The simplest one is the situation where d and ` (or ω̂) are fixed, and the only
thing that can vary is the overall phase φ. In such a case, an ordinary superfluid mass
flow similar to that in 4 He appearsvs (R) =~∇φ(R),
2m(3.56)

where 2m comes from the formation of the Cooper pairs. The only special point is that in
the A phase the superfluid density ρs is a tensor with its axis defined by ` (ρs⊥ > ρsk , sincemore quasi-particles are excited parallel to ` since there are nodes in the ±` direction in
the ABM state. See Sec. 3.2.1.). Just as in the superfluid 4 He, we can also expect vortices
to appear.
The second case is where φ and ` are fixed but d is spatially varying (or φ fixed but ω̂varying). In this situation, the up-spin Cooper pairs are flowing in the diﬀerent direction
to the down-spin Cooper pairs. Therefore, there is no mass current, but there is a spin
supercurrent.
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The final case is where in the A phase d is fixed but ` (and φ) varies in space. Thissituation is more complicated than the others. Since F (r) ∼ f (r)sinθk exp(iφk ), theoverall phase rotation is equivalent to the rotation around the `-axis. However, if ` varies
in space, such rotations are not holonomic, and ∇ × vs 6= 0 in general [8].

3.10.2Mermin–Ho vortex and topological singularities

If we continue to define vs in terms of infinitesimal rotations around `, i.e., byvs ≡then we find
∇ × vs =

~
∇φ,
2m

~ ∑
ijk `i ∇`j × ∇`k .
4m ijk

(3.57)(3.58)

This leads to the possibility of “coreless” vortices (Fig. 3.9.). If we start from a situation inwhich `ˆ is constant (say parallel to ẑ) in space and the condensate phase has a circulationof 4π around the annulus, then by rotating ` locally around an axis in the xy-plane whichis perpendicular to the radial vector at that point, we achieve a fixed state in which ` iseverywhere antiparallel to the z-axis and the phase is constant. Thus, we have achieved
the phase which has no circulation around the contour in the middle.
A topological insulator in two dimensions is basically under the exactly same situation
we get now. A very useful tool for analyzing the possible varieties of texture in 3 He-A
and B is the branch of mathematics known as the homotopy theory.
Energetics will always pin ` normal to the walls and experiments seem to be consistent with this hypothesis. For the A phase in a simply connected geometry, topologicalargument shows the ABM phase order parameter must have at least two topological singularities somewhere on the surface (most likely boojums, see Fig. 3.10.), and also, apartfrom these topological singularities, have all sorts of nonsingular textures (of d, `, ω).

no circulation

4π of circulation
Fig. 3.9. Schematic illustration of the Mermin–Ho vortex.

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Fig. 3.10. Schematic illustration of the boojum texture.

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References[1] D. Vollhardt and P. Wölfle, The superfluid phases of helium 3 (CRC, 1990).
[2] J. C. Wheatley, Rev. Mod. Phys. 47, 415 (1975).
[3] Y. A. Vdovin, Application of Methods of Quantum Field Theory to Problems of Many
Particles (Gosatomizdat, Moscow, 1963).
[4] R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553 (1963).
[5] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).
[6] P. W. Anderson and W. F. Brinkman, Phys. Rev. Lett. 30, 1108 (1973).
[7] Question from a student: Does this estimation depend on whether the anisotropic
Fermi superfluid has the macroscopic angular momentum or not?
Answer : It is thought that all pairs away from the Fermi surface are independent as
regards to their spins. Therefore, only the few close to the Fermi surface can aﬀect
this estimation. So the answer is “it is the same”.
[8] Question from a student: What kind of current is it? It is neither superfluid masscurrent nor superfluid spin current.
Answer : No. It turns out that if ` is varying in space, in general there is a masscurrent, but the formula for it is more complicated.

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Lec. 4 Definition and diagnostics of“exotic” superconductivity
What do we mean by “exotic” superconductors? To some extent, it is a matter ofconvention or a definition, but crudely speaking, one can isolate quite a large numberof properties, which are common to the old-fashioned superconductors and not foundin some of recent ones. The principal and common characteristics of the “classical”, or
old-fashioned superconductors are(1) Tc ≤ 25 K.
(2) The normal state is well described by the Fermi-liquid theory.
(3) The mechanism of Cooper-pair formation is the phonon-induced attraction.
(4) The symmetry of the order parameter (the pair wave function) is the s-wave.
All of these properties are common to a wide class of materials, such as aluminum, lead,tin, etc. In fact, until 1986 one could say that all the superconductors have the transition
temperature Tc less than 25 K, and they are all classical superconductors. Note that
there are two exceptions to this “definition”: BKBO and MgB2 . BKBO and MgB2 havetransition temperatures Tc ' 30 K, and Tc ' 40 K, respectively, much higher Tc thanthat for the typical classical superconductors. Except for this extraordinarily high Tc ,however, they seem to behave exactly like classical BCS superconductors, so we rather
often classify them as classical superconductors.
In addition to the above four characteristics, there are three other properties which areusually but not always satisfied among many superconductors:(5) The crystal structure is essentially three-dimensional.
(6) The superconducting state is not close to other broken symmetry phases (e.g., antiferromagnetic).
(7) For alloys, the superconductivity is not particularly sensitive to stoichiometry.
“Exotic” superconductors, in contrast, fail to satisfy at least one of (1)-(7), as we cansee from Table 4.1. The cuprates are the most non-BCS like of all the superconductors
known today. That is why I will spend most of the time of the lectures to the cuprates.
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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

Table 4.1. The list of the common or usual properties for the superconductors.

Property

Classical Heavy-fermions Organics Ruthenates Fullerenes Ferropnictides Cuprates√√√√
1. Tc < 25 K
( )
×
×
×
√
√
√
2. Fermi liquid normal state
×
×
×
( )
×
√
√
3. Phonon mechanism
×
?
?
×
×
√
√
4. Order parameter s-wave
?
?
×
(×)
×
√
√
5. Crystal structure 3D
×
×
×
×
×
√
√
√
√
6. No neighboring transition
×
×
×
√
√
√
√
7. Stoichiometry-insensitive
××
×
Maximum Tc [K]
40
2
15
2
40
56
150

How do we know whether the above properties are satisfied? The conditions for (1)the transition temperature, (5) crystal structure, (6) broken-unbroken symmetry phases,and (7) stoichiometry can be checked more or less by direct inspections. Whether (2)the normal state is well described by the Fermi liquid theory can be seen by Fermi liquid
like signals: the specific heat is linear in temperature, the conductivity satisfies BlochGrüneisen formula, etc. Therefore, one can judge more or less from experiments and alittle theory.
How about (3) and (4), the mechanism of the Cooper pair formation and the symmetryof the pair? I think the most obvious evidence for the phonon-induced attraction is theisotope eﬀect, which can be seen from the quantitative agreement between the BCS theory
and the experiment. A valid evidence for the symmetry of the pair to be s-wave can beobtained if many experimental quantities such as the specific heat fall oﬀ exponentiallywith temperature. This shows that the gap must have non-zero value everywhere on the
Fermi surface. The certain and most obvious way that can happen is the pairs forming
the isotropic state.

4.1

Diagnostics of the non-phonon mechanism

4.1.1

Absence of isotope eﬀect

How do we find the mechanism to be phonon or non-phonon induced ones? The firstconsideration, as you might expect, has to do with the isotope eﬀect. Let us have a brief
review of the isotope eﬀect in the BCS theory.
In the original BCS theory, the transition temperature is proportional to M −1/2 , where
M is the isotope mass. This can be seen easily by assuming a constant interaction aroundthe shell of the Fermi surface, recalling that the width of that shell is proportional to the
Debye frequency, which is proportional to M −1/2 .

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

A theory of Tc for a realistic phonon-plus-Coulomb interaction is complicated and requires solutions of Eliashberg equations [1, 2]. McMillan [3] argued that a good analytic
approximate solution of the Eliashberg equation is
(
])
)
( [
ΘD
1.04(1 + λ)
Tc =
,
(4.1)
exp −
1.45
λ − µ∗ (1 + 0.62λ)∫ ωD 2α (ω)F (ω)
dω,
(4.2)
λ≡2
ω
0
where α(ω) is the phonon coupling function and F (ω) is the phonon density of state,and ΘD is the Debye temperature. The quantity λ is proportional to the static local
compressibility of the lattice, which is independent of the isotope mass M since it isirrelevant to the ionic motion. Then where, apart from the parameter Θ, does the isotope
eﬀect comes in? In this formula µ∗ represents the Coulomb pseudopotential defined asµ∗ ≡N (0)hV̂c i
1 + N (0)hV̂c i ln(εF /ΘD )

,

(4.3)

where hV̂c i is the averaged Coulomb interaction. In an approximation of ignoring ΘD dependence of µ∗ , Tc is simply proportional to ΘD and then proportional to M −1/2 .
∂(ln Tc )Therefore, the “isotope exponent” αI ≡ −becomes∂(ln M )
1
α= ,
(4.4)
2
which is just the “textbook” BCS result.
If we take into account the dependence of µ∗ on ΘD with the formula (4.3), the isotope
exponent becomes[]
1
1.04µ∗ 2 (1 + λ)(1 + 0.62λ)
1
αI =
1−
(4.5)
< .
∗
2
2
[λ − µ (1 + 0.62λ)]2Thus even if µ∗ is small, αI can deviate appreciably from 0.5 (and can even be negative!)in the BCS theory. Crudely and qualitatively speaking, therefore, we can reasonably say
the following things:
αI ' 0.5 ⇒ probably phonon-mediated;αI  0.5 ⇒ probably but not certainly non-phonon-mediated;
αI > 0.5 ⇒ possibly phonon-mediated but the BCS theory certainly is inapplicable(no cases known so far).
This is a fairly a standard argument, which is usually given in literatures. However
the above argument does not take into account other possible ways in which isotopic
substitution could modify Tc , such as lattice distortion. For example, let us substitute thedeuterium 2 D into the hydrogen 1 H in a superconductor. Since the deuterium has mucha smaller zero-point energy than the hydrogen, it tends to modify the lattice structure.
Therefore, even if we see the large isotope eﬀect, we must not conclude the interaction
must be phonon-mediated. It is possible that something more sophisticated occurs.
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4.1.2

LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

Absence of phonon structure in tunneling I-V characteristics

Recall that in the BCS theory the gap equation takes the form∑∆k0
∆k = −
Vkk0
.
2E
k0
0
k

(4.6)

However, in a realistic theory of a phonon-mediated interaction, the interaction potential
Vkk0 , even in a normal state, is energy dependent:
Vkk0 = |gkk0 |2

εk0 + ωph (k − k0 ).
(εk0 + ωph (k − k0 ))2 − ε2k(4.7)

In this situation, it turns out that we have to modify the BCS calculation by a little bit
and need the Eliashberg theory [1, 2]. I think the Eliashberg equation is for many people
very non-intuitive, but most people believe that it describes the electron-phonon interactions well. If one solves the Eliashberg equation, the “gap” (oﬀ-diagonal field) becomes
frequency-(energy-)dependent ∆ → ∆(ω) (but nearly independent of momentum). At
T = 0, the gap equation becomes{}∫ ∞∆(ω 0 )
10dω Re∆(ω) =
Z(ω) 0(ω 02 − ∆2 (ω 0 ))1/2{∫ ∞}2(ω 0 + Ω)
2∗×dΩα (Ω)F (Ω) 0
−µ .
(4.8)
(ω + Ω)2 − ω 20Here Z(ω) is the renormalization function, given by the second Eliashberg equation.
McMillan and Rowell [4] pointed out that if we actually look at a tunnel junction
between superconductors and normal metals the diﬀerential tunnel conductance measures∆(ω) by{}ω(∂I/∂V )S= Re, ~ω = eV.
(4.9)(∂I/∂V )N(ω 2 − ∆2 (ω))1/2
Using this Eliashberg equation, we can reconstruct α2 (Ω)F (Ω) from ∆(ω) obtained inthis tunneling experiment. On the other hand, the phonon density of state F (Ω) canbe directly measured by the neutron scattering, and α(Ω) can be obtained by the fairly
well calculations. Therefore, we can compare α2 (Ω)F (Ω) obtained by these two diﬀerent
methods, and see whether they give a consistent value. For the BCS superconductors,
these two are known to agree very well.
In a qualitative point of view, if the origin of Cooper pairing is the phonon-mediatedattraction, then peaks in neutron scattering spectrum (i.e., in F (Ω)) must be reflectedin the tunneling I-V characteristic. Conversely, absence of such a characteristic usually
implies the non-phonon mechanism.
Of the existing groups of the “exotic” superconductors, the alkali fullerides and possibly
the organics are the only ones where the mechanism seems to be the phonon-mediated one.
All the other groups in the Table 4.1 have the evidence for the non-phonon mechanism.
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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

4.2

General properties of the order parameter

4.2.1

Definition of the order parameter

Since the many-body wave function is unlikely to be the simple BCS type, to discuss
this subject we need a more general definition of the order parameter or the pair wave
function [5]. The works of the superfluidity and superconductivity are started oﬀ froman early work by Penrose and Onsager [6] on the superfluid 4 He. Here, we parallel the
discussion on the superfluid 4 He and on the superconductors to get a general way todescribe the pairing in the superconductors.
Consider the two-particle reduced density matrix
ρ2 (r1 σ1 , r2 σ2 : r10 σ10 , r20 σ20 : t) ≡ hψ̂σ† 1 (r1 t)ψ̂σ† 2 (r2 t)ψ̂σ20 (r20 t)ψ̂σ10 (r10 t)i∑ ∫= N (N − 1)ps
d3 r3 · ·d3 rN
s,σ3 ···σNΨ∗S (r1 σ1 , r2 σ2 , ··, rN σN : t)ΨS (r10 σ10 , r20 σ20 , r3 σ3 · ·, rN σN : t).
(4.10)
Here ps is the probability of the occupation of the many body state s. Crudely speaking,this is the best possible representation of “the eﬀective behavior of two electrons averaged
over behavior of remaining N − 2”. Note all two-particle expectation values can be
expressed in terms of ρ2 . For example, if we consider the two-particle potential energyV̂ ≡1∑V (r̂i − r̂j ),2 ij

its expectation value is given as∑∫ ∫hV̂ i(t) =d3 r1 d3 r2 V (r1 − r2 )ρ2 (r1 σ1 r2 σ2 : r1 σ1 r2 σ2 : t).

(4.11)(4.12)σ1 σ2Furthermore, the one-particle reduced density matrix can be given by
∫
1 ∑
0 0
ρ1 (rσ, r σ : t) =d3 r2 ρ2 (rσ, r2 σ2 : r 0 σ 0 , r2 σ2 : t) + O(N −2 ).
N σ(4.13)

2

Thus, basically, if we know the two-particle reduced density matrix, we know all thephysically meaningful quantities of the system. This, of course, does not mean we know
all about the system. It is possible that the many-body wave functions are totally diﬀerent
but still give the same reduced density matrix1 , and for a certain physical quantity theygive diﬀerent values. However, for most practical purposes in the traditional condensed
matter physics, the two-particle density matrix really gives all we need to know.
1

I know no particular case this occurs, but, for example, in the topological quantum computing thatquestion can arise.

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

As a function of the variables (r1 σ1 , r2 σ2 ), the two-particle reduced density matrix ρ2
is Hermitian by construction. Therefore, it can be diagonalized, i.e., always written in
the form∑ρ2 (r1 σ1 , r2 σ2 : r10 σ10 , r20 σ20 : t) =ni (t)χ∗i (r1 σ1 , r2 σ2 : t)χi (r10 σ10 , r20 σ20 : t),(4.14)i∑ni (t) = N (N − 1),(4.15)iwhere {χi } is an orthonormal set (χi , χj ) = δij . Note that as Yang proved [5], themaximum value of any one ni is O(N ), not O(N 2 ). Another thing to notice is that ρ2
must be antisymmetric under (r1 σ1
r2 σ2 ) etc. due to Fermi statistics, so must be
eigenfunctions χi . The two-particle density matrix is a very intuitive way of looking at abehavior of the many-body system. If we just take two particles at random and look at
the behavior averaged over all the other N −2 particles, then we have a certain probability
proportional to ni (t) that we find the particular two-particle state of this form.
From now on we assume the states to be time-independent since we would like toconsider the equilibrium state. In principle, there are three possibilities:
1. All eigenvalues ni ∼ o(1) (normal state) (e.g., free Fermi gas).
2. Two or more eigenvalues are O(N ), and the rest eigenvalues are o(1) (“fragmented”
pseudo-BEC, usually disregarded2 ).
3. Only one eigenvalue is O(N ), and the rest eigenvalues are o(1) (“simple” pseudoBEC, i.e., Cooper pairing).
In the case 3, there is the “special” state which has the largest eigenvalue of ρ2 . Let us
denote the largest eigenvalue as N0 ∼ O(N ), and define√
F (r1 σ1 , r2 σ2 ) ≡ N0 χ0 (r1 σ1 , r2 σ2 ),(4.16)
which is the order parameter or the pair wave function. For the BCS case this definition
reduces to the one given in Lec. 2. Note that from the orthonormality condition on the
χi , we have∑∫d3 r1 d3 r2 |F (r1 σ1 , r2 σ2 )|2 = N0 .
(4.17)σ1 σ2

Therefore, N0 can be regarded as the “number of Cooper pairs” just as in the BCS case.
A possible alternative definition of F is
F (r1 σ1 , r2 σ2 ) ≡ hN + 2|ψ̂σ† 1 (r1 )ψ̂σ† 2 (r2 )|N i,(4.18)

2
See discussion in Sec. 3.9 of spin triplet pairing in the limit gD → 0. For 3 He, if the dipole energy
and the number of the particles are small enough, the Cooper pairs are not the coherent superpositionof ↑↑ and ↓↓ but half of them are ↑↑ and the half are ↓↓. Such state is a fragmented state. However, thisis unusual because non-fragmented states almost always take over in the thermodynamic limit.

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

where |N i, |N + 2i are ground states (often “abbreviated” hψ̂σ† 1 (r1 )ψ̂σ† 2 (r2 )i with implicit
assumption of N -non-conservation).
If we define the center of mass coordinate and the relative coordinate by
r1 + r2, r = r1 − r2 ,2

(4.19)F ≡ F (R, r, σ1 , σ2 ).

(4.20)R=then

For this lecture, we will usually be interested in the case in which F is independent of R,i.e.,
F = F (r, σ1 , σ2 ).
(4.21)
This implies the translational invariance of the system. This is not quite accurate in a
crystal, but crudely speaking, even for that case we will usually be more interested in thedependence on the relative coordinate than in dependence of the center of mass coordinate.
Talking about the “symmetry of the order parameter”, we mean the dependence of F onr, σ1 , σ2 .

4.2.2

Order parameter in a crystal3

For the moment, I am going to assume we can forget about spin-orbit coupling. I will
come back to that at the end of this lecture. It turns out forgetting about the spin-orbit
coupling does make the problem considerably simpler. In addition, we restrict ourselves
for the moment to the case where only a single band4 intersects the Fermi surface5 .
Let us take the Fourier transform of Fαβ (r) in the Bloch basis, i.e.,Fαβ (k) ≡ ha†kα a†−kβ i = −Fβα (−k),(4.22)

where k is the Bloch wave (not plane wave) and the last relation follows from the Fermi
statistics. Because we neglect spin-orbit coupling, we can deal with the spin and theorbital wave function separately, i.e.,Fαβ (k) = sαβ f (k),(4.23)

where f (k) is the orbital wave function and sαβ is the spin wave function. As you know,
we can adopt s(`s , ms )αβ (`s = 0 or 1, ms ≤ `s ) as a basis of the spin wave function,

3

For exhaustive discussion, see Ref. [7].
You might object here that I am talking implicitly in free electron language. This is indeed a very
delicate point but I assume we can get away with this. The evidence is in certain exotic superconductors
we can. Thus the idea of identifying diﬀerent bands, even though it is, strictly speaking, a sort ofnon-electron kinds of scenario, still makes sense mostly.
5
Not true for Sr2 RuO4 , ferropnictides. etc.
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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

where s(`, m)αβ ’s are defined ass(1, 1)αβ s(1, 0)αβs(1, −1)αβ





 s(0, 0)αβ

= δα↑ δβ↑ ,1= √ (δα↑ δβ↓ + δα↓ δβ↑ ),2= δα↓ δβ↓ ,1= √ (δα↑ δβ↓ − δα↓ δβ↑ ).
2(4.24)

Note that s(1, ms )αβ is symmetric, while s(0, 0)αβ is antisymmetric. By combining thisfact with Eq. (4.22), we can conclude
sαβ : spin-singlet⇔

f (k): even parity,sαβ : spin-triplet⇔

f (k): odd parity.

Spin symmetry
In the spin-singlet case,√Fαβ (k) = −(iσy )αβ f (k) = − 2s(0, 0)αβ f (k)(4.25)

with f (−k) = f (k). In the spin-triplet case, we can define, just as for 3 He,d(k) ≡ −(iσy σ)αβ Fβα (k) = −tr[(iσy σ)F (k)].

(4.26)In the last equation, we regard Fαβ (k) as a two by two square matrix. Inversely, we canrepresent Fαβ (k) in terms of d(k):Fαβ (k) = i[d(k) · σσy ]αβ /2(4.27)√1= [(−d1 (k) + id2 (k))s(1, 1)αβ + 2d3 (k)s(1, 0)αβ + (d1 (k) + id2 (k))s(1, −1)αβ ],2with d(−k) = −d(k). If d(k) is real, which is true for most of the cases we are interestedin, d(k) represents the direction along which pair (k, −k) has Sz = 0. (i.e., d · S|F i = 0.)
In principle, we could take linear combinations of spin-singlet and triplet states. However, in the absence of spin-orbit coupling, we have to chose very pathological values ofthe parameters to realize such an ground state. Therefore, at least in the absence ofspin-orbit coupling, the question of spin symmetry is very straightforward.

Orbital symmetry
What about the orbital symmetry? There is a rather delicate point here. Strictly
speaking, once we get away from the simple BCS-like picture, we should not really assume
the quantity Fαβ (k), which is the order parameter or the eﬀective wave function of pairs,
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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

has much to do with the energy gap ∆αβ (k). However, let us assume, at least as a minimal
assumption, that the dependence of Fk on the magnitude of k is not qualitatively diﬀerent
from the BCS form ∆k /2Ek , where ∆k is roughly independent of |k|. If that is true, thenthe angular dependence of Fαβ (k) on the Fermi surface is much the same as that of∆αβ (k).
The crucial point is that the possible forms of Fk are classified by transformation
properties under the crystal symmetry operations, such as the reflection at k = 0, rotation
around a certain axis. If Fk transforms according to the identity representation, we callthe superconductivity is “conventional”, or “s-wave” like (need not to be totally sphericalsince in a crystal we have only discrete rotational operation), otherwise “exotic”.

4.3

Diagnostics of the symmetry of the order parameter

Let me move on to the question: how one can diagnose the symmetry of the orderparameter? It is diﬃcult to obtain the whole form of the order parameter only by a single
type of measurements. Each kind of experiments provides only partial information, suchas its spin state, orbital state, node, etc. In this section, I introduce various methods todetermine the symmetry of the order parameter.

