SPIN FLUCTUATION THEORY FOR UNCONVENTIONAL SUPERCONDUCTIVITY INANTIFERROMAGNETIC METALS

By

WENYA W. ROWE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2014

c⃝ 2014 Wenya W. Rowe

2

ACKNOWLEDGMENTS

I thank my advisor Professor Peter J. Hirschfeld for his continued support and

encouragement which have help me to grow. His patience, humor, enthusiasm, and

broad knowledge have been invaluable for my doctoral study.

I am also grateful for the time and support from members of my committee, Profes-

sors K. Muttalib, P. Kumar, G. Stewart, and S. Phillpot. I would like to thank Professors.

D. Maslov, K. Muttalib, and K. Ingersent for their lucid and rich lectures, their patience

and extra guidance.

My work has mostly been done in collaboration with researchers in other institutes.

I express my special appreciation to Professor Ilya Eremin at the Ruhr University in

Bochum for his guidance over the years. Thanks to Dr. J. Knolle who helped me with the

calculations of the spin susceptibility and the mean field energy. Thanks to Professor

B. M. Andersen for his help with the potential calculations. And thanks to A. Rømer for

her meticulous comparison of the results. I would like to thank the Ruhr University in

Bochum for their hospitality during my short visits and during the last year of my doctoral

study in Germany.

I thank Drs. G. Boyd, A. Kemper, V. Mishra and M. Korshunov, former members of

the Hirschfeld group. They provided encouragement and informative advices during the

beginning of my research years. I thank Dr. A. Kreisel, Y. Wang and P. Choubey for their

helpful discussions about all matters.

3

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Spin fluctuation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.1 Kondo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.2 Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.3 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Spin fluctuations and superconductivity . . . . . . . . . . . . . . . . . . . 141.3 Unconventional superconductivity . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.2 Iron-pnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Heavy fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.4 Organic and fullerene superconductors . . . . . . . . . . . . . . . . 23

2 ANTIFERROMAGNETIC STATE . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1 Ferro- and Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Itinerant electron magnetism . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Mean field phase diagram including AF and superconductivity . . . . . . . 28

3 DYNAMIC SPIN SUSCEPTIBILITY . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1 Theory and calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.1 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Spin excitations in the pure antiferromagnetic state . . . . . . . . . . . . . 393.2.1 The dynamic spin susceptibility in the antiferromagnetic state . . . 393.2.2 The effect of next-nearest hopping, t ′ on the spin excitations . . . . 423.2.3 The effect of the dopants on spin excitations . . . . . . . . . . . . . 44

3.3 Spin excitations in the coexistence state . . . . . . . . . . . . . . . . . . . 48

4 THE PAIRING INTERACTION ARISING FROM ANTIFERROMAGNETIC SPINFLUCTUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 The pairing interaction in the antiferromagnetic background . . . . . . . . 554.2 The pairing symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Angular dependence of the coherence factors . . . . . . . . . . . . 63

4

4.2.2 Angular dependence of the pairing potentials . . . . . . . . . . . . 674.2.2.1 Charge and longitudinal interaction . . . . . . . . . . . . 674.2.2.2 Transverse interaction . . . . . . . . . . . . . . . . . . . . 694.2.2.3 Interband interactions . . . . . . . . . . . . . . . . . . . . 70

4.2.3 LAHA expansion of gap equation . . . . . . . . . . . . . . . . . . . 714.2.4 Comparison with numerical evaluation . . . . . . . . . . . . . . . . 74

5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

APPENDIX

A MEAN FIELD QUANTITIES IN THE PURE ANTIFERROMAGNETIC STATE . 81

A.1 Antiferromagnetic order parameter equation: derivation . . . . . . . . . . 81A.2 The electron filling: derivation . . . . . . . . . . . . . . . . . . . . . . . . . 82

B DERIVATIONS IN THE COEXISTENCE STATE OF ANTIFERROMAGNETISMAND SUPERCONDUCTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.1 Antiferromagnetic order parameter equation in the coexistence state withsuperconductivity: derivation . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.2 Filling level of electrons in the coexistence state: derivation . . . . . . . . 83B.3 Mean field energy in the coexistence state: derivation . . . . . . . . . . . 85

C DERIVATTIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE PURE AN-TIFERROMAGNETIC STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

C.1 Transverse dynamic spin susceptibility in the antiferromagnetic state:derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

C.2 Umklapp term for the transverse dynamic spin susceptibility . . . . . . . . 92C.3 The longitudinal dynamic spin susceptibility . . . . . . . . . . . . . . . . . 94C.4 The longitudinal Umklapp susceptibility . . . . . . . . . . . . . . . . . . . 96C.5 Analytic proof for the formation of the Goldstone mode . . . . . . . . . . 98

D DERIVATIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE COEXISTENCESTATE OF ANTIFERROMAGNETIC AND SUPERCONDUCTIVITY . . . . . . 99

D.1 Derivations of transverse dynamic spin susceptibility in the coexistencestate of antiferromagnetism and superconductivity . . . . . . . . . . . . . 99

D.2 The Umklapp term for the transverse dynamic spin susceptibility . . . . . 106D.3 The longitudinal dynamic spin susceptibility . . . . . . . . . . . . . . . . . 110

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5

LIST OF TABLES

Table page

4-1 Coherence factors, p2(k, k′) and n2(k, k′) expanded around hole pockets fork = (π

2, π2) and k′ = (±π

2, π2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4-2 Coherence factors, p2(k, k′) and n2(k, k′) expanded around hole pockets fork = (−π

2, π2) and k′ = (±π

2, π2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4-3 Coherence factors, p2(k, k′) and n2(k, k′) expanded around electron pockets . 66

4-4 Coherence factors, p2(k, k′) and n2(k, k′) expanded around electron and holepockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4-5 Potentials from the charge- and longitudinal spin-fluctuation contribution ex-panded around hole pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4-6 Potentials from the transverse spin-fluctuation contribution, −2Γs expandedaround hole pockets in the limit of khF → 0 . . . . . . . . . . . . . . . . . . . . . 70

4-7 Potentials from the charge- and longitudinal spin-fluctuation interband contri-bution expanded between electron and hole pockets . . . . . . . . . . . . . . . 71

4-8 Angular dependence of the s-wave and dx2−y2-wave symmetries on the holepockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4-9 Angular dependence of the s-wave and dx2−y2-wave symmetries on the elec-tron pockets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6

LIST OF FIGURES

Figure page

1-1 The Feynman diagram in the Berk-Schrieffer approximation to the effectiveelectron-electron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1-2 The relative signs of superconducting gap on a cuprate-like Fermi surface . . . 16

1-3 Phase diagrams of hole-doped and electron-doped cuprates . . . . . . . . . . 18

1-4 Crystal structures of the electron-doped R2−xSrxCuO4 and the hole-dopedLa2−xSrxCuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1-5 Crystal and spin structures of the electron-doped R2−xSrxCuO4 and the hole-doped La2−xSrxCuO4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-1 The Fermi surface and the band structure of electron-doped cuprates . . . . . 35

2-2 The doping-temperature phase diagram of electron-doped cuprates . . . . . . 35

3-1 Neutron scattering on Pr1−xLaCexCuO (PLCCO) . . . . . . . . . . . . . . . . . 37

3-2 The band structures and imaginary part of transverse dynamic spin suscepti-bility at half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3-3 Three possible types of Fermi surface topology in the antiferromagnetic statein layered cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3-4 Calculated imaginary part of transverse χ+−RPA(q,q, Ω) . . . . . . . . . . . . . . . 46

3-5 Calculated Imaginary part of the transverse χ+−RPA(q,q, Ω) spin excitation spectra 50

3-6 Calculated imaginary part of the longitudinal susceptibility, χzzRPA(q,q, Ω) spinexcitation spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4-1 General structure of the Fermi surface of layered cuprates . . . . . . . . . . . . 64

4-2 Comparison of the analytical calculations up to (keF )2 for the longitudinal and

transverse pairing potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

SPIN FLUCTUATION THEORY FOR UNCONVENTIONAL SUPERCONDUCTIVITY INANTIFERROMAGNETIC METALS

By

Wenya W. Rowe

May 2014

Chair: Peter J. HirschfeldMajor: Physics

The understanding of unconventional superconductivity is still a challenge for con-

densed matter physicists. To understand the interplay between antiferromagnetic order

and superconductivity is crucial for the development of unconventional superconductivity

theory, not only because the antiferromagnetic state coexists with superconductivity in

many materials such as cuprates, iron-based pnictides, heavy fermions and organic

superconductors, but also because spin fluctuations near a magnetically ordered phase

have been proposed to possibly mediate superconductivity. In Chapter 1, we introduce

the mechanism of spin fluctuations and review important categories of unconventional

superconductors. In Chapter 2 we review elements of the relevant theory of magnetism.

We discuss the differences between localized and itinerant approaches to study mag-

netism. We focus on cuprates which have a simple one-band Fermi surface. We use the

Hubbard model to describe the band structure of the cuprates, and introduce the mean

field phase diagram of electron-doped cuprates. In Chapter 3, we study the dynamic

susceptibility of cuprates in the pure antiferromagnetic state and in the coexistence state

of antiferromagnetism and superconductivity. We identify the key features of particle-

hole spin excitations which are affected by the next-nearest neighbor hopping, t ′. We

compare the different spin wave features between electron- and hole-doped cuprates.

We conclude that the long range commensurate antiferromagnetic state is unstable

on the hole doped side within the self-consistent-mean field theory due to the negative

8

spin-stiffness. In the coexistence state, we see the spin resonance peak caused by

superconductivity as well as the Goldstone mode in the spin excitation spectrum. Last

we present the instability analysis of superconductivity from spin fluctuations in the

antiferromagnetic state. We derive the superconducting pairing potentials and the gap

equations for the spin singlet and triplet pairings. We separate the singlet potentials

into longitudinal and transverse channels and expand the them around the pocket’s

center in the small pocket limit. Our result shows on the electron-doped side the leading

symmetry is dx2−y2−wave and on the hole-doped side it is p−wave. This implies that

from the paramagnetic state to the antiferromagnetic state, the superconducting gap

has a smooth transition on the electron-doped side whereas the gap has to change

symmetry on the hole-doped side. We conjecture that this may account for the lack of

bulk coexistence of antiferromagnetic and superconducting order on the hole-doped side

of the cuprates phase diagram.

9

CHAPTER 1INTRODUCTION

1.1 Spin fluctuation models

For localized spin systems, magnetic properties can be fairly well described by

a variety of theoretical approaches[1]. For weakly ferromagnetic systems, which are

itinerant, predictions are not accurate, since statistical fluctuations of the charge and

screening of the spin moment have to be included for the itinerant electron systems.

Both thermal and quantum fluctuations can be important. Thermal fluctuations vanish

when temperature is zero, and then increase with temperature, but quantum fluctuations

are present even when the temperature is zero. Here we consider quantum spin

fluctuation theory which has been developed with Green’s functions. Spin fluctuation

theory is really a collection of methods based on a small set of models[2], the main

ones being the Kondo, Anderson and Hubbard models, each of them having several

variations.

The theoretical treatment of magnetism in metals began with Stoner theory in the

1930s. At that time, people had doubts about the possibility of describing spin waves

in itinerant systems. The RPA was later developed by Doniach and Engelsberg[3],

and applied it to Pd metal which is nearly ferromagnetic. Anderson and Brinkman [4]

used the same theory to understand the stability of the 3He A-phase liquid. Berk and

Schrieffer [5] made the important observation that spin fluctuations would suppress

s-wave superconductivity.

I now briefly review the models which have been discussed in the context of spin

fluctuations.

1.1.1 Kondo Model

The Kondo model has been used to describe isolated impurities in metals and

quantum dot systems.The magnetism comes from the partially filled d− or f− shell,

10

which results in an uncompensated moment Sj . The classic material is magnetic Mn

impurity in Cu.

The model is described by the following Hamiltonian:

H =∑k,σ

εkc†k,σck,σ

− JN

∑k,p,j

e iRj ·(k−p)[(c†k,↑cp,↑ − c†k,↓cp,↓)S

(z)j + c

†k,↑cp,↓S

(−)j + c†k,↓cp,↑S

(+)j ]

(1–1)

The first term is the kinetic energy of the conduction electrons. The other terms are the

scatterings of the electrons from the local spin at Rj . J < 0 is the antiferromagnetic

exchange coupling. Condensed matter physicists observed singular effects of magnetic

impurity in nonmagnetic metals. The electron scattering contribution to the resistivity is

temperature independent for the nonmagnetic impurity. But for nonmagnetic metals with

magnetic impurity, there is a minimum in resistivity at low temperature. The minimum is

called the Kondo effect, representing a singularity in the many-body scattering amplitude

of the electrons from the localized spin. The first-order self-energy of the electron is

independent of wave vector or energy. The second-order perturbation of the Green’s

function which has to be evaluated is

G(2)αβ (k, τ) = −12

∫ β

0

dτ1

∫ β

0

dτ2⟨Tck,α(τ)V (τ1)V (τ2)c†k,β(0)⟩ (1–2)

where V is the interaction in 1–1,

V = − JN

∑kpαβ

exp[iRj · (k− p)]σαβ · Sc†kαcpβ. (1–3)

For calculating the anomalous behavior of the resistivity, the third order in perturbation

theory has to be considered. The third order in the Green’s function is

G(3)(k, τ) = −cJ3

N2

∑pq

∫ β

0

dτ1

∫ β

0

dτ2

∫ β

0

dτ3G(k, τ − τ1) (1–4)

×G(p, τ1 − τ2)G(q, τ2 − τ3)G(k, τ3)L(τ1, τ2, τ3)

(1–5)

11

where Gis the bare Green’s function in the pure system and

L(τ1, τ2, τ3) =∑νµλ

σ(ν)αs σ(µ)ss′ σ

(λ)s′α⟨TS

(ν)(τ1)S(µ)(τ2)S

(λ)(τ3). (1–6)

In third order, one finds a log divergence of the Green’s function. Kondo derived the

temperature dependence of resistivity from perturbation theory to third order[6], and got

ρ(T ) = ρ(0)[1 + 4Jg(0) ln(kBT/W )] + bT5. (1–7)

The term with T 5 is due to the lattice vibrations. The term with ln(T ) is the Kondo effect,

which creates a minimum at finite temperature. This divergence signals the breakdown

of perturbation theory at low temperatures, The exact solution to the problem[7, 8] as

well as the renormalization groups (RG) show that the moment S is screened below a

temperature known as the Kondo temperature, which is approximately

TK ≈ εF√J exp (−1/N0J), (1–8)

where εF is the Fermi energy, J is the exchange interaction strength and N is the density

of states.

1.1.2 Anderson model

The Anderson model is a model describing a localized state when the state is far

below the Fermi level interacting with conduction electrons. The Hamiltonian is

H =∑k,σ

[εkc†k,σc

†k,σ +

Vk√N(c†k,σfσ + f

†σ ck,σ)] (1–9)

+εf∑σ

f †σ fσ + U∑µ>σ

nσnµ

where nσ = f †σ fσ. fσ is the destruction operator for the local state and c is the deconstruc-

tion operator on the conduction state. Vk is the hybridization. U is the electron-electron

interaction between the localized electrons. The electron can move from a local level

with energy εf to a conduction state and vice versa. In the limit where εF << Γ where

12

γ = πN0V2 is the bare hybridization width, the model reduces to a Kondo impurity model

with J = 2V 2/εF [9].

1.1.3 Hubbard model

The Hubbard model has a very simple form[10]:

H = −∑⟨ij⟩σ

tijc†jσciσ + U

∑i

ni↑ni↓ (1–10)

where ⟨ij⟩ are lattice sites and ⟨ij⟩ means nearest neighbor bonds. The first part is the

hopping term between lattice sites. The second part is the interaction term. U is the

on-site Coulomb interaction. An electron can hop from site to site. In the non-interacting

limit, U << t, the Hubbard model is just a tight-binding model with itinerant electron

band. But at half-filling with U >> t, the electrons have too large a Coulomb interaction

to overcome, and the system becomes a Mott insulator. With comparable values of U

and t, there will be metal-to-insulator transition.