4.3.1

Diagnostics of the spin state

The spin susceptibility (via the Knight shift)
The simplest way to observe the spin state of electrons in solid is to measure the Paulisusceptibility via the Knight shift. As shown in Fig. 4.1 (a), for the spin-singlet state, justlike the BCS one, χ/χn falls to 0 as T → 0 because the opposite spins are pairing up andit takes a finite pairing energy to break the pairs. We can apply the same argument to
the ESP triplet states with d(n)kH. However, for the ESP triplet states with d(n) ⊥ H,
χ/χn remains approximately constant. To understand this behavior, let us take the ESP
state with d(k) = d(k)êz as an example. From Eq. (4.27), the pair wave function is givenbyd(k)(δα↑z δβ↓z + δα↓z δβ↑z ).
(4.28)Fαβ (k) =2
Therefore, if we apply a magnetic field in the z-direction (d(n)kH), we have to sacrifice
the pairing energy to gain the Zeeman energy, from the reason I explained just above.
However, if we change the spin axis from z to x, the same state can be re-expressed as
0
(k) =
Fαβ

d(k)
(δα↑x δβ↑x − δα↓x δβ↓x ).
2

69(4.29)

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

(a)

(c)
ESP triplet ()(b)
isotropic triplet

singlet
ESP triplet ()

Fig. 4.1. (a) The Pauli spin susceptibility via the Knight shift. (b) A magnetic field
generates a diﬀerence on the radius of the Fermi sphere between up and down spins. (c)The “Hebel–Slichter” coherence peak in the NMR.

Since the fermions with the same spin are pairing up or down in the x-direction, when
we apply a magnetic field in the x-direction (d(n) ⊥ H), the field does not interfere withthe pair formation, and the susceptibility in this case is not be much diﬀerent from thatin the normal state. On the other hand, for a non-ESP triplet, (e.g., the BW-type state),some pairs are formed in a spin antiparallel configuration seen in the magnetic field axis,
so that 0 < χ/χn < 1. We can therefore conclude thatχ(T = 0) 6= 0⇒

Spin-triplet,χ(T = 0) = 0⇒

Either the spin-singlet, or the ESP triplet with d k H.

The Chandrasekhar–Clogston (CC) limit on the upper critical field Hc2 [8, 9]
Almost all exotic superconductors are extreme type-II. In an old-fashioned BCS-type
superconductor, the upper critical field may be set by the Meissner eﬀect. However,if the superconductor becomes very dirty, the upper critical field Hc2 predicted from anaive “Meissner” eﬀect becomes really enormous, on the order of 103 Tesla. Under thosecircumstances, the actual Hc2 may be set by the second limiting eﬀect. This eﬀect isdue to the fact that if we apply the large enough field, it is energetically advantageous tosimply forget about the pairing and obtain the Zeeman energy, leaving the system in thenormal state (Fig. 4.1 (b)). This may occur at the point when the field H becomes theorder ∆(0)/µB . If the pairing is spin-singlet or the ESP state with d k H, we expect thiseﬀect. Thus, we crudely conclude that
Absence of the CC limit⇒

Spin-triplet.

Presence of the CC limit⇒

Spin-singlet or the ESP triplet with d k H.

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

The “Hebel–Slichter” (HS) coherence peak in NMR
The prediction from the BCS theory is that if we measure the relaxation rate of nuclear
spins of metal, in the normal phase, it is proportional to T . The spectacular predictionof the BCS theory, assuming a singlet pairing, is that if we go into the superconducting
phase as we lower the temperature, it first rises due to the singularity (∝ (E 2 − ∆2 )−1/2 )in the density of states at the edge of the gap, and then it drops (Fig. 4.1 (c)) [10]. The
presence of this HS peak requires(a) Singularity in DOS at gap edge.
(b) Absence of canceling factor (∝ (E 2 − ∆2 )) in matrix element, due to coherence.
Unfortunately, (b) should still hold for the spin-triplet case, and thus the presence orabsence of the HS peak does not give much information about the spin state. The peakis predicted to be half “canonical” size, but it is diﬃcult to check this.

4.3.2

Diagnostics of the orbital state

Before I start oﬀ explaining the diagnostics of the orbital state, let me give two cautions:(1) Even if the order parameter transforms according to the identity representation of
the crystal group, it may still have nodes. This is called “extended s-wave” (seeFig. 4.2 (a) as an example).
(2) The order parameter Fk is a two-particle (“bosonic”) quantity, whereas the energy
gap ∆k is single-particle (“fermionic”) quantity, since it is the actual gap in the
single-particle spectrum. Within the BCS theory, the two are closely proportional
as regards the angular dependence since
Fk = ∆k /2Ek ,(4.30)(b)(a)

Fig. 4.2. (a) An example of the extended “s-wave” type gap. (b) An example of the 2D
Fermi surface and the gap with the point node.

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY

and in most of the range of k, Ek does not have large dependence on the directionof k. However, this need not necessarily be true in a more general theory. Thus,
strictly speaking, one should try to distinguish carefully between those experiments
which measure the energy gap ∆k and the order parameter Fk .
Measurements of |∆k |The most obvious thing we can measure is the actual magnitude of the gap ∆k 6 , since
there are several ways in doing this. As far as I know, there are only very indirect ways
in measuring the actual complex quantity ∆k itself. On the other hand, there are some
nice ways of measuring the complex quantity Fk 7 . The main thing I am thinking in thiscontext is so-called Josephson-type experiments. They have been very important in theanalysis of the cuprates, and in somewhat (much less) important in the analysis of otherexotic superconductors. I will discuss this in detail when I talk about the cuprates inLec. 7.
Let us move on to the measurement of |∆k |.
(a) The most direct measurements of |∆k | are various kinds of spectroscopic measurements, such as ARPES and STM. In principle, this kind of experiments do directly
measure |∆k | as a function of k and therefore as a function of position.
(b) The more indirect thermodynamic measurement is via measurement of the densityof states, Ns (E). Since Ns (E) is represented as
∫
∑
dΩ
E
√
δ(E − Ek ) = N (0)Ns (E) =,(4.31)
2 − |∆(n̂)|24πE|∆(n̂)|≤Ek
it is proportional to the area of the Fermi surface S(E) where |∆(n̂)| ≤ E is satisfied [11].
Figure 4.2 (b) illustrates an example of a 2D Fermi surface and a gap with the pointnode. We see that S(E) ∝ E if the gap function crosses its zero linearly at the node.
This situation is generalized to• 3D, point node: Ns (E) ∝ E 2 .
• 3D, line node: Ns (E) ∝ E.
• 2D, point node: Ns (E) ∝ E.
If Ns (E) ∝ E n , we know the low temperature behavior of physical quantities as follows(see chapter 7 of Ref. [12]):Specific heat: CV ∝ T n+1 .
6
7

In the triplet case, this is “total” gap and is proportional to |d(n)|.
This is something only realized comparatively recently.

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Penetration depth: λ(T ) − λ(0) ∝ T n .
Nuclear spin relaxation rate: T1−1 ∝ T 2n+1 .
Knight shift: Ks ∝ T n (for spin-singlet state).
Therefore, by measuring these quantity experimentally, we can determine the type of thenode.
Note that in three dimensions, if the orbital angular momentum ` is larger than 1 (i.e.,` ≥ 2), the energy gap must always have nodes. For example, remember that when wetalking about superfluid 3 He, I mentioned that in early days, people thought it is probably
d-wave pairing. There is a general theorem which claims if it is d-wave, there must be
at least two nodes on the Fermi surface. On the other hand, in two dimensions, it needs
not have node for any `, because we can have the order parameter Fk ∼ exp(i`ϕ) withangular momentum `, but the magnitude of this is just a constant.

4.3.3

Eﬀect of impurities

A digression: impurity scattering in the BCS (s-wave) superconductors
Let us consider the following Hamiltonian,Ĥ = Ĥ0 + V̂ ,Ĥ0 = K̂ + U (r, σ),(4.32)

where Ĥ0 (V̂ ) is the single (two)-particle Hamiltonian, and K̂ is the kinetic term andU (r, σ) is an impurity potential. Let us forget about lattice potential for the moment.
We denote eigenenergies and eigenfunctions and of Ĥ0 as εn and χnσ (r, σ).
(A) Nonmagnetic impurities
If the impurity potential is independent of σ, i.e., U (r, σ) = U (r) , then the system
would be invariant under time-reversal. In that case, χ∗n (r, −σ) is also an eigenfunction, with same energy εn . We can therefore pair electrons in time-reversal states(Anderson’s theorem). Pairing between the exact eigenstates ensures that the expenditure of Ĥ0 is minimal. In this case, the order parameter evaluated at the same∑
point F (r, r) ∼ n |χn (r)|2 is rather large because it is a sum of positive values,and hence we still get a large value for −hV̂ i even under the impurity. Thus, crudely
speaking, the outcome of Anderson’s theorem is that non-magnetic impurity do not
do anything significant to s-wave superconductivity8 .
(B) Magnetic impurities
If the impurity potential depends on σ (i.e., U = U (r, σ)), time-reversal invarianceis broken. Then we have two choices:8

In fact, if anything, it tends to raise Tc but I will not explain it further here.

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LEC. 4. DEFINITION & DIAGNOSTICS OF EXOTIC SUPERCONDUCTIVITY(a) Pairing in the exact eigenstates εn of Ĥ0 .
Using the exact eigenstates still ensures the minimum expenditure of kinetic
energy Ĥ0 . However, F (r, r) is much reduced, because the Cooper pairs are nolonger formed in the time-reversal pairs. This is usually highly disadvantageousand thus this choice is hopeless.
(b) Pairing in the eigenstates of K̂.
Firstly, we forget about the impurity and reconstitute the original “impurityfree” Fermi sea. This is advantageous since it gain the original BCS condensation energy Ec , originated from the large (not reduced) order parameter.
However, it sacrifices the energy due to U of Ĥ0 :1∆EU = + N (0)Γ2U ,2(4.33)

where ΓU is, crudely speaking, the relaxation rate of the time-reversal operator.
1
It becomes energetically disadvantageous when ∆EU > Ec = N (0)∆20 , where2∆0 is the gap for a pure system. Hence, we expect the superconductivity todisappear (at T = 0) at
ΓU = ∆0 .
(4.34)
Surprisingly, this very simple hand-waving argument is confirmed by the exact
Abrikosov–Gor’kov theory [13].
Generalization to the exotic order parameter9
Let us generalize the above argument to the exotic order parameter. Even if impurities
are nonmagnetic, exact eigenstates χn of Ĥ0 will now be complicated superposition of
k’s from all over the Fermi surface. Thus, if we pair in χn ’s, hV̂ i will be very small,which is very disadvantageous. Therefore we must again “reconstitute” impurity-freeeigenfunctions. It cost some energy due to the impurity, ∆EU = 21 N (0)Γ2` , (supposing
there is a dominant ` in Vkk0 ; Γ` is the relaxation rate of `-symmetry distortion). We
compare the energy loss with the gain of the condensation energy. By analogy with theabove argument, superconductivity disappear when Γ` = ∆0 .

4.4

Addendum: the eﬀect of spin-orbit coupling

The discussion of the spin-triplet pairing in Lec. 2 and above implicitly assumes that the
eigenstates of the single-particle Hamiltonian Ĥ0 are mostly10 invariant under the spinrotation and the orbital crystal group separately. Under these conditions, the pairing
910

For more details cf. pp. 347-348 of Ref. [14].
Except for dilute magnetic impurities, etc.

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state can usually be classified into the spin-singlet and the triplet state, as discussed inSec. 4.2.2. Mixing of these two possibilities requires rather pathological values of theparameters.
If Ĥ0 is not invariant under spin and orbital operations separately, on the other hand,
there may be strong mixing of the spin and orbital degrees of freedom. The most obvious
origin of this is spin-orbit coupling. Because it is scaled as Z 3 , it rapidly becomes importantin heavy metals, e.g. U compounds. Under these conditions the only symmetries left areparity P and time-reversal T (both of which are preserved by spin-orbit coupling ∝ ` · s),
and we must use only these to classify the irreducible representations (see Ref. [7] forfurther details).

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References[1] G. Eliashberg, Sov. Phys. JETP 11 (1960).
[2] D. Scalapino, In: Parks RD (eds) Superconductivity, vol 1 (Marcel Deklcer Inc, New
York, 1969).
[3] W. L. McMillan, Phys. Rev. 167, 331 (1968).
[4] W. McMillan and J. Rowell, Phys. Rev. Lett. 14, 108 (1965).
[5] C. Yang, Rev. Mod. Phys. 34 (1962).
[6] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
[7] G. E. Volovik and L. P. Gor’kov, Sov. Phys. JETP 61, 843 (1985).
[8] A. M. Clogston, Appl. Phys. Lett. 1, 7 (1962).
[9] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
[10] L. C. Hebel and C. P. Slichter, Phys. Rev. 113, 1504 (1959).
[11] P. W. Anderson and P. Morel, Phys. Rev. 123, 1911 (1961).
[12] J. F. Annett, N. Goldenfeld, and S. R. Renn, Physical Properties of High Temperature
Superconductors II (World Scientific, New Jersey, NJ, USA, 1990).
[13] A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 12, 1243 (1961).
[14] A. J. Leggett, Quantum Liquids (Oxford University Press, New York, NY, USA,2006).

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Lec. 5 Non-cuprate exotic
superconductivity
5.1

Alkali fullerides

5.1.1

Structure

Formula of this compound is A3 C60 , where A is some alkali element (Na, K, etc.) andC is a carbon. C60 molecule has a famous soccer-ball pattern, with 20 hexagons and 12
pentagons; its symmetry is icosahedron. While the carbon atom has a molecular orbital
1s2 2s2 2p2 in C60 molecule or solid, and 2s and 2p orbitals hybridize to form four sp2 states.
Of these, three are used up in the σ bonding (in-plane bonding), leaving one electron per
carbon atom, i.e., 60 electrons per C60 molecule, in the pz -like state. It is this last electron,
which sticks out of the surface of the soccer ball, that controls the low-energy properties.
Obviously the theory of the electronic states of the C60 molecule is quite complicated, so
we will not go into in its details and we only briefly summarize its important properties;
C60 molecule has various molecular orbitals as usual. The highest occupied molecularorbital (HOMO) state is 5-fold orbitally degenerate, and the lowest unoccupied molecularorbital (LUMO) is 3-fold degenerate. In the single molecule, the HOMO-LUMO splittingis ∼0.6 eV.

5.1.2

Fullerene crystals

When a large number of C60 molecules are put together, they form a so-called fullerene
crystal. Its lattice has the face centered cubic structure; the cubic lattice parameter is

P
H
H
P

Fig. 5.1. Lattice structure of C60 .
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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITYDOS

Energyband gap
alkali ns state

top of HOMO band

Fig. 5.2. Density of states of fullerene crystals.

∼14.2 Å, which roughly corresponds to the close-packing of C60 molecules. The complicated point which arises in connection with the crystals is that individual molecules canhave two diﬀerent orientations. Despite the fact that the molecules look pretty spherical,
they are not exactly spherical, so there are two slightly diﬀerent orientations for eachmolecule. Quite often, by just quenching the C60 into a solid state, one finds that theorientation of each molecule may be tuned to either of these two orientations. This phenomenon is called merohedral disorder, which is quite important for the detailed theoryof transport, etc., but we do not go into its details here.
If C60 molecules are combined, they form a band structure. The density of states (DOS)of the band which evolves from the LUMO state is shown in Fig. 5.2. One finds a dip in
the middle of the band, which is sensitive to the lattice parameter. This fact is importantwhen one considers the origin of superconductivity.

5.1.3

Alkali fullerides

A pure fullerene crystal, a simple collection of C60 molecules, is not superconducting but
a fairly good band insulator with a band gap ∼ 1 eV. When alkali atoms are introduced
to the crystal, there are certain positions in the crystal where alkali atoms are likely
to locate. Since the energy levels of the intercalated alkali atoms are slightly above the
LUMO bond (Fig. 5.2), they will donate their s-electrons to the LUMO band. Thereafter,
the ionized alkali atoms, the cores of the rare-earth metals, just serve as spectators, playing
no important role in the behavior of the system (cf. charge reservoir atoms in cuprates).
The normal-state properties of fullerene crystal are roughly consistent with textbook
picture of half-filled band: the specific heat Cv ∼ T , the Pauli spin susceptibility χ ∼const., and the spin-lattice relaxation time T1 ∼ T −1 etc. Note that experiments suchas photoemission and plasmons confirm that the DOS at the Fermi energy N (0) ≡78

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( )
1 dn
, an important value for transport in normal and superconducting states,
2 dε ε=εF
is strongly dependent on the lattice constant a.

5.1.4

Superconducting state

Superconductivity in Ax C60 occurs only when the filing of alkaline atoms x is almost
equal to 3, but then has Tc up to ∼ 40 K (Cs3 C60 ). This high transition temperature isquite surprising because the intercalated graphite, for example KC8 , has Tc lower than
1 K. Tc increases as the lattice spacing a is increased by the pressure or substitution;
typically, the increase in Tc by the lattice spacing is given by∂Tc∼ 33 K/Å.
∂a

(5.1)

This is consistent with the BCS result
Tc ∼ ωD exp(−1/N (0)V0 ),(5.2)

under the assumption that V0 comes mainly from the intramolecular origin hence insensitive to a, and that the density of states N (0) increases with a.
A3 C60 ’s are strongly type-II. In fact, the pair radius is ξ(0) ∼ 26 Å, which is less than
twice the lattice spacings, and the London penetration depth is λ(0) ∼ 2400-4800 Å.
What are the characteristics of the pairing state? Behaviors of the spin-lattice relaxation
time T1 and the infrared reflectivity seem to suggest that the quasi-particle density ofstates is small when |E| < ∆ ≈ 1.76kB Tc , as it should be in the BCS theory. The Hebel–
Slichter peak is seen in the muon spin resonance (µSR) and 13 C NMR, which strongly
indicate the pairing state has an s-state as in the BCS superconductors. So far, everythingis consistent with the ordinary BCS superconductors except for one slight puzzling thing:
the deviation of the London penetration depth from its zero temperature value does notfollow an exponential law but some power law:∆λ(T ) ∼ T α , α ∼ 3.

(5.3)

What about the pairing mechanism? The isotope substitution 12 C by 13 C yields a large
isotope exponent ≈ 0.4; remember that the isotope exponent value of the ideal BCS-type
superconductor is exactly 0.5, and in real life there are various reasons for underestimating
the value. This fact is therefore consistent with the phonon mechanism. However, one
finds rather unusual situation in the ratio between the Debye frequency and the Fermi
energy. Since carbon atoms are light and tightly bound in the solid, we have a large Debye
frequency, and the Fermi energy is not that high. Then one obtains the ratioωD /εF ∼ 0.3-0.6.
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which is far larger than that of the BCS superconductors, typically ωD /εF ∼ 10−2 . Therefore, even if the mechanism has phonon as its origin, details seem to be rather diﬀerent
from the BCS theory. The prevalent explanation is that alkali fullerides are pretty welldescribed by the BCS theory, but that the strong molecular structure enables them toavoid the limit on Tc (cf. MgB2 ).

5.2

Organics

There are various kinds of organic superconductors. Most of the extensively studied materials are quasi-2D crystals based on ET(BEDT-TTF) (bis(ethylene-dithio)-tetrathiafulvalene)(Fig. 5.3). Their structures have the form (ET)2 X, where X is a monovalent anion (I−3,−
Cu(NCS)2 ...). The structure is slightly counterintuitive (Fig. 5.4 (a)): conducting layers
consist of (ET)2 , and the blocking layers consist of anions. Conventionally in the literatures, a-axis is taken parallel to the stacking direction of anion layers. These layers
are pretty insulating, which makes diﬃcult the electronic transportation along the a-axis.
Pretty flat (ET)2 molecules line up along the b-axis, constructing metallic layers in thesystem. The distance between the insulating layers is quite large, ∼ 50 Å.

5.2.1

Normal state

In the normal state, the conduction electron density is quite low (∼ 1021 cm−3 ). Thisfeature is characteristic of quite a few classes of exotic superconductors including cuprates.
They have a strong a-axis anisotropy1 : ρa /ρbc ∼ 102 -103 , which is comparable to cuprates.
Interestingly, clean samples made possible the de Haas-van Alphen (dHvA) type oscillation
experiments. The dHvA-eﬀects are electronic oscillatory eﬀects under strong magnetic
field. As a result of the fact that diﬀerent orbits constitute the Fermi surface, various
physical quantities such as susceptibility, resistivity etc. are periodic in the inverse of
the magnetic field 1/B. The dHvA experiment requires quite clean samples; crudely
speaking, the period of cyclotron motion in the magnetic field must be smaller than
the scattering time. This condition, although practically quite stringent, is satisfied in
the organic superconductors. Figure 5.4 (b) shows a typical Fermi surface; everything is

Fig. 5.3. Ethylene-dithio molecule.
1

Confusingly, the conventional notation calls the hard direction a, not c as in the cuprates.

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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

(b)

(a)
a
“Blocking” anion
layers

b

c

c

b
holes

Fig. 5.4. (a) Crystal structure of organic superconductors. (b) Fermi surface of (K(ET)2 Cu(NCs)2 ).
strongly two-dimensional, i.e., cylindrical in three dimensions. This implies the conduction
electrons cannot penetrate the blocking anion layers.

5.2.2

Superconducting state

The superconducting state typically has Tc ∼ 10-12 K. Normally, superconductors withTc ∼ 10 K are not regarded as high-Tc . However, there is an argument as follows: thecalculated in-plane hopping matrix element is small. This means that the bands which
form in-plane are narrow and the Fermi energy is small. If one expresses Tc not in termsof Kelvin but rather as the fraction of Fermi energy/Fermi temperature, then the Tc value
divided by TF is comparable to that of cuprates. In this sense, the organic superconductors
can be regarded as high-temperature superconductors.
The organic superconductors have the following physical properties:• Extremely type-II (Hc1 ∼ a few mT, Hc2 ∼ 8-15 T).
• Estimated pair radius ξ|| ∼ 50 Å in the easy plane, while ξ⊥ ∼ 5 Å ( interlayer
distance ∼ 50 Å) along the hard axis.
• Symmetry: CV (T ) ∼ e−∆/T for T → 0 =⇒ characteristic s-wave.
• Nuclear spin-lattice relaxation time: T1−1 ∼ T 3 =⇒ substantial low-energy DOS.
No Hebel–Slichter peak.
• Isotope eﬀect: the substitution 12 C → 13 C has αI ∼ 0.1, but the substitution 1 H →2D produces inverse isotope eﬀect (αI < 0). This is consistent with the BCS theory,
but the origin is more likely to be the eﬀect of lattice deformation.
In my opinion, to be honest, no one is quite sure about the mechanism of high-Tc superconductivity in the organics. The simplest default option is that the organics are
somewhat BCS-like, but electronic eﬀects may be competitive.

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5.3

LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

Heavy fermions

Heavy fermions are actually the oldest class of exotic superconductors, discovered in1979 [1]. Heavy fermion systems [2] are, in general, compounds which contain rare-earth
elements (usually Ce) or actinide elements (usually U, and Pt). Heavy fermions have an
extraordinary large specific heat even in the normal phase. In fact, the specific heat for
the heavy fermion systems is 10 to 100 times larger than “textbook” values of ordinary
metals: hence the name “heavy fermions”2 . All these materials are three-dimensional,
and as far as I know, none of the heavy fermion material is strongly layered. This isactually somewhat remarkable because, as we will see, all the other classes of exotic
superconductors really are strongly layered.

5.3.1

Normal-state behavior

For the normal-state behavior, there is a problem: at T ∼ 300 K, the behavior of theheavy fermion systems, generally speaking, is quite diﬀerent from that of textbook metals.
It is not even universal in the class. For example, the resistivity ρ(T ) looks metallic for
UPt3 [3], while they behave as semiconducting for most others. However, the following
property typically hold:• χ ∝ 1/T
• T1−1 = const.,• CV = const.,• neutron scattering showing a simple Lorentzian peak centered at T = 0,
where χ is the magnetic susceptibility, and T1−1 is the NMR relaxation time, and CV is the
specific heat. For typical rare-earth or actinide elements in heavy fermion systems, suchas Ce3+ : 4f1 , U4+ : 5f2 , there are characteristic spare f -electrons. We would assume thatthese f -electrons are fairy tightly bound. Then, it is attractive to think, with the above
properties in mind, a model with spare f -electrons of rare-earth or actinide elements,
crudely speaking.
As T is lowered, on the other hand, it seems that in all cases, a crossover to a Fermiliquid-like regime occurs3 . In fact, at low temperature,• CV = γT ,• T1−1 ∝ T ,2
Note that this large eﬀective mass is seen not only in the specific heat, but also in other quantities
as well, in particular in the dHvA experiments.
3
The temperature at which this crossover occurs depends on the system we are talking about. It canbe as high as 100 K, or as low as 5 K.