The one-dimensional Hubbard model has been solved exactly[11] but no exact

solution is available in higher dimensions. For the large U/t limit, the Hubbard model

at half filling can be mapped onto the Heisenberg model, and shows the relation

J = 4t2/U. Away from 1/2-filling, J is the antiferromagnetic exchange coupling in the

t − J model that describes strongly correlated electron systems.

The one-band Hubbard model has been generalized to accommodate different

systems. The three-band Hubbard model describes 2 − d CuO2 layers in cuprates. This

model has three different U ’s - Cu(3dx2−y2) state with Ud , O(2px ,y ) state with Up and the

nearest-neighbor interaction with Upd .

The Hubbard Hamiltonian in real space can be transformed into momentum space.

The Hamiltonian is

H∑k,σ

εkc†k,σck,σ +

U

2

∑k,k′,q,σ

c†k,σck+q,σc†k′,σck′+qσ (1–11)

13

Figure 1-1. The Feynman diagram in the Berk-Schrieffer approximation to the effectiveelectron-electron interaction. There are non-spin-flip processes, (a) andspin-flip processes, (b) and (c)

where ε = −2t(cos kx + cos ky) + 4t ′ cos kx cos ky − µ if we only consider nearest- and

next-nearest-hopping in a two-dimensional lattice. µ is the chemical potential to decide

the doping of the system. This model has been adapted to describe the hole-doped

cuprates with antiferromagnetic order[12–14].

1.2 Spin fluctuations and superconductivity

Spin fluctuations suppress singlet superconductivity in electron-phonon mediated

superconductors, but later it was shown that spin fluctuations may give rise to p-wave

pairing in the superfluid phase of 3He and also superconductivity in heavy fermions and

other high-Tc superconductors.

14

In spin fluctuation theory, we can sum over a subclass of all Feynman diagrams.

The random phase approximation (RPA) shown in Figure 1-1 retains an infinite subset

of these diagrams. It contains non-spin-flip processes in Figure 1-1 (a) and spin-flip

processes in Figure 1-1 (b) and (c). The summation of Figures 1-1(a), (b) and (c) are

written as:

− U3χ201− U2χ20

=− 12U[ 1

1− Uχ0− 1

1 + Uχ0

](1–12a)

U3χ201− U2χ20

=1

2U[ 1

1− Uχ0+

1

1 + Uχ0− 2]

(1–12b)

U2χ01− Uχ0

=U[ 1

1− Uχ0− 1]

(1–12c)

where χ0(q,ω) is the bare dynamic susceptibility of the metal,

χ0(q,ω) =∑k

f (εk+q)− f (εk)ω − εk+q + εk + i0+

. (1–13)

Here is an example from Scalapino et al. [15] showing how spin fluctuations can

give rise to d−wave pairings in cuprates. We separate the interaction into singlet (s) and

triplet (t) channels and obtain

Vs =U2χ01− Uχ0

+U3χ2

1− U2χ20(1–14a)

Vt =− U2χ01− U2χ20

. (1–14b)

In a model band εk = −2t(cos kx + cos ky) + 4t ′ cos kx cos ky , it may be shown that

χ = χ0/(1 − Uχ0) is strongly peaked at (π, π). If we solve the BCS gap equation at

T = 0,

∆(k) = −∑p

V (k− p)2Ep

∆(p), (1–15)

with Ep the quasiparticle energy, the largest contribution would be the wave vector of

(π, π) which spans the Fermi surface at the so called ”hot spot” as shown in Figure 1-2.

If we are solving the superconducting gap at an arbitrary point k, then the interaction

15

Figure 1-2. The relative signs of superconducting gap on a cuprate-like Fermi surface.The blue line is the Fermi surface. The brown dashed line is the position ofgap nodes.

couples to the superconducting gap at point p which is (π, π) away from point k as

shown in Figure 1-2. V (π, π) = Vs(π, π) and Ep are both positive. So we need the gap

to change sign in order to satisfy the gap equation. Therefore the superconducting gap

at point k and p have opposite signs. With this relation and the periodic conditions of the

Fermi surface, we can get a d-wave superconducting gap as indicated in Figure 1-2 for a

cuprate-like Fermi surface. This result is for the interaction in the paramagnetic state.

1.3 Unconventional superconductivity

After the emergence of BCS theory in 1957, physicists thought the mystery of

superconductors had been solved. However in 1979, Steglich reported the discovery

of superconductivity in heavy fermion CeCu2Si2[16]. Scientists continue discovering

superconductivity in materials where superconductivity id not phonon mediated. They

are categorized as unconventional superconductors[17].

16

Since then theorists and experimentalists invest their efforts studying the pairing

symmetries of different materials, and expect it can help us reach the understanding

of the pairing mechanism. Experimentalists have various kinds of ways to extract infor-

mation about the materials. The most common methods include: resistivity, magnetic

susceptibility, and specific heat to confirm a phase transition; ARPES (Angle-Resolved

Photoemission Spectroscopy) to map out the Fermi surfaces which play an important

role in superconductivity, and the gap magnitude itself; phase-sensitive experiments

like Josephson tunneling to distinguish one pair symmetry from another; and electro-

magnetic response properties such as optical conductivity and Raman scattering to

determine the superconducting gap energy. From the first discovered unconventional

superconductors to the most recent, we can generally group them into several main

branches: heavy-fermion SC, organic SC and high Tc SC(cuprates and iron pnictides).

1.3.1 Cuprates

Superconductivity in cuprates was discovered by Bednorz and Muller[18]. The

cuprates superconductors have the highest critical temperatures, which makes the

cuprates the most popular materials for higher temperature superconductor applications.

HgBa2Ca2Cu3O8 has the highest Tc , around 150K under pressure. All cuprates have a

layered structure with CuO2 planes. The parent compounds are antiferromagnetic Mott

insulators. Upon doping, the antiferromagnetism is destroyed abruptly, especially on the

hole-doped side as in Figure 1-3.

To understand the cuprates, we should take a closer look at the doping-temperature

phase diagram. The phase diagrams of hole-doped and electron-doped cuprates

are quite different, as shown in Figure 1-3. The antiferromagnetic region on the hole-

doped side is almost five times narrower than on the electron-doped side (see gray

region of Figure 1-3). The electron-doped cuprates [19] have a robust commensurate

antiferromagnetic phase but a much narrower SC range. It is believed that there is a

coexistence of superconductivity and antiferromagnetism on the electron-doped side.

17

Figure 1-3. Phase diagrams of hole-doped and electron-doped cuprates. T ⋆ is thepseudogap transition temperature. TN is the Neel tempetature and Tc is thesuperconducting temperature. Reproduced with permission from Armitage,N. P. and Fournier, P. and Greene, R. L., Progress and perspectives onelectron-doped cuprates. Rev. Mod. Phys., 82(3):24212487, Sep 2010(Page2422, Figure 2). c⃝(2010) by The American Physical Society

The experimental evidence includes neutron scattering experiment on Nd2−xCexCuO4

which shows coexistence region up to the optimal doping level[20, 21]. NMR (Nuclear

Magnetic Resonance) could be used as a probe in determining the coexistence state

and superconductivity in the electron cuprates. But the rare-earth atoms in the spacer

layer of the electron-doped cuprates give a large magnetic response; therefore it is

hard to interpret the data[19]. On the hole-doped side, superconductivity coexists with

only striped or glassy disorder induced magnetic order according to NMR (Nuclear

Magnetic Resonance)[22] and neutron scattering measurements[23]. For a review,

see Ref. [24]. One aspect that reflects the asymmetry in the phase diagram is that the

hole-doped cuprates generally have incommensurate antiferromagnetic order while the

electron-doped ones always have commensurate order with momentum vector (π, π).

18

One of the most puzzling regions of the phase diagram is the so-called ”pseudogap

phase”, which reprents a region where the system displays a partial gapping of low-

energy excitations, although no antiferromagnetic or superconducting long-range order

is present. Long thought to be a crossover phase with no broken symmetries, the

pseudogap transition T∗ has recently been shown to coincide with a weak breaking of

time reversal symmetry, although the nature of the phase is still not clear[25].

Figure 1-4. Crystal structures of the electron-doped R2−xSrxCuO4 and the hole-dopedLa2−xSrxCuO4. Here R is one of the rare-earth ions-Nd, Pr, Sm or Eu.Reproduced with permission from Armitage, N. P. and Fournier, P. andGreene, R. L., Progress and perspectives on electron-doped cuprates. Rev.Mod. Phys., 82(3):24212487, Sep 2010 (Page 2422, Figure 1). c⃝(2010) byThe American Physical Society.

Understanding the particle-hole asymmetry in the phase diagram may be funda-

mental to elucidating the nature of the cuprate superconductors and their relation to

the Mott insulating phase at half-filling. The electron-doped and hole-doped cuprates

have slightly different crystal structures. Figure 1-4 shows the crystal structures of

19

La2CuO4 (LCO) and of an electron-doped Nd2−xCexCuO4 (NCCO). The electron-doped

compounds have a T ′ crystal structure which lacks oxygen atom in the apical position.

The superconducting cuprates generally have d-wave symmetry. This has been

confirmed by STM (Scanning Tunneling Microscopy ), London penetration depth,

ARPES and tricrystal-grain boundary experiments[26, 27]. Although both electron-

Figure 1-5. Crystal and spin structures of the electron-doped R2−xSrxCuO4 and thehole-doped La2−xSrxCuO4. Here R is one of the rare-earth ions-Nd, Pr, Smor Eu. Reproduced with permission from Armitage, N. P. and Fournier, P.and Greene, R. L., Progress and perspectives on electron-doped cuprates.Rev. Mod. Phys., 82(3):24212487, Sep 2010 (Page 2435, Figure 16).c⃝(2010) by The American Physical Society.

doped and hole-doped cuprates have the same magnetic periodicity, the magnetic

moments for electron-doped case point along the Cu-O bond directions, whereas for

hole-doped case they point at roughly 45 to the Cu-O bond directions as in Figure 1-

5[19, 23].

Cuprates have been studied by using the t − J model due to the antiferromagnetic

Mott insulator parent compounds[28–31]. At half-filling (or no doping), the magnetic

20

moments are treated as localized spins. Upon doping, electrons are introduced to

the system with antiferromagnetic background. This is justified for the hole-doped

cuprates which exhibit many properties of doped Mott insulators. Although cuprates

are considered to be strongly correlated systems, experiments have shown that in the

electron-doped side, the system is metallic in part of the antiferromagnetic state. [19]

Furthermore DMFT calculations of undoped n-type cuprates with T ′ crystal structure

have argued the insulating property is due to the presence of magnetic long-range order

rather than the Mott charge transfer gap physics[32]. This suggests that a weak-coupling

approach for the magnetism in electron-doped cuprates may be more appropriate.

1.3.2 Iron-pnictides

Kamihara et al.[33] of Hosono’s group discovered the first iron-pnictide super-

conductor LaFePO with Tc = 6K in 2006. Later, with more and more discoveries of

materials in this category, people realized that iron-based superconductors come in

many forms just like the cuprates. The iron-based superconductors can be categorized

as LaFeAsO (1111), MFe2As2 (122), MFeAs (111), FeSe (11), Sr2MO3FePn (21311,

M=Sc, V, Cr; Pn=P, As) and the defect A0.8Fe1.6Se2 structure (122*, A=K, Rb, Cs, Tl)

[34].

There is also long range antiferromagnetism in the iron-pnictides, found to be

suppressed by electron-, hole- and isovalent-doping and also by applying pressure.

The superconductivity appears around the point when AF order disappears. The crystal

structures typically undergo a tetragonal to orthorhombic transition upon cooling. The

structural phase transition does not necessarily coincide with the Neel temperature,

however. For the parent compound, these two points are close together. Otherwise

the antiferromagnetic transition always happens at a slightly lower temperature. The

temperature difference gets larger with larger doping or more impurities. Some iron-

pnictides exhibit coexistence state of superconductivity and antiferromagnetism in

both electron- and hole-dopings[35]. The various forms of the coexistence state have

21

been studied for different subcategories of iron-pnictides by using NMR[36–38]. Co-

doped BaFe2As2(122) and other materials display coexistence of superconductivity and

antiferromagnetism at the atomic scale[36]. But Ru isovalent doping in 122 results in

superconducting clusters in antiferromagnetic background[37]. In the 245 iron-selenide

RbFeSe, the system exhibits phase separation of these two states[38].

Unlike the cuprates, the Fermi surfaces of iron-based superconductors have multi-

orbital character[39]. They have more complicated band structures across different

subcategories of materials. Generally the Fermi surfaces of the various families have

hole pockets around Γ point and electron pockets around (±π, 0) or (0,±π). Exceptions

include KFe2As2, which has no electron pockets, and KFe2Se2 and monolayer FeSe,

which have no hole pockets.

Doping can affect the number of pockets, ellipticity (nesting) of the pockets as well

as the quasi-two dimensional nature of the Fermi surface. The question of whether iron-

based superconductors are more localized or itinerant is under debate. Experiments

have shown that the spread in the degree of correlation and localization of magnetic

states is pretty wide for the iron-based superconductors[40].

1.3.3 Heavy fermions

The first heavy fermion superconductor to be found was CeCu2Si2 by Steglich et

al. [16] in 1978. It was also the first unconventional superconductor to be discovered.

The superconducting transition temperatures for heavy fermions are pretty low, only

several Kelvin maximum. The materials in this category contain elements from the

lanthanide or actinide series which have incomplete f shells. This system is in a balance

between the strong Coulomb interaction which tend to form localized moments and the

hybridization with extended band state which tend to induce itinerant electrons. For

some heavy fermions, doping level or pressure can change not only the magnitudes of

superconducting and magnetic orders, but also the symmetries of these orders.

22

Antiferromagnetism is common among heavy fermions. However in some materials,

the lower temperature phase shows weak magnetic moments which is confirmed by

muon spin resonance measurements such as CeAl3 and CeCu6[41]. The supercon-

ducting pairs in UPt3 and UBe3 have the possibility of being p−wave symmetry with

parallel spins from the specific heat measurements[42, 43]. There are also transport

and thermal measurements showing that the symmetry could be d−wave[44]. Although

heavy-fermions were the first unconventional superconductors to be discovered , with

strong interactions and complex phase diagram, the microscopic explanation for super-

conductivity remains a mystery. There are experimental[42, 45] and theoretical [46, 47]

evidences which suggest superconductivity mediated by spin fluctuations .

1.3.4 Organic and fullerene superconductors

Lastly, another remarkable set of unconventional superconductors is the organic su-

perconductors. The highest Tc at ambient pressure is 33 K in the alkali-doped fullerene

RbCs2C60. The superconductivity can be induced by pressure or doping. Organic super-

conductors have very rich phase diagrams. In addition to superconductivity, they also

contain metal-insulator transition, antiferromagnetic order, charge-, spin- density-wave

phases and dimensional crossover. There is also coexistence state of superconductivity

and spin density waves in organic superconductors such as the quasi-one-dimensional

(TMTSF)2PF6 compounds[48]. With its rich phase diagram, organic superconductors

represent a great testing ground for competing order studies.

23

CHAPTER 2ANTIFERROMAGNETIC STATE

Some of the material I presented here has appeared as ”Spin excitations in layered

antiferromagnetic metals,” W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld, Phy. Rev.

B, 86, 134513 (2012).

2.1 Ferro- and Antiferromagnetism

The classical theory of magnetism started from a localized picture of mag-

netic moments. With this concept, Langevin explained the Curie’s law of magnetic

susceptibility[49] which was first discovered by Pierre Curie experimentally. Langevin’s

result for the magnetic susceptibility is inversely proportional to T

χ = N0m2/3kBT = C/T (2–1)

where N0 is the number of atoms in the crystal, m is the magnetic moment and C is

the Curie constant. Later Weiss introduced interaction between the atomic magnetic

moments which is averaged from a molecular field and then added to the external field

in Langevin’s calculation[50]. He obtained the result above the Curie temperature for the

susceptibility as

χ = C/(T − TC). (2–2)

This is called the Curie-Weiss law. The relation is common for most ferromagnets above

the Curie temperature. The experimental data can be compared with the Curie-Weiss

law to see whether the studied materials have localized spins other than itinerant

electrons in the systems. From the microscopic point of view, there exist two challenges

to this classical description. One is the explanation of the source of the magnetic

moment. The other is to explain the Weiss molecular field which is too small if one

calculates the average magnetic dipole-dipole interaction comparing with the values

from the observed Tc . These two problems were later addressed with the help of

quantum mechanics.

24

According to quantum mechanics, the electron state can be described by the orbital

angular momentum L and spin S. The magnetic moment of an electron is described by

M = µB(L + 2S) with the Bohr magneton, µB = e~/2mc . The physical values in the

Langevin equation can be replaced by their quantum-mechanical counterparts. Then the

magnetic susceptibility can be re-written as

χ = N0g2JµB j(j + 1)/3kBT (2–3)

where gJ is the Lande g factor, j is the quantum number of the total angular momentum.

The susceptibility has the same temperature dependence as in the classical result and

we have Curie constant, C = N0g2JµB j(j + 1)/3kB .

The explanation of the Weiss molecular field was given by Heisenberg[51]. He ar-

gued that the origin of the field was from the quantum-mechanical exchange interactions

between the atoms. He described the magnetic system with the atomic spin operators:

H = J∑i ,j

Si · Si (2–4)

With the interatomic exchange interaction constant, J < 0, this Hamiltonian describes a

ferromagnetic state, whereas with J > 0, it describes an antiferromagnetic state. In the

antiferromagnetic case, in order to minimize the energy, |Si − Sj | = 0, the adjacent spins

have to be anti-parallel.

Antiferromagnetism is common in correlated systems. The adjacent spins like to

align anti-parallel to lower the energy due to the exchange coupling in the systems.

This state, similar to the ferromagnetic state, occurs at lower temperatures. Above a

certain critical temperature, the spin ordering would be destroyed by thermal fluctuations

and the system turns into paramagnetic state. Louis Neel was the first to identify this

antiferromagnetic ordering, therefore this critical temperature is also called the Neel

temperature.

25

Below the Neel temperature, the net magnetization is still zero as in the paramag-

netic state. Therefore it is not easy to detect. The first experimental confirmation of this

state was by Shull in 1949 with neutron scattering[52]. It can also be tested by the uni-

form susceptibility, χ(T ) which would show a weak maximum at the Neel temperature,

or by measurement of the staggered susceptibility χ(q,T ) which shows a divergence by

neutron scattering. The NMR probe can also determine the structure by the symmetry

analysis of the hyperfine coupling tensor[53].

The common description of this state is by assuming two ferromagnetic sublattices

A and B which have opposite spin orientation. The two main models are Heisenberg

model for localized spins and the Hubbard model for itinerant electrons.

2.2 Itinerant electron magnetism

The Heisenberg model has success in describing ferro-, antiferro- and ferrimag-

netisms in systems with localized spins. It led to the discovery of spin waves and the

develop of magnetic resonance techniques. Since the Heisenberg model assumes

localized spins, in principle it can only explain the physics of insulating magnets. But we

know that materials like transition metals which are magnetic metals sometimes display

weak moments and cannot be described by the Heisenberg model. Bloch proposed the

possibility of ferromagnetism arising from electron gas with the help of the Hartree-Fock

(HF) approximation[54]. As discussed in the introduction, electron doped cuprates ap-

pear to display itinerant antiferromagnetic behavior close to the superconducting phase.

We would like to understand the nature of this magnetic state in order to describe its

influence on superconductivity.

The bare susceptibility at wave vector q is

χ0(q) =∑k

f (εk+q) − f (εk)εk − εk+q

. (2–5)

26

Within the RPA the interacting susceptibility is then

χ(q) =χ0

1− Uχ0. (2–6)

The Stoner criterion for the spin density wave instability at q is

Uχ0(q) = 1. (2–7)

We can see that χ0(q) is sensitive to the topology of the Fermi surface, especially if

there is a ”nesting”. Nesting means that there exists segments of the Fermi surface

which can be translated by a vector q and align with the original Fermi surface. The

nesting often results in a singular behavior of the susceptibility which signals an insta-

bility to a new magnetic phase. Such systems are called itinerant magnets. According

to the usual criterion of band theory, if one or more bands are partially filled, the system

would be a metal.

Localized and itinerant models represent opposite approaches. The former has

the electron states localized in real space whereas the later has the states localized

in momentum space. The magnetism in a localized system due to strong interactions

usually make the system insulating. And the magnetic moment in an itinerant system

can move freely, therefore the systems remain metallic. The classification of metal

and insulators according to band theory was quite successful but fails, for instance in

NiO, which is an insulator but is supposed to be a metal according to its calculated

band structure. Mott later proposed an explanation to solve the puzzle[55]. The Mott

insulators are antiferromagnetic. The study of the Mott insulators at half-filling and with

additional electrons or holes doping in the system is important for the superconductivity

study, especially in systems like the hole-doped cuprates.

27

2.3 Mean field phase diagram including AF and superconductivity

The Hubbard model in real space is expressed as

H =∑ij ,σ

tijc†i ,σcj ,σ +

U

2

∑i ,σ

niσni σ. (2–8)

where i , j are lattice sites, niσ is the number operator on site i with spin σ which is

opposite to spin σ and tij is the hopping matrix element between sites i and j . In Fourier

space, we have

H =∑k,σ

εkc†k,σck,σ +

U

2

∑k,k′,q,σ

c†k,σck+q,σc†k′,σck′+qσ (2–9)

We consider a two-dimensional system with normal state tight-binding energy dispersion

εk = −2t(cos kx+cos ky)+4t ′ cos kx cos ky −µ. Although the Hubbard model is apparently

very simple, it has no exact solution in higher dimensions. It is the simplest model to

display the Mott phenomenon, since when U is large, double occupation is forbidden

and a half-filled system cannot conduct.

In the antiferromagnetic state, the unit cell in real space doubles, therefore the unit

cell in momentum space is half of the size of that of the paramagnetic state. The sum of

k over the full Brillouin zone therefore has to be folded into the reduced Brillouin zone as

in Figure 2-1 (a). We use a mean-field approximation to decouple the spin density wave

term by defining the antiferromagnetic order parameter as

W = U/2∑kσ

< c†k+Qσckσ > sgn(σ). (2–10)

with ordering momentum Q = (π, π). We apply the Bogoliubov transformation,

ckσ = ukαkσ + vkβkσ

ck+Qσ = sgn(σ)(−vkαkσ + ukβkσ)(2–11)

to the Hamiltonian and determin uk, vk to diagonalize it. We get the Hamiltonian with the

28

new basis of αk and βk as

H =∑k,σ

′Eαk α

†k,σαk,σ + E

βk β

†k,σβk,σ (2–12)

The prime over the summation indicates the sum over the reduced Brillouin zone. The

quasiparticle energies for the electron band, α and the hole band β are:

Eα,βk =

εk + εk+Q2

±√(εk − εk+Q2

)2 +W 2 (2–13)

The spin density waves (SDW) coherence factors are

u2k =1

2

(1 +

ε−k√(ε−k )

2 +W 2

)(2–14)

v 2k =1

2

(1− ε−k√

(ε−k )2 +W 2

)(2–15)

where ε±k = (εk ± εk+Q)/2. The antiferromagnetic order forces the Fermi surface to

reconstruct. It splits into an electron band (α-band) and a hole band (β-band) as in

Figure 2-1 (b). In Figure 2-1, the green line shows the original Fermi surface in the

normal state with no antiferromagnetic order. Then in the antiferromagnetic state the

Fermi surface splits in to electron pockets around (±π, 0) and (0,±π) (red lines) and

hole pockets around (±π/2,±π/2) (blue lines). The position of the dash line in Figure

2-1 (b) at Fermi level can be adjusted by changing µ for certain doping levels.

From the definition of W, Equation 2–10, the magnitude of the magnetic moment for

a given U should be calculated self-consistently by

W = U∑k

′ W√(εk − εk+Q)2 + 4W 2

[tanh

( Eαk

2kBT

)− tanh

( Eβk

2kBT

)](2–16)

The derivation is included in Appendix A.1.

The electron band filling is defined by n = 1 + x =∑k,σ⟨c

†k,σck,σ⟩. To obtain certain

doping levels, we can change the chemical potential, µ such that it satisfies the following

29

equation:

n = 1 + x = 2−∑k

′[tanh

( Eαk

2kBT

)+ tanh

( Eβk

2kBT

)](2–17)

The derivation is included in Appendix A.2.

The sizes of the hole pockets and electron pockets depend on the magnitude of

W . WhenW increases, the size of the hole pocket shrinks and the size of the electron

pocket expands on the electron-doped side.

In order to study the situation in the coexistence state of antiferromagnetic and

superconducting states, we add a phenomenological superconducting term to the

Hamiltonian. The total Hamiltonian is written as:

H =∑kσ

εkc†kσckσ +

∑k,k′,σ

U

2c†kσck+Qσc

†k′+Qσck′σ

+∑k,p,q,σ

Vq c†k+qσc

†p−qσcpσckσ (2–18)

Now we perform a mean field decomposition on the V term (superconducting term) in

Equation 2–18, assuming the spin singlet superconducting order parameter

∆k = V ⟨ck↑c−k↓⟩ (2–19)

The full mean field Hamiltonian then becomes

H =∑k,σ

′Eαk α

†k,σαk,σ + E

βk β

†k,σβk,σ − ∆αα

†k,σα

†−k,σ − ∆ββ

†k,σβ

†−k,σ (2–20)

We perform a BCS Bogoliubov transformation to further diagonalize the coexistence

state of superconductivity and spin density waves with the BCS transformation in the

spin density wave state:

αk↑ = uαk γ

αk0 + v

αk γ

α†kl

α†−k↓ = −vαk γα

k0 + uαk γ

α†kl .

(2–21)

30

For the β operator, the transformation is the same with exchange of α with β. The BCS

coherence factors are

(uαk )2 =1

2

(1 +Eαk

Ωαk

)(vαk )

2 =1

2

(1− E

αk

Ωαk

) (2–22)

in the coexistence state. We now get new quasiparticle energy dispersions:

Ωγk =

√(E γk )2 + (∆γ

k)2 (γ = α, β). (2–23)

The SC gaps ∆α,βk are determined self-consistently from two coupled gap equations,

derived previously by Ismer et al[56]. The gap functions are

∆αk =−

∑p∈R

[(Vk−pF

u,vk,p − Vk−p+QF v ,uk,p )

∆αp

2Ωαp

+ (Vk−pNv ,uk,p − Vk−p+QNu,vk,p )

∆βp

2Ωβp

]

∆βk =−

∑p∈R

[(Vk−pN

v ,uk,p − Vk−p+QNu,vk,p )

∆αp

2Ωαp

+ (Vk−pFu,vk,p − Vk−p+QF v ,uk,p )

∆βp

2Ωβp

] (2–24)

where Nx ,yk,p ,Fx ,yk,p = u

2kx2p ± 2ukvkupvp + v 2k y 2p and x , y = u or v . The sum over p is limited

to |E γp | ≤ ~ωD , with ~ωD being the Debye frequency on its analog in an electronic pairing

model.

For the dx2−y2−wave case, we may choose Vk−k′ = Vd(cos kx − cos ky)(cos k ′x −

cos k ′y)/4 = Vdφkφk′/4, and we find that the superconducting order parameter takes the

form

∆γk = φk(∆

γ0 + ukvk∆

γ1). (2–25)

The coexistence of commensurate antiferromagnetic and superconducting states can

therefore generate a higher harmonic component ∆1, with the dx2−y2-wave potential[56].

This harmonic is proportional to the magnitude of the antiferromagnetic order and super-

conducting gaps. It arises due to Umklapp Cooper-pairing terms like ⟨ck,↑c−k−Q,↓⟩. These

expectation values appear in the coexistence phase as the wavevector Q becomes the

new reciprocal wave vector of the lattice in the antiferromagnetic state. At the same time,

31

due to additional breaking of the spin rotational symmetry associated with the antifer-

romagnetic transition, the Umklapp Cooper-pairing terms formally belong now to the

spin-triplet component of the Cooper-pair wave function with mz = 0 as was discussed

previously by several authors[57–59]. This indicates that the appearance of the ∆1 is

associated with an additional phase transition in the coexistence phase with a change of

the underlying symmetry of the mean-field Hamiltonian.

The equations for calculating antiferromagnetic order parameter,W and the doping

level, x have to be modified in the coexistence state according to both the SDW and the

superconducting coherence factors.

W = U∑k∈R

W

2√

ε−k +W20

[Eαk

Ωαk

tanh( Ωα

2kBT

)− E

βk

Ωβk

tanh( Ωβ

2kBT

)](2–26)

The derivation ofW in the coexistence state is included in Appendix B.1.

n = 1 + x =∑k

2− Eαk

Ωαk

tanh(Ωαk

2T)− E

βk

Ωβk

tanh(Ωβk

2T) (2–27)

The derivation of n is included in Appendix B.2.

With the above equations, we can solve the order parameters at different dopings

and temperatures self-consistently on the electron-doped side. The mean field phase

diagrams can be constructed as in Figure 2-2 with a dx2−y2−wave superconducting

pairing potential[56]. The antiferromagnetic moment decreases as the doping level

increases. The superconductivity order parameter has non-zero solutions from n = 1

to 1.2. The mean field energy was calculated in order to determine the stable state

solutions. The mean field energy is:

EMF = ⟨H⟩ =∑kσ

′(Eαk −Ωα

k + Eβk −Ωβ

k +∆α2k

2Ωαk

tanh(Ωα

k

2T

)+∆β2k

2Ωβk

tanh(Ωβ

k

2T

)+2Ωα

k f (Ωαk ) + 2Ω

βk f (Ω

βk ) +

W 2

U

) (2–28)

32

The derivation is included in Appendix B.3. This results in a first order superconducting

transition at doping around n ≈ 1.05 in the phase diagram in Figure 2-2.

For our phase diagram, we assume the superconducting gap appears with an

antiferromagnetic background, i.e. in the limit of TSC < TNeel . This allows us to perform

the unitary transformation to SDW quasiparticles followed by a subsequent Bogoliubov

transformation to the coexistence state quasiparticles of antiferromagnetism and

superconductivity. There is a study which is performed in the other limit of TSC >

TNeel [61]. By comparing with a 4 × 4 transformation [58] where antiferromagnetic

and superconducting orders are on equal footing, we can be sure that the results are

justified. The quasiparticles are the same between our approach and the 4× 4 ones with

∆1 = 0. For ∆1 = 0 the eigenenergies agree only up to very small terms of the order

∼ O(∆1) but start to differ for the higher order terms. To recover the same quasiparticle

energy, one needs to take into account the Cooper-pair terms ⟨α†k,↑β

†−k,↓⟩. They may

have to be taken into account when the antiferromagnetic order becomes small and the

interband Cooper-pairing may become important. Furthermore, the particular form of the

pairing interaction (s−wave, d−wave or others) in momentum space can further modify

the structure of the superconducting gap equations in the coexistence state.

In Figure 2-2, we present the mean-field phase diagram in the coexistence state

of commensurate antiferromagnetism and d−wave superconductivity with ∆1 = 0.

Although there are similar results in the literature[58],there are some important features

that are important to mention. Within a pure antiferromagnetic phase at finite doping,

there is a Lifshitz transition (blue curve) separating phases with a different FS topology

with either one or two types of FS pockets. At higher temperatures, both electron and

hole type of pockets are present at the FS, while below the Lifshitz transition only the

electron pockets are present. The open circle at half-filling, n = 1 is the transition

between a semimetal (higher T ) and an insulator (lower T )

33

Another interesting feature concerns the character of the phase transition into the

coexisting state at low temperatures. After analyzing the free energy, we find that the

transition from antiferromagnetism to the coexistence state is first order as a function of

doping, while it becomes second order as a function of temperature. Furthermore, we

notice that our total energy analysis shows that the stationary solutions for coexistence

of antiferromagnetism and d−wave superconductivity in the electron-doped cuprates

have always slightly lower free energy in the case ∆1 = 0, in other words, when the

triplet component of the Cooper-pairing is absent. In our approach, the symmetry of

the problem remains SU(2) × U(1) in the coexistence regime. We believe that it is

connected to the fact that we ignored the contribution from the interband Cooper-pair

averages ⟨α†k,↑β

†−k,↓⟩ in the coexistence phase. Although these values are small, they

could change the balance of the free energy towards the coexistence state with finite

’triplet’ component of the Cooper-pairing. Furthermore, a modification of the momentum

dependence of the Cooper-pairing interaction may also change the balance of the

mean-field states. One apparent disadvantage of the prediction of the phase diagram

shows that the system is metallic with the exception of half-filling at temperatures below

about T = 0.05t, while in experiments the parent compound cuprates are found to

be insulators up to the Neel temperature. However, in electron-doped cupartes, the

system appears to be metallic for dopings above x ≈ 0.125, whereas Neel order does

not disappear until about x ≈ 0.15. It is in precisely the doping range where optimal

superconductivity occurs.