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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY• ρ ∝ A + BT 2 ,where ρ is the resistivity. Note that the constant term in the resistivity comes from an
impurity scattering, while the T 2 term is what we get from electron-electron Umklapp
scattering4 . These behaviors look much like those of a Fermi liquid, but what is special
to the heavy fermion system is that the coeﬃcient γ in CV = γT is enormous, up to ∼1600 mJ/mole K2 (CeCu6 [4, 5]) (contrast “textbook” metal, CV ∼ a few mJ/mole).
Hence the question for a normal state of the heavy fermion system arises: are we sure
this specific heat is due to mobile electrons or not (since we can get this linear specific heatalso from a localized electron)? I think that once the superconducting phase transition
occurs, then it is rather clear that it has to be that of mobile electrons, since the jump in
the specific heat at Tc is roughly the same as the BCS value. Thus, it is rather plausiblethat the electrons contributing to the large specific heat in the normal state and thoseforming the Cooper pairs in the superconducting phase are both mobile electrons, rather
than localized ones.
Thus, the heavy fermion system at low temperature behaves as a standard Fermi liquid,
but with the large eﬀective mass. Note that this large eﬀective mass is also confirmedby the measurement of the large Pauli susceptibility χ, typically 10 to 1000 times larger
than that of textbook metals, and in the de Haas-van Alphen experiment.
We can imagine a naive model for the normal state as follows: suppose that f -electrons,
fairly tightly bound, form very narrow band, with its width ∆ ∼ a few K. For kB T  ∆,all states in the band will be almost equally populated, which is equivalent to sayingthat electrons are localized on lattice sites independently. Then, it behaves as χ ∝ 1/T ,T1−1 = const, and small CV . For kB T . ∆, we need a proper “band” picture with a largem∗ (∝ ∆−1 ), which seems to account for the crossover to a Fermi-liquid behavior with alarge eﬀective mass.
At first sight, this argument sounds reasonable, and gives a good qualitative explanation
for the normal-state behavior. However, unfortunately, there is a rather important point
missing in it: it ignores the conduction (s or d) electrons. In fact, the conduction electronsinteract with f -electrons and in general have complicated eﬀects, in particular the Kondo
eﬀect [6, 7]. The Kondo eﬀect occurs when we have conduction electrons moving in
the presence of a localized single impurity with spin. The tendency for the conduction
electrons and the impurity spin to form a singlet bound state gives rises to interesting
phenomena which are studied in a vast amount of literature. In the heavy fermion system,
the situation is much more complicated than this. The Kondo eﬀect favors the singlet
state between the conduction and the localized f -electrons. On the other hand, there
is also the interaction between f -electrons mediated by a polarization of the conduction
electrons (RKKY interaction [8, 9, 10]), which favors a magnetic ordering of f -electrons.
4
Note that the electron-electron scattering in a free space does not give us a finite resistivity, since
the momentum of colliding electrons, and thus the current must be conserved. In a crystal, however, thetotal momentum of the colliding electrons can change by modulus of the crystal lattice vector by the
Umklapp process, so that it can have a resistivity of ∝ T 2 .

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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

In fact, in many heavy fermion systems (even in some superconducting states), they show
an antiferromagnetism at T . 20 K.

5.3.2

Superconducting phase

General remarks
We can classify heavy fermion systems into four classes as follows:
1) No phase transition occurs down to T = 0 (e.g., CeAl3 [11]).
2) Show only a magnetic phase transition (e.g., CeCu6 [4, 5]).
3) Show only a superconducting transition (e.g., UPt3 [3], CeCu2 Si2 [1], UBe13 [12]...).
4) Both magnetic and superconducting transitions occur (e.g., UPdAl3 [13], URu2 Si2[14], UGe2 [15, 16] ...). In this class, magnetic order and superconductivity coexist(contrary to established “textbook” wisdom!).
The last class is, in some sense, the most interesting one. This is very surprising since
up to the 1970s, it is strongly believed, as a kind of dogma, that a magnetic order andsuperconductivity cannot coexist. A qualitative reason for this, although it is a partial
one, is that the magnetic ordering occurs for tightly bound electrons, such as f -electrons,
whereas the superconductivity occurs for hybridized and delocalized conduction electrons.
In all cases known of the superconductivity in the heavy fermion system, the transition
temperature can be never larger than of the order of a few Kelvin5 : Tc . 2 K. This isactually quite remarkable because, as we have already seen and we will see further, there
are other classes of exotic superconductors where Tc is much higher.
As to the mechanism, the most crucial observation is that, as far as we know, no heavy
fermion superconductor shows any appreciable isotope eﬀect. This strongly suggests,
although it does not actually prove, that the mechanism for the superconductivity is notphononic but should be “all-electronic” one. I think that everyone would agree that,whatever may be the case for the other exotic superconductors, the mechanism is notphonon mechanism for heavy fermion superconductors.

Pairing states
The pairing states of the heavy fermion superconductors can be characterized by various methods, such as low temperature behavior of the specific heat CV , nuclear relaxation

In the class 4, TN ∼ 10-50 K. We also note that there is some exceptions for this estimate of Tc . For
example, PuCoGa5 , which is normally regarded as a heavy-fermion system, has a transition temperature
Tc = 18.5 K.
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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

Table 5.1. Properties of some heavy fermion superconductors, determined from experiments. AF stands for an antiferromagnetism, F for a feromagnetism, and P for a paramagnetism. TN is the Nèel temperature.
SystemMagnetism

Tc (K)

Parity Gap nodes

UPt3 [3]CeCu2 Si2 [1]UBe13 [12]UPdAl3 [13]
CeCoIn5 [17]UNiAl3 [18]URu2 Si2 [14]UGe2 [15, 16]

P
P
P
AF
P
AF
AF
F

0.56
0.65
0.9
2.0
2.3
1.0
0.8
0.6

−
+(?)
−(?)
+
+
−
−
−

X
X
X
X
X
?
?
?

Comments

TN = 14.5 K
probably dx2 −y2
TN = 4.6 K
TN = 17.5 K
TCurie = 30 K (at point of max Tc )rate T1−1 , electronic thermal conductivity κel , Knight shift, upper critical field Hc2 , sensitivity of Tc to nonmagnetic scattering, or by an existence of multiple phases6 . Some
representative heavy fermions and their (suggested) symmetries are shown in Table 5.1.
Note that for the heavy fermion system, the spin-orbit coupling is not negligible andthus, we cannot separate the orbital and spin part as we did in BCS-like superconductors.
In fact, for rare earth elements such as U or Ce, the spin-orbit coupling is, in fact, verylarge. In such a case, we have to classify them by the parity and time-reversal symmetry,
as we discussed in Sec. 4.4. We also note that there is no evidence for the violation of thetime-reversal symmetry in any heavy fermion, which is a rather interesting fact. Thus,
the only important symmetry seems to be the parity, and it diﬀers for diﬀerent materials.
Most of the heavy fermion superconductors, as we can see, do appear to have a node in
their gap, which can be seen, for example, by the power-law behavior of quantities suchas CV , or T1−1 at low temperatures. Thus, the form of the gap is, at least, not simplest. It
is conceivable that they are so-called extended s-wave superconductors, but it is generally
thought that these are indeed exotic superconductors. Beyond that, it is really not thateasy to say very much about the symmetries.
The thing I would like to comment on is that in Table 5.1 they do not appear to havea lot in common apart from the fact that they are not simple s-wave. That is rather
surprising because, after all, they do have the normal state behavior in common, thelow temperature Fermi-liquid-like phase. However, when they go into a superconducting
phase, they seem to behave diﬀerently.
As for the magnetic properties, most of them are paramagnetic or antiferromagnetic.
There is one case, recently discovered, which is ferromagnetic: UGe2 [15, 16]. This is quite
surprising because it has been used to be thought that a ferromagnetism is inconsistent6

For a simple s-wave picture, it is fairly diﬃcult to account for multiple phases, so if there are multiple
phases, it can be a strong evidence for the anisotropic pairing85

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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

with a superconductivity. It looks like the order parameter has an odd parity, which
seems to be consistent with the ferromagnetism. An odd parity state suggests, crudely
speaking, that the Cooper pairs are mostly triplet, so that a strong magnetic field does
not suppress the spin-triplet pairing.
Note that not all of these have been entirely settled down. In fact, for example asuperconductivity of UGe2 is found around 2000 [15, 16], and there are a lot of work evenrecently.

5.4

Strontium ruthenate: Sr2RuO4

5.4.1

History

For the cuprates, the superconductivity starts to occur at the transition temperature
up to Tc ∼ 150 K [19, 20]. Typical crystal structure of the cuprates is illustrated in
Figs. 5.5 (a) and (b). There are characteristic CuO2 planes, and between them there arelanthanoides. The oﬀ-plane oxygens are diﬀerent from the in-plane ones in that they are,in some sense, not directly connected to the superconductivity. It does seem rather clearthat the CuO2 planes are essential to the high-Tc superconductors. For all superconductors
without CuO2 planes, Tc is below 100 K. Therefore, at the moment when the cuprates
were discovered, people started asking whether it is essential to have Cu, or we can replace
it by other elements. An obvious replacement with Au, Ag did not work.
The crystal structure for Sr2 RuO4 is shown in Figs. 5.6 (a) and (b). As we can see, it
is extremely similar in its structure to the cuprates: the only diﬀerences are firstly that
the Cu atom is replaced by Ru, and secondly the spacer is not Ba, but Sr.
It turns out, however, that the normal-state behaviors of Sr2 RuO4 are quite dissimilarfrom those of cuprates. In addition, the transition temperature is quite diﬀerent. For
La2−x Bax CuO4 Tc ∼ 40 K, while it turns out that Tc ∼ 1 K for Sr2 RuO4 regardless of
Typical (original) cuprate : La2−x Bax CuO4 (Tc ∼ 40 K)

(a)

(b)

Fig. 5.5. (a) Crystal structure of cuprates for ac-plane (side view). (b) Crystal structureof cuprates for ab-plane (top view).

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Sr2 RuO4(a)(b)

Fig. 5.6. (a) Crystal structure of Sr2 RuO4 for ac-plane (side view). (b) Crystal structureof Sr2 RuO4 for ab-plane (top view).
the optimistic hope that we would get a high-Tc superconductor from its similarity in the
crystal structure.

5.4.2

Experimental properties of Sr2 RuO4

We briefly review some important experimental properties of Sr2 RuO4 . For more details, see Ref. [20].

Normal Phase
In the normal state for Sr2 RuO4 , below T ∼ 25 K, it appears to behave as a Fermi
liquid in all three directions. Thus, the specific heat is a combination of the contributions
from the electronic and phonon parts
CV ∼ γT + βT 3 .

(5.5)

For the susceptibility, we observe the Pauli susceptibility:χ ∼ const.

(5.6)

For both the ab-plane and c-axis, the resistivity isρ ∼ A + BT 2 ,(5.7)

which is a characteristic form for a coherent Bloch wave transport limited by the impurity and electron-electron Umklapp scattering: the first term comes from the impurity
scattering, while the second term from the Umklapp scattering. The value ρab itself isconsiderably smaller compared with that of the cuprates, ρab ∼ 1µΩ cm. This means thatthe samples are very pure systems, and indeed it turns out that one can make a very puresample of Sr2 RuO4 .
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Table 5.2. Some of the Fermi liquid parameters for Sr2 RuO4 . Note that the me is thebare electron mass, while mband is the band mass.
α-band β-band γ-band−1

kF (Å )m∗ /mem∗ /mband

0.3
3.3
3.0

0.6
7.0
3.5

0.75
16.0
5.5

Another important feature of Sr2 RuO4 is that it is a strongly anisotropic Fermi liquid.
In fact, ρc /ρab ∼ 103 , and this is comparable to those of typical cuprates. This strong
anisotropy can be also seen in its band structure. We can reconstruct the Fermi surface
from the de Haas-van Alphen kinds of experiments7 , or we can calculate the band structureby the local density approximation (LDA). Both of them suggest that the Fermi surface
consists of three strongly two-dimensional sheets: one hole-like sheet (called α-sheet), andtwo electron-like sheets (β- and γ-sheets). This is what we may expect from the strong
anisotropy in the resistivity, which indicates that the hopping matrix element along thec-axis is far smaller than that for the inner-plane hopping. In fact, the deviation from theideal two-dimensional Fermi surface is fairly small, of the order of ∼ 1%.
The behavior of electrons for each of the sheets is shown in Table 5.2. The Fermi
momentum is of the order of a typical value, and it is not so much an exciting value. The
eﬀective mass is several times larger than the band mass, which is estimated by neglecting
the electron-electron interaction and just solving an one electron problem. This indicatesthat electrons are strongly correlated, although it can be well described by the Fermi
liquid.

Superconducting phase
People have expected that the transition temperature for Sr2 RuO4 would be as high as
those of the cuprates, but it turns out that Tc is not so high, although it does undergo asuperconducting phase transition. The transition temperature for Sr2 RuO4 is only about
1.5 K, and in that sense it was disappointing.
It turns out, however, that the susceptibility appears to be χ ∼ const. in the superconducting state for all directions, which seems to indicate a triplet ESP state. I think
this is fairly firm, and most people believe that the superconducting state of Sr2 RuO4 isindeed a triplet ESP state.

7
Although there are various ways to measure the band structure, if we are faced with some newmaterials and want to figure out what their band structures are, the most reliable way is the de Haas-van
Alphen kinds of experiments, measuring a quantum oscillation with respect to the external magnetic
field.

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(a)

(b)

Deviation from perfectcylinderFig. 5.7. (a) Crystal structure of Sr2 RuO4 . (b) Schematic illustration of the Fermi surfaceof Sr2 RuO4 .
With that in mind, let us suppose that the order parameter of Sr2 RuO4 is indeed of
the ESP form:
F (k; σ1 , σ2 ) = F (k; σ1 , σ2 ) = f (k)(| ↑↑i + | ↓↓i).
(5.8)
A crucial question arises: what is the orbital wave function of the pairs f (k)? Since it isa spin-triplet state, it has to be an odd parity state. This does appear to be consistent
with some Josephson experiments, where it is shown that the order parameter seems tochange its sign if we reflect it.
The most interesting question is whether the order parameter is real or complex. In
the real case, such as f (k) ∼ kx , the time-reversal symmetry is not broken. On the other
hand, if it is complex, for example f (k) = kx + iky 8 , it can be broken. In the BCS theory,
we know a priori, that the amplitude of order parameter |OP|2 want to be as uniform as
possible over the Fermi surface, although this is not always the case in a more general
theory9 .
It turns out that there are various experiments which are in favor of the time-reversal
symmetry broken order parameter. Some of the experiments are as follows:• Muon spin rotation
In the muon spin rotation experiment, we essentially measure an eﬀective magnetic
field where the muon is sitting after it is trapped to somewhere. Generally speaking,in the normal state under a zero external magnetic field, the muon sees indeed no
local magnetic field. If, on the other hand, we have a system where the time-reversal
symmetry is spontaneously broken, then you would expect to see that the muon spinrotates, and indeed we will see it. What is peculiar for Sr2 RuO4 is that the signal of
8This state has an angular momentum along z-axis, and you can easily see that the time-reversal
symmetry is broken.
93
He is one counterexample for this, where we could not exclude the possibility for the polar phase,
which does not breaks the time-reversal symmetry.

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the time-reversal symmetry breaking starts to appear at the superconducting phasetransition [21]. This experimental fact suggests that the time-reversal symmetry is
spontaneously broken in the superconducting state.
There is one worrying factor there: the direction of the local magnetic field. An
obvious option for this is in the direction along the symmetry axis, that is along thec-axis. However, this is not the case, and in fact we have a finite component in theab-plane. This is what remains to be understood.
• Magnetic field dependence and telegraph noise in Ic of Josephson junctions interpreted in terms of switching of domains.
If the time-reversal symmetry is broken as kx + iky , there is also an equal possibility for having kx − iky state. There is an argument based on a macroscopic
electromagnetic eﬀect that it is likely to have various domains of them. Experimentalists interpreted the telegraph noise in Ic of Josephson junctions as the switching
between these two states [22].
• Kerr eﬀect in zero magnetic field.
Kerr eﬀect is a measurement of the rotation of the polarization of the light when it
passes through the material. If this rotation occurs, that is again a direct evidenceof the time-reversal symmetry breaking. An experiment has observed the rotationand has confirmed this time-reversal symmetry breaking [23].
So far, so good. For an “ideal” kx + iky state, the energy gap has no node |∆| ∼ |F | ∼const., so that the number of quasi-particles for T  Tc should be exponentially small.
This should be seen in the measurement of the specific heat, thermal conductivity andso on. However, unfortunately what we find is a power-law behavior for most of thesequantities. Whether this power-law is due to the existence of nodes in the gap is somewhat
controversial. In principle, there is a critical test for this: Josephson junction test (seeFig. 5.8), which I shall talk in more detail in Sec. 7.4.

Sr2RuO4
Al
Fig. 5.8. Schematic illustration of the Josephson experiment.

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5.5

LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

Ferropnictides

Ferropnictides are discovered in January 2008 [24]; please note that much experimental
data may not be definitive10 .

5.5.1

Composition

There are two major elements, each containing a transition metal (usually Fe) and apnictide, element in nitrogen column of periodic table (usually As). We have two mainclasses of parent compounds:
1111: (RE)(TM)(PN)O (example: LaFeAsO)
122: (AE)(TM)2 (PN)2 (example: BaFe2 As2 )where RE=rare earth, TM=transition metal, PN=pnictide and AE=alkaline earth. To
realize superconductivity at high temperature, one has to dope carriers into the system; forexample, LaFeAsO1−x Fx for 1111, and Ba1−x Kx Fe2 As2 for 122. The third class is LiFeAs,
FeSe (11 system), etc. Most works on (1111) in this section refer to LaFeAsO1−x Fx .

5.5.2

Structure (1111 compounds)

The left figure in Fig. 5.9 shows the lattice structure of LaFeAsO. This material iscomposed of LaO layers and FeAs layers. Crudely speaking, the valence state in theparent compound is probably (La3+ O2− )+ (Fe2+ As3− )− ; La3+ and O2− have doubly closed
shell, Fe2+ has 3d6 , and As3− has closed shell. In some sense, it looks like the iron isoxidized, while Fe 3d and As 4p orbitals hybridize. If we substitute fluorine for oxygen,the embedded extra electron moves to the FeAs layer; if one substitutes by 10 percent(LaFeAsO0.9 F0.1 ), the carrier density is ∼ 1021 cm−3 , which again is comparable to thatof cuprates.
One interesting thing about the iron-arsenide compounds is that they obviously havevarious things in common with the cuprates. The most important point is the strong
two-dimensionality of the system. Another thing they have in common is the charge
reservoir, the atoms which originally supply an extra charge. In LaFeAsOF, for example,
the fluorines are substituted with the oxygens in the LaO planes, and they supply the extra
charge. On the other hand, superconductivity itself almost certainly occurs in metallic
FeAs layers. The charge reservoir is separated from the layers where superconductivity
occurs. That again we see is characteristic of the cuprates. 1111s are electron-doped,
which means the compound has surplus electrons, while 122s are hole-doped; this fact isinteresting in comparison with the cuprates.
10

For a review of high-temperature superconductivity in ferropnictides, see Ref. [25].

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Fig. 5.9. Lattice structures of ferropnictides.

5.5.3

Phase diagram

Figure 5.10 (a) shows the phase diagrams of ferropnictides. The parent compounds are
antiferromagnetic metal. In case (a), the crystallographic phase transition appears slightly
over the antiferromagnetic phase transition. In case (b), the 1st order phase transition
exists at the doping level p = 0.05, where the doping per unit formula p is equal to thestoichiometry x in this case (see Sec. 6.2 for more details). The carrier doping suppresses
the antiferromagnetic order, and superconductivity appears at the doping level x ' 0.05.
In case (a), near x ' 0.05, the superconductivity and antiferromagnetism coexist at low
temperature, while in case (b) there is no coexistence. The dependence of Tc on doping
is weak; maximum in Tc (x) at x ∼ 0.12-0.15 is shallow.
The phase diagrams of ferropnictides look quite similar to that of the cuprates (Fig. 5.10(b)). One also has a superconducting dome and an antiferromagnetic phase where thecarrier density is low. However, there are clear diﬀerences. In the cuprates, as we shall see
later, the antiferromagnetic phase is not a metal but rather a Mott insulator. Furthermore,in the underdoped region, an anomalous pseudogap phase exists.

5.5.4

Experimental properties (normal state)• CV ∼ αT + βT 3 ,• χ ∼ A + BT ,• ρ (DC conductivity) ∼ A + BT 2 ,• Hall coeﬃcient ∼ A + BT .

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LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

(a)

(b)
Crystallographictransition

Normalmetal

UD

Normal
metal
AF
metal

AF
metal
S
0.05

S

AF

OD

S
p

0.05

p

p

SG

optimaldopingFig. 5.10. (a) Phase diagrams of ferropnictides. (b) Phase diagram of the cuprates.
What are the properties of the normal phase? Crudely speaking, the specific heat CV
has the electron contribution αT and the phonon contribution βT 3 . The susceptibility χ
has a constant A plus BT . For the DC conductivity, so far, the anisotropy of resistivity
has not been measured carefully. People have measured a parent anisotropy rate, but
it is indirect and not totally reliable. Since this is a quite delicate point, it is advisable
to check up the up-to-date literature. For some reasons it seems diﬃcult, at least todate, to prepare these metals in a large single crystal form, and there has not been any
anisotropy measured. However, overall, the DC conductivity in the ab-plane is given by
A+BT 2 , where A is typical of impurities and B of the electron-electron Umklapp process.
Furthermore, at room temperature, the DC resistivity is ∼ 3 mΩ, which is comparable tothat of cuprates. The Hall coeﬃcient goes like A + BT , and its value is not dramaticallydiﬀerent from textbook values.

5.5.5

Band structure

What about the band structure? If one performs angle-resolved photoemission spectroscopy (ARPES), quantum oscillation experiments, and the local density approximation(LDA) calculation, one finds the Fermi surface shown in Fig. 5.11 (a). Two holelike Fermi
surfaces, which are not qualitatively diﬀerent, are around the Γ point (0,0); two electronlike Fermi surfaces are around the M point (π, π)11 (be careful about notation!). There isalso one fairly 3D band around Γ point.
There is one important point here. In 1111 systems, the Fermi surface is quite twodimensional (barrel-like), while in 122 systems, it is much more three-dimensional. Thissuggests that the Fermi surface changes its shape along the z-direction (Fig. 5.11 (b)).
Additionally, magnetism is much stronger in 122.

11

The number of Fermi surfaces varies depending on compounds, carrier concentrations, etc.

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M
electronlike
holelike

2D(1111)

3D(122)

Fig. 5.11. (a) Fermi surface of ferropnictides. (b) Fermi surfaces in three dimensions.

5.5.6

Superconductivity

When I asked if a metal containing iron could be a good high-temperature superconductor, in the past most people answered in the negative. In fact, iron does have magnetic
moment ∼ µB as is measured by neutron scattering, and this magnetic moment would bedestructive to superconductivity. However, the ferropnictides show superconductivity; Tc
rises up to ∼ 56 K in (doped) 1111, 38 K in 122, 20 K in 11. We also note that Tc is only
weakly dependent on the carrier concentration x (e.g., LaFeAsO1−x Fx ).