34

Figure 2-1. Left panel: the Fermi surface of electron-doped cuprates in normal state(green lines), and in the antiferromagnetic state with electron pocket (redlines) and hole pocket (blue lines). Right panel: the band structure.

1 1.05 1.1 1.15 1.20

0.05

0.1

0.15

0.2

0.25

Tem

p (

t)

doping, n

Néel temp.

Hole pocket transitionfor the pure SDW state

TSC

Figure 2-2. The doping-temperature phase diagram of electron-doped cuprates with thesuperconducting transition (red line) and Neel temperaturs (black line).Thereis a Lifshitz transition of both pockets present to only electron pocket present(blue lines). The open circle at half-filling, n = 1 is the transition between asemimetal (higher T ) and an insulator (lower T ) [60]. Reproduced withpermission from W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spinexcitations in layered antiferromagnetic metals and superconductors. Phys.Rev. B, 86:134513, Oct 2012 (Page 134513-7, Figure 4). c⃝(2012) by TheAmerican Physical Society.

35

CHAPTER 3DYNAMIC SPIN SUSCEPTIBILITY

Some of the material I present here has appeared as ”Spin excitations in layered

antiferromagnetic metals,” W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld, Phy. Rev.

B, 86, 134513 (2012).

3.1 Theory and calculations

The superconductivity in layered oxides[62, 63] and iron-pnictides[64, 65] is

proposed to arise from spin fluctuations. Therefore an understanding of the spin

excitations is important for the study of unconventional superconductivity. The symmetry

of the superconducting gap as well as the electronic structure of the materials can

influence the spin excitations. The spin excitations can be detected with neutron

scattering experiments[66].

3.1.1 Neutron scattering

The neutron scattering is important for measuring spin fluctuations. It can determine

the crystal and magnetic structure and also the motion of the atoms. The neutron

scattering experiment focus neutron beams with certain momentum and energy on

the material, then the neutrons are scattered from the crystal. The detectors collect

the neutrons and the scattered diffraction pattern shows the positions of the atoms in

the material. The neutrons may also create phonons or magnons when they penetrate

the material. Their energy changes due to this inelastic process. The quantity which is

measured by inelastic neutron scattering is the dynamic structure factor. It is related to

the spin susceptibility via the relation,

S(q,ω) = χ′′(q,ω)

1− e−~ω/kBT. (3–1)

There have been many neutron scattering experiments on layered cuprates[20], many of

which exhibit a feature called the ”neutron resonance” or ”spin resonance” for T < Tc .

The magnetic resonance has been observed in the hole-doped[67, 68] and electron

36

doped cuprates[69]. Figure 3-1 shows the spin resonance of Pr1−xLaCexCuO (PLCCO)

in the superconducting state. In Figure 3-1 (a), the temperature difference spectrum

between 2 and 30 K suggests a resonance-like enhancement at ≈ 11 meV. In Figure

3-1 (b), black squares show temperature dependence of the neutron intensity (≈ 1 h per

point) at (1/2, 1/2, 0) and 10 meV. In Figure 3-1 (c), Q-scans at ~ω = 10 meV obtained

with Ef = 14.7 meV at 2 K and at 30 K The spin resonances in the superconducting

Figure 3-1. Neutron scattering on Pr1−xLaCexCuO (PLCCO). (a) The temperaturedifference spectrum between 2 and 30 K suggests a resonance-likeenhancement at ≈ 11 meV. (b) Black squares show temperaturedependence of the neutron intensity (≈ 1 h per point) at (1/2, 1/2, 0) and 10meV. (c) Q-scans at ~ω = 10 meV obtained with Ef = 14.7 meV at 2 K and at30 K. Reproduced with permission from S. D. Wilson, P. Dai, S. Li, S. Chi, H.J. Kang, and J. W. Lynn, Resonance in the electron-dopedhigh-transition-temperature superconductor Pr0.88LaCe0.12CuO4−δ. Nature,442(7098):5962, 07 2006.(Page 61, Figure 3(e) and Figure 4(d,e)). c⃝(2006)by Nature Publishing Group

37

state have been studied in materials such as YBa2Cu3O7 (YBCO)[70], Sr2RuO4[71] and

iron-pnictides[66, 72]. The superconducting resonance can be traced to the sign change

in the d−wave superconducting order parameter ∆k under momentum transfer k→ k+ q

with k ≈ Q = (π, π), due to a coherence factor (1− ξkξk+q/EkEk+q) which appears in the

dynamical susceptibility.

3.1.2 Spin waves

There are mainly two approaches for the study of spin waves in cuprates. One

is the strong-coupling approach representing by the t − J model and the t − J1 − J2

model. The t − J model is derived from Hubbard model in the limit of large U and the

t − J1 − J2 model is an extension of Heisenberg model including interactions between

next-nearest-neighbor spins.

The other approach for studying spin waves is the weak-coupling approach, which

assumes electrons are itinerant. Using different approaches to study the spin excitations

gives us different spin excitation spectra. This can help us categorize wether the spins

in a specific material are more localized or more itinerant by comparing the theoretical

results with the experimental data. In some limits, these two approaches can yield the

same general features. For instance, the Hubbard model reduces in the large U limit

to the t − J model. Ideally we hope there is a way to describe the spin excitations with

accuracy and without depending on assumptions of sensitive parameters. The pursuit of

a unified model for general magnetic systems continues to be an intriguing challenge.

Here we start from a itinerant approach, using the Hubbard model, which was

introduced in Equation 2–9. The derivation of the dynamic susceptibility starts with the

linear response theory[73, 74].

38

3.2 Spin excitations in the pure antiferromagnetic state

3.2.1 The dynamic spin susceptibility in the antiferromagnetic state

We employ the RPA formalism with a single-band Hubbard model. The dynamical

spin susceptibility for the longitudinal, χzz , and the transverse, χ+−, are defined as

χlm(q,q′, Ω) =

∫dt

[i

2N⟨TS lq(t)Sm−q(0)⟩

]eω+iδt , (3–2)

with lm = zz or +−. The spin operators can be written in terms of raising and lowering

operators,

S+q (τ) =∑

k

c†k+q↑(τ)ck↓(τ),

S−q (τ) =

∑k

c†k+q↓(τ)ck↑(τ),

Szq (τ) =∑kσ

σc†k+qσ(τ)ckσ(τ).

(3–3)

The antiferromagnetic ordering at Q = (π, π) which corresponding to the magnetic

order in the cuprates doubles the unit cell and requires accounting for the breaking of

translational symmetry[12, 14]. The Brillouin zone in momentum space would be half

of the full zone. As a result, the total susceptibility in the transverse channel is a 2 × 2

matrix with off-diagonal Umklapp terms

χ+−0 =

χ+−(q,q,ω) χ+−(q,q+Q,ω)

χ+−(q+Q,q,ω) χ+−(q+Q,q+Q,ω).

(3–4)

The spin susceptibility would be enhanced if we consider the RPA proccess. By solving

the Dyson equation, we get the susceptibility [12] as

χ+−RPA =(1− Uχ+−0

)−1· χ+−0 , (3–5)

39

and the bare components are given by

χ+−0 (q,q,ω) = −12

∑k,γ

′

1 + ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γk )

ω + iδ − E γk+q + E

γk

−12

′∑k,γ =γ′

1− ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γ′

k+q)− f (Eγk )

ω + iδ − E γ′

k+q + Eγk

,

(3–6)

with γ = α, β. f (Ek) is the Fermi function and the prime refers to the sum over the

magnetic (reduced) Brillouin Zone. The detailed derivation is included in Appendix C.1.

We can use the same equation for the calculation of χ+−0 (q+Q,q+Q,ω)

For the Umklapp term, the susceptibility is

χ+−0 (q,q+Q,ω) =

W

2

∑k

′

1√(ε−k+q

)2+W 2

− 1√(ε−k)2+W 2

( f (Eαk+q)− f (Eα

k )

ω + iδ − Eαk+q + E

αk

−f (Eβ

k+q)− f (Eβk )

ω + iδ − Eβk+q + E

βk

)

−

1√(ε−k+q

)2+W 2

+1√(

ε−k)2+W 2

( f (Eβk+q)− f (Eα

k )

ω + iδ − Eβk+q + E

αk

−f (Eα

k+q)− f (Eβk )

ω + iδ − Eαk+q + E

βk

).

(3–7)

The detailed derivation is included in Appendix C.2. The other element in the suscepti-

bility matrix can be obtained by the relation, χ+−0 (q,q+Q) = χ+−0 (q+Q,q). We clearly

see that this term has coherence factors (which are the coefficients in the front of each

term explicitly depending on ε−k andW ) proportional to the antiferromagnetic order

parameterW . If there is no magnetic order, we wound not have translational symmetry

breaking and this term would be zero.

For the longitudinal part of the spin susceptibility, the calculation is similar. Since

there is no breaking of spin rotational symmetry in the xy−plane, we would get the

Umklapp susceptibility to be zero. This can also be proved by direct calculation which is

40

included in Appendix C.4. Therefore the calculation of the RPA susceptibility is a simple

equation which is given by

χzzRPA(q,q,ω) =χzz0 (q,q,ω)

1− Uχzz0 (q,q,ω). (3–8)

The bare longitudinal spin susceptibility is

χzz0 (q,q,ω) = −12

∑k,γ

′

1 + ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γk )

ω + iδ − E γk+q + E

γk

−12

′∑k,γ =γ′

1− ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γ′

k )

ω + iδ − E γk+q + E

γ′

k

. (3–9)

The detailed derivation is included in Appendix C.3. Note that the coherence factors are

different (opposite sign in front ofW 2) for longitudinal and transverse spin susceptibili-

ties.

The structure of the dynamic spin susceptibility in the antiferromagnetic state with

ordering momentum Q has been studied[12, 14] in the context of t ′ = 0 and on the

hole-doped side. These works did not include the effects of non-zero next-nearest-

neighbor hopping, t ′ and the case with electron dopants These two points, t ′ = 0

and electron dopings, which can effect the spin excitation spectrum significantly will

be discussed separately in this thesis. The main features that we also observed are

the breaking of the spin-rotational symmetry in the anisotropy in the susceptibility,

χ+− = χzz and the gapless Goldstone mode as expected. The imaginary part of the

transverse susceptibility is gapless and displays the Goldstone mode at the ordering

vector Q = (π, π) and ω → 0 in the antiferromagnetic state. The Goldstone mode is

guaranteed by the fact that the condition of the pole formation in the RPA part of the

transverse spin susceptibility coincides with the mean-field equation forW and is valid

for any doping level as soon as the equations are calculated self-consistently. Our

41

analysis is included in Appendix C.5. For the longitudinal part, the susceptibility at Q is

gapped by twice the antiferromagnetic gap magnitude,W .

3.2.2 The effect of next-nearest hopping, t ′ on the spin excitations

When the next-nearest-neighbor hopping t ′ is turned on, the behavior of spin

excitations away from Q is less known. At half-filling (x=0), it is possible to have a

compensated metal at finite temperatures. At zero temperature, the Fermi surface is

gapped by a value which depends onW and t ′ and is determined by the self-consistent

calculation of the chemical potential. To study the effect of t ′ on the system, we plotted

the spin susceptibilities of the transverse channel in Figure 3-2 for the half-filled case.

The excitations in the transverse channel of the Stoner insulator are spin waves -

collective spin modes of the antiferromagnetic ground state[75].

The band structures and imaginary part of transverse dynamic spin susceptibility

at half-filling are shown in Figure 3-2 with (a), (b) t ′ = 0.0t, (c), (d) t ′ = 0.2t and (e), (f)

t ′ = 0.35t. All panels have U = 2.80t andW = 0.75t. The Fermi surface is fully gapped

for the half-filled case at low temperature, as shown in Figure 3-2 (a), (c) and (e). The

particle-hole Stoner excitations and the spin waves are therefore separated in energy

and may interact only around the particle-hole continuum frequency, ωp−h(q). For t ′ = 0,

the onset of the particle-hole continuum is gapped at least up to ωp−h(Q) = 2W . This is

because the top of the lower β-band and the bottom of the upper α-band are located at

the reduced Brillouin zone boundary, i.e. cos kx + cos ky = 0, at energies −W and +W ,

respectively as shown in Figure 3-2 (a). Therefore, there exists a degenerate manifold

of q wave vectors for which ωp−h(q) = 2W . As a result, the spin waves do not interact

with the particle-hole continuum for sufficiently large values ofW and look identical

to those obtained within a Heisenberg model of localized spins which interact via an

antiferromagnetic exchange between nearest neighbors, J1 ∼ t2

U, see Figure 3-2(b).

With the non-zero values of next nearest hopping, t ′, there are non-degenerate positions

of the top of the β−band and bottom of the α-band as clearly seen in Figure 3-2 (c) and

42

Figure 3-2. The band structures and imaginary part of transverse dynamic spinsusceptibility at half-filling. with (a), (b) t ′ = 0.0t, (c), (d) t ′ = 0.2t and (e), (f)t ′ = 0.35t. All panels have U = 2.80t andW = 0.75t [60]. Reproduced withpermission from W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spinexcitations in layered antiferromagnetic metals and superconductors. Phys.Rev. B, 86:134513, Oct 2012 (Page 134513-3, Figure 2). c⃝(2012) by TheAmerican Physical Society

(e). Therefore it reduces the overall magnitude of the indirect gap in the particle-hole

continuum and shifts it to lower energies at the (π2, π2) point of the Brillouin zone.

For any non-zero t ′, the bottom of the upper α-band is located at (±π, 0) and

(0,±π) or Y points of the Brillouin zone at energy −4t ′ + W − µ > 0, whereas

the top of the lower β−band is located at (±π2,±π

2) points of the Brillouin zone at

energy −W − µ < 0. As a result, the smallest indirect gap between these two bands

which determines also the lowest position of the particle-hole continuum occurs at

43

ωp−h(q) = 2W − 4t ′ for q = (±π2,±π

2). For increasing t ′/t ratio and a constant

value ofW , the spin waves are bounded from above at momentum q = (±π2,±π

2)

and form a local minimum at energies below 2W − 4t ′ > 0. In particular, in Figure

3-2(d) it occurs below 0.7t withW = 0.75t and t ′ = 0.2t and is shifted to much lower

energies for t ′ = 0.35t for a fixedW = 0.75t as shown in Figure 3-2(f). For zero doping

we always find either an insulating antiferromagnetic state or a normal state metal

at low temperature. If we have the fictitious case of a compensated metal and have

2W − 4t ′ < 0 in the band structure, the real part of transverse susceptibility will become

negative at q = q and ω = 0. This would be an unstable solution, therefore we never get

a self-consistent solution in this case.

The local minimum for a finite t ′ at q is due to the interaction of spin waves with the

particle-hole continuum. This is a feature of weak-coupling which allows 2W to be of the

same order as 4t ′. This effect would not occur for the localized model such Heisenberg

or J1 − J2 models, where J2 refers to the antiferromagnetic exchange between the

next-nearest neighbors. J2 only lowers the position of the maximum of the spin wave

dispersion at the Y point of the Brillouin zone, an effect clearly reproduced in the weak-

coupling calculations as well, comparing with Figures 3-2 (d) and (f). At the same time,

within the localized model the particle-hole excitations always remain gapped by the

large value of U andW . Correspondingly the local minimum in the spin susceptibility at

q never forms in the localized picture.

3.2.3 The effect of the dopants on spin excitations

For the doped case, we study three different scenarios: hole-doped with only

hole pockets, electron-doped with only electron pockets and electron-doped with both

electron and hole pockets. These three cases can be realized by using doping level at

x = −0.05, 0.10 and 0.12 respectively. The topology of the Fermi surfaces are shown

in Figure 3-3. Figure 3-3 (a) is the case for hole doping. (b) and (c) are both electron

doping. The original Fermi surface in the normal state is shown as green curve. The

44

kx

ky

−π 0 π −π

0

π

(a)

−π −π

0

π

kx0 π

(b)

ky

−π −π

0

π

ky

kx0 π

(c)

Figure 3-3. Three possible types of Fermi surface topology in the antiferromagnetic statein layered cuprates[60]. (a) is the case for hole doping. (b) and (c) are bothelectron doping. The original Fermi surface (green curve) in the normalstate, The hole pockets (blue curves) centered around (±π

2,±π

2) and

electron pockets (red curves) centered around (±π, 0) [(0,±π)] points of theBrillouin zone. For larger doping and smaller sizes of the antiferromagneticgap both types of the pockets can be present. As argued in the text, thecommensurate antiferromagnetic order becomes unstable once the holepockets appear around (±π

2,±π

2) points of the Brillouin zone. Reproduced

with permission from W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spinexcitations in layered antiferromagnetic metals and superconductors. Phys.Rev. B, 86:134513, Oct 2012 (Page 134513-2, Figure 1). c⃝(2012) by TheAmerican Physical Society.

hole pockets (blue curves) centered around (±π2,±π

2) and electron pockets (red curves)

centered around (±π, 0) [(0,±π)] points of the Brillouin zone. For larger doping and

smaller sizes of the antiferromagnetic gap both types of the pockets can be present. The

commensurate antiferromagnetic order becomes unstable once the hole pockets appear

around (±π2,±π

2) points of the Brillouin zone.

The spin wave dispersion is symmetric with respect to the (0, 0) and (π, π) points,

which reflects the fact that both are equivalent symmetry points of the magnetic (re-

duced) Brillouin zone. At the same time, the absolute intensity of the spin waves is

different and is determined by the antiferromagnetic coherence factors which are sup-

pressed around the Γ-point. We can see clearly from Equation 3–6 that at low frequency,

the non-vanishing contribution to the intensity comes from the interband (α → β and

45

vice versa) transitions which are proportional to the antiferromagnetic coherence factor

c interk,q =

(1− ε−k ε

−k+q−W

2√(ε−k )

2+W 2

√(ε−k+q)

2+W 2

). For q ∼ Q one finds ε−k+Q ≈ −ε−k and c interk,q∼Q ∼ 2,

whereas it is c interk,q∼0 ∝ (2W 2

(ε−k )2+W 2 for q ∼ 0. This apparently affects the intensity of the

susceptibility greatly at (0, 0) and (π, π) points, although they are considered the same

points in the magnetic momentum space.

Figure 3-4. Calculated imaginary part of transverse χ+−RPA(q,q, Ω), (left panel) andlongitudinal, χzzRPA(q,q, Ω) (right panel)[60]. Reproduced with permissionfrom W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spin excitations inlayered antiferromagnetic metals and superconductors. Phys. Rev. B,86:134513, Oct 2012 (Page 134513-4, Figure 3). c⃝(2012) by The AmericanPhysical Society.

The spin excitation spectra Ω vs. q for the metallic antiferromagnetic state are

shown in Figure 3-4 with t ′/t = 0.35 and U = 1.3875t. (a),(d) refer to the hole doping,

x=-0.05,W = 0.61t, µ = −0.8819t, (b),(e) refer to the electron doping , x = 0.10,

W = 0.4404t, µ = −0.4284t, and (c),(f) refer to the electron doping x = 0.14,

46

W = 0.12t, µ = −0.302t. The corresponding Fermi surface topology is shown in Figure

3-3. The intensity in states/t is shown on a log scale. The white arrows in Figure 3-4

(a) denote the incommensurate momentum. Note that the intensity maps are different

between left and right panels.

The incommensurate mode in the hole doped side shows that our mean field

assumption of momentum ordering at (π, π) is not a stable magnetic structure. The

instability is related to the appearance of the small FS hole pockets, and to the spin

stiffness of the commensurate spin excitations at Q. In contrast to the undoped case,

there is an additional contribution to the spin stiffness which arises due to intraband

β − β transitions which are now gapless. We expand the dispersion of the lower β−band

for U >> t around (±π/2,±π/2) points which yields Eβk = −µ −W −

p2||2m||

− p2⊥2m⊥

, where

p|| = (kx − ky)/2, p⊥ = (kx + ky)/2 and m|| = (8t′)−1, and m⊥ = (16t

2/W − 8t ′)−1.

Following the analysis of the denominator of the transverse spin susceptibility at Q, we

see that the spin stiffness ρs acquires a finite correction in the doped antiferromagnetic

metal[14, 76, 77] as ρs = ρ0s (1 − z) where z = 2U√m⊥m||π

is proportional to the Pauli

susceptibility of the β-band, and ρ0s is the bare spin stiffness in the undoped case. The

correction z > 1 for large U indicates that the commensurate antiferromagnetic order

is unstable upon hole doping. On another hand, for the opposite case with U << t, the

expansion yields Eβk = −µ−

p2||2m||

−v⊥p⊥+p2⊥2m⊥

where here m⊥ = m|| = (8t′)−1 and v⊥ ∼ t.

This indicates that for t ′ < t the dispersion along p⊥ is essentially linear. As a result the

static susceptibility of the β-band will have singular behavior at 2kF . We also analyzed

the behavior of the denominator of the RPA spin susceptibility and found that for the

case of Figure 3-4(a) we have z > 1. Therefore, the instability of the commensurate

antiferromagnetic order for U ∼ t and hole doping occurs due to negative corrections

to the spin stiffness. The instability of the commensurate magnetic structure may be

an explanation of why in hole-doped side cuprates the long-term magnetic order is

vulnerable upon doping while on the electron-doped side the magnetic order is robust

47

up to five times of the critical doping level on the hole-side when the antiferromagnetic

level disappears. In the experiments the incommensurate mode has been found on

the lightly hole-doped La2CuO4 and coexists with the commensurate mode at certain

dopings[23]. The question of how to describe such a phenomenon with a theoretical

model is intriguing.

3.3 Spin excitations in the coexistence state

For the calculation of the dynamic spin susceptibility in the coexistence state of

antiferromagnetism and superconductivity, in addition to the unitary transformation,

we also need a BCS Bogoliubov transformation to get the correct quasiparticles. The

detailed derivation of the transverse spin susceptibilities is included in Appendix D.1.

The diagonal transverse spin susceptibility in the coexistence state is

χ+−0 (q,q,ω) =

−∑k,γ=γ′

′ 1

4

1 + ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +Eγk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

]f (Ωγ

k+q)− f (Ωγk )

ω + iδ −Ωγk+q +Ω

γk

+1

2

[1−Eγk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

](f (Ωγ

k+q) + f (Ωγk )− 1

ω + iδ +Ωγk+q +Ω

γk

+1− f (Ωγ

k+q)− f (Ωγk )

ω + iδ −Ωγk+q −Ω

γk

)

−∑k,γ =γ′

′ 1

4

1− ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +Eγk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

]f (Ωγ′

k+q)− f (Ωγk )

ω + iδ +Ωγ′

k+q −Ωγk

+1

2

[1−Eγk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

](f (Ωγ′

k+q) + f (Ωγk )− 1

ω + iδ +Ωγ′

k+q +Ωγk

+1− f (Ωγ′

k+q)− f (Ωγk )

ω −Ωγ′

k+q −Ωγk

). (3–10)

The superconducting gaps ∆αk and ∆β

k are defined in chapter 2. We assume a dx2−y2-

wave superconducting gap which was observed in most cuprates experimental

studies[26, 27].

48

The Umklapp transverse spin susceptibility is

χ+−0 (q,q+Q,ω) =

W

4

∑k,γ

′

1√(ε−k+q

)2+W 2

− 1√(ε−k)2+W 2

±(E γk+q

ωγk+q

+E γk

Ωγk

)f (Ωγ

k+q)− f (Ωγk)

ω −Ωγk+q +Ω

γk

±(E γk+q

Ωγk+q

− Eγk

Ωγk

)(1− f (Ωγ

k+q)− f (Ωγk)

ω −Ωγk+q −Ω

γk

+f (Ωγ

k+q) + f (Ωγk)− 1

ω +Ωγk+q +Ω

γk

)

+W

4

∑k,γ =γ′

′

1√(ε−k+q

)2+W 2

+1√(

ε−k)2+W 2

±(E γk+q

Ωγk+q

+E γ′

k

Ωγ′

k

)f (Ωγ

k+q)− f (Ωγ′

k )

ω −Ωγk+q +Ω

γ′

k

±

(E γk+q

Ωγk+q

− Eγ′

k

Ωγ′

k

)(1− f (Ωγ

k+q)− f (Ωγ′

k )

ω −Ωγk+q −Ω

γ′

k

+f (Ωγ

k+q) + f (Ωγ′

k )− 1ω + iδ +Ωγ

k+q +Ωγ′

k

). (3–11)

The ± sign corresponds to γ = α and γ = β respectively.

The longitudinal spin susceptibility has the form

χzz0 (q,q,ω) =

∑k,γ

′ 1

4

1 + ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +Eγk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

]f (Ωγ

k+q)− f (Ωγk )

ω + iδ −Ωγk+q +Ω

γk

+1

2

[1−Eγk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

](f (Ωγ

k+q) + f (Ωγk )− 1

ω + iδ +Ωγk+q +Ω

γk

+1− f (Ωγ

k+q)− f (Ωγk )

ω + iδ −Ωγk+q −Ω

γk

)

∑k,γ =γ′

′ 1

4

1− ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +Eγk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

]f (Ωγ′

k+q)− f (Ωγk )

ω + iδ −Ωγ′

k+q +Ωγk

+1

2

[1−Eγk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

](f (Ωγ′

k+q) + f (Ωγk )− 1

ω + iδ +Ωγ′

k+q +Ωγk

+1− f (Ωγ′

k+q)− f (Ωγk )

ω + iδ −Ωγ′

k+q −Ωγk

). (3–12)

The derivations of the dynamic susceptibilities are included in Appendix D.

The Umklapp term of the longitudinal part is again zero as in the pure antiferromag-

netic case for the same reason that the symmetry is not broken along the z direction. It

has the same coefficient which is comprised of antiferromagnetic coherence factors as

in the pure antiferromagnetic state, therefore remains zero.

49

Figure 3-5. Calculated Imaginary part of the transverse χ+−RPA(q,q, Ω) spin excitationspectra for three different electron dopings, n = 1.06, n = 1.09, and n = 1.12,(from upper to lower panel) for the coexistence state and ∆1 = 0 (rightpanel). For comparison the left panel shows the results for the pureantiferromagnetic state. The blue lines denote 1 = UReχ+−0 (q, Ω) condition.The intensity is shown on the log scale. The following parameters are usedin the units of t for (a) µ = −0.5362,W = 0.5617, for (b) µ = −0.4573,W = 0.4617, for (c) µ = −0.3694,W = 0.3456, for (d) µ = −0.5349,W = 0.5530, ∆0 = 0.0727, for (e) µ = −0.4509,W = 0.4555, ∆0 = 0.0705,and for (f) µ = −0.3624,W = 0.3458, ∆0 = 0.0618 [60]. Reproduced withpermission from W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spinexcitations in layered antiferromagnetic metals and superconductors. Phys.Rev. B, 86:134513, Oct 2012 (Page 134513-8, Figure 5). c⃝(2012) by TheAmerican Physical Society.

50

The spin excitation spectra for three different electron dopings with n = 1.06,

n = 1.09, and n = 1.12 in Figure 3-5 (from upper to lower panels) for the pure antifer-

romagnetic state (left panels) and the coexistence state with ∆1 = 0 (right panels). For

comparison the left panels show the results for the pure antiferromagnetic state. The

blue lines denote 1 = UReχ+−0 (q, Ω) condition. The intensity is shown on the log scale.

The following parameters are used in the units of t for (a) µ = −0.5362,W = 0.5617, for

(b) µ = −0.4573,W = 0.4617, for (c) µ = −0.3694,W = 0.3456, for (d) µ = −0.5349,

W = 0.5530, ∆0 = 0.0727, for (e) µ = −0.4509,W = 0.4555, ∆0 = 0.0705, and for (f)

µ = −0.3624,W = 0.3458, ∆0 = 0.0618.

The Goldstone mode in the transverse channel remains robust and gapless in

the coexistence regime. This can be proved by an analytical check using the self-

consistent equation forW and U, as in the case of the pure antiferromagnetic state. The

excitations in the transverse channel are dominated by the renormalized spectrum of the

spin waves.

At the same time, we find that the excitations in the longitudinal channel include a

resonance mode at the commensurate momentum close to (π, π) due to the supercon-

ducting gap.

The spin velocity is also calculated in the coexistence state. To evaluate the spin

wave velocity, we expand the denominator of Equation 3–5 around q ≈ Q with ω = 0 up

to quadratic order. This procedure leads to the spin wave velocity, c of the form

c2 =yt2(1/U −W 2z)

W 2x2 + (v)(1/U −W 2z)(3–13)

where

x =∑k

′ 1√(ε−k)2+W 2

(Ωα

k +Ωβk

)2(Eα

k

Ωαk−Eβ

k

Ωβk

)(3–14)

v =∑k

′ 1(Ωα

k +Ωβk

)3(1− E

αk E

βk − ∆2

ΩαkΩ

βk

)(3–15)

51

Figure 3-6. Calculated imaginary part of the longitudinal susceptibility, χzzRPA(q,q, Ω) spinexcitation spectra Ω vs. q for three different electron dopings, n = 1.06,n = 1.09, and n = 1.12 (from upper to lower panels). [60]. Reproduced withpermission from W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spinexcitations in layered antiferromagnetic metals and superconductors. Phys.Rev. B, 86:134513, Oct 2012 (Page 134513-9, Figure 6). c⃝(2012) by TheAmerican Physical Society.

z =∑k

′ 1((ε−k)2+W 2

)(Ωα

k +Ωβk

) (1− Eαk E

βk + ∆

2k

ΩαkΩ

βk

). (3–16)

The coefficient y is comprised of two terms. The first one arises from the intraband

contribution

52

y1 =∑kγ=α,β

′ ∆2k

2(Ωγ

k

)3 W 2 sin2 kx((

ε−k)2+W 2

)2 − (cos kx + cos ky)2(ε−k)2+W 2

− 3ε−k ∆

2kE

γk((

ε−k)2+W 2

)(Ω

γk)5

2 sin2 kx cos ky ± ε−k sin2 kx√(

ε−k)2+W 2

. (3–17)

and the other one from the interband contributions

y2 =1

4

∑k,γ =γ′

′ 2W 2 sin2 kx((ε−k)2+W 2

)2 (Ωγk +Ω

γ′

k

) (1− Eγk E

γ′

k − ∆2kΩγkΩ

γ′

k

)

− 2

ΩγkΩ

γ′

k

(Ωγk +Ω

γ′

k

) t ′tcos(kx + ky )±

(cos2 kx + cos kx cos ky

)√(ε−k)2+W 2

∓ W 2 sin2 kx((ε−k)2+W 2

)3/2Eγ′

k +∆2k2t2

− 1(Ωγk

)3Ωγ′

k

(Ωγk +Ω

γ′

k

) ((E ′γk

)2Eγ

k Eγ′

k − 2∆2k∆20sin2 kxt2

)

+

(Eγ

k Eγ′

k + ∆2k

)(Ωγk

)5Ωγ′

k

(Ωγk +Ω

γ′

k

) ((E ′γk

)2 (Eγ

k

)2+ 2∆2k∆

20

sin2 kxt2

)

− 1(Ωγk

)2 (Ωγk +Ω

γ′

k

)3(1−Eγ

k Eγ′

k − ∆2kΩγkΩ

γ′

k

)((E ′γ

k

)2 (Eγ

k

)2+ 2∆2k∆

20

sin2 kxt2

)

− 1(Ωγk

)2Ωγ′

k

(Ωγk +Ω

γ′

k

)2((E ′γ

k

)2Eγ

k Eγ′

k − 2∆2k∆20sin2 kxt2

)

+

(Eγ

k Eγ′

k − ∆2k)

(Ωγk

)4Ωγ′

k

(Ωγk +Ω

γ′

k

)2((E ′γ

k

)2 (Eγ

k

)2+ 2∆2k∆

20

sin2 kxt2

)

−

(Eγ

k Eγ′

k +∆2k

)(Ωγk

)2Ωγ′

k

(Ωγk +Ω

γ′

k

) −(1−Eγ

k Eγ′

k − ∆2kΩγkΩ

γ′

k

)1(

Ωγk +Ω

γ′

k

)2

×[∆2k (E ′γ

k

)2+(Eγ

k

)2∆20 sin

2 kx/t2(

Ωγk

)3− 4

Ωγk

t ′tcos(kx + ky )±

(cos2 kx + cos kx cos ky

)√(ε−k)2+W 2

∓ W 2 sin2 kx((ε−k)2+W 2

)3/2Eγ

k +∆2kt2

].(3–18)

53

This result can be reduced to the spin wave velocity in the pure antiferromagnetic state

and be compared with the spin velocity calculations in [13, 78]. The spin velocity is con-

sistent with the dispersions in Figures 3-5 (a), (b) and (c) for the pure antiferromagnetic

state and in Figures 3-5 (d), (e) and (f) for the coexistence state. The spin waves are

strongly modified by the resonance created by superconducting gap. The effect is par-

ticularly strong around 2∆0 where the spin waves in the coexistence region exhibit a kink

structure due to the interaction with the particle-hole continuum of the intraband tran-

sitions. They are enhanced due to d−wave symmetry of the superconducting gap, i.e.

due to the fact that one finds ∆γk = −∆γ

k+qifor incommensurate momenta qi > (0.8, 0.8)π.