5.5.7

Experimental properties (superconducting state)

All ferropnictides are strongly type-II, with extrapolated Hc2 (0) ∼ 55T. This value
exceeds the Chandrasekhar–Clogston limit. The anisotropy is relatively small (∼ 23). The zero-temperature penetration depth and the zero-temperature pair radius are
λab (0) ∼ 1600-2400 Å, and ξab (0) ∼ 20 Å, respectively. ARPES shows that, on all sheets,
the superconducting gap is only weakly k-dependent.
What about the low-temperature behaviors of NMR and the penetration depth? Unfortunately, again experimental data in the literature are mutually inconsistent, but favor
power law. Importantly, the Knight shift goes to zero as T → 0 for all directions ofmagnetic field; this fact strongly indicates spin-singlet pairing. Isotope-eﬀect experiments
are mutually inconsistent, but most recent ones give a small value of αI ; this suggests thatthe mechanism is probably non-phonon.
Another reason why people tend to believe rather strongly it is not a phonon mechanism is that by now the actual theoretical techniques for calculating electron-phononsuperconductivity are pretty well evolved. Most experts in this area are fairly confident
that phonon mechanism by no means gives the critical temperature 55 K.

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5.5.8

LEC. 5. NON-CUPRATE EXOTIC SUPERCONDUCTIVITY

Pairing state

The Knight shift fairly strongly suggests the singlet spin state. Since the crystal has
a tetragonal symmetry, main candidates are nodeless s-state and dx2 −y2 -state with nodes
along the diagonal directions. We will come back to this point much more in detail in thecontext of cuprates. ARPES data suggests nodeless state, i.e., s-wave state. At low orzero temperature, however, various quantities such as T1 and λ favor nodes.
Again, the extra complication is that in this case there are several bands, which causes
the second Fermi surface. There is no particular reason why the gap must have the same
sign on all sheets of the Fermi surface. Theory based on spin-fluctuation mechanism(most obvious non-phononic mechanism) predicts that the gap changes sign between theelectronlike and holelike sheets (s± -wave). Experimental evidence claimed in favor of thisassignment: Josephson-like experiment on polycrystalline sample shows half flux quanta[26], which is characteristic to s± state. STM again showed indirectly that s± -state isfavored [27]. Unfortunately, there has been no true Josephson experiment yet; one needs
more carefully prepared single crystals. We may have them in the future.
Since ferropnictide superconductors have only 3-years history right now12 , one needs
more time to settle the mechanism of the pairing state. One thing I could say pretty sureis that it is spin-singlet. Beyond that, we have yet to answer.

12This lecture was done at The University of Tokyo in 2011.

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Lec. 6 Cuprates: generalities, andnormal state properties
When the cuprate superconductors were originally discovered in 1986, it was a great
surprise; they were the first materials to show reproducible superconductivity above 100
K. We know that at least 300 cuprates are superconductors and many of them, several
dozens, actually have transition temperatures Tc ∼ 110-120 K. One important point isthat, though most cuprate materials are superconductors often with very high transition
temperatures, there is a small class of cuprates which does not show superconductivity
under any condition.

6.1

Basic chemical properties

6.1.1

Composition

The chemical composition of the cuprates is very characteristic and typical. We here
introduce a very natural (but unconventional) notation:(CuO2 )n An−1 X(e.g., YBa2 Cu3 O7−δ ; n = 2, A = Y, X = BaCuO3−δ )A(intercalant)= alkaline earth elements (or rare earth elements, Y or La)

X(charge reservoir) = combination of elements, always including O
This notation reflects a characteristic structure of the cuprates.

6.1.2

Structure

Figure 6.1 (a) shows the side view of the cuprates with n = 2, in which we can seethat there are two copper oxide planes. Between them there are various materials X,
which always contain oxygens, and these oxygens are frequently called apical oxygens.
The reason for this nomenclature is that if we consider the positions of the apical oxygens
in the three-dimensional picture they sit in the apices of the octahedron with the oxygens
included in the CuO2 plane. The intercalants A sit between the planes in the case of morethan one copper oxide planes per unit cell.
A rather important characteristic of the cuprate superconductors is that we very often
find homologous series. The homologous series is a sequence of materials with the same98

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

(a)

(b)

Fig. 6.1. (a) Side view of cuprates with n = 2. (b) Top view of cuprates with n = 2.
intercalant A and the same charge reservoir X but with a diﬀerent number of layers (e.g.,
Tl2 Sr2 CuO6 , Tl2 Sr2 CaCu2 O8 , Tl2 Sr2 Ca2 Cu3 O10 ,· · · ), which is schematically shown in theFig. 6.2 (a).
The superconductivity in the cuprates occurs in the neighborhood of what is calledparent compounds. We can often start (for a given X) with a stoichiometric compound(“parent” compound) when valences balance, i.e., the total charge becomes neutral;
La2 CuO4 = (La3+ )2 Cu2+ (O2− )4 ,YBa2 Cu3 O6 = Y3+ (Ba2+ )2 (Cu2+ )2 Cu+ (O2− )6 .
The state of the copper in the plane is Cu2+ : the copper starts with a (3d10 )(4s)1 electronstate and loses two electrons and then becomes a 3d9 , so that there is one hole in thed-shell per formula unit. This is typical for all the parent compounds of the cuprates.
If we have a structure of this type, then we have an odd number of electrons per unit
cell. Naively thinking, the system ought to be a metal. However, the system becomes anantiferromagnetic Mott insulator (Fig. 6.2 (b)): the electrons are localized at the copper
atoms and order antiferromagnetically.

6.2

DopingSuppose one starts with a “parent” compound, say La2 CuO4 , and replaces a fraction xof La with Sr per unit cell, i.e.,
La2 CuO4 → La2−x Srx CuO4 .
Since La has valence 3+ and Sr has 2+, the eﬀect of this replacement is to add −x
electrons, i.e., x holes, per formula unit. In this case, the holes have nowhere to go but
to the CuO2 plane, and hence we are fairly confident that p, the doping per CuO2 unit,
is equal to the chemical stoichiometry x.
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(a)

(b)

Fig. 6.2. (a) Homologous series. (b) Antiferromagnetic Mott insulator.
Alternatively, let us consider adding δ oxygen atoms per formula unit,
La2 CuO4 → La2 CuO4+δ .
In this case, since the valence of O is 2−, the eﬀect is to add 2δ holes per CuO2 unit, i.e.,p = 2δ. In multiplane materials (n > 1), one must remember that any added holes have tobe shared between the n planes per unit cell. Hence, naively, for example YBa2 Cu2 O6+δ ,we expect p = 21 (2δ) = δ. However, in this case there are other complications because
the added holes can go elsewhere than to the CuO2 planes (e.g., in the “chains”). The
chains are very specific for this compound, and then this is a complication special to thismaterial.
What can we guess about the orbit(s) occupied by theholes in the CuO2 plane? There is a general belief basedon the quite reliable band structure obtained from atomic
physics calculations; both the original one hole per CuO2
unit in the parent state and any added holes by the doping
occupy mainly the 3dx2 -y2 orbital. However, this orbital may
be somewhat hybridized with the px or py states on theneighboring oxygens (Fig. 6.3).
A vast majority of the cuprate superconductors are holeFig. 6.3. Hybridization
doped; if we start from the parent compound we add notof the orbits.
extra electrons but extra holes. However, there are alsoa few electron-doped cuprates. The best known electrondoped cuprates is NCCO (Nd1−x Cex CuO4 ). Properties of the electron-doped materials
are qualitatively very similar to the hole-doped ones [1, 2].

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T
Antiferromagnetic

Underdoped
Overdoped
Superconductivity

0.04 0.05

0.16

Spin glass

Optimal doping

0.27

p

Fig. 6.4. Phase diagram of cuprates. The doping dependence is universal, and the temperature dependence (i.e., Tc,max ) is strongly material-dependent.

6.3

Construction of phase diagram

There are various factors controlling the interesting behavior of the cuprates, but thetwo most important ones are temperature T and the number of holes p per CuO2 unit in
the plane. How do we fix p? There are two possibilities:(a) All added holes are known to go to CuO2 planes (e.g., La2−x Srx CuO4 ).
(b) The destination of the added holes are ambiguous (e.g., YBCO).
In the former case, we can simply fix p from x (or δ). However, in the latter case, weshould fix p pragmatically. In most cases, what one usually does in practice is, if there
exists another cuprate corresponding to the case (a) and an accessible region in the phase
diagram has some overlap with it, we fix p so that we get the same p-dependence. This isimportant because there is no cuprate which can be explored in the whole range of p thatis interesting to us. Therefore, we have to overlay diﬀerent materials in the phase diagram.
This might seem to be a little bit skew, but so far this is consistent with experiments.
Phase diagram of the cuprates
The phase diagram of the cuprates is shown in Fig. 6.4. As one goes from a Mott
insulating parent compound to small values of the doping p, it remains antiferromagnetic.
However, at the doping p ' 0.04, the system comes into a mysterious phase, usually
thought of a spin glass phase. Then the superconducting phase appears roughly between
0.05 < p < 0.27. To some extent, this universality of the p-dependence is a matterof construction because we simply decided to measure p so that the p-dependence isconsistent. At least everything goes well and is consistent in this way; the p-dependence
is universal and always the superconducting dome starts at p = 0.05 and terminates at
p = 0.27. On the other hand, the dependence on the T -axis is rather non-universal;
for Bi-2201 Tc,max ∼ 10 K, while for Hg-2223, one with the highest Tc , the transition101《講義ノート》

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

strangemetal

T

Antiferromagnetic䐠pseudogap䐟Superconducting

0.04 0.05

0.16

0.19

Fermi
liquid0.27 pFig. 6.5. T ∗ -line: locus of “crossover” behavior in various experimental quantities (CV ,χ, ρc ,· · · ).
temperature can be as large as Tc,max ∼ 160 K under 20 GPa. This problem is a huge
mystery.
In the region p > 0.27, superconductivity does not appear and the system is welldescribed by the genuine Fermi-liquid theory. On the other hand, in the region 0.04 <p < 0.05, it is far from the Fermi-liquid state.
Here is an interesting question: let us consider a virtual path in the phase diagram whichgoes above the superconducting dome and connects these two regions (the dotted arrow in
Fig. 6.5). Is there any point of a phase transition or any steep crossover? Apparently, there
is no actual phase transition, but there is a so-called T ∗ -line. This roughly corresponds
to a relatively sharp crossover line.
Although there is much controversy, most people claim that the Fermi-liquid region
first changes to the strange metal region, and the pseudogap regime appears between theantiferromagnetic phase and the superconducting dome. Roughly speaking, there is a
sharp crossover in various physical quantities between the strange metal regime and thepseudogap regime.
There are some interesting and controversial questions related to this T ∗ -line. The firstone is about the property of the T ∗ -line;
1. Does it really correspond to a hidden phase transition not seen apparently or just acrossover?
Another one is about the low-T behavior;
2. Does the T ∗ -line (i) intersect the superconducting dome, or (ii) join smoothly onto
the Tc (p) line?
This question has arisen because it was very diﬃcult to change p until quite recently. Now
there is an argument which strongly favors the option (i) [3].

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Table 6.1. Transition temperatures of various homologous series.

6.4

Common nameX groupn=1

Hg-12,n − 1, n
Tl-22,n − 1, n
Tl-12,n − 1, n
Pb-22,n − 1, n
Bi-22,n − 1, n

HgBa2 O3+δTl2 Ba2 O4
TlBa2 O3−δPbSr2 O3+δBi2 Sr2 O4

98
126 135 125 110
95
118 125 112 10570(?) 103 123 112 107∼40 97(?) 122 107
10-20
89
107

2

3

4

5

Determinants of Tc

As noted, Tc,max varies enormously between diﬀerent cuprates. What else determines
Tc other than the doping p? Correlations with• the degree of the orthorhombic anisotropy,• the degree of buckle of the planes,• the distance to apical the oxygens,• the chemical environment of the planes,
have been suggested, but none of them is “overwhelming”. On the other hand, in thehomologous series, we can see a universal dependence of the maximum transition temperature Tc,max on the number of CuO2 planes n, as shown in Table 6.1. As n goes to twofrom one and to three from two, Tc,max also goes up. Beyond n = 3, Tc,max decreases as nincreases. Note that there is often a relation ∆Tc(2−3) ' 31 ∆Tc(1−2) .
I think probably it is not significant that Tc falls beyond n = 3. For all diﬀerent
members in the homologous series, we need to dope all of the planes, and if we get too
many planes then they are just run out of the doping. Therefore, I suspect that thefeature of the drop in Tc beyond n = 3 may be rather trivial.
There is one digression I just want to make about “the dogs which did not bark in thenight-time”. This refers to the class of the cuprates which never become superconducting.
There is a little-noticed fact that I think very interesting; without exception, all of these
non-superconducting cuprates are either bi/tri-layer with alkaline-earth spacers (Sr, Ba)
which are heavier than Ca. I think this is not due to the basic eﬀect of the replacement

Fig. 6.6. Schematic illustration of the intruder O atoms.

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Table 6.2. Notable features of various cuprates.

CompoundCommon name

Notable features

La2−x Srx CuO4
YBa2 Cu3 O6+δ

LSCO
YBCO

Bi2 Sr2 Can−1 Cun O(n+2+δ)BSCCO

HgBa2 Cun Can−1 O2(n+1)
Nd1−x Cex CuO4
Ca1−x Srx CuO2

HgBCO
NCCO
∞-layer(almost) earliest HTS, large crystals
first liquid-nitrogen temperature superconductors whose Tc ∼ 90 K, “E. coli” ofcuprates“baklava”, good for ARPES, EELS, etc.
most anisotropic high-Tc superconductor
known.
record holder for Tc (∼ 160 K at 20 GPa).
electron-doped
no reservoir group X, infinite set of CuO2
planes seperated by alkaline earth ions
Ca/Sr.

of Ca with Sr or Ba, but rather due to the eﬀect of what I call the “intruder” oxygens.
This is shown schematically in Fig. 6.6.
Let us start from the bilayer La compound where the intercalant is Ca. This is asuperconductor with Tc ∼ 55-60 K. Then, we replace Ca by Sr, which is slightly bigger
than Ca. The larger Sr intercalant atoms widen the distance between the CuO planes,
and the oxygens are more likely to enter the aperture between the copper oxide planes,
which depresses the transition temperature Tc . However, no one knows the reason why
the intruder oxygens are so destructive to superconductivity.

6.5

Other remarks: carrier density and list of cuprate
superconductors

At the “optimal” doping (p ' 0.16), the density of the (extra) holes in the plane isabout 1.5 × 1014 cm−2 . Hence the three-dimensional density is around 1-2 × 1021 cm−3
and interestingly that is quite comparable to strontium-ruthenates, ferrropnictides,· · · .
This is rather interesting, since the high-temperature superconductivity does seem torequire a rather low electron density.
Before we go into experimental techniques for the cuprates, here we list some interesting
cuprates in the Table 6.2.

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6.6

LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

Experimental properties of the normal state: general discussion

First of all, unlike most of the other superconductors we are familiar with, the propertiesof the cuprates do seem to be very strongly dependent, not just on the doping, but on
the impurities. For example, if we deliberately put Zn into a place of some of the copperin the CuO planes, this in fact has an enormous eﬀect on the superconductivity. In theearly days, arguments about some of the early experiments were completely spoilt by the
impurity eﬀect. We have to always bear in mind that, in particular with new experimental
techniques, the experimental facts presented here are subject to revision.
Secondly, it is easy to vary temperature T at a fixed doping p, but it is much morediﬃcult to vary p at a fixed T . Quite recently, people have started taking up a new kind oftechniques; they prepare thin films of the cuprates, and then try to dope them electrically.
However, one has to be cautious because the doping in this case only extends to the first
few layers, and we may be probing the surface property rather than the bulk property.
Thirdly, many types of experiments can only be done on particular cuprates. For
example, the ARPES is typically done on BSSCO and YBCO, and the neutron scatteringon LSCO and YBCO. We always have to bear in mind that even if we have measured
some spectacular properties on one cuprate, we are not sure whether they apply to all.
Generally speaking, people tend to assume it does, but there is no warranty and sometimes
naive application is quite dangerous.
It turns out that some regions of the phase diagram are relatively conventional. First,
the least controversial phase is the Mott antiferromagnetic insulator phase in the lowdoping region. Our knowledge suggests that this is just a standard antiferromagnetic
insulator described by the nearest-neighbor Heisenberg Hamiltonian∑Ŝ i · Sˆj ,(6.1)
Ĥ = J
hiji

with Jin-plane ∼ 1000 K and Jbilayer ∼ 200 K. This is consistent with experimentallyobserved TN ∼ 300 K at p=0 for LSCO, 500 K for YBCO. YBCO has a higher Nèeltemperature than LSCO, probably because YBCO has two CuO layers, while LSCO has
only one.
Second, in the overdoped regime again, most people agree that the system is approximately a standard Fermi liquid; the system becomes more conventional as one dopes
more in the overdoped regime. Finally, in the narrow spin-glass region of the doping
0.04 < p < 0.05, a few experiments suggest the resistivity ρ(T ) increases without limit as
one decreases the temperature; if one could reach T = 0, which is impossible in real life,
the system would be a perfect insulator. This behavior is rather similar to what we findin disordered granular films.

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6.7

Experimental properties at the optimal doping

6.7.1

Electronic specific heat

Let us move on to the controversial regimes. First of all, let us consider the optimal
doping p ∼ 0.16, where Tc takes its maximum value. What about the thermodynamic
properties? It is known that the electronic part of the specific heat behaves as CVel = γTat least up to the room temperature [4]. The coeﬀcient γ per CuO2 unit is about γ ' 6.5
mJ/mol(CuO2 )K2 , which is almost the same for LSCO, YBCO, and Tl-2201. If one
interprets the value with the Fermi liquid theory, the density of states is given by
N (0) ' 1.4 eV−1 spin−1 (CuO2 unit)−1 .

(6.2)

This value is about four times the two-dimensional free electron value ma2 /2π~2 . The
specific heat above Tc does appear to be consistent with the Fermi liquid model with themass enhancement m∗ /m ∼ 4. Given the tight-binding structure of the cuprates, thismass enhancement seems quite reasonable.

6.7.2

Magnetic properties

The Pauli spin susceptibility χ, determined mostly from the Knight shift on YBCO,
is approximately independent of T [5]. There is one interesting point here; the in-plane
Knight shift for Cu is much larger than that for Y or O. This is significant for people
who believe in the spin fluctuation theory. The nuclear spin relaxation rate T1 is inversely
proportional to the temperature, i.e., T1−1 ∝ T , which is known as the Korringa relation.
So far, all the properties of the normal phase at the optimal doping appear to be prettymuch Fermi-liquid like.

6.7.3

Transport

Cuprates are very strongly layered materials. Almost all the transport properties arestrongly anisotropic. Quoted results below are values in the ab-plane unless otherwise
stated [5, 6].
If one measures the DC resistivity of the cuprates at the optimal doping, all the cuprates
have ρ ∝ T from T ∼ 800 K all the way down to Tc (down to T ∼ 10 K for Bi-2201).
Above 800 K the situation becomes more complicated because the oxygens get disturbed,
and thus one usually does not think about that regime. Except for this high-temperature
regime, the resistivity appears to be exactly linear to the temperature, which is quite aremarkable fact.
What happens if one goes away from the optimal doping? It turns out that the resistivity generally behaves as ρ ∝ T α , where α varies continuously. In the overdoped limit,106

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α = 2, which is consistent with the Fermi-liquid theory if the electron-electron Umklapp
scattering is taken into account. If one moves from the overdoped regime to the optimal
doping regime, α changes from 2 to 1. If one moves further into the underdoping regime,
α becomes smaller than 1 and finally less than 0.
If one thinks about the conductivity rather than the resistivity, since the conductivity
is simply a sum of contributions from diﬀerent planes, one can define the conductivity
per CuO plane. One obvious question is whether this value is universal for all cuprates.
The answer is generally no. However, for higher-Tc materials, the resistivity per plane,i.e., the inverse of the conductivity per plane, seems to be approximately universal
RR.T. ∼ 3 kΩ ∼ 0.12RQ ,(6.3)

where RR.T. and RQ are the resistivity at the room temperature and the quantum universal
resistance, respectively1 . In the lower-Tc materials, on the other hand, the conductivity,
in general, is much smaller. This may be rather trivially understood from the following
argument: almost universally2 in all lower-temperature cuprates, the dopants sit close to
CuO2 planes. Therefore, they provide extra scattering mechanisms.
Can the ρ ∝ T law at the optimal doping be explained by the phonons? It is wellknown that at least at high temperatures, ordinary metals are described by the Fermiliquid theory and show a linear-T dependence for T & ΘD . This resistivity is caused by
the scattering of electrons by phonons. Is this what is going on at the optimal doping?
Almost certainly not. First of all, the Debye temperature of most cuprates is typically
about the room temperature, but the linear behavior is still exactly linear down to far
lower temperatures than the Debye temperature3 . Furthermore, in a naive calculation
based on what we know about the phonon spectrum and the electron-phonon couplingin the cuprates, it looks as if the phonon contribution should give us a resistivity larger
than this. Thus, the phonon mechanism seems to be unable to explain the linear law, andsomething more subtle is going on here.
As for the AC conductivity σ(ω), on the other hand, the following Drude form is wellknown for ordinary metals
ne2 τ /mσ(ω) ∼.
(6.4)
1 + iωτWe can therefore ask whether the resistivity of cuprates fits well with this formula. The
answer is yes, only if the relaxation time τ is allowed to be a function only of ω, with the
behavior
τ (ω) ∼ max(ω, kB T /~).
(6.5)
1

In in two dimensions, the resistivity has the same dimension as the resistance.
Bi-2201 seems to be a special exception.
3
One should be cautious here; it is known that in some textbook metals like Re, the linear law ofresistivity persists well below ΘD . Re has the Debye temperature 300 K and the linear behavior persists
down to 75 K. Thus even if the linear behavior is observed down to a far lower temperature than the
Debye temperature, it is not a suﬃciently convincing argument against phonons.
2107

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIESThe early model of the normal state of cuprates, the so-called marginal Fermi liquid
theory, is essentially based on the same assumption.
For the Hall angles for pure samples in the high-field ∼ 8 T, one normally finds
cot ΘH ∝ T 2 .

(6.6)

A rather odd thing to be remarked; if one plots the thermoelectric power at low temperatures, it does not look so spectacular. If one plots it as a function of the doping, however,
the room temperature value, as a function of the doping, crosses zero almost exactly atthe optimal doping for all the known cuprates [7]. In fact, if one wants to know whether acertain material at the room temperature becomes a superconductor, the thermoelectric
power at the room temperature can thus be a good criterion.
So far, we have talked about the ab-plane transports. What about the c-axis? The
c-axis resistivity at the optimal doping is rather interesting. It always appears to vary as
ρ(T ) ∝ T α .

(6.7)

However, the power α can range from −1 to +1 for diﬀerent cuprates, which is quite
puzzling. There have been a large number of papers trying to explain this fact.