This leads to an enhancement of the intraband particle-hole continuum of both bands

for Ω ≈ 2∆0. As the electron band around (±π, 0) and (0,±π) points always crosses

the Fermi level in the antiferromagnetic state, the enhancement of the particle-hole

continuum of this band around 2∆0 is responsible for the kink structure seen in the

spin waves. In other words, the damping effects of the particle-hole continuum on the

spin waves are present in both pure metallic antiferromagnetic and coexistence states.

However, in the coexistence region there is also an effect of the strong renormalization

of the spin wave due to the 2∆0 structure of the particle-hole continuum of the intraband

susceptibility, which then yields the renormalization of the spin wave velocity around

2∆0. Another interesting feature is that d−wave superconductivity stabilizes the com-

mensurate antiferromagnetic state by partial gapping the particle-hole continuum in the

coexistence state. Observe, for example, that the spin waves computed for n = 1.12 in

the pure antiferromagnetic state show a tendency towards incommensurability, while in

the coexistence state the spin excitations are still commensurate.

54

CHAPTER 4THE PAIRING INTERACTION ARISING FROM ANTIFERROMAGNETIC SPIN

FLUCTUATIONS

This chapter presents the analysis of the pairing instability created by the spin

fluctuations in the antiferromagnetic state. The study follows the theory proposed by

Schrieffer, Wen and Zhang[12]. Some of the material I presented here has appeared

as ”Doping asymmetry of superconductivity coexisting with antiferromagnetism in spin

fluctuation theory,” W. Rowe, I. Eremin, A. Rømer, B. M. Andersen and P. J. Hirschfeld,

arXiv:1312.1507.

4.1 The pairing interaction in the antiferromagnetic background

The effective interaction arises from the spin correlation proposed by Berk and

Schrieffer[5]. They studied the paramagnon-mediated interaction in a nearly ferromag-

netic Fermi liquid. This approach was adapted by Nakajima to study liquid He3[79] and

by Fay and Appel[80] to study p−state superconductivity with itinerant ferromagnetism.

The pairing interaction arises from summing over the RPA diagrams. The effective

Hamiltonians obtained from sum over all the possible RPA processes as in Figure 1-1 in

the charge-fluctuation channel is[5]

Hc =1

4N

∑k,k′,q

∑s1,s2

[2U − Vc(k− k′)]c†k′s1c†−k′+qs2c−k+qs2cks1, (4–1)

in the longitudinal spin-fluctuation channel

Hz = − 14N

∑k,k′,q

∑s1,s2,s3,s4

Vz(k− k′)σ3s1,s2σ3s3,s4c†k′s1c

†−k′+qs3c−k+qs4cks2, (4–2)

and in the transverse spin-fluctuation channel

H+− = − 14N

∑k,k′,q

∑s1,s2,s3,s4

V+−(k− k′)(σ+s1,s2σ−s3,s4+ σ−

s1,s2σ+s3,s4)

× c†k′s1c†−k′+qs3c−k+qs4cks2

(4–3)

55

where

Vc(q) =U2χ000 (q)

1 + Uχ000 (q), (4–4a)

Vz(q) =U2χzz0 (q)

1− Uχzz0 (q), (4–4b)

V+−(q) =U2χ+−0 (q)

1− Uχ+−0 (q), (4–4c)

and χ000 , χzz0 and χ0+− are bare static (ω = 0) susceptibility for charge, longitudinal spin

and and transverse spin. These results were obtained by Schrieffer et al.[12]. In the

paramagnetic state, spin rotational invariance implies χzz0 =12χ+−0 = χ000 . However, in the

antiferromagnetic state with staggered magnetization along z-axis, χzz0 = 12χ+−0 , and the

only remaining degeneracy is χzz0 = χ000 . In the superconducting state, all symmetries

are broken. We study the instability in the antiferromagnetic background, therefore we

consider the susceptibility in the pure antiferromagnetic state. The expressions for the

bare susceptibilities are shown in Equations 3–6, 3–7 and 3–9. The relative magnitude

of the charge potential, Vc , longitudinal, Vz and transverse V+− can be estimated. The

renormalization for the charge potential is not strong as for the longitudinal spin one. We

have always Vz > Vc for the same momentum. V+− has a singularity at the ordering

vector Q therefore it is a dominant contribution.

In the antiferromagnetic state, the spin fluctuations are modified due to the breaking

of rotational symmetry. We apply the unitary transformation described in Equation 2–11

56

to the effective Hamiltonian, and get

Hc =1

4N

∑k,k′

∑s1,s2,s3,s4

[2U − Vc(k− k′)]l2(k, k′)δs1,s2δs3,s4

+ [2U − Vc(k− k′ +Q)]m2(k, k′)σ3s1,s2σ3s3,s4

[α†k′s1

α†−k′s3α−ks4αks2 + β†

k′s1β†−k′s3β−ks4βks2]

+[2U − Vc(k− k′)]p2(k, k′)δs1,s2δs3,s4 + [2U − Vc(k− k′ +Q)]n2(k, k′)σ3s1,s2σ3s3,s4

[α†k′s1

α†−k′s3β−ks4βks2 + β†

k′s1β†−k′s3α−ks4αks4],

(4–5)

Hz =− 1

4N

∑k,k′

∑s1,s2,s3,s4

[Vz(k− k′)]l2(k, k′)σ3s1,s2σ3s3,s4+ [Vz(k− k′ +Q)]m2(k, k′)δs1,s2δs3,s4

[α†k′s1

α†−k′s3α−ks4αks2 + β†

k′s1β†−k′s3β−ks4βks2]

+ [Vz(k− k′)]p2(k, k′)σ3s1,s2σ3s3,s4+ [Vz(k− k′ +Q)]n2(k, k′)δs1,s2δs3,s4

[α†k′s1

α†−k′s3β−ks4βks2 + β†

k′s1β†−k′s3α−ks4αks2],

(4–6)

and

H+− =− 1

4N

∑k,k′

∑s1,s2,s3,s4

[V+−(k− k′)]n2(k, k′)− [V+−(k− k′ +Q)]p2(k, k′)

(σ+s1,s2σ−s3,s4+ σ−

s1,s2σ+s3,s4)[α

†k′s1

α†−k′s3α−ks4αks2 + β†

k′s1β†−k′s3β−ks4βks2]

+[V+−(k− k′)]m2(k, k′)− [V+−(k− k′ +Q)]l2(k, k′)

(σ+s1,s2σ−s3,s4+ σ−

s1,s2σ+s3,s4)[α

†k′s1

α†−k′s3β−ks4βks2 + β†

k′s1β†−k′s3α−ks4αks2],

(4–7)

57

with the coherence factors

m(k, k′) = ukv′k + vku

′k (4–8a)

l(k, k′) = uku′k + vkv

′k (4–8b)

p(k, k′) = ukv′k − vku′k (4–8c)

n(k, k′) = uku′k − vkv ′k. (4–8d)

In Schrieffer, Wen and Zhang’s study[12], they considered the hole-doped-case with

only hole pockets and ignored all the terms except the ones only with the hole band

operators, β. Here our effective Hamiltonian is general but reduces to their result in this

limit.

The total mean field Hamiltonian corresponds to the effective Hamiltonian includ-

ing the kinetic term, Hubbard term and the superconducting effective interaction in

Equations 4–1, 4–2 and 4–3 can be expressed in terms of the SDW quasiparticles,

H =∑kγ

E γγ†kγk −

∑kγσσ′

∆γ∗σ′σγ−kσγkσ′ −

∑kγσσ′

∆γσσ′γ

†kσγ

†−kσ′. (4–9)

Here γ = α, β are the indices of the bands. The gap function ∆σσ′(k)is a matrix in spin

space,

∆(k) =

∆↑↑(k) ∆↑↓(k)

∆↓↑(k) ∆↓↓(k)

=−dx(k) + idy(k) ∆s(k) + dz(k)

−∆s(k) + dz(k) dx(k) + idy(k).

(4–10)

We obtain the diagonal term of the matrix in spin space as

∆γσσ(k) = − 1

4N

∑k′

[Γρ(k, k

′) + Γzs (k, k′)]⟨γ−k↑γk↑⟩ (4–11)

+[Γ′ρ(k, k

′) + Γz ′s (k, k′)]⟨γ′

−k↑γ′k↑⟩,

58

where γ and γ′ are opposite bands, and the off-diagonal terms as

∆γσσ(k) = − 1

4N

∑k′

[Γρ(k, k

′)− Γzs (k, k′)]⟨γ−kσγkσ⟩+

[Γ′ρ(k, k

′)− Γz ′s (k, k′)]⟨γ′

−kσγ′kσ⟩

+ 2[Γ⊥s (k, k

′)]⟨γ−kσγkσ⟩+ 2

[Γ⊥′s (k, k

′)]⟨γ′

−kσγ′kσ⟩.

(4–12)

The Γs are defined as

Γρ(k, k′) = [2U − Vc(k− k′)]l2(k, k′)− [Vz(k− k′ +Q)]m2(k, k′), (4–13a)

Γ′ρ(k, k′) = [2U − Vc(k− k′)]p2(k, k′)− [Vz(k− k′ +Q)]n2(k, k′), (4–13b)

Γzs (k, k′) = [2U − Vc(k− k′ +Q)]m2(k, k′)− [Vz(k− k′)]l2(k, k′), (4–13c)

Γz ′s (k, k′) = [2U − Vc(k− k′ +Q)]n2(k, k′)− [Vz(k− k′)]p2(k, k′), (4–13d)

Γ⊥s (k, k′) = −[V+−(k− k′)]n2(k, k′) + [V+−(k− k′ +Q)]p2(k, k′), (4–13e)

Γ⊥′s (k, k

′) = −[V+−(k− k′)]m2(k, k′) + [V+−(k− k′ +Q)]l2(k, k′), (4–13f)

where the primed vertices indicate the inter-band interactions. To separate the gap

functions into singlet channel and triplet channels, we follow the standard definition,

∆(k) =

−dx(k) + idy(k) d0(k) + dz(k)

−d0(k) + dz(k) dx(k) + idy(k),

(4–14)

and we get

d0(k) =1

2(∆↑↓ − ∆↓↑) (4–15a)

dx(k) =1

2(−∆↑↑ +∆↓↓) (4–15b)

dy(k) =−i2(∆↑↑ + ∆↓↓) (4–15c)

dz(k) =1

2(∆↑↓ + ∆↓↑). (4–15d)

59

The gap equations for the triplet gaps are

dγx/y(k) = − 14N

∑k′

[Γρ(k, k

′) + Γzs (k, k′)]dγi (k′)2Ωγk′tanh

(Ωγk′

2T

)(4–16)

+[Γ′ρ(k, k

′) + Γz ′s (k, k′)]dγ′

i (k′)

2Ωγ′

k′

tanh(Ωγ′

k′

2T

)(4–17)

and

dγz (k) = − 14N

∑k′

[Γρ(k, k

′)− Γzs (k, k′) + 2Γ⊥s (k, k′)]∆γ0(k

′)

2Ωγk′tanh

(Ωγk′

2T

)+

[Γ′ρ(k, k

′)− Γz ′s (k, k′) + 2Γ⊥′s (k, k

′)]∆γ′

0 (k′)

2Ωγ′

k′

tanh(Ωγ′

k′

2T

),

(4–18)

and for the singlet gaps are

∆γs (k) ≡ d

γ0 (k) =− 1

4N

∑k′

[Γρ(k, k

′)− Γzs (k, k′)− 2Γ⊥s (k, k′)]∆γs (k

′)

2Ωγk′tanh

(Ωγk′

2T

)+[Γ′ρ(k, k

′)− Γz ′s (k, k′)− 2Γ⊥′s (k, k

′)]∆γ′s (k

′)

2Ωγ′

k′

tanh(Ωγ′

k′

2T

).

From the definition of the coherence factors,m2, l2, p2 and n2, Equations 4–8, we

can write down the general expressions,

m2(k, k′) =1

2

[1− ε−k ε

−k′ −W 2√

(ε−k )2 +W 2

√(ε−k′)

2 +W 2

], (4–19a)

l2(k, k′) =1

2

[1 +

ε−k ε−k′ +W

2√(ε−k )

2 +W 2√(ε−k′)

2 +W 2

], (4–19b)

p2(k, k′) =1

2

[1− ε−k ε

−k′ +W

2√(ε−k )

2 +W 2√(ε−k′)

2 +W 2

], (4–19c)

n2(k, k′) =1

2

[1 +

ε−k ε−k′ −W 2√

(ε−k )2 +W 2

√(ε−k′)

2 +W 2

]. (4–19d)

60

By using the condition ε−k+Q = −ε−k , we have the relations m2(k + Q, k′) =

m2(k, k′ + Q) = l2(k, k′) and p2(k + Q, k′) = p(k, k′ + Q) = n2(k, k′). This results in a

periodicity condition for the potentials. We have m2(k+Q, k′ = m2(k, k′ +Q) = l2(k, k′).

This implies Γρ(k, k′ + Q) = Γzs (k, k′) and Γ⊥s (k, k′ + Q) = −Γ⊥s (k, k′). These relations

guarantee the antiperiodic condition of

Vs(q+Q) = −Vs(q), (4–20)

where Vs is the singlet gap potential Vs(k−k′) = Γρ(k, k′)−Γzs (k, k′)−2Γ⊥s (k, k′)[12]. The

antiperiodicity of the potential leads to the antiperiodicity of the superconducting gap,

∆s(q+Q) = −∆s(q) (4–21)

in order to fulfill the gap equation.

We can check that the gap equation in the antiferromagnetic state reduces to the

gap equation in the paramagnetic superconducting state. We start with singlet gap

function

∆αs (k)

= − 14N

∑k′

′[2U(l2 −m2)− [Vc(k− k′)− Vz(k− k′)]l2 + [Vc(k− k′ +Q)

− Vz(k− k′ +Q)]m2

+ 2V+−(k− k′)n2 − 2V+−(k− k′ +Q)p2]∆αs (k

′)

2Ωαk′tanh

(Ωαk′

2T

)+[2U(p2 − n2)−

[Vc(k− k′)− Vz(k− k′)]p2 + [Vc(k− k′ +Q)− Vz(k− k′ +Q)]n2

+ 2V+−(k− k′)m2 − 2V+−(k− k′ +Q)l2]∆βs (k

′)

2Ωβk′

tanh(Ωβk′

2T

).

(4–22)

In the paramagnetic state, we only have one band, therefore the superconducting

gap has only one form ∆α/β → ∆s . By combining the inter- and intra-band interactions in

61

the original antiferromagnetic state, the expression reduces to

∆s → − 14N

∑k′

′[− [Vc(k− k′)− Vz(k− k′)] + [Vc(k− k′ +Q)− Vz(k− k′ +Q)]

+2V+−(k− k′)− 2V+−(k− k′ +Q)]∆s(k′)2Ek′tanh

(Ek′2T

). (4–23)

Here we use the antiperiodicity ∆s(k) = −∆s(k+Q) in the gap and get

∆αs (k) = − 1

4N

∑k′

′[− [Vc(k− k′)− Vz(k− k′)] + 2V+−(k− k′)

]∆s(k′)2Ek′

tanh(Ek′2T

)+ [Vc(k− k′ +Q)− Vz(k− k′ +Q)]− 2V+−(k− k′ +Q)

]−∆s(k′ +Q)2Ek′+Q

tanh(Ek′+Q2T

).

(4–24)

Using the condition that Vz(k, k′) = V+−(k, k′) ≡ Vs(k, k′) in the paramagnetic state, and

restricting the sum to the full Brillouin zone, we have

∆(k) = − 12N

∑k ′

′[32Vs(k − k ′)−

1

2Vc(k − k ′)

]∆s(k ′)2Ek ′

tanh(Ek ′2T

)+3

2[Vs(k − k ′ +Q)−

1

2Vc(k − k ′ +Q)]

]∆s(k ′ +Q)2Ek ′+Q

tanh(Ek ′+Q2T

)= − 12N

∑k ′

[32Vs(k − k ′)−

1

2Vc(k − k ′)

]∆s(k ′)2Ek ′

tanh(Ek ′2T

).

(4–25)

This gap equation is equivalent to the result in studies of superconductivity in the

paramagnetic state[15]. Note that the sign of the overall interaction 32Vs − 1

2Vc is

repulsive, i.e. > 0.

4.2 The pairing symmetries

The singlet superconducting gaps are two non-linear coupled equations for the

α and β bands. In the weak coupling limit, we can assume that the interaction only

happens around the Fermi surface. Then we can solve them self-consistently with

numerical methods. The pairing potential is a complicated function of k − k′. It is not

possible to analyze the symmetry of the full potential without further approximation.

Although we know the antiperiodicity of the potential, this is not enough to determine

the symmetry of the gaps. Without the knowledge of the symmetry of the gaps, the

62

calculation would be expensive. There are also numerical difficulties associated with the

description of the singularity in the transverse RPA susceptibility. These obstacles to a

complete solution can be overcome, but it is useful to have an analytical solution in a

well-defined limit to guide the calculation.

Here we take the limit of small pocket size which appears under small doping with

large antiferromagnetic orderW . We estimate the leading symmetry of the gaps by

expanding the coherence factors and the RPA susceptibilities around the centers of the

electron pockets, k = (±π, 0), (0,±π) and the hole pockets, k = (±π/2,±π/2) assuming

small circular pocket size. The ellipticity of the pocket size or any deviation from perfect

circularity should only change the weights of the harmonics but not the overall symmetry

of the gap. Then we compare the expanded potential with the projected gap symmetries

to determine the leading symmetry of the superconducting gaps.

4.2.1 Angular dependence of the coherence factors

First, in order to expand the coherence factors around hole pockets, we take

Equation 4–19, and expand k and k′ around the pockets centers. Here we use the

example of an intraband expansion around k = k′ = (π/2,π/2) to explain the process.

We assume small quantities δk and δk′ such that k = (π2, π2) + δk and k′ = (π

2, π2) + δk′.

We use Mathematica to expand the coherence factors in Equation 4–19. We get, for

instance, the leading terms

m2(k, k′) ≈ 1− t2

W 2(δkx + δky − δk ′x − δk ′y)

2 (4–26)

Then we replace δk and δk′ with the angular dependent expression along the small

pockets, δkx + δky =√2khF cos θ and δkx − δky =

√2khF sin θ where khF is the Fermi

momentum or pocket radius on the hole pockets. The angles for hole pockets is θ and

for electron pocket is ϕ, as shown in Figure 4-1. The presence of the hole pockets

centered around (±π/2,±π/2) points and the electron pockets around (±π, 0) and

(0,±π) points of the Brillouin zone depend on the types (electron or hole) and the

63

Figure 4-1. General structure of the Fermi surface of layered cuprates in thecommensurate antiferromagnetic state for electron or hole doping.

amount of doping. Note that the definition of the hole angle at zero degrees is 45 from

the x−axis. This is for simplifying the expression on the hole pockets.

The expansions of the coherence factors p2(k, k′) and n2(k, k′) around small hole

pockets with the leading terms are presented in Table 4-1 with k = (π2, π2) and Table 4-2

with k = (−π2, π2). The other two coherence factors can be obtained from the relations

Table 4-1. Coherence factors, p2(k, k′) and n2(k, k′) expanded around hole pockets fork = (π

2, π2) and k′ = (±π

2, π2). The coherence factors l2 and m2 can be

obtained from l2(k, k′) = 1− p2(k, k′) and m2(k, k′) = 1− n2(k, k′).k′ (π

2, π2) (−π

2, π2)

p2(k, k′)t2(khF )

2

W 2 (2− 4 cos θ cos θ′ + cos 2θ + cos 2θ′)t2(khF )

2

W 2 (2− 4 cos θ sin θ′+cos 2θ − cos 2θ′)

n2(k, k′)t2(khF )

2

W 2 (2 + 4 cos θ cos θ′ + cos 2θ + cos 2θ′)

t2(khF )2

W 2 (2 + 4 cos θ sin θ′

+cos 2θ − cos 2θ′)

l2(k, k′) = 1 − p2(k, k′) and m2(k, k′) = 1 − n2(k, k′). The expansion of k′ around

64

Table 4-2. Coherence factors, p2(k, k′) and n2(k, k′) expanded around hole pockets fork = (−π

2, π2) and k′ = (±π

2, π2). The coherence factors l2 and m2 can be

obtained from l2(k, k′) = 1− p2(k, k′) and m2(k, k′) = 1− n2(k, k′).k′ (π

2, π2) (−π

2, π2)

p2(k, k′)t2(khF )

2

W 2 (2− 4 sin θ cos θ′ − cos 2θ + cos 2θ′)t2(khF )

2

W 2 (2− 4 sin θ sin θ′− cos 2θ − cos 2θ′)

n2(k, k′)t2(khF )

2

W 2 (2 + 4 sin θ cos θ′ − cos 2θ + cos 2θ′) t2(khF )

2

W 2 (2 + 4 sin θ sin θ′

− cos 2θ − cos 2θ′)

(−π2,−π

2) and (π

2,−π

2) can be obtained by using the relation p2(k, k′) = n2(k, k′ + Q).

where Q = (π, π).

For the expansions of the coherence factors around electron pockets, we have the

results shown in Table 4-3. We only have even order harmonics in the leading terms for

the electron pockets. And we can see clearly, the relation p2(k, k′) = n2(k, k′ + Q) is

satisfied in Table 4-3.

The interband interaction in the gap equations may be important when we have both

pockets present. The coherence factors of the interband expansions between electron

and hole pockets are shown in Table 4-4.

65

Tabl

e4-

3.C

oher

ence

fact

ors,p2(k,k

′ )an

dn2(k,k

′ )ex

pand

edar

ound

elec

tron

pock

ets

withk=(±

π,0)

andk′=(±

π,0)

and(0,±

π).

The

cohe

renc

efa

ctor

sl2

andm2

can

beob

tain

edfro

ml2(k,k

′ )=1−p2(k,k

′ )an

dm2(k,k

′ )=1−n2(k,k

′ ).

k′=

(±π,0)

(0,±

π)

p2(k,k

′ )t2(ke F)4

4W2

[ 1−2cos2ϕcos2ϕ′+1 2(cos4ϕ+cos4ϕ′ )]

t2(ke F)4

4W2

[ 1+2cos2ϕcos2ϕ′+1 2(cos4ϕ+cos4ϕ′ )]

n2(k,k

′ )t2(ke F)4

4W2

[ 1+2cos2ϕcos2ϕ′+1 2(cos4ϕ+cos4ϕ′ )]

t2(ke F)4

4W2

[ 1−2cos2ϕcos2ϕ′+1 2(cos4ϕ+cos4ϕ′ )]

Tabl

e4-

4.C

oher

ence

fact

ors,p2(k,k

′ )an

dn2(k,k

′ )ex

pand

edar

ound

elec

tron

and

hole

pock

ets

withk=(±

π,0)

andk′

arou

nd(−

π 2,−

π 2)

and(π 2,−

π 2).

The

cohe

renc

efa

ctor

sl2

andm2

can

beob

tain

edfro

ml2(k,k

′ )=1−p2(k,k

′ )an

dm2(k,k

′ )=1−n2(k,k

′ ).

k′=

(π 2,π 2)

(−π 2,π 2)

p2(k,k

′ )t2 W2

[ (kh F)2(1+cos2θ′)−

√2kh F(ke F)2(cos2ϕcosθ′)]

t2 W2

[ (kh F)2(1

−cos2θ′)+√2kh F(ke F)2(cos2ϕsinθ′)]

n2(k,k

′ )t2 W2

[ (kh F)2(1+cos2θ′)+

√2kh F(ke F)2(cos2ϕcosθ′)]

t2 W2

[ (kh F)2(1

−cos2θ′)−√2kh F(ke F)2(cos2ϕsinθ′)]

66

4.2.2 Angular dependence of the pairing potentials

Now we have the leading term expansion of the coherence factors. In order to get

the approximated potentials, we replace the full coherence factors with their angular

dependent approximations. In Schrieffer el al.’s study[12], they ignore the contribu-

tions from the transverse susceptibility, appealing to the Adler principle[31, 81], the

suppression of the divergent pairing interaction at the ordering wave vector by ver-

tex corrections. Frenkel and Hanke[81] and others[14, 77] showed, however that the

transverse contributions (spin waves) to the interactions were of the same order as

the longitudinal excitations. Here we would like to investigate the relative importance

between transverse and longitudinal channels. We separate the singlet pairing potential

into the charge- and longitudinal spin-fluctuation part and the transverse spin-fluctuation

part. We discuss the situation with both electron and hole pockets present below. To

simplify the analysis, we assume only the intraband interaction for the moment.

4.2.2.1 Charge and longitudinal interaction

The charge- and longitudinal spin-fluctuation contribution to the potential for singlet

channel is

Γρ(k, k′)− Γzzs (k, k′) =

2U(l2 −m2)−[Vc(k− k′)− Vz(k− k′)

]l2 +

[Vc(k− k′ +Q)− Vz(k− k′ +Q)

]m2.

(4–27)

Within this channel, we can see the antiperiodicity is still valid. Replacing the coherence

factors with the approximated angular expressions in Tables 4-1 and 4-2, we get the

longitudinal potential as in Table 4-5. The symbols in the table are V = Vc(Q)−Vc(0) +

Vz(0) − Vz(Q) and V ′ = Vz(Q) − Vc(Q) + Vz(0) − Vc(0). Note that the bare potentials

are all positive and the largest value would be Vz(Q). Therefore we have V < 0 and

V ′ > 0. The expansions around k = or k′ = (−π2,−π

2), (π

2,−π

2) can be obtained by

Γρ(k, k′)− Γzzs (k, k′) = −Γρ(k+Q, k′) + Γzzs (k+Q, k′) = −Γρ(k, k′ +Q) + Γzzs (k, k′ +Q).

67

Table 4-5. Potentials from the charge- and longitudinal spin-fluctuation contribution,Γρ(k, k

′)− Γzzs (k, k′) expanded around hole pockets.HHHHHHk

k′(π2, π2) (−π

2, π2)

(π2, π2) V

[1− 2t2(khF )

2

W 2

]2U

t2(khF )2

W 2 8 cos θ sin θ′

+4t2(khF )

2

W 2 (4U + V′) cos θ cos θ′ −8[Vc(π, 0)− Vz(π, 0)]

−V t2(khF )

2

W 2 (cos 2θ + cos 2θ′) × t

2(khF )2

W 2 cos θ sin θ′

(−π2, π2) 2U

t2(khF )2

W 2 8 sin θ cos θ′ V

[1− 2t2(khF )

2

W 2

]−8[Vc(π, 0)− Vz(π, 0)]

t2(khF )2

W 2 sin θ cos θ′ +

4t2(khF )2

W 2 (4U + V′) sin θ sin θ′

−V t2(khF )

2

W 2 (cos 2θ + cos 2θ′)

The scattering between the pocket itself or to the pocket shifted by Q is much stronger

than the scattering which is shifted by (±π, 0) or (0,±π).

For the scattering within the same electron pocket, the potential from the charge-

and longitudinal spin-fluctuation contribution in Equation 4–27 is then approximately

Γρ(k, k′)− Γzzs (k, k′)

≈V − (V − 8U)t2(keF )

4

4W 2+ V ′ t

2(keF )4

2W 2(cos 2ϕ cos 2ϕ′) + V

t2(keF )4

8W 2(cos 4ϕ+ cos 4ϕ′).

(4–28)

For scattering between different electron pockets, we use the antiperiodic relation

Γρ(k, k′) − Γzzs (k, k′) = −Γρ(k, k′ + Q) + Γzzs (k, k′ + Q). This relation can simplify the

electron multi-pocket problem into a one electron pocket problem. Later we will see that

when solving the superconducting gap equation, we can use these symmetries to allow

us to consider only one pocket for the electron and two pockets for the hole case.

Although the charge- and longitudinal spin-potential does not diverge and can be

viewed as a constant in the expansions, for consistency to order kF , we also expand

these two terms around small q. We have

V+−(q) ≈ V+−(0) +1

2(U ∓ V+−(0))

U(χzz ′′0 (0)

1± Uχzz0 (0)q2 (4–29)

68

and

V+−(q+Q) ≈ V+−(Q) +1

2(U ∓ V+−(Q))

U(χzz ′′0 (Q)

1± Uχzz0 (Q)q2. (4–30)

where V+ and V− correspond to Vc and Vz , respectively and χ′′0(x) is the second

derivative of χ with respect to q and evaluated at x .

4.2.2.2 Transverse interaction

The contribution from the transverse spin-fluctuations is

−2Γ⊥s (k, k′) =2[V+−(k− k′)]n2 − 2[V+−(k− k′ +Q)]p2.

(4–31)

Here we have singularities when k = k′ in the evaluation of V+−(k − k′ + Q) or when

k = k′ + Q. But the coherence factors p2(k, k′) and n2(k, k′) are zero when k = k′ and

k = k′ + Q, respectively. Therefore we have to expand the transverse susceptibility

and the coherence factors to obtain the limit at the singular point. We follow Frenkel and

Hanke’s study[29, 31, 77, 81, 82], and get

V+−(Q) ≃1

t2yq2=

1

t2y(keF )2

[(2− 2(cos θ cos θ′ + sin θ sin θ′)

]−1, (4–32)

where y = 2t2∑′k

sin2 kx (1−3(ε−Q )2/2((ε−Q )

2+W 2))− 12cos2 kx− 12 cos kx cos ky

((ε−Q )2+W 2)3/2

− 4t ′2∑′ksin2 kx cos2 ky

((ε−Q )2+W 2)3/2

.

We use the above expression for V+−(Q) together with the expansion for the co-

herence factors and get the angular expansion of the transverse interaction around hole

pockets in Table 4-6. The large contribution still comes from intra-pocket scattering.

The transverse susceptibility expansion in terms of angles on the electron pockets

around q = Q is

V+−(Q) ≃1

t2yq2=

1

t2y(keF )2

[(2− 2(cosϕ cosϕ′ + sinϕ sinϕ′)

]−1. (4–33)

69

Table 4-6. Potentials from the transverse spin-fluctuation contribution, −2Γs expandedaround hole pockets in the limit of khF → 0

HHHHHHkk′

(π2, π2) (−π

2, π2)

− 2yW 2 (1− cos θ cos θ′ + sin θ sin θ′)

(π2, π2) +2V+−(0)

t2(khF )2

W 2 (2 + 4 cos θ cos θ′ 16V+−(π, 0)

t2(khF )2

W 2 cos θ sin θ′

+cos 2θ + cos 2θ′)− 2yW 2 (1 + cos θ cos θ

′ − sin θ sin θ′)(−π2, π2) 16V+−(π, 0)

t2(khF )2

W 2 sin θ cos θ′ +2V+−(0)

t2(khF )2

W 2 (2 + 4 sin θ sin θ′

− cos 2θ − cos 2θ′)

The transverse interaction expansion around the electron pockets is then

−2Γ⊥s (k, k′) = 2V+−(0)t2(keF )

4

4W 2

[1 + 2 cos 2ϕ cos 2ϕ′ +

1

2(cos 4ϕ+ cos 4ϕ′)

]− (k

eF )2

2yW 2

[1 + cosϕ cosϕ′ + sinϕ sinϕ′ − 1

2(cos 3ϕ cosϕ′

− sin 3ϕ sinϕ′ + cosϕ cos 3ϕ′ − sinϕ sin 3ϕ′)− cos 2ϕ cos 2ϕ′ + sin 2ϕ sin 2ϕ′].