6.7.4

Spectroscopic probes: Fermi surface

If one is dealing with some materials, the obvious question is whether they have any
Fermi surface and, if they have, what the Fermi surface looks like. Two most usefulprobes of the Fermi surface in a metal are angularly resolved photoemission spectroscopy(ARPES) and quantum-oscillation phenomena.
In the late 1980s, it was recognized that ARPES should be a particularly nice way for
examining cuprates4 . In ARPES, one shines light with a given frequency, and measures themomentum and energy of the electrons kicked out. If one deals with a three-dimensional
metal, there would be a slight problem there; although the momentum is conserved in
the transverse direction, the component normal to the surface is likely to change whenthe electron emerges from the metal. In the cuprates, very luckily, one can essentiallytreat it as a two-dimensional problem, and the above problem does not appear. ARPES
essentially measures the spectral function A(k, ε), the probability of finding an electronwith its momentum k and energy ε in the thermal equilibrium state. For non-interacting
electrons, it would be proportional to δ(ε − εk ), because if one electron has a Bloch wave
k it has the definite energy corresponding to the wave vector. One expects that thecoeﬃcient of the delta function is essentially 1 when k is inside the Fermi surface, and 0
for the other. However, this is not the case, in general, for interacting cases including thecuprates.
4

For a review of ARPES studies on cuprate superconductors, see Ref. [8].

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIESThe quantum-oscillation phenomena occur in thin materials under a high magnetic
field, in which various quantities such as the magnetization and the resistivity oscillate
as a function of the magnetic field. The oscillation is not periodic in the field itself but
rather in the inverse of the field. Let us refer to the class of these eﬀects as de Haas-van
Alphen (dHvA) eﬀects. The quantum oscillations were, for a long time, thought to bealmost impossible in the cuprates. However, in the last three years or so, researchers
have succeeded in doing dHvA experiments. The dHvA-type experiments measure thearea(s) of those parts of the Fermi surface corresponding to the closed orbits, the classicalmotion in the magnetic field. One cannot say anything about their shape or position
through dHvA experiments. Because of the strongly two-dimensional (layered) nature of
the cuprates, the magnetic field is always applied along the c-axis, and it simply measures
the areas of the two-dimensional Fermi surface(s), which is a much simpler situation thanin three-dimensional metals.

6.7.5

Results of ARPES experiments at the optimal doping

First of all, A(k, ε) does not look like δ(ε − εk ) at all; an incoherent background seems
to be ∼ 90% of the total weight. We do get the peak but it is just 10% of the total
weight. Crudely speaking, this result suggests that if indeed the cuprates at the optimal
doping are well described by the Fermi liquid theory, they are very bad ones. However, theenergy-integrated function, which indicates whether the state k is occupied, does show
a jump (∼ 10%) as a function of |k| for a given direction n̂. Since the Fermi surface
is defined as the locus of the points where the discontinuity occurs, this fact gives us awell-defined Fermi surface. This function for cuprates at the optimal doping is shown in
Fig. 6.7. Filled states are located at the center of the first Brillouin zone, and the hole-like
Fermi surface is located at the zone corner (π, π). Luttinger’s theorem states that thevolume enclosed by the Fermi surface is directly proportional to the particle density, andit allows us to calculate the hole density: nh ∼ 1.19 (∼ 1 + p, as naively expected). This is(π, π)

Filledstateskykx
Fig. 6.7. Energy-integrated function of the cuprates.

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

a rather important result; at least, at the optimal doping, it is consistent with the Fermi
liquid picture. Moreover, the number of electrons occupying the Fermi sea is exactly 1
plus p, where the former contribution (1) is originally there in the parent compound,while the latter one (p) is added by the doping. It turns out that dHvA experiments ona somewhat overdoped side are consistent with this result.

6.7.6

Neutron scattering

Neutrons couple mainly to electron spins, and the cross-section σ(q, ω) measures the
spin-fluctuations. For fixed q, the cross-section as a function of ω in the normal state
has no marked structure. However, for fixed ω, the cross-section as a function of q has
a marked peak at q = (0.5π/a, 0.5π/a) [9]. Importantly, the q value is precisely equal
to the Bragg vector of the magnetic superlattice in the antiferromagnetic phase, which ofcourse is measured independently. This suggests that the strong antiferromagnetic spin
fluctuations, which are there in the original pure antiferromagnetic phase, do seem topersist quite strongly into the non-magnetic phase.

6.7.7

Optics (ab-plane)

Most of the direct experiments measure the optical reflectivity R(ω) [9]. Unfortunately,
the optical reflectivity measures a raw and nasty combination of the real and imaginaryparts of the dielectric function:R(ω) =(1 − Re ε(ω))2 + (Im ε(ω))2.
(1 + Re ε(ω))2 + (Im ε(ω))2(6.8)

To obtain anything about the individual part, one can exploit the fact that the real andimaginary parts are related by the Kramers–Kronig relations:∫ ∞dω 0 Im ε(ω 0 )Re ε(ω) − 1 = P
,
(6.9)
0
−∞ π ω − ω∫ ∞dω 0 Re ε(ω 0 ) − 1
Im ε(ω) = −P.
(6.10)
ω0 − ω
−∞ π
Using either of these equations, and with a knowledge of R(ω) in all frequencies, one can,in principle, obtain Re ε and Im ε.
There is, however, a more direct way to measure these optical quantities. In the last
few years, two or three groups have done ellipsometric measurements. In this method, we
can directly measure the complex dielectric constant ε(ω) individually without using the
Krammers-Kronig relation.
The most important quantity in the optical data is the loss function,
L(ω) = − Im
110

1
.
ε(ω)

(6.11)

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(a)

0.3

LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

(b)

“MIR”
peak

1

~250K
Underdoped

T

T*
Antiferromagnetic

0.1

materialdependent

10

Pseudogap
Superconductingp(log scale, eV)
Fig. 6.8. (a) Loss function L(ω). (b) Phase diagram in the underdoped regime.
Can we expect this quantity to be universal between cuprates? First of all, note thatε(ω) is a three-dimensional quantity, and therefore, it is sensitive both to the CuO2 planedensity and the charge reservoir contribution to Re ε. The charge reservoir is, generally
speaking, pretty insulating. It is not likely to make much contribution to the measured
properties. The shape of the loss function in logarithmic scale is given in Fig. 6.8 (a).
Surprisingly, it looks as if the mid-infrared (MIR) peak, which is always seen in optics inthe cuprates, is only weakly material-dependent.
The behavior is not much interesting below 0.1 eV. Between 0.1-1 eV, one always obtains
a strong and broad MIR peak; this peak is very characteristic of all cuprates, including
non-superconducting ones. Then L(ω) drops drastically. The energy value at which L(ω)
takes the minimum value, typically 1-2 eV for the cuprates, roughly corresponds to theplasma frequency. After that, L(ω) again rises; this behavior is material-dependent.
Not just optics show this behavior. Incidentally, this behavior is consistent with theelecron energy-loss spectroscopy (EELS) experiments [10], which directly measure theloss function L(q, ω) of impinged electrons through the interaction with a solid. In thenormal phase, one sees the MIR peak up to q ∼ 0.1 Å−1 , beyond which it looks somewhat
attenuated.

6.8

Experimental properties at the underdoped regime

6.8.1

Pseudogap

Let us move on to the underdoped regime [11]. The pseudogap region as shown in
Fig. 6.8 (b) is, from an experimental point of view, not superconducting. Presumably, the111

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

system does not have any long-range order5 . However, there are considerable evidencesthat below the T ∗ -line the fermionic excitation spectrum has an energy gap similar tothat of the superconducting phase. Some of the evidences are listed below:(a) Specific heat: CV ∼ γT for T T ∗ , but for T ≤ T ∗ it drops well below the T -linear
behavior [12].
(b) Static spin susceptibility from the Knight shift: χ ∼ const. for T  T ∗ , and it
drops below that value for T ≤ T ? . Interestingly, if we define the so-called Wilson
ratio S/χT (rather than CV /χ), it is almost independent of T for all T .
(c) Nuclear spin relaxation rate: T1−1 ∝ T (Korringa law) for T  T ∗ , and it falls below
the T -linear behavior for T ≤ T ∗ .
(d) Transport: both the in-plane resistivity ρab (T ) and the c-axis resistivity ρc (T ) dropfor T ≤ T ∗ at frequency ω ≤ 500 cm−1 (corresponds to ∼ 750 K).
What do all of these suggest? Remember that the resistivity is certainly always limitedby the electron-electron scattering. If the resistivity drops, this means less scattering,or less carriers, which then suggests a reduced DOS below E ∼ a few 100 K. However,this does not directly suggest an actual energy gap. The direct observation of the finite
energy gap requires diﬀerent kinds of experiments. If we perform experiments at the
optimal doping, we find nothing spectacular, but experiments at the underdoped regime
give us something unconventional:(e) Tunneling: a gap-like feature is seen in tunneling characteristics, above Tc and up
to the room temperature. The tunneling characteristic of metals between normal
metals like Cu and cuprates in the underdoped regime looks somewhat similar to

(a)

(b)
Bi2212

0

N(ε)

N(ε)

ε
εF
Normal state∆εεFSuperconducting stateFig. 6.9. (a) Tunneling result for the underdoped cuprates. (b) Gap opening in the densityof states at εF .
5

I will come back to the topic about the precise relationship between the superconductivity and thelong-range order in Sec. 8.2.

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the standard tunneling result of normal-metal superconducting materials (Fig. 6.9(a)).
(f) ARPES (see below): can detect the feature more directly.
There are some recent and somewhat controversial evidences for the spontaneous breaking of the rotational π2 invariance in the pseudogap regime. We may get a stripe-like structure then [13]. Also, controversially, some researchers claim the spontaneous violation oftime-reversal symmetry.

6.8.2

ARPES in the pseudogap regime: the puzzle of the Fermi
surface

Let us discuss ARPES in the pseudogap regime. How the Fermi surface behaves in
the pseudogap regime? In the superconducting state at the optimal doping, the diﬀuse
normal-state ARPES spectrum develops into a sharp peak which is pulled well back fromthe Fermi energy (Fig. 6.9 (b)). The n̂-dependent diﬀerence is usually taken as a measureof the gap; it is of dx2 −y2 form, i.e., 0 at (π, π) and largest along the crystal axes. We will
come back to this point in the next lecture. At the optimal doping and on the overdopedside, the ∆(n̂, T ) so defined appears to vanish for T > Tc . However, in the underdoped
regime, a gap of similar dx2 −y2 form is seen in the superconducting state and persists
above Tc [14, 15]; ∆(n̂, T ) appears to be more or less T -independent, but the amplitudeof the peak decreases, and it disappears around the room temperature.
There is a puzzle here; ARPES data appear to show (modulo the gap) that even in
the psudogap regime the Fermi surface is well-defined and is quite similar to that seen at
the optimal and overdoped values of p, i.e., the large (∼ 1 + p) Fermi surface (Fig. 6.7).
Unfortunately, the dHvA oscillations seem to indicate equally definitively that the Fermi
surface(s) is very small. Supposing 4-fold rotational symmetry, because dHvA experiments do not provide information about the exact locus of the Fermi surface in the
momentum space, one possible case is that the small pocket-like Fermi surfaces appear
around (± π2 ,± π2 ) (Fig. 6.10). Even under this assumption the total hole concentration in
the Fermi sea would be only ∼ p, not 1 + p.
One possible resolution is that, because dHvA has to be conducted at high magnetic
field (∼ 20 T), it might qualitatively change the nature of the ground state. Rememberthat in the overdoped regime, dHvA and ARPES show a large Fermi surface. This fact
seems to imply that there is a qualitative change in the Fermi surface when one sweeps
across the phase diagram from the overdoped regime to the underdoped regime. At least
under large fields, there must be a discontinuity or a crossover in the qualitative natureof the ground state as a function of the doping.
Here are obvious questions; where exactly does the change occur? What kind of change
is this, a general phase transition or a crossover? It is diﬃcult to avoid the following113《講義ノート》

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LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES

kykx
Fig. 6.10. A possible Fermi surface in the underdoped regime according to dHvA experiment.

conclusion: unfortunately, to the best of our knowledge, the existing dHvA experiments
cannot answer them. This is certainly something which researchers must be working onin the next few years.

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References[1] Question from a student: You said that there is a sequence with diﬀerent n and thenthe conducting layers will be accumulated and get more three dimensionality. Then,
how would Tc change? Does the more three dimensionality indicates higher Tc ?
Answer: Yes, indeed. You are absolutely right. I will come back to this later in moredetail, but, generally speaking, almost always if I add more layers it does increase Tc
up to n = 3 and after it goes down.
[2] Question from a student: So far, in the high-Tc superconductors such as the cuprates
and the ironpnictides, there are two layers, the superconducting layer and the insulating layer. Is that a requirement for the high-Tc ?
Answer: A nice question. I would say that we know not only the cuprates and theferropnictides but also the organics, which are high-temperature superconductors insome sense. It may be important that the metallic planes are separated by the insulating layers, but I think in some sense more important point is that the places which
supply the doping are separated from the CuO2 planes or the equivalent ones for theother compounds. For example let us go back to LSCO, which is not a very goodhigh-temperature superconductor: it has the transition temperature only about 40K.
I think the reason is, at least for the large part, that the doped Sr sits actually rightabove the CuO2 planes. Therefore, it acts as the impurities which scatter the electrons. In most of the rather high-temperature material, such as bismuth, thallium,
mercury compounds, the doping sits away from the copper oxide planes. I suspect
this is a rather trivially important point for the high-Tc .
[3] Question from a student: Do you mean that there are two phases in the superconducting dome because of T ∗ -line intersecting the superconducting phase?
Answer: No. As far as we know at least, the T ∗ -line corresponds to the relatively
sharp quantitative crossover but not the phase transition. Unfortunately, many of theproperties which undergo discontinuities here tend to vanish in the superconducting
phase. So it is not easy to follow this transition. What one can do is to apply thestrong magnetic field6 in order to suppress superconductivity, and for the low Tc
compounds that is possible. Then, you will rather see a crossover as a function of thedoping.
One can also track the T ∗ -line in the superconducting dome by adding zinc instead of applying themagnetic field to suppress the superconductivity [16].
6115《講義ノート》

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A. J. Leggett

LEC. 6. CUPRATES: GENERALITIES, AND NORMAL STATE PROPERTIES[4] J. Tallon, T. Benseman, G. Williams, and J. Loram, Physica C 415, 9 (2004).
[5] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. Endoh, Rev. Mod. Phys. 70,
897 (1998).
[6] S. Ono, Y. Ando, T. Murayama, F. F. Balakirev, J. B. Betts, and G. S. Boebinger,
Phys. Rev. Lett. 85, 638 (2000).
[7] S. D. Obertelli, J. R. Cooper, and J. L. Tallon, Phys. Rev. B 46, 14928 (1992).
[8] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
[9] M. R. Norman and C. Pépin, Rep. Prog. Phys. 66, 1547 (2003).
[10] N. Nücker, H. Romberg, S. Nakai, B. Scheerer, J. Fink, Y. F. Yan, and Z. X. Zhao,
Phys. Rev. B 39, 12379 (1989).
[11] J. L. Tallon and J. W. Loram, Physica C 349, 53 (2001).
[12] J. W. Loram, K. A. Mirza, J. M. Wade, J. R. Cooper, and W. Y. Liang, Physica C
235-240, 134 (1994).
[13] C. V. Parker, P. Aynajian, E. H. da Silva Neto, A. Pushp, S. Ono, J. Wen, Z. Xu,
G. Gu, and A. Yazdani, Nature 468, 677 (2010).
[14] C. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki, and O. Fischer, Phys. Rev. Lett.
80, 149 (1998).
[15] A. Matsuda, S. Sugita, and T. Watanabe, Phys. Rev. B 60, 1377 (1999).
[16] S. H. Naqib, J. R. Cooper, J. L. Tallon, R. S. Islam, and R. A. Chakalov, Phys. Rev.
B 71, 054502 (2005).

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Lec. 7 Cuprates: superconducting
state properties
I want to spend most of this lecture on the properties of cuprates in the superconducting
phase. This has been the major topic of research over the last 25 years. One should bearin mind that like the normal state properties, the properties in the superconducting states
do tend to be somewhat doping-dependent. They are actually less doping-dependent than
the normal state properties. I will quote results at the optimal doping unless otherwise
stated. To give you an orientation, I will compare them with the BCS predictions wherever
possible.

7.1

Experimental properties

7.1.1

Structural and elastic properties and electron density distribution

These quantities are measured, for example, by the X-ray scattering. As far as weknow, they do not change substantially at Tc or below, as in the BCS prediction.

7.1.2

Macroscopic electromagnetic propertiesi) Strongly type-II.
ii) Strongly anisotropic.
The penetration depth along the c-axis1 is much greater than that in the ab-plane,
λc  λab . This means that the pair radius, or the coherence length, is far less in thec-axis direction ξc  ξab .
We can actually measure the upper critical field fairly close to Tc [1]. If we extrapolate2
it to T = 0, their values are Hc2 (0) ∼ 50 T for H || c, and ∼ 400 T for H ⊥ c. If
we interpret these results by the BCS theory, the corresponding coherence lengths areξab (0) ∼ 15-30 Å, and ξc (0) ∼ 2-3 Å. Note that ξc (0) ∼ 2-3 Å is smaller than the1
The penetration depth in the c-axis corresponds to the current flowing along the c-axis. That does
not mean the penetration of the magnetic field in the c-axis direction.
2
This is much greater than the field we can obtain with the existing magnets, so these are justextrapolations.

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

actual spacing of the unit cell dimension in the c-axis. Notice, incidentally, that there isanother way within the BCS theory of getting the coherence length from the transition
temperature: ξab (0) = 0.18~vF /kB Tc . The estimate done by this formula agrees well withthe one obtained from the upper critical field (if we use m∗ /m = 4).

7.1.3

Specific heat and condensation energy

There is a jump in the specific heat at Tc , as expected by the BCS theory. The amountCs − Cnof the jump is(Tc ) ∼
= 1.6-2 [2], which is somewhat larger than the BCS value 1.4.
Cn
At the optimal doping, the shape of the peak is relatively sharp, while in the underdoped
and overdoped regimes, the peak is considerably more rounded. That is to be expectedon the underdoped side, but it is puzzling on the overdoped side.
For T < Tc , the specific heat falls oﬀ sharply, as in the BCS theory. In contrast, for
T → 0 it goes as a power law, probably T n . It is probably consistent with n = 1 (but I
think it is dubious. I will come back this point later).
The condensation energy is rather striking. The condensation energy as a function of
the doping has a sharp peak at p ≈ 0.19. Remember that this doping value is not the“optimal” value p ≈ 0.16 corresponding to the maximum Tc , but it is rather the point, atleast approximately, where the T ∗ -line intersects the axis if we extrapolate it. This may
be significant, since the T ∗ -line is often interpreted as an indication of the quantum phasetransition at T = 0, and if that idea is right, the point p ≈ 0.19 is exactly where thequantum phase transition occurs. Note, however, that at finite temperature, the T ∗ -line
itself represents a crossover rather than a phase transition.
It is interesting to work out the condensation energy in terms of the CuO2 unit. It turns
out that the maximum condensation energy is ∼ 33 J/mol at p ≈ 0.19, corresponding
to ∆Uns (0) ∼= 2 K/CuO2 unit. This value does seem to be relatively independent of thecompounds.

7.1.4

NMR

The relaxation time T1−1 drops precipitately below Tc , and appears to behave as T1−1 ∝T 3 as T → 0, which is consistent with the point node [3]. χ behaves rather like that of
the BCS theory in T & 0.5Tc . Below that temperature, the value is larger than the BCS
value, again probably consistent with the power law.

7.1.5

Penetration depth

The penetration depth is one of the most frequently measured quantities. There areseveral diﬀerent methods for measuring the penetration depth:118

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES(a) Magnetization-related
There is a method related to the magnetization. Unfortunately, because of the anisotropy,
the theory is not in a simple form, and thus it is not so reliable.
(b) Fraunhofer diﬀraction in Josephson junction
This method has been used for some classical superconductors, but it is not widely usedin the cuprates so far.
(c) µSR (muon spin rotation)
It is generally believed that this method is reliable.
(d) Microwave surface impedance
I think this is the most reliable method to measure the penetration depth. In particular,
the group of Bonn and Hardy at the university of British Columbia has specialized in thismeasurement, and it does appear to agree, at least approximately, with the muon spinresonance result. Therefore, I have a certain amount of confidence in this method.
Let us discuss the superconducting fraction ρs /ρ, which is proportional to λ−2 , rather
than discussing the penetration depth directly. In the ab-plane, near Tc , it is probablyρs /ρ ∝ (Tc − T )2/3 . This is interesting: this behavior is exactly what we obtain in the
3D XY-model, a complex scaler field theory. In 4 He, this power-law dependence, with
its power not exactly 2/3, but nearly 2/3, has been extremely well verified. Therefore,
it seems that the superconductivity in the cuprates has the same universality class with
4
He.
For T → 0, things become slightly complicated. In pure samples, the correction appears
to behave linearly ρs /ρ = 1 − αT , while in dirty samples, the correction is quadratic
ρs /ρ = 1 − α0 T 2 . These facts are consistent with a gap with a point node. The value ofthe penetration depth at T = 0 is λ(0) ∼ 1000 Å for YBCO at the optimal doping in theb-axis direction, while it can be as large as λ(0) ∼ 4000 Å for LSCO.

Fig. 7.1. Temperature dependence of the superfluid density.

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

It is interesting to ask how well do the data on the ab-plane penetration depth λab (0)agree with the “naive” (quasi-London) prediction
λ−2 (0) = n3D e2 µ0 /m∗ .

(7.1)

We can fit with the data m∗ = 4m, which is consistent with the eﬀective mass in thenormal state value of the specific heat, and with n3D = 1022 cm−3 × peﬀ , where peﬀ is theeﬀective number of carriers per CuO2 unit. We must take here not peﬀ = p, but rather
peﬀ = 1 + p.

(7.2)

In other words, we must count not only the excess ones over the parent compound, butrather all the holes in the Cu 3d9 band. This is a rather important conclusion. Therefore,in terms of the superconductivity, it is not just the added electrons, but rather all theelectrons in the d-band are playing the role.
There is one puzzling thing: in YBCO, λa (0) ∼= 1600 Å, while λb (0) ∼= 1000 Å. If wesquare them up to get the superfluid density, the value in the b-axis is 2.5 times larger
than that in the a-axis. At first sight, that appears to say that the chains, which arepointing along the b-axis, are carrying 3/2 as much contribution as both planes! That isreally quite surprising, and as far as I know, there has been no solution to this puzzle.
The c-axis penetration depth is far larger than that in the ab-plane: from λc (0) ∼ 11, 000
Å for YBCO, to ∼ 100 µm (=0.1 mm!) for Bi-2212. These values are much larger thanwhat we find in the typical BCS superconductors. When we underdope them, these values
increase very rapidly. This is not surprising: particularly, in the case of YBCO, as one
underdopes, one certainly depletes the chains, which are forming the bridges between theplanes. Thus, we decrease the contact along the c-axis.
The temperature dependence of the c-axis penetration depth at low temperature ismuch weaker than in the ab-plane, and it may be fit by ρs /ρ ∼ 1 − αT 5 .
An interesting question we may ask is as follows: what will happen if we model theinterlayer contact as a Josephson junction and apply the standard Ambegaokar–Baratoﬀ
formula Ic (0)Rn = π∆(0)/2e, which relate the critical current Ic of the Josephson junction
and the energy gap. With a little bit of algebra, we find
λ−2c (0)ρc dint /∆(0) = const.,(7.3)

where dint is the mean spacing between the multilayers. This agrees reasonably well withexperiments, which suggests that the idea of modeling the inter-plane contact as a set ofJosephson junction works.

7.1.6

AC conductivity

Both in the BCS superconductors and in the cuprates, the behaviors of the AC conductivity are really quite complicated, but there are two qualitative diﬀerences:120《講義ノート》

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIESi) In the cuprates, σ(ω) is appreciable for ω < 2∆(0), while for the BCS superconductors
σ(ω) = 0 since there is no single electron state there.
ii) For finite ω, σ(ω) rises immediately below Tc , and thereafter drops, which suggeststhat σ(ω) is limited by the electron-electron scattering, and the number of electrons taking
part in the scattering will later drop below Tc . That is self-consistent with the behaviorof the thermal conductivity, as we shall see next.

7.1.7

Thermal conductivity

We can use the kinetic-theory formula1
κ = CV v`
3(7.4)

for both the phonon and electron contributions to the thermal conductivity. From thisformula, we obtainκph /κel ∼ (T /ΘD )2 (`ph /`el )(7.5)

for p ∼ 1, where ΘD is the Debye temperature, `ph and `el are the mean free paths ofelectrons and phonons. We have used cs /vF ∼ ωD /εF here.
For the BCS superconductors, κph /κel  1 at Tc , so that κ ∼ κel . Therefore, thethermal conductivity is proportional to the number of normal electrons, which falls oﬀ
rapidly below Tc .
For the cuprates, since the transition temperature is high and the electron mean freepath is short, it turns out that κph /κel & 1. In other words, the phonon contribution isquite important near Tc . As an experimental fact, we know that κ rises below Tc andthereafter drops. We can argue, probably, that the phonons are dominant, which is limitedby the scattering with electrons. This eﬀect of scattering then drops below Tc , since thenumber of normal electrons decreases. However, it is also possible that even though theelectrons are dominant, we still get this scattering. Therefore, it is not entirely clear how
much is due to electrons and phonons.

7.1.8

Tunneling

We can see a very characteristic diﬀerence in tunneling experiments [4, 5, 6]. In the BCS
model, the diﬀerential conductance measured in the tunneling experiment G ≡ ∂I/∂V isflat in the normal state, while in the superconducting state,εθ(ε − ∆),
Gs (E)/Gn = √2
ε − ∆2121(7.6)

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(a)

(b)

1

0

0

Fig. 7.2. (a) Tunneling density of states for a BCS superconductor. (b) Tunneling densityof states for a typical cuprate (ab-plane).
which is illustrated in Fig. 7.2 (a). It is zero for the energy smaller than the gap, while
for the energy larger than the gap, there is a characteristic peak from the increase in
the number of states. In the cuprates, first of all there is some energy dependence as
Gn (E) ∼ a + b(E), while in the superconducting state it has an interesting feature as
illustrated in Fig. 7.2 (b). As you can see, there is (i) a “dip” beyond the peak which is
not present in the BCS superconductor. Secondly, (ii) Gs (E = 0) 6= 0, which would rather
strongly indicate that there is a non-zero density of electronic state at the zero energy.
Finally, (iii) the peak-to-peak distance, which is “2∆” in the BCS theory, is typically∼ 7-8kB Tc for the ab-plane tunneling, whereas in the BCS theory it is only 3.5kB Tc . It
turns out, however, that it is closer to the BCS theory for the c-axis tunneling, and also
for the overdoping. The fact that the overdoping region is close to the BCS theory is
consistent with a rather general observation that perhaps all the cuprates become more
BCS-like toward the overdoping (the normal state becomes more normal Fermi liquid-like,and it becomes BCS-like in the superconducting state).

7.1.9

ARPES

A typical behavior of the ARPES spectra [7, 8, 9] is illustrated in Fig. 7.3 (a). First of all,we do have both in the normal and superconducting states a very apparent discontinuity,
or something looking like more or less a discontinuity, at what we identify as the Fermi
energy. However, the discontinuity is rather small both in the normal and superconducting
states, and it is only something like 10% of the whole value; it appears that the backgroundis not zero. Another interesting feature is that there is a dip beyond the peak just as in
the tunneling spectrum. Furthermore, as we go into the superconducting state, there
appears a “pullback” ∆ε of the spectrum from the normal state Fermi energy, typically∆ε(π, 0) ∼ 4-5kB Tc . The most interesting feature of this pullback is the dependence of∆ε(n) or the peak height on the angle on the Fermi surface. Both the pullback and thepeak height take their maxima in (π, 0) and (0, π) directions in the Brillouin zone, while
they take minima, in fact they go to zero, in (π, π) direction. This would then indicate
122

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(a)

(b)
10 %
N state

45

。

90 %
0

。

45

90

。

Fig. 7.3. (a) ARPES spectrum of a typical cuprate in the anti-nodal direction. (b) First
Brillouin zone of the cuprates, and the orientation dependence of the energy gap (dx2 −y2symmetry).
that, at least as seen in the ARPES data, the “gap” has a node in (π, π) direction. People
have tried to fit the gap numerically to simple formulae, but I do not think that is the
crucial point. One sometime worries if it does not fit some simple formulae, but I think
there is no reason to expect that.

7.1.10

Neutron scattering (YBCO, LSCO, and Bi-2212)

Remember that in the normal state, the neutron scattering cross section σ(q, ω) is
fairly strongly peaked as a function of q at q = (0.5π/a, 0.5π/a) (magnetic superlatticevalues), but featureless as a function of frequency. In the superconducting state in YBCO
[9, 10, 11] and Bi-2212 [12], nothing changes in the “even” channel (both planes in thec-axis contribute in phase). On the other hand, in the “odd” channel (both planes in thec-axis contribute out of phase, i.e., qz = π/d, where d is the interlayer spacing), there
is a striking peak seen around q = (0.5π/a, 0.5π/a) with ω ∼ 41 meV. It is only seenin the odd channel, not seen in LSCO [9], which only has a single plane and there is no
odd channel for it. Thus, we can ask a very obvious question: is this peak is peculiar tobilayer cuprates? I have been trying hard to find any recent data on that (the above isquite early observation, first done in the late 1990s), and I am rather sure that people
have been trying to do experiments on a single layered material. However, I have notbeen able to find any published result for this experiment. Thus, we do not know whether
it is peculiar to bilayer cuprates. My guess would be that it strongly is, considering thefact that it appears only in the odd channel. In any case, no one is quite sure what issignificant for this strong peak, although it is a very striking feature.

7.1.11

OpticsIn the BCS theory, the superconductivity aﬀects only those single electron states with
their energy of the order of ∆ ∼ kB Tc , so that the optical behavior at frequency ω  ∆is not so aﬀected by the superconductivity. In fact, with a calculation based on the BCS
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theory, we can indeed show that the eﬀect is of the order of (kB Tc /~ω)2 . If we put thevalues for the cuprates and ~ω ∼ 1 eV, the result is ∼ 10−4 , which is probably too small
to be seen. In reality, the changes of the spectra in superconducting state are ∼ 2-5%
[13, 14, 15]. Even at 5 eV, we still see appreciable changes in the optical spectrum at Tc
and below Tc .
More precisely, the original measurements are done for the reflectivity R(ω), which isthe easiest kinds of measurements we can do about the optics. If we define a quantity
η(ω) asRs (ω)
η(ω) ≡− 1,(7.7)
Rn (ω)
we obtain a curve drawn in Fig. 7.4. What is striking is that the zero crossing of η(ω)
occurs almost exactly where the normal state reflectivity Rn (ω) has the minimum. Thatthen might suggestη(ω) = −δ(∂Rn /∂ω)
(7.8)
where δ is the average downward shift of the “initial” state (no change in the final state).
Unfortunately, the relation does not work for the following reasons:(a) It requires δ ∼ ∆, which in turn implies, apparently, that the initial states arestrongly concentrated over the energy range ∼ ∆ near εF . It does appear that there is no
obvious reason for this to happen.
(b) While the above assumption does seem qualitatively to fit the data on the reflectivity, recent ellipsometric experiments [14, 15] have shown that it does not seem to fit
ε1 (ω) and ε2 (ω).

0.5

(%)
1

Fig. 7.4. Behavior of η(ω) as defined in Eq. (7.7).

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7.1.12

LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIESElectron energy loss spectroscopy (EELS)

Unfortunately, for reasons I do not understand, only a few EELS experiments weredone in the early days of the cuprates, in the late 1980s and early 1990s [15, 16]. We
actually have a lot of data in the normal state and the mid-infrared peak can be seen. For
the superconducting states, on the other hand, the published data can be found only at
low energy ω . 100 meV. This is probably because they expected to see the energy gap,
which I think is, a priori, a bad idea. In fact, we may not see the gap with the EELS.
Crudely speaking, the low energy EELS data is strongly overscreened and it is possiblethat we do not see anything interesting there. Unfortunately they did not go into the mid
infrared regime (ω ∼ 0.5-1 eV), although I wish they had. Now my colleagues are goingto do that.

7.2

What do we know for sure about superconductivity in the cuprates?

In the last twenty five years, there have been hundreds of microscopic theories on what isgoing on in the cuprates. No one can agree which of these theories are correct. Thus, let us
discuss here what we can say for sure about the superconductivity in the cuprates, without
adopting any particular microscopic theory, but rather just by looking at experimental
properties of the cuprates with some very basic theoretical considerations.
1. “Cooper pairs” are formed in the cuprate superconductors.
For the BCS superconductors, it is an almost universal belief that the onsetof superconductivity coincides with the onset of the oﬀ-diagonal long range order(ODLRO). In other words, superconductivity is due to the formation of Cooper pairs.
More technically, the basic “topology” of many-body wave function isΨN ∼ A[φ(r1 σ1 ; r2 σ2 )φ(r3 σ3 ; r4 σ4 ) · · · φ(rN −1 σN −1 ; rN σN )],(7.9)

where φ is the same “molecular” wave function for all pairs (quasi-BEC of Cooper
pairs). For most purposes, it is more convenient to work in terms of closely related(but diﬀerent) quantity
F (r1 , r2 , σ1 , σ2 ) = hψσ† 1 (r1 )ψσ† 2 (r2 )i. (“pair wave function”)(7.10)“Macroscopic wave function” of Ginzburg and Landau, Ψ(R), is just F (r1 , r2 , σ1 , σ2 )for σ1 = −σ2 = +1, and r1 = r2 = R, i.e., the wave function of the center of massof Cooper pairs.
Why do we assume these forms “pairing hypothesis” for old-fashion superconductors? One good reason is that one spectacular prediction of this hypothesis is that125

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flux is quantized in hc/2e. With the flux quantization and Josephson experiments(see Sec. 7.4 for more details.), it is shown that this pairing hypothesis to be exactly satisfied for old-fashion superconductors. Those experiments are also done forthe cuprates, and sure enough we obtain the same quantization, and the pairing
hypothesis is also exactly satisfied.
2. “Universality” of high-Tc cuprate superconductors with very diﬀerent chemicalcompositions, etc.
Almost all agree that the main actors in the superconductivity are electrons in
CuO2 planes. It is not necessarily true that CuO2 planes are the main actor justbecause all the cuprates have them in common3 . However, it is widely believed thatCuO2 planes are the main actors.
We know that charge reservoirs are very important but not all the cuprates havecharge reservoirs (the “infinite layer” cuprates do not have charge reservoirs but
they are still very good high-Tc superconductors).
3. Spin state of the Cooper pairs is singlet.
For simplicity, we assume that we can neglect the spin-orbit interaction, because
oxygen and copper are light (atomic masses: 16 and 29) and thus the spin-orbit
interaction is small at least within CuO2 planes. Then, we can do the simple analysis
in terms of the spin wave function and the orbital wave function.
What do we know, first of all, about the spin wave function? We discussed before
that at least in the BCS theory, the Pauli spin susceptibility drops rapidly for spinsinglet and remain constant for spin-triplet, as shown in Fig. 7.5. Experimental
results in the NMR (χs , T1 , ...) imply the latter. Therefore it is fair conclusion thatthe spin state of the cuprates is singlet (not triplet), i.e.,1
F (r1 , r2 , σ1 , σ2 ) = √ (↑↓ − ↓↑)F (r1 , r2 ).
2(7.11)

4. Copper pairs are formed out of electrons in the same band.
Some years ago, J. Tahir–Kheli made an interesting proposal [17]. He said why
do we always assume that Cooper pairs are formed from time-reversal states thatare from the same band. There may be another possibility that they are formedby taking one electron from one band and second electron from another. He claims
that the theory may explain some of the experimental properties. In particular, it
might give a natural explanation on why superconductivity is so strongly peaked as afunction of doping. However, according to his theory, we always expect a substantial
3

As an example, one of my colleagues pointed out that whole class of airplane have wheels in common
but they are not the main part of flight.

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

Fig. 7.5. Spin susceptibility as a function of temperature.

FIR absorption just above the gap edge, which is certainly not seen experimentally.
Thus, I do not believe the theory is totally correct, and we can conclude with agood confidence that the Copper pairs are formed from the states with the opposite
spins and opposite momenta (i.e., the time-reversed states) in the same band.
5. Small Cooper pair size, and large eﬀect of fluctuations.
As we discussed in Sec. 7.1.2, the order-of-magnitude estimates from Tc and Hc
shows that the in-plane “radius” of Cooper pairs ξ0 is typically 15-30 Å, and it is
the order of a few lattice spacings a: ξ0 /a ∼ 3-10. Compared with value for the BCS
superconductors, ξ0 /a ∼ 104 for Al for example, it is far smaller. This implies thatthe fluctuations ought to be much more important in the cuprates than in the BCS
superconductors. This is certainly true. For example, while the resistivity is smoothin the normal phase and drops vertically4 when it enters the superconducting phase
for Al, the transition is slightly rounded for BSCCO. When we apply a stronger
magnetic field (∼ 10 T), it becomes so rounded that it is very diﬃcult to say whereexactly the Tc is.
6. Irrelevance of the inter-layer tunneling.
In the normal state, the c-axis resistivity is very high. One can interpret theresistivity in terms of electrons’ hopping from one plane to the next, being somehow
disturbed by fluctuations and decay. The average time taken for electrons to hop
between planes is much greater than ~/kB T . This has a significant implication:
crudely speaking, ~/kB T is of the order of the characteristic time associated withsuperconducting state. Thus, the large hopping time strongly suggests that thepairs in diﬀerent multilayers can be regarded eﬀectively as independent. In otherwords, in order to analyze the superconducting behaviors of the cuprates, it issuﬃcient, at least in the first approximation, to focus on a single plane. This has
not always been the universal view; in the first ten to fifteen years of the subject,
Anderson’s interlayer tunneling theory, which rejected this point of view and claimed
that the hopping in the c-axis is crucial, was rather appreciated. However, one
4

Basically, it is very diﬃcult to measure the width of superconducting transition in Al because it is
too small.

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

critical experiment in 2000 or so has led most of us, including Anderson himself, todisbelieve his theory. Therefore, the c-axis hopping is negligible.
7. Absence of substantial isotope eﬀect and “folk-theorems” on Tc .
At least in higher-Tc cuprates, no substantial isotope eﬀect has been seen. In
lower-Tc cuprates, and even in higher-Tc ones in strongly overdoped or underdoped
regimes, we do have the isotope eﬀect. This tends to suggest that phonons areplaying very minor roles. The isotope eﬀect appears only when electronic eﬀects arestrongly suppressed.
A “folk-theorem” on Tc is another evidence for the non-phonon mechanisms. Band
theorists, using such methods as the local density approximation (LDA), claim tobe very good at predicting Tc with the BCS scheme: i.e., with the hypothesis of theelectron-phonon coupling. They say that we will never get higher Tc than 40 K in
the phonon mechanism. Thus, basically5 the phonon alone is not enough to explain
the higher Tc .

7.3

Symmetry of the order parameter (gap)In the BCS theory, the fermionic gap ∆(n) has almost the same n-dependence as theorder parameter Ψ(n). In a more general theory, however, this is not guaranteed, but it
seems very unlikely that the symmetries of these two quantities are totally diﬀerent.
As we have seen, NMR experiments indicate that the order parameter is spin-singlet,i.e.,1
F (r1 , r2 , σ1 , σ2 ) = √ (↑1 ↓2 − ↓2 ↑1 )F (r1 , r2 ),2(7.12)

where F (r1 , r2 ) and Fk are even-parity. We assume relevant symmetry is that of CuO2
plains, i.e., (approximately) tetragonal. Then symmetry group is C4v (symmetry groupof square) with the fundamental operations below:(a) Rotation through π/2 around z-axis (R̂π/2 ).
(b) Reflection in crystal axis, e.g. (100) (Iˆaxis ).
(c) Reflection in a 45◦ axis, e.g. (110) (Iˆπ/4 ).
Note that the operations (a)-(c) are not independent:Iˆaxis Iˆπ/4 R̂π/2 ≡ 1.
5

(7.13)

There is one or two exceptions against this folk-theorem. Tsuei et. al. suggested we could somehow
get a phonon mechanism without the isotope eﬀect, but there is no positive evidence of this mechanism
[18].

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Table 7.1. Four allowed symmetries of the order parameter in the cuprates.
Informal nameGroup theoretic
notationR̂π/2IˆaxisRepresentative
state

s+

A1g

+1

+1

1

s− (‘g’)

A2g

+1

−1

xy(x2 − y 2 )dx2 −y2

B1g

−1

+1

x2 − y 2

dxy

B2g

−1

−1

xy

2
2
It is rather trivial to see Iˆaxis≡ Iˆπ/4≡ 1. Moreover, since we restrict ourselves to the2case where Fk is even parity, R̂π/2 Fk = F−k = Fk for any Fk under consideration, which2means6 R̂π/2= 1.
As a result, all of these operators can only have eigenvalues ±1, which determines
only four possible irreducible representations (all one-dimensional). Table 7.1 gives theoverview of these possible cases. You should not take the figures so seriously; for example,
it is still possible that the s-wave gap has nodes in the directions of 30◦ and 60◦ . When
we talk about the symmetry of the gap, we should be thinking of only how the orderparameter behaves under symmetry operations.
Here is an interesting question: are we actually sure that the order parameter belongs toa single irreducible representation? Could it not belong to more than one representations?
In general, yes it could. However, with the following argument, we exclude this possibility.
If i labels the diﬀerent irreducible representations, the free energy can be expanded as

F (T ) =∑ij

αij (T )ψi∗ ψj +1∑βijk` (T )ψi∗ ψj∗ ψk ψ` ,2 ijk`(7.14)

with a help of the global U(1) symmetry. Quite generally, invariance of the free energy
under the above lattice symmetry operations implies αij = αj δij . Thus, the first terms
should be αi |ψi |2 . For the second term, the order parameters of the diﬀerent representa6

In two dimensions, π rotation around z-axis is identical with the parity operation.

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tions can couple, so that it can make a complicated behavior in general.
If we assume more than one ψi 6= 0 contribute to the free energy, we automatically have
two phase transitions. Crudely speaking, the order parameter with a stronger attraction
starts to take non-zero values in the first phase transition, and the other one with a weakerattraction becomes non-zero in the second phase transition. This is certainly not seen
experimentally. (For these two phase transitions to be coincide, we need very pathologicalfine-tuning of parameter.) Thus we can conclude thatthe order parameter belongs to a single irreducible representation.
According to the above argument, the order parameter must be just one of the four
forms, s, dx2 −y2 , dxy and s− . However, people usually concentrate on the first two, becauses-wave is favored by some types of theory (e.g., phonon) and dx2 −y2 is favored by the spinfluctuation theory.
How to tell?
(a) dx2 −y2 , dxy , s− must have (at least) four nodes on Fermi surface. s need not.
Thus, exponential decrease of quasiparticle-associated quantities (χ, T1−1 , ∆λ(T )...)
certainly implies s-wave. Experimentally, all these quantities have power-law dependences consistent with two-dimensional point node. Does this inevitably excludes-wave? Unfortunately not, because even the s-wave state may have nodes allowedby symmetry, i.e, the “extended s-wave” (see Fig. 7.6).
(b) More specifically, dx2 −y2 (and s− ) must have nodes at (π, π), whereas s (anddxy ) would have nodes at (π, π) only by pathology. Thus, the observation of nodes
suggests dx2 −y2 . The ARPES data indeed indicates such nodes. However,i)

The “gap” seen in ARPES may not simply be the superconducting gap (cf.
pseudogap regime).

ii)

The “extended s-wave” state of the form F (θ) = A + B cos 4θ (0 < B − A A) may be diﬃcult in practice to distinguish from dx2 −y2 (Fig. 7.6).

Therefore, we need experiments which are directly sensitive to the sign (or more generallyphase) of order parameter, such as the Josephson (“phase-sensitive”) experiments.

Fig. 7.6. Extended s-wave state, described as F (θ) = A + B cos 4θ (0 < B − A  A).

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7.4

LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

Josephson experiment in cuprate (and other exotic) superconductors

The Josephson experiment as a tool to determine the phase of the order parameter was
first suggested in the context of the p-wave superconductivity [19]. The first experiment
done in this direction was in 1993 with my colleagues [20].
The Josephson eﬀect occurs for any geometry, including a superconductor-normalsuperconductor junction (Fig. 7.7), and a SQUID geometry (Fig. 7.7 (b)). For the former
case, to the lowest order in Ψ, the Josephson energy or the simple s-wave case is given by
EJ ∝ −const.(Ψ∗1 Ψ2 + Ψ∗2 Ψ1 ) ∼ −const. cos ∆ϕ.

(7.15)

where ∆φ is the phase diﬀerence between the two superconductors. In the SQUID geometry, the critical current shows an interesting behavior: the critical current is maximum
at Φ = nΦ0 , and minimum at Φ = (n + 1/2)Φ0 . This can be understood easily by using∑∆ϕi = 2πΦ/Φ0(7.16)iwhere ∆ϕi is the change of phase across the junction, and Φ is the flux through trappedin the circuit7 . For Φ = nΦ0 , we can consider the simplest case Φ = 0. In this case, thephase drops in the two paths are the same. On the other hand, for Φ = (n + 1/2)Φ0 , thereis π-phase diﬀerence, so that the Josephson current interferes destructively and takes its
minimum.
Let us start to see how these goes for the Josephson experiment in the cuprates. Basically, the Josephson experiment are based on the following two principles:(a) If two bulk superconductors separated by the Josephson junction, and they are described by Hamiltonian Ĥ, which is invariant under symmetry group G, then theJosephson coupling energy must be similarly invariant under G.

Fig. 7.7. Various geometry of the Josephson experiments.

7Ψ and ∆ϕi are defined consistently here.

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LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES(b) For a circuit, as shown in Fig. 7.8 (b), we again find the fundamental equation
Eq. (7.16).
We always assume the lowest-order Josephson energy (which is testable via Fraunhofer
diﬀraction pattern etc.) is given by Eq. (7.15).
Application (a) alone Let us consider π/2 rotation around z-axis on the system shownin Fig. 7.8 (a). The Pb order parameter ΨPb is invariant, i.e., ΨPb → +ΨPb . If YBCO iss-wave (s or s− ), ΨYBCO → +ΨYBCO , then the lowest-order Josephson eﬀect allowed. On
the other hand, if YBCO is d-wave (dx2 −y2 or dxy ), ΨYBCO → −ΨYBCO , the lowest-order
Josephson eﬀect forbidden (otherwise, the free energy cannot be invariant under the π/2
rotation around the z-axis). Thus, this diﬀerence can be a crucial test for the symmetryof the order parameter of exotic superconductors.
Application (b) alone Since the order parameter in Pb ΨPb is invariant under π/2
rotation, ΨPb takes the same value at junctions 1 and 2. If YBCO is s-wave, the situationis the same as in conventional DC SQUID, so that Ic (Ψ) is maximum at Ψ = nΨ0 , andminimum at Ψ = (n + 1/2)Ψ0 . On the other hand, if YBCO is d-wave, the sign of Ψchanges between junction 1 and 2 in YBCO. This means that we eﬀectively add π to theright hand side of Eq. (7.16), which then implies that the critical current Ic is maximum
at Ψ = (n + 1/2)Ψ0 , and minimum at Ψ = nΨ0 . The latter is what precisely D. Wollman
et. al. have found in the experiment [20]; there have been about twenty phase-sensitive
experiments by diﬀerent groups as well. The conclusion from all these experiments isthat:
the order parameter of the cuprates is dx2 −y2 .
Some experiments prepared beautiful “tricrystal ring”, and exploit in some sense bothprinciples (a) and (b) (see pp. 345-346 of Ref. [21] for more detail), and got the sameconclusion.

(b)

(a)

Fig. 7.8. (a) Pb and YBCO are separated by Josephson junction which is invariant under
π/2 rotation around z-axis. (b) The setup for π-junction experiment [20].
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7.5

LEC. 7. CUPRATES: SUPERCONDUCTING STATE PROPERTIES

What is a “satisfactory” theory of the high-Tc
superconductivity in the cuprates?

It is interesting to ask a rather sociological question: how will we know when we havea “satisfactory” theory of the high-Tc superconductivity in the cuprates? Right now, if
I go to my dozens of colleagues and ask, “do we understand high-Tc superconductivity”,
every one of them says “Yes!”. If we then ask “OK, then what is the correct theory?”, we
get totally diﬀerent theories. That is not satisfactory at all.
How should we get out of this? By definition, we have a satisfactory theory if 90% or
95% of people agree. Apart from that, do we have a more objective criterion? I think weshould, at least, be able to:(A) give a blueprint for building a robust room-temperature superconductor,
OR (B) assert with confidence that we will never be able to build a (cuprate-related)room temperature superconductor,
OR (C) say exactly why we cannot do either (A) or (B).
The above two (A) and (B) may sound demanding because strictly speaking, we cannot
do them even for the BCS theory. However, in the BCS theory, we can do (A) and (B),given a particular material with its certain information, such as the phonon spectrum,
phonon matrix element, band structures etc. Right now, to best of my knowledge, there
is no theory which satisfies any of these three conditions.
Let us ask a few more specific questions:(1) Are the cuprates unique in showing high-Tc superconductivity8 ?
(2) If so, what is special about them?
(e.g., band structure, two-dimensionality, antiferromagnetism, etc.)(3) Should we think of high-Tc superconductivity as a consequence of the anomalous
normal-state properties, or vice versa?
(4) Is there a second phase transition associated with the T ∗ -line? If so, what is thenature of the low temperature (“pseudogap”) phase9 ?
(5) If yes to (4), is the phase transition relevant to high-Tc superconductors or a completely
unconnected phenomenon?
(6) Why does Tc depend systematically on n in homologous series?

8
This depends on what we call the high-Tc superconductors, but nowadays people would say no since
the recent discovery of the ferropnictides, at least, is an counterexample.
9
Right now, there is no positive evidence for the phase transition. Everything looks, at least, consistent
with the idea of a smooth crossover. However, some people believe that it corresponds to a sort of hidden
phase transitions, and suggest that the pseudogap phase breaks the time-reversal symmetry, rotational
symmetry, or some other kinds of symmetries.

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[18] C. C. Tsuei, D. M. Newns, C. C. Chi, and P. C. Pattnaik, Phys. Rev. Lett. 65, 2724
(1990).
[19] V. B. Geshkenbein, A. I. Larkin, and A. Barone, Phys. Rev. B 36, 235 (1987).
[20] D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett,Phys. Rev. Lett. 71, 2134 (1993).
[21] A. J. Leggett, Quantum Liquids (Oxford University Press, New York, NY, USA,2006).

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Lec. 8 Exotic superconductivity:
discussion
In this last lecture, I certainly do not want to try to review the whole field of exotic
superconductivity that would be quite impossible. I would like to do mainly three things:(1) Discuss what properties are shared by all the exotic superconductors.
(2) Give a brief review of existing theoretical approaches.
(3) General considerations on energy saving in all “electronic” (non-phonon) superconductors.
In this last part of the lecture, I will be talking largely about some ideas I have beenplaying with over last more than ten years. Some of those are published and some arenot. This idea is, though not fully recognized in the community, quite interesting since it
is a rather diﬀerent approach from many of conventional approaches in the literatures.

8.1

Common properties of all exotic superconductors

Trivially, the most obvious common property is the superconductivity itself. What doesthe “superconductivity” exactly mean, and what does it imply? In the original experimentby H. K. Onnes, he applied constant current through, aluminum and measured voltage
drop across it (Fig. 8.1 (a)). He found that the resistivity drops sharply to zero at Tc .
This original experiment is rather simple from an experimental point of view, but this is
not the simplest experiment, theoretically speaking. There are two kind of much simpler

Fig. 8.1. (a) Cu is in the normal state, while Al is in the superconducting state. We apply
constant current through them and measure the voltage drops across Al. (b) Experimental
setup for the Meissner eﬀect (c) Experimental setup for the persistent current.

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ones. The first one is the Meissner eﬀect. In this experiment, one takes a simply connected superconducting sample and hangs it up in zero magnetic field initially (Fig. 8.1(b)). If one applies a magnetic field in the normal state (above Tc ), the magnetic line
just goes straight through it. When one cools down the sample below Tc , then the magnetic field lines suddenly appear to be totally expelled from the metal. In other words,
superconductors show perfect diamagnetism.
It is important to realize that the Meissner eﬀect is a true thermal-equilibrium eﬀect.
We have not changed anything other than temperature. Thus, the Meissner eﬀect cannot
simply be the consequence of zero-resistivity and it goes beyond that. The zero-resistivity
would tell us that the system may not come into equilibrium with the boundary conditions,but here what is happening is coming out of equilibrium.
The second well-known phenomenon on the superconductivity is the persistent current.
One first takes a system with ring geometry and applies a magnetic filed through it(Fig. 8.1 (c)). Then, makes the sample into superconducting state and sets up somekinetic current by turning the magnetic flux down from finite value to zero. What one
finds is that the current circulate for an astronomically long time. It is, however, rather
easy to demonstrate that this cannot be an equilibrium eﬀect and this is in fact anastronomically long-lived metastable eﬀect.
There is no a priori guarantee that these two phenomena always go together, but it
seems to come together, in all superconductors known to date.

8.2

Phenomenology of superconductivity

The following phenomenology are established mainly by London, Landau and Ginzburg
during 1938-50. The theory postulates that the superconducting state is characterized
by Schrödinger-like “macroscopic wave function” Ψ(r), which can be analyzed into themagnitude and the phase:Ψ(r) = |Ψ(r)| exp(iφ(r)).
(8.1)
The crucial point is that the phase must be single valued modular 2π like the Schrödinger
wave function.
It further postulates that the current is given in a similar form to what we obtain in
the single particle quantum mechanics:
J (r) ∝ |Ψ(r)|2 (∇φ(r) − e∗ A(r))(8.2)

where e∗ is an eﬀective charge (e∗ = 2e in the BCS theory) and A is a gauge potential.
Under these two simple assumptions, let us see how far we can explain the two major
phenomena of superconductivity.

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8.2.1

LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

Meissner eﬀect

The Meissner eﬀect turns out to be the exact analog of atomic diamagnetism (for example, xenon gas). We apply the magnetic vector potential to a simply connected systemand see what happens. It basically induces a diamagnetic current, which is opposite inside
to the vector potential.
First of all, because we are thinking of a bulk superconductivity in a simply connected∫
geometry, ∇φ(r) · dl = 0. Thus, it is allowed to assume ∇φ(r) itself remains 0 evenne2
when we apply magnetic field. Then, we obtain J = −A, which directly leadsm∇2 B = λ−2
L B ⇔ B = B0 exp(−z/λL ),

(8.3)

√
where λL = m/µ0 ne∗2 is the London penetration length and z is the distance from thesurface. Thus if we apply the magnetic field to the surface, it falls exponentially.
This argument can be applied to both a single atom and superconductors. However,there is a big diﬀerence between them. If we put the typical values and evaluate λL , it isalmost the same for both the single atom and superconductors (perhaps the order of ∼1000 Å). On the other hand, the size of the atom is much less than that of superconductors.
Thus, in the case of atom, it shields only a small fraction of the applied field, while
superconductors show almost complete diamagnetism.

8.2.2

Persistent current∫
We can define a winding number n ≡ (2π)−1 ∇φ(r) · dl. n is conserved unless|Ψ(r)| = 0 across some cross section, which is highly unfavorable energetically. As a
direct consequence, J ∼ n is conserved for an astronomically long time.

8.2.3

Summary

What was essential in the above argument? For these two phenomena to occur, there
must exist a complex order parameter Ψ(r) such that(a) Nonzero values of |Ψ(r)|2 are (locally) stable.
(b) Spatial gradients of the phase of Ψ(r) correspond to charge currents.
An overwhelmingly natural guess at the nature of such an order parameter is as follows:Ψ(r) represents macroscopically occupied eigenfunction of n-particle density matrix, i.e.,the system possesses oﬀ-diagonal long range order (ODLRO). More rigorous arguments(Yang [1], Kohn and Sherrington [2]) claim to show rigorously thatODLRO is a necessary and suﬃcient condition for superconductivity.
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Fig. 8.2. Experimental setups for flux quantization (left) and AC Josephson junction(right).

In strict two dimensions, we know from other exact theories that we cannot get the
ODLRO but the system can go through a Kosterlitz–Thouless (KT) phase transitionand essentially behave like a superconductor. Thus this is not a totally reliable theorem
but I do not know any counterexample in three dimensions. Historically, in the 1980s,“anyon superconductivity” was claimed to be a superconductivity without the ODLRO.
However, it turned out that anyon superconductors are just a kind of the standard BCS
superconductors and thus not an counterexample.
Even if this “theorem” is true, it says nothing about the value of n. Since electrons are
fermions, n must be even. Yet, in principle, n could be 4, 6, ... Then why can we assume
n = 2? Two kinds of experiments are particularly persuasive (Fig. 8.2):(a) In a (thick) ring geometry, total trapped flux Φ through the hole is quantized inunits of h/ne.
(b) In the AC Josephson eﬀect, the principal frequency of current, when we apply
voltage V between Josephson junctions, should be ω = neV /~.
Both of these experiments have been done not only on old-fashioned superconductors but
also on exotic superconductors, and all results seem to be entirely consistent with n = 2.
There is no evidence for any value of n other than 2 in any superconductor. Thus, we canreasonably conclude thatsuperconductivity is due to the formation of Cooper pairs.
That is the one thing all superconductors most surely have in common.

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8.3

LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

What else do exotic superconductors have in common?

Apparently, exotic superconductors do not have much in common, apart from superconductivity itself. Even if we exclude alkali fullerides1 ,(a) Not all are non-phonon mechanism (still controversial: some evidences suggestthat the mechanism of organics is phonon, and others show against that).
(b) Not all are quasi-two-dimensional (most heavy Fermions are three-dimensional).
(c) Not all are close to an antiferromagnetic phase (some heavy Fermions, Sr2 RuO4
do not have the phase).
However, if we restrict ourselves to “high-temperature” superconductors (the cuprates,
ferropnictides, organics), then we get the following common features:(a) All are strongly two-dimensional.
(b) All have an antiferromagnetic phase close to the superconducting phase.
(c) All have charge reservoirs well separated from superconducting layers.

8.4

Theoretical approaches (mostly for the cuprates)

1. Generic “BCS-like” approach.
Crudely speaking, what BCS did was to try to isolate a particular physical eﬀect
which they believe, in some sense, is the key to the superconductivity in metals ofaluminium and lead. This fundamental physical eﬀect was the eﬀective attraction
between the electrons supplied by the exchange of virtual phonons. Next, they wrote
down the eﬀective low energy Hamiltonian, which they believed encapsulated andexpressed this fundamental physical feature. This is, of course, the famous BCS
Hamiltonian, which was spectacularly successful.
Recently, there have been lots of attempts to do something very similar on exotic
superconductors. Typically one, tries to isolate what he/she thinks might be important eﬀects (for example, the exchange of some kinds of excitons, and the formationof d-density wave, bipolarons, and whatever). One then, arbitrarily writes downsome kinds of postulated low energy eﬀective Hamiltonian, which involves theseparticular physical eﬀects, and then move on to calculations. There are thousands
1
Remember that when we discussed alkali fullerides in Lec. 5, we concluded that all the evidencesof alkali fullerides indicate they are probably just standard BCS superconductors due to the phononmechanism, and the only diﬀerence is the strong molecular structure that may explain the high-Tc in thiscase. Thus, they are not genuine exotic superconductors.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSIONof papers in the literature which carry out this kinds of properties. Of course, theyusually claim that they can explain at least some certain experimental data.
I think there are a number of problems in this kind of approaches. Obviously,
we have no particular reason to believe any one of these eﬀective Hamiltonians isgoing to be the right description. For the BCS theory, we were very lucky; before
BCS did it, there was no particular reason to believe that this approach worked forelectron-phonon interaction, but it did. Furthermore, none of these (to the best of
my knowledge, except spin-fluctuation theory which is a bit diﬀerent and I will treat
it somewhat diﬀerently) has ever made any prediction ahead of experiments.
2. Approaches based on the Hubbard model (which involves less guess than the firstclass).
There are a large number of theoretical approaches in the literature based onsome forms of the Hubbard model. The simplest form of the Hubbard model is∑∑a†iσ ajσ + Uni↑ ni↓ ,(8.4)
Ĥeﬀ = −tσ,(i,j)∈n.n.

iwhere n.n. in the summation stands for the nearest neighbors. As long as we focus
on one relevant electronic orbital per site, then we do not have to worry about
parallel spin interaction because of the Pauli principle. This is very simple looking
Hamiltonian. However, as far as we know, it is not analytically soluble even in two
dimensions. Therefore, we have to resort to some possible strategies:(a)

Digital numerical simulations: even for most powerful computers available,
we can solve the problem typically up to only ∼ 10 × 10 sites, and it is not
obvious this is enough to give us a reasonable thermodynamic result.

(b) Analog simulation: last ten years lots of amazing things have happened in
the experiments on ultracold atoms in optical lattices. It turned out that atoms
in optical lattices under appropriate conditions are marvelous simulations of
the Hubbard model (much better than electrons in real solids). People havejust about started right now using this technique.
(c)“Guesses” at analytic solution: one kind of rather plausible results is that
we start from the BCS type wave function, and then project oﬀ all terms
having doubly occupied sites by the Gutzwiller projection operator, ΨN =P̂G ΨBCS , reflecting the strong repulsion between two electrons on the samesite. Some people claim qualitatively good explanation on experiments taking
this approach.
One problem of the Hubbard model in general is that the model may omit
important physical eﬀects (e.g., long-range part of Coulomb interaction).

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3. Antiferromagnetic (AF) spin fluctuation exchange
In all high-Tc superconductors, the superconducting phase occurs close to the
AF phase (Fig. 8.3). Moreover, both the NMR and the neutron scattering data inthe cuprates suggest that the in-plain spin susceptibility χ(q, ω) in normal phase isfeatureless as a function of ω, while they are strongly peaked as a function of q at
q = Q ≡ (±π/a, ±π/a), the superlattice Bragg vector in the AF phase, as shownin Fig. 8.4.
This is often expressed as a possible ansatz for χ(q, ω) (≡ χNAFL (q, ω))2 [3]. Pines
et. al. [3] postulated a crude phenomenological ansatz that, far from pseudo-Bragg
vectors Q, χNAFL (q, ω) has Fermi liquid-like form:
χNAFL (q, ω) ∼
=

χq
χ0
∼
.
=
1 − iω/Γq
1 − iω/Γ0(8.5)

This is not particular interesting. However, when we are very close to a pseudoBragg vector,χQi ( χq )
χNAFL (q, ω) ∼(8.6)=
1 + (Qi − q)2 ξ 2 (T ) − iω/ωSF
where ωSF ( Γ0 ) is the AF fluctuation frequency, and ξ(T ) is the AF correlation
length. What this approach emphasizes is that the particular nearly antiferromagnetic spin fluctuations are playing an important role.
Ansatz3 : single electrons couple strongly to AF spin fluctuations, whose exchange thengenerates an eﬀective electron-electron attraction. This is similar to what is going on in
3
He, except that in 3 He, what they are trying to exchange is paramagnon, related to theclose ferromagnetic phase.
Unfortunately, the question of what is correct coupling constant is rather controversial.
People usually put the value of the constant by hand or try to do some approximate calculations to evaluate it. However, there has been an argument in the literature which states

Fig. 8.3. The well-known phase diagram for the cuprates. S is the superconducting phaseand AF is the antiferromagnetic phase. p represents the density of holes.
2
3

NAFL stands for Nearly antiferromagnetic Fermi liquid.
Not directly testable in experiment.

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Fig. 8.4. (a) The AF spin order in real space. (b) The superlattice Bragg vector Q in
k-space.

that the coupling constant is actually zero. There are certain amount of controversies
about that, but let us here suppose that the coupling constant is not zero and see thestriking prediction of the spin-fluctuation theories (rather generic):(a) The points on the Fermi surface most strongly connected by Qi are roughly at(π/a, 0), (0, π/a) etc. (Fig. 8.4). Then, we can expect the gap becomes maximum
there. Thus we expect the gap is singlet, as indicated by ARPES.
(b) The sign of the pair wave function Fk : scattering processes should as far as
possible leave Fk invariant. However, the emission of the virtual spin fluctuations
flips the spin, changes momentum by Q. If a state is singlet, spin-flip is equivalent
with multiplying (−1). Hence to preserve Fk , momentum change by Q (A → B in
Fig. 8.4) must also multiply (−1).
Hence, from (a) Fk must be large at (π/a, 0) (b) Fk must change sign under R̂π/2 . Of
the four even-parity irreducible representations of C4v (s, s− , dx2 −y2 and dxy ), only dx2 −y2works4 . Thus,
spin-fluctuation theories unambiguously predict dx2 −y2 symmetry.
As I said, that is the only prediction I know, in the whole history of the high temperature
superconductivity, where theorists made a non-trivial prediction which was later confirmedby experiments. This is an appreciable credit of the spin-fluctuation theories. The main
criticism directed to the spin-fluctuation theories is that we just have too many fitted
parameters.

8.5

Which energy is saved in the superconducting
phase transition?

Let us start on the last part. I believe that the arguments in this section are at a moregeneral level than much of the theories I have talked about so far. Let me give a review of
4

See Lec. 7.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

Fig. 8.5. Antiferromagnetic spin fluctuations connect the point on the Fermi surface closeto (π/a, 0), (0, π/a).

this interesting theory, where some of these are published [4] and some not published yet.
We start with a very general question: do we know anything about the energy which issaved or not in the superconducting phase transition? Actually, it turns out that this isjust the case for diﬀerent questions, and it can be applied to any other phase transition.

8.5.1

Virial theorem

We start with a really basic level, non-relativistic limit of the famous Dirac Hamiltonian:Ĥ = K̂ + V̂ ,∑ p̂2∑ P̂ 2
i
α
K̂ =
+
,
2m
2M
α
i
{
}
∑
∑ (Ze)2
∑ Ze2
e2
1
V̂ =
+
−2
.
8πε0 i,j |ri − rj | α,β |Rα − Rβ ||ri − Rα |
i,α

(8.7)
(8.8)
(8.9)

Let us imagine the following thought experiment5 : let us imagine we cool the systemdown to zero temperature, but forbid it to form Cooper pairs. Then, basically speaking,we have got the “best” normal ground state of fermions. Let us relax that constraint andallow them to form Cooper pairs. We know that this is what we see in experiments. Why
is it? It is because, trivially, it saves the energy by forming the Cooper pairs. In otherwords, trivially, the energy of the best superconducting ground state must be below thatof the best normal state.
There is a very simple argument done by Chester [5] in the context of the old-fashioned
superconductivity, and he simply uses the sum rules. At zero pressure, we obtainhĤi = hK̂i + hV̂ i,
1
hK̂i = − hV̂ i25This argument works not only for the superconductivity.

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from the virial theorem. Remember that the virial theorem works because we have asimple form of the interaction. Thus, the expectation value for the total Hamiltonian is1
hĤi = hV̂ i.
2(8.12)

Therefore, since Econd ≡ hĤiN − hĤiS > 0, the total Coulomb energy (electron-electron,
electron-nuclei, and nuclei-nuclei) must be smaller in the superconducting phase:hV̂ iS < hV̂ iN .

(8.13)

This is a very general statement. Conversely, according to this relation, the total kinetic
energy (the kinetic energy of electrons plus the kinetic energy of nuclei) must be increasedin the the superconducting phase transition. That is a theorem, and we cannot get out of
that. This is a rather strong constraint to certain kinds of theories, in which they claimthat the kinetic energy is actually saved in the the superconducting phase transition. Suchtheories can never be true unless we redefine the kinetic energy in some ways: otherwise,
this theorem kills them.
Let us now go on further from the Dirac Hamiltonian, and go on to an intermediate-leveldescription. Let us do the partition of electrons into “core” electrons and “conduction”
electrons. Since there are experiments showing that the phonons do not play a major rolein the cuprates, let us ignore the phonons. Then, we have an eﬀective Hamiltonian forthe conduction electrons as follows:Ĥ = K̂eﬀ + V̂eﬀ
∑ p̂2iK̂eﬀ =+ Û (ri )
2m
i
1 ∑
e2
V̂eﬀ =,
8πε0 i,j ε|ri − rj |(8.14)(8.15)(8.16)

where Û is the potential of the static lattice, and ε is a high-frequency dielectric constant(contribution to the dielectric constant from the core ions). We ignore the c-axis now. I
will later argue that almost certainly the only part of the kinetic energy we have to worry
about is in the ab-plane: basically forget about the c-axis, since that contribution turns
out to be very small. It is important to be aware, though it is not often remarked in theliteratures, that the interaction term is not the bare Coulomb interaction. Why not? It isbecause it is going to be screened by atomic cores, and we know from experiments, that
this eﬀect is not small. In fact, typically, the high-frequency dielectric constant of thecuprates is somewhat 4 to 5. Therefore, it is a big eﬀect.
Let us assume that U (ri ) is independent of ε. This is not obvious, and it could berelaxed, but for the moment, let us just play with it. Then, the only point where ε comesin is the interaction part. If this is right, we can compare two systems which have the same145

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form of U (r) and carrier density, but diﬀerent ε. Let us imagine that we can somehow
do that comparison. What do we then expect? Is it better for the high-temperature
superconductivity to have a large value of ε, that is a weak Coulomb interaction, or the
opposite, small ε, and therefore a strong Coulomb interaction? Naively speaking, one
would say “look, the repulsion is not good for the superconductivity, and it is better to
set the repulsion down. In that way, we might be able to get an eﬀective attraction.”
This is, in fact, not true. What we will get is the opposite. We can just apply the
Hellman-Feynman theorem, and obtain
*
* +
+
∂ V̂
V̂
∂hĤi
=
=−
.
(8.17)∂ε∂εεTherefore, provided hV̂ i decreases in the normal to superconducting phase transition,∂Econdthen< 0. In other words, “other things” (U (r), n) being equal, it is advantageous∂εto have as strong a Coulomb repulsion as possible. At first sight, it is counterintuitive. I
think it is somewhat plausible to look in the following way: forget everything you haveever known about the BCS theory, eﬀective attraction. What is happening is that there isan enormous amount of the repulsion in the normal phase, but by forming Cooper pairs,
this repulsion somehow manages to decrease. The more repulsive energy we have in thenormal phase, the more chances of it to be released. Intuitively speaking, that is how it
works. This is a rather remarkable conclusion, and other things being equal, we want tohave a stronger Coulomb repulsion as possible for the high-temperature superconductivity.
Here is an example, which I think a possible manifestation of this. Let us consider asingle plane Hg-1201, and compare that with a central plane of Hg-1223. What do weknow about these two materials? Of course, there are other eﬀects which distinguishthese two, but let us focus here on the screening of the Coulomb repulsion. Since bothhave the same CuO2 plane, the screening by Cu and O is essentially the same. For theinter-layer materials, there is a diﬀerence. For Hg-1201, there is BaO2 sitting between theplanes, which has a large polarizability, contributing to a large dielectric constant. Thus,
for Hg-1201 the Coulomb repulsion is well screened. On the other hand, for Hg-1223,
there is Ca++ between the layers, which has a little polarizability since it forms a closed
shell structure. Thus, the Coulomb interaction in the central plane is not screened. Thus,
the conclusion from the above argument is that the transition temperature for Hg-1223

Fig. 8.6. Schematic illustration of the crystal lattice for Hg-1201 and Hg-1223.

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is higher than that of Hg-1201: this is true indeed. I do not claim that this is the only
reason for the diﬀerence in the value of Tc , but at least it is going in the right direction6 .

8.5.2

Energy consideration in “all-electronic” superconductors

Let us now turn to the general question: what do we know the way in which the energy
is saved or not saved in “all-electronic” superconductors? Again, I will start from theintermediate level description (neglect phonons, inter-cell tunneling):
Ĥ = T̂(k) + Û + V̂c ,(8.18)

where T̂(k) is the in-plane electron kinetic energy, Û is the potential energy of conduction
electrons under the static field of the lattice, and V̂c is the inter-conduction electron
Coulomb energy (intra-plane and inter plane). That is all we assume: we do not add spin
fluctuations, excitons, anyons... If they are there, they should come out from the above
Hamiltonian automatically. We know, from the previous section, that at least one of hT̂ i,
hÛ i, hV̂c i must be decreased by the formation of Cooper pairs. The default option for thisis hV̂c i: this is not necessarily true, but just to have a starting point, let us assume so.
An important observation, although it is not noted in much of literatures, is that there
is a rigorous relation (sum rule):
{
}
∫
∫
1
d
hV̂c i ∼ − d q dω Im,(8.19)
1 + Vq χ0 (q, ω)
where Vq is a Coulomb interaction (repulsive), and χ0 (q, ω) is a bare density response
function. For the three-dimensional case, this is∫∫3
hV̂c i3D ∼ d q dω (−Im ε(q, ω)−1 ),(8.20)
where Im ε(q, ω)−1 is called the loss function. Thus, this sum rule is a relation between
the interaction energy and the loss function.
Let us raise a question: where in the space of (q, ω) is the Coulomb energy saved (ornot)?. An interesting point is that quite independent from any microscopic theory, inprinciple, this can be answered by experiments. It can be answered to an extent by
an optics, or X-rays, but the best answer will be obtained from the electron energy-loss
spectroscopy (EELS) (I will come back to this point in Sec. 8.5.6).
Let us discuss the property of Eq. (8.19), especially in the case Vq χ0 (q, ω)  1 andVq χ0 (q, ω)  1.
A. Vq χ0 (q, ω)  1
(eﬀ)

−1

Suppose that Vq χ0 (q, ω)  1. This typically means q  qTF ∼ min(kF , kTF ) ∼ 1 Å .
6

This argument may be too naive because for Hg-1223 it neglects the polarizability of the other CuO2
planes.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

By expanding the denominator, and we find∫∼hV̂c iq = +Vq dω Im χ0 (q, ω) = Vq hρq ρ−q i0 .

(8.21)

This means that in order to decrease hV̂c iq , hρq ρ−q i0 must decrease. However, if wehave anything like the BCS theory, the change in the density correlation function in the
superconducting phase transition δhρq ρ−q ipairing can be written as∑δhρq ρ−q ipairing ∼∆p+q/2 ∆∗p−q/2 .
(8.22)
p

Now we arrive at an interesting point: if ∆p has the same sign over the Fermi surface, thisterm should be positive. That means that the formation of the Cooper pairs increases the
Coulomb energy, which is exactly what we do not like. That gives us a strong argumentthat the gap should change its sign over the Fermi surface. From this point of view, it
becomes not a big surprise that so many of the exotic superconductors appear to havealso exotic pair wave functions, i.e., gap symmetries.
B. Vq χ0 (q, ω)  1(eﬀ)

Let us consider the opposite case Vq χ0 (q, ω)  1. This typically means q  qTF . Then,we find)1 (hV̂c iq ∼−Imχ0 (q, ω)−1 .
(8.23)=
Vq
This tells us that in order to to decrease hV̂c iq (i.e., decrease Tc ), we should increaseImχ0 (q, ω), which is consistent with an increase in hρq ρ−q i0 . In other words, an increased
correlation leads to an increased screening, which in turn means a decrease of the Coulomb
energy.

8.5.3

Eliashberg vs. Overscreening

In the standard Eliashberg approach, crudely speaking, we generate an eﬀective attraction between pairs of electrons, as in Fig. 8.7 (a), the electron with k and −k will forma pair with that attraction. The attraction may be an exchange of some kinds of bosons,such as phonons, spin fluctuations, or excitons, but it must be attractive.
In the overscreened picture, as illustrated in Fig. 8.7 (b), the main eﬀect of pairing
is not coming from the electrons interacting in the opposite momentum, but from all
momenta k1 , k2 . The contribution comes from the modification of the interaction energy
between those electrons. The great advantage of this picture is that it does not require
an attractive interaction in the normal phase. Thus it is a diﬀerent picture from theEliashberg types of theories.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

(a)

(b)

Fig. 8.7. (a) Feynman diagram for the Eliashberg type of interaction between electrons.
(b) Feynman diagram for the overscreened interaction between electrons.

8.5.4

Role of two-dimensionality

Let us turn to a generic question: can we get any insight into why so many, or all, ofthe high-temperature superconductors are strongly two-dimensional? To consider this,use the sum rule again:∫}
1 ∑ dω {1
hV̂ i = −Im
2 q
2π
1 + Vq χ0 (q, ω)(8.24)
∫ ∞
}
{
1
1
1
.
=−
dd qIm2 (2π)d+1 0
1 + Vq χ0 (q, ω)
Here, note that Vq depends on the dimension. In three dimensions, Vq ∼ q −2 , and fromthe definition, 1 + Vq χ0 (q, ω) ≡ ε|| (q, ω), we find
∫
∫
{
1 }
2
hV̂ i ∼ q dq dω − Im.
(8.25)
ε|| (q, ω)Note that we have the loss function in the right-hand side. Therefore, the “small” q regionis strongly suppressed in the integral7 .
In two dimensions, on the other hand, Vq ∼ q −1 , so thatdVq χ0 (q, ω) ∼ q (ε(q, ω) − 1),2
where d is an inter-plane spacing. Then, we obtain8 ,∫{}1
hV̂ i ∼ qdq − Im
1 + q d2 (ε3D (q, ω) − 1)
∫
{
}
1
1
∼
dq − Im.
d
ε3D (q, ω)(8.26)(8.27)

7
The similar argument is used in the original paper of Hubbard [6] to justify that the small q region
is not essentially important, so that we can neglect the long-range part of the interaction.
8Note that the high-frequency dielectric constant for the cuprates is rather large, so that the termdq 2 (ε3D (q, ω) − 1) can be dominant even for a low momentum.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

Thus, the linear q factor in the integration is canceled by the linear q dependence on theright-hand side of Eq. (8.26), and the small q contribution may be as important as large
q part. That raises an important question: in 2D-like high-temperature superconductors(cuprates, ferropnictides, organics...), is the saving of the Coulomb energy, mainly at small
q?

8.5.5

Constraints on the Coulomb saved at small q

What do we know about constraints on the saving of the Coulomb energy at small q[4]? To answer this question, let us recall∫
Vq ∞Imχ(q, ω)dω.
(8.28)hV̂ i = Vq hρq ρ−q i =2π 0Here and from now on, we are using the “full” density response χ(q, ω), not the “baredensity response”. Note that the dimension can be arbitrary in the following discussion.
From the sum rule for the full density response function, we obtain∫
2 ∞ dωJ−1 ≡Kramers–Kronig relation,(8.29)Imχ(q, ω) = χ(q, 0)
π 0 ω
∫
2 ∞
nq 2
J1 ≡
ωdωImχ(q, ω) =f-sum rule,(8.30)
π 0
m
∫
q2
2 ∞ 3
n2
ω dωImχ(q, ω) = 2 hAi + q 4 2 Vq + o(q 4 ).
J3 ≡(8.31)
π 0mm
The third equation is a generalized Mihara–Puﬀ relation [7], andhAi ≡ −

1∑(k · q̂)2 U−k ρk .
π k

(8.32)

There is, in fact, a strong argument that hAi should be positive here. Note that in two
dimensions, the term in hAi is dominant at small q compared with the Vq term. From thegeneral Cauchy–Schwartz inequalities (for arbitrary dimension), we obtainor equivalently,

1√ 2
1√ 2 3
Vq J−1 J1 ≥ hV̂ iq ≥Vq J1 /J3 ,2
2(8.33)~ωp~ωp1√+ o(q 2 ) ≥ hV̂ iq ≥+ o(q 2 ).
hAi
2
2
1+

(8.34)nmωp2

It is interesting to see from this inequality that for hAi = 0 (for example, the “jellium”
model9 ), there is no saving of the Coulomb energy for q → 0 up to o(q 2 ). In other9

The jellium model is a model where we assume a uniform background of positive charge, rather thana periodic potential produced by the ion.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

words, if indeed a lot of the Coulomb energy is saved in the low-energy part for thehigh-temperature superconductors, it is crucial to have the lattice!
Here, let me show a further argument on why the small q region may be important.
Let us get back to the question which, in principle, can be answered by experiment:
Where in the space of q and ω is the Coulomb energy saved (or not)?
Let us, at this point, go over to a question at first totally diﬀerent:
Why does Tc depend on n, the number of layers?
As we have discussed for Ca-spaced homologous series, Tc rises with n at least up ton = 3 (this point is noncontroversial). This rise may be fitted by the formula (for “nottoo large” n)(1)
Tc(n) − Tc(1) ∼ const 1 −.
(8.35)nNote that this formula is controversial. A possible explanation is as follows:

A . Boring explanation
The superconductivity is a single-plane phenomenon, but multi-layering aﬀects properties
of individual planes, such as doping, band structure, screening by oﬀ-plane ions...
B . “Interesting” explanation: inter-plane eﬀects
Inter-plane eﬀects play an important roles. Some of the examples are listed below:
1. Anderson inter-layer tunneling model10 .
2. Kosterlitz–Thouless theory.
3. Inter-plane Coulomb interactions (we know that they are present!).
Primarily, what about the third possibility? For sure, we know that they are there.
The point is whether they are important or not. We have to have the electron-electroninteraction between neighboring planes if they are not negligible. If we take the eﬀectof the Coulomb interaction from electrons in the nearest plane, and perform the Fouriertransform, we obtain
Vint (q) ∼ q −1 exp(−qd),(8.36)
where q is the in-plane wave vector, and d is the intra-multilayer spacing, which is typicallyd ∼ 3.5 Å.
If the option 3 is right, then even in the single-plane materials, a dominant region of q
is q < d−1 !

10Although this idea is abandoned, but it is still interesting.

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8.5.6

LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

Mid-infrared optical and EELS spectra of the cuprates

Then, where in ω is energy saved? Here I use the WILLIE SUTTON principle11 : wecan only save the energy for making the superconducting phase transition in regions ofq and ω where there is a lot of energy stored up in the normal phase. Thus, to achievea high-temperature superconductor, we have to specify where the energy is saved in thenormal phase. This can be investigated in experiments.

Optics
Optics show that there is a lot of energy saved in the long-wavelength, mid-infrared
regime, as we can see from the measurement of the loss function L(ω) ≡ −Imε−1 (ω)
shown in Fig. 8.8.

EELS
EELS measures q → 0 shape of the loss function, and verifies that (roughly) the same
shape persists for finite q. (at least up to ∼ 0.3 Å).
Thus, it looks like that that’s where the money is! That’s where the energy is saved in
the normal phase!

0.001 eV

0.01 eV

0.1 eV

1 eV

10 eV

(logarighmic scale)Fig. 8.8. Schematic illustration of the loss function L(ω) measured by an optical measurement. The large peak corresponds to the mid-infrared peak, while the contribution
in the higher-energy part is a rather material-dependent part, coming from each atoms.

11

WILLIE SUTTON is a famous bank robber in the US in the 1930s. One of the reason he is notorious
is when journalists interview him, “why do you rob banks?”. Then he replies, “Because that’s where themoney is!”.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

Then, that is the obvious guiding principle we can use for the realization of high-temperature
superconductors.
If the saving of the Coulomb energy is mainly in the low-q, mid-infrared regime, thenwe can expect that a certain change in the shape of spectrum should appear in thenormal-to-superconducting phase transition: this should be able to be observed in the
EELS spectrum. In fact, from the sum rule, we expect that Imε−1 (must) decrease in theδεphase transition in the mid-infrared regime. This means that Im ε2 > 0, where δεnis the change of the dielectric constant through the phase transition. To estimate thisquantity from experiments, we should know both the real part and imaginary part of thedielectric constant. Unfortunately, in most experiments, they only show such quantities
as the reflectivity, eﬀective plasma frequency, and so on, and there are few papers whichshow both the real and imaginary part explicitly. There is one paper which does show
both12 , and there is a very interesting piece of data in it [8]: both in the real part and theimaginary part, they have a strong frequency dependence. However, for any frequency,
they are exactly the opposite to each other. That is, the real part of ε is exactly minusof the imaginary part:ωp2
εn (ω) ∼(8.37)= 2 (−1 + i),ωwhere ωp is the plasma frequency. This equation means
ε−2
n ∼

ω4
i.
2ωp4

(8.38)

If this is right, we can conclude that a condition Reδε < 0 is required in the mid-infrared
regime. We now apply the Kramers–Kronig transform to see what it implies for theimaginary part, and we obtain∫ ∞ {1
ωe + ω 0 ωe }
ω 04log
− 0 Imδχ(q, ω 0 )dω 0 < 0 (ωe ∼ ωp ).
(8.39)02
ωe − ω
ω0
It is important to note that the bracket part in this equation is positive for ω 0 > ω e , whereω e is some characteristic frequency of the order of the plasma frequency ω e ∼ ωe ∼ ωp .
On the contrary, it is negative for ω 0 < ω e . If indeed a lot of energy is saved in the
low-momentum, mid-infrared regime, what we would expect from this is that the spectral
weight transfers from ω > ωp to ω < ωp : a transfer into the mid-infrared regime. The
experimental data and the presentation in the literatures are complicated, but I believethat this in fact is entirely consistent with what they actually see in the experiment.
Even if it works, however, I do not want you to care too much. The reason we should
not worry is that the optics measure q  ξ −1 , whereas the saving of the Coulomb energy should be mainly from ξ −1 < q . qTF . Therefore, what we really need is the EELS
experiment! Although there have been EELS experiments in the normal phase in the12Admittedly, they measured the reflectivity, and then used the Kramers–Kronig relation to obtain the
dielectric constant, but it seems that they are right.

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LEC. 8. EXOTIC SUPERCONDUCTIVITY: DISCUSSION

mid-infrared regime, and there have been also done in the superconducting regime, there
has been no experiment which has shown the changes in going from the normal to superconducting phase in the mid-infrared regime. I have been leaning on various experimentalgroups for 15 years to do these experiments, and at last my colleagues P. Abbamonte,
and J. Zuo at UIUC are going to do this experiment. Hopefully, in a year or two, we willknow the result of the experiment. That will be a really crucial test for this argument.

8.6

How can we realize room temperature superconductors?

Let us just assume that the above idea is checked in the EELS experiment, and ask
what are good ingredients for enhancing Tc . The possible ways are as follows:
1. Two-dimensionality (a weak tunneling contact between layers, but a strong Coulomb
contact).
2. Strongest possible Coulomb interaction (intra-plane and inter-plane).
3. Strong Umklapp scattering may give rise to a wide and strong mid-infrared peak.
What we have actually shown is that in the absence of the lattice potential, we will not
get the saving of the energy. Thus, the lattice is crucial. One of the eﬀect of the lattice
is to give rise to the Umklapp scattering, although there are of course many other eﬀects.
If there is a strong Umklapp scattering, it gives rise to a wide and strong mid-infrared
peak. Since the wide and strong mid-infrared peak is a very characteristic thing in the
cuprates, it seems that this argument works. However, note that although we know forsure that there is such a peak, we are not sure about its origin, and the Umklapp process
is only one possibility. For example, a strong antiferromagnetic fluctuation may be theorigin for the mid-infrared peak.
Anyway, the EELS experiment will show whether the above argument works or not,and these guidelines are correct or not.
Finally, I will make my bet on robust room-temperature superconductors. I do notthink we will get it in my lifetime unless I am very lucky (∼ 10%). But, I suspect inyour lifetime, we will probably get a robust room temperature superconductor (> 50%).
Hopefully, this kind of argument may be somewhat helpful in finding new materials andso forth.
Thank you154

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物性研究・電子版 Vol. 1, No. 1, 011201 （2012年5月号）

References[1] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
[2] W. Kohn and D. Sherrington, Rev. Mod. Phys. 42, 1 (1970).
[3] P. Monthoux and D. Pines, Phys. Rev. B 50, 16015 (1994).
[4] M. Turlakov and A. J. Leggett, Phys. Rev. B 67, 094517 (2003).
[5] G. V. Chester, Phys. Rev. 103, 1693 (1956).
[6] J. Hubbard, Proc. R. Soc. A 276, 238 (1963).
[7] N. Mihara and R. D. Puﬀ, Phys. Rev. 174, 221 (1968).
[8] A. El Azrak, R. Nahoum, N. Bontemps, M. Guilloux-Viry, C. Thivet, A. Perrin,
S. Labdi, Z. Z. Li, and H. Raﬀy, Phys. Rev. B 49, 9846 (1994).

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Index
3

He
A phase of —, 42, 43
A1 phase of —, 42, 44
B phase of —, 42, 45
Cooper pair in —, 32
Normal liquid —, 29, 42
superfluid —, 43

ABM state, 43, 46, 47, 55
Abrikosov–Gor’kov theory, 74
Alkali fullerides, 78
Ambegaokar–Baratoﬀ formula, 120
Anderson pseudospin representation, 16Anderson’s interlayer tunneling theory, 127,
151
Anderson’s theorem, 73
Anderson–Brinkman theory, 48
Anyon superconductivity, 139
ARPES, 108— in the cuprates, 109, 122— in the pseudogap regime, 113
BCS wave function, 11, 12
BKBO, 61
Bloch state, 25, 67
Bloch-Grüneisen formula, 62
Boojum texture, 58
Broken pair (BP) states, 18
BW state, 45–47
Chandrasekhar–Clogston limit, 70
Charge reservoir, 78, 98
Classical superconductor, 61
Coherence length, 118
Condensation energy, 74— in the cuprates, 118
156

Conductivity— in the cuprates, 120
Critical field
Upper — in the cuprates, 117
Cuprates
AC conductivity in —, 120
ARPES in —, 109, 113, 122
Condensation energy in —, 118
Crystal structure of —, 86, 99
dHvA experiment in —, 113
Doping in —, 99
EELS in —, 125
Ellipsometric measurements in —, 110,124
Hall angle in —, 108
Isotope eﬀect in —, 128
Knight shift in —, 106, 112, 118
List of —, 104Loss function in —, 110
Luttinger’s theorem in —, 109
Microwave surface impedance in —,119
Muon spin rotation in —, 119
Neutron scattering in —, 110, 123
Nuclear spin relaxation rate in —, 106,
112, 118
Optical reflectivity in —, 110, 124
Pair radius in —, 117, 127
Penetration depth in —, 117, 118, 120
Phase diagram of —, 93, 101
Resistivity in —, 106, 112
Specific heat in —, 112, 118
Superconducting fraction in —, 119
Thermal conductivity in —, 121

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Index

Thermoelectric power in —, 108
Tunnel conductance in —, 112, 121
Upper critical field in —, 117

Gutzwiller projection, 141
Hall angle— in the cuprates, 108
Hartree term, 13
Healing length, 24Ginzburg–Landau —, 27
Heavy fermion, 82— superconductor, 62
Normal state of —, 82
Hebel–Slichter coherence peak, 71
Hellman-Feynman theorem, 146
Homologous series, 98, 103, 133, 151
Hubbard model, 141

d-vector, 37
de Haas-van Alphen (dHvA) eﬀect, 80,
88, 109— in the cuprates, 113
Debye frequency (temperature), 21, 63
Density matrix
Reduced density matrix, 65
Dipole-dipole interaction— in 3 He, 50–52
Electron energy-loss spectroscopy (EELS),
111, 152— in the cuprates, 125
Eliashberg equation, 26, 63, 64
Ellipsometric measurements— in the cuprates, 110, 124
Equal spin pairing (ESP) state, 36, 69, 88
Excited pair (EP) state, 18
Exotic superconductor, 61
Common properties of —, 136
Extended s-wave state, 71, 130

Intercalant, 98
Isotope eﬀect, 21, 62— in the cuprates, 128
Isotope exponent, 63
Josephson experiment, 72, 89, 125
AC Josephson eﬀect, 139
Principles of —, 131
Kerr eﬀect, 90
Knight shift, 69, 73— in the cuprates, 106, 112, 118
Kondo eﬀect, 83
Korringa relation, 106
Kosterlitz–Thouless (KT) phase, 139, 151
Kramers–Kronig relation, 110, 150

Fermi liquid theory, 25, 29, 62
Ferropnictides, 62, 91
Far infra-red (FIR) absorption, 126
Flux quantization, 125, 139
Fock term, 13, 23
Fragmented BEC, 56, 66
Fullerene crystal, 77

London penetration length, 138
Loss function— in the cuprates, 110
Luttinger’s theorem, 109

Gap equation, 20— at zero temperature, 17— at finite temperature, 20— for ESP state, 36— for general triplet states, 37
Ginzburg–Landau theory, 38
Generalized —, 46, 129
Groud pair (GP) state, 18

Macroscopic angular momentum, 34
Meissner eﬀect, 137, 138
Mermin–Ho vortex, 58
MgB2 , 61, 80
Microwave surface impedance— in the cuprates, 119
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Index

Mid-infrared (MIR) peak, 111, 152
Mihara–Puﬀ relation, 150
Muon spin rotation, 89— in the cuprates, 119

RKKY interaction, 83
Screening, 26, 146, 148, 151
Overscreening, 148
Sommerfeld model, 10, 25
Specific heat, 62, 72— in the cuprates, 112, 118
Spectators, 78
Spin fluctuation
Persistent — in 3 He, 31— feedback, 48— theory in the cuprates, 142
Spin-orbit couplingEﬀect of —, 74— in heavy fermions, 85
Spontaneously broken spin-orbit symmetry (SBSOS), 51
Strontium Ruthenate, 62, 86
Sum rule, 147
f-sum rule, 150— in 3 He, 50
Superconducting fraction— in the cuprates, 119
Supercurrent, 57
Mass —, 57
Spin —, 57

Neutron scattering— in the cuprates, 110, 123
NMR
Hebel–Slichter peak in —, 71— in 3 He, 49, 55
Nuclear spin relaxation rate, 73— in the cuprates, 106, 112, 118
Oﬀ-diagonal long range order (ODLRO),
125, 138
Old-fashioned superconductor, 61
Optical lattice— in ultracold atoms, 141
Optical reflectivity— in the cuprates, 110, 124
Organic superconductor, 62, 80
Pair radius, 14, 24, 25— in the cuprates, 117, 127
Pair wave function, 23, 66— at zero temperature, 13, 18— at finite temperature, 20
Paramagnon, 31
Parent compound, 99
Penetration depth, 73, 118
London —, 138— in the cuprates, 117, 118, 120
Persistent current, 137, 138
Planar (2D) state, 47
Polar (1D) state, 47
Pseudogap regime, 102, 111

T ∗ -line, 102, 133
Thermal conductivity— in the cuprates, 121
Thermoelectric power— in the cuprates, 108
Tunnel conductance— in the classical superconductors,
64— in the cuprates, 112, 121

Quasi-particle, 19, 33

Umklapp scattering, 83, 87, 93, 107, 154

Random-phase approximation (RPA), 26
Reduced density matrix, 65
Resistivity— in the cuprates, 106, 112

Virial theorem, 145
WILLIE SUTTON principle, 152

158