(4–34)

4.2.2.3 Interband interactions

The interaction between electron and hole pockets is expected to be small due to

the fact that the connecting vector is away from Q and most of the time we only have

one kind of pocket present. In the rare case when we have both pockets, we can still

estimate the contribution from interband scattering. To calculate the potential for the

scattering between hole and electron pockets, we have to consider a different interband

potential. The charge and the longitudinal interband (primed) potential is

Γ′ρ(k, k′)− Γzz ′s (k, k′) =

2U(p2 − n2)−[Vc(k− k′)− Vz(k− k′)

]p2 −

[Vz(k− k′ +Q)− Vc(k− k′ +Q)

]n2.

(4–35)

70

The transverse interband potential is

−2Γ⊥′s (k, k

′) = +2V+−(k− k′)m2 − 2V+−(k− k′ +Q)l2.

(4–36)

The approximated interband potential is shown in Table 4-7 There is a four-

Table 4-7. Potentials from the charge- and longitudinal spin-fluctuation interbandcontribution, Γ′ρ(k, k′)− Γ′zzs (k, k′)− 2Γ⊥′

s (k, k′) expanded between electron

and hole pocketsPPPPPPPPPk =

k′ =(π, 0)

(π2, π2) −2

√2[2U − Vc(π2 ,

π2) + Vz(

π2, π2) + 2V+−(

π2, π2)]khF (k

eF )2 cos θ cos 2ϕ

(−π2, π2) 2

√2[2U − Vc(π2 ,

π2) + Vz(

π2, π2) + 2V+−(

π2, π2)]khF (k

eF )2 sin θ cos 2ϕ

fold symmetrical property of V (π/2, π/2) = V (−π/2,−π/2) = V (π/2,−π/2) =

V (−π/2, π/2). We can see that the interband potential is quite small up to third power of

the pocket size khF (keF )2.

4.2.3 LAHA expansion of gap equation

In order to determine the gap symmetry, we use the leading angular harmonics

approximation (LAHA) method[83]. The LAHA method is a simplified version of the BCS

gap equation in Equation 1–15. That becomes exact in the limit of small pocket size.

The simplified form

∑j

∫ 2π0

dΨ′

2πNF ,jΓi ,j(Ψ,Ψ

′)∆α,j(Ψ′)L = −λα∆α,i(Ψ) (4–37)

is a eigenvalue problem where i and j are the band or orbital indices, α is the symmetry

of the gap, Γij is the interaction, NF ,j is the density of state at Fermi surface and L is a

constant proportional to lnEF/TC , Ψ = θ,ϕ represents angles on either hole or electron

pockets. The interaction of each channel is restricted within the leading harmonics.

From the antiperiodicity of the potentials V (q) = −V (q + Q), possibilities include

extended s−wave, which forces the superconducting gap to change sign on the reduced

Brillouin zone boundary; dx2−y2-wave, which does not change sign on the boundary but

71

which has nodes along the 110 directions; and also odd-parity p−wave symmetry[12].

dxy -wave pairing is excluded because it does not fulfill the antiperiodicity condition.

Here we expand the s-wave and dx2−y2-wave symmetries on the hole pockets to get the

angular dependence of the gaps as in Table 4-8.

Table 4-8. Angular dependence of the s-wave and dx2−y2-wave symmetries on the holepockets.

∆(k) cos kx + cos ky (extended s-wave) cos kx − cos ky (dx2−y2-wave)(π2, π2) −

√2khF cos θ

√2khF sin θ

+√236(khF )

3(5 cos θ − 2 cos 3θ) +√236(khF )

3(5 sin θ + 2 sin 3θ)

(−π2, π2) −

√2khF sin θ

√2khF cos θ

−√236(khF )

3(5 sin θ + 2 sin 3θ) −√236(khF )

3(5 cos θ − 2 cos 3θ)

The leading term angular dependent gap on the first and second hole pockets as

labeled in Figure 4-1 can be written as

∆sh1(θ) = ∆sh cos θ, ∆

sh2(θ) = ∆

sh sin θ

(4–38)

for the extended s−wave symmetry, and

∆dh1(θ) = ∆dh sin θ, ∆

dh2(θ) = ∆

dh cos θ

(4–39)

for the dx2−y2-wave symmetry.

Table 4-9. Angular dependence of the s-wave and dx2−y2-wave symmetries on theelectron pockets.

∆(k) cos kx + cos ky (extended s-wave) cos kx − cos ky (dx2−y2-wave)(π, 0)

(keF )2

2cos 2ϕ− (keF )

4

24cos 4ϕ −2 + (keF )

2

2− (keF )

4

96(3 + cos 4ϕ)

In addition, we have the leading gap symmetry Ansatze for the electron pockets,

∆se(ϕ) = ∆se cos 2ϕ, (4–40)

72

∆de (ϕ) = ∆de (1 + αde cos 4ϕ). (4–41)

We also have the possibility of p−wave symmetry[12]. On each type of pockets, the

gaps have the angular dependence as

∆ph1(θ) = ∆ph(1 + αph cos 2θ),

∆ph2(θ) = ±∆ph(1 + αph cos 2θ).

(4–42)

The ± sign refers to two distinct p-wave states, with signs + + −− or + − −+ on hole

pockets h1, ..., h4.

By comparing the gap angular dependence with the interactions, Γs, we can find

the leading symmetry. From Table 4-5 and Equation 4–28, we can get the charge and

longitudinal contribution of the potentials, V ℓ ≡ Γρ − Γzs . The expansions up to order k2F

on each type of pockets for the charge- and longitudinal- potentials are expressed as

V lh1h1(θ, θ′) ≈ch + ah cos θ cos θ′ + bh cos θ cos θ′

+ ch(cos 2θ + cos 2θ′), (4–43a)

V lh2h2(θ, θ′) ≈ch + ah sin θ sin θ′ + bh cos θ cos θ′

+ ch(cos 2θ + cos 2θ′), (4–43b)

V lee(ϕ,ϕ′) ≈ce + de(cosϕ cosϕ′ + sinϕ sinϕ′), (4–43c)

where ch ≡ V +[V − 2t2V

W 2

]khF2, ah ≡

[− V + 4t2V

W 2

]khF2, bh = V khF

2, ce ≡ V + V keF2, de ≡

V keF2, and Y (x) = 4U χ′′

zz (x)Vz (x)

1+Uχzz (x)

(1+ Vz (x)2U−2

(1+Uχzz (x))2

), V ≡ [Vz(0)− Vc(0) + (Vc(Q)− Vz(Q))],

and V ≡ Y (0) − Y (Q). V is positive since Vz(Q) is the dominant term in the definition.

This implies that the leading contribution to the intra-pocket interactions are attractive

for both electron and hole cases. But due to the antiperiodicity, it does not give rise

to a conventional s−wave gap. Comparing the electron potential, V lee(ϕ,ϕ′) with the

expansion of dx2−y2− gap, Equation 4–41, we can see both expressions have leading

73

terms being constant and. This leads to a leading d−wave instability for an electron-

doped system with antiferromagnetic order. For the hole pocket case, on the other hand,

only p− wave has a constant leading attractive interaction. Therefore the charge- and

longitudinal spin-potential gives rise to a p− wave symmetry in the superconducting

gap. There is a sub-dominant contribution from the s−wave symmetry. The s−wave

contribution is quite small and scaled with (khF )2.

The expansion for V tr ≡ −2Γ⊥s then has the following form:

V trh1h1(θ, θ′) ≈ Ah(1− cos θ cos θ′ + sin θ sin θ′) + Bh(2 + 4 cos θ cos θ′ + cos 2θ + cos 2θ′),

(4–44a)

V trh2h2(θ, θ′) ≈ Ah(1 + cos θ cos θ′ − sin θ sin θ′) + Bh(2 + 4 sin θ sin θ′ − cos 2θ − cos 2θ′),

(4–44b)

V tree (ϕ,ϕ′) ≈ Ae

[1 + cosϕ cosϕ′ + sinϕ sinϕ′

− 12(cos 3ϕ cosϕ′ − sin 3ϕ sinϕ′ + cosϕ cos 3ϕ′ − sinϕ sin 3ϕ′)

− cos 2ϕ cos 2ϕ′ + sin 2ϕ sin 2ϕ′], (4–44c)

where Ah ≡ − 2yW 2 , Bh ≡ V±(0)

(tkhFW

)2, and Ae ≡ − keF

2

2yW 2 . For the transverse channel,

the hole pockets have a stronger pairing potential compared with the electron pockets.

The potential for the hole pocket has a leading constant term while the leading term for

electron pockets scales with (keF )2. For the hole pockets, the transverse potential still

supports p−wave pairing whereas for the electron pockets, it supports dx2−y2−wave

pairing.

4.2.4 Comparison with numerical evaluation

We compare the full expression of the potentials with the approximate expansion

of the potentials by plotting these two along the electron pocket shown in Figure 4-2

with doping, x = 0.03. For the hole pockets we will have to overcome the problem of

74

Figure 4-2. Comparison of the analytical calculations up to (keF )2 for the longitudinal (left

panel) and transverse (right panel) pairing potentials, V lee and V tree on theelectron pockets for the doping level of n = 1.03 (black curves), together withthe full numerical evaluation of Γρ − Γzs , and −Γ⊥s (blue points).

the incommensurate mode by modifying the interband contribution in the transverse

susceptibility. This is saved for future investigation. The blue dots are calculated with

ϕ′ = 0 according to the charge- and longitudinal spin-fluctuations in Equation 4–27 in

Figure 4-2 (a) and to the transverse spin-fluctuations in Equation 4–31 in Figure 4-2 (b).

Then the dots are fitted by the leading harmonics,

F (ϕ,ϕ′) = a + b cosϕ cosϕ′ + c cos 2ϕ cos 2ϕ′ + d cos 3ϕ cos 3ϕ′, (4–45)

with a, b, c and d as fit parameters, giving the red curves in Figure 4-2. The approximate

potentials given in Equations 4–43c and 4–44c are plotted as black curves. We find

ce = −1.534, de = −0.611 and Ae = −0.308 (in units of t). The red curves denote the fit

when ce , de , and Ae are not computed analytically but fitted to the numerical results with

the least square method (ce = −1.457, de = −0.547 and Ae = −0.339 (in units of t)).

Here, we use U = 2.775t which givesW = 0.6537t. It is evident that the approximated

potentials agree with the full expression on the general symmetry and the magnitude.

The errors may come from restriction of the finite order of the expansion, the evaluation

75

of the value of y which is derived assuming half-filling in the transverse channel, and the

finite size of the electron pockets.

The implication of dx2−y2−wave symmetry on the electron pockets in the coexistence

state and p−wave on the hole pockets provides a natural explanation of why we do not

have robust coexistence state on the hole-doped side of the cuprate phase diagrams.

In is well-known that on both side of the cuprates we have dx2−y2− superconducting

gaps. In the crossover from pure superconducting state to the coexistence state, the

electron-doped cuprates do not need to change symmetry whereas the hole-doped

cuprates have to change symmetry to a nodeless singlet p−wave to avoid nodes on

the pocket. Note that the triplet p−wave gap gives rise to a nodal structure on the hole

pockets, therefore is less favored. Recent experiments have shown a fuliy gapped

superconducting gap in the deeply underdoped cuprates [84–88], to which our work may

apply.

For future work, a new phase diagram could be generated based on the same

model. The previous calculation of spin excitations in the coexistence state of AF

and superconductivity can be included to create a self-consistence pairing interaction

calculation. Therefore we do not need the assumption of phenomenological order

parameter for superconductivity. It would be interesting to see the symmetry of the

superconducting gap across the phase diagram and to compare the gap structure

whether we will see as interband pairing contribution which arise due to the Umklapp

processes in the AF state[56].

76

CHAPTER 5CONCLUSION

The mean field phase diagram in the electron-doped cuprates has been studied

with a one-band square lattice Hubbard model. In the antiferromagnetic state, the Fermi

surface is reconstructed and the energy dispersion splits into an α (electron) band and a

β (hole) band. This creates electron and hole pockets on the Fermi surface. We derived

the self-consistent equations for the calculation of the antiferromagnetic order parameter

with a given on-site Coulomb interaction U. We changed the the chemical potential µ

to adjust the doping level and added a phenomenological superconducting Hamiltonian

to study the coexistence state of antiferromagnetism and superconductivity. The mean

field energy was also calculated to determine the favorable state for a given doping and

temperature. From the half-filling region, we have a superconducting phase starting at

around x = 0.05 doping. This transition across the doping is of first order due to the

mean-field energy, unlike the transition across temperature which is second order. For

this calculation, we assumed a dx2−y2−wave pairing in the superconducting state. The

superconducting gap in the coexistence state has the possibility of a triplet Sz = 0 term

which is a higher harmonic correction and is proportional toW . Within the mean field

calculation, the phase without triplet corrections has lower energy.

We next studied the spin dynamic susceptibility in the half-filled antiferromagnetic

state with different next-nearest hopping t ′. For t ′ = 0, the spin wave has no softening

at (π2, π2) which is consistent with the strong coupling results. But with increased t ′,

we found a softening at (π2, π2). The ”denting” of the dispersion is related to the non-

degenerate points introduced by t ′. At half-filling, with t ′ = 0 the α band and the β

band are well separated by 2W . But with finite t ′, the indirect gap (the distance between

lowest α band the the highest of the β band) become 2W − 4t ′. The minimum at

q ≈ (π2, π2) is due the the interaction between spin waves and the particle-hole spectrum.

This is a feature of the weak-coupling approach which allows W to be the same order as

77

t ′. If we continue to increase t ′ and 2W − 4t ′ < 0, the real part of the RPA susceptibility

becomes negative. This makes the ordered system unstable and it returns to the normal

state.

With finite dopings and t ′/t = 0.35 in the pure antiferromagnetic state , we studied

the cases of hole-doped systems with only hole pockets, electron-doped systems with

only electron pockets and electron-doped systems with both pockets. We recovered the

Goldstone mode in the transverse spin susceptibility. The longitudinal spin susceptibility

is gapped at q = (π, π). In the cases with hole pockets, there is an incommensurate

mode, indicating the breakdown of the mean field commensurate assumption. With the

calculation of the spin-stiffness and the real part of the RPA susceptibility, we found that

the assumption of long range commensurate antiferromagnetic order is not suitable

for the hole-doped case. This may also explain why the antiferromagnetic order on the

hole-doped side is less robust than on the electron-doped side.

We also studied the dynamic spin susceptibility in the coexistence state of an-

tiferromagnetic and superconductivity. We assumed a dx2−y2−wave pairing in the

superconducting state. The Goldstone mode is still robust and gapless. We find that

the spin waves are modified by the resonance which is created by the superconducting

gap. The result is a kink due to the interaction with the particle-hole continuum in the

spin wave dispersion. We also calculated the spin-wave velocity in the coexistence state

as observed in the transverse susceptibility. For the longitudinal susceptibility, we found

a resonance mode at the incommensurate momentum close to q = (π, π) due to the

sign-changing superconducting gap.

The above studies about superconductivity are based on a phenomenological

mean field d−wave pairing interaction. In order to understand the microscopic theory

of superconductivity in the presence of an ordered antiferromagnetic state, we adapted

the ”bag” theory proposed by Schrieffer, Wen and Zhang and generalized it to the

electron-doped case. We also included the transverse part of the spin fluctuation pairing

78

vertex which was ignored by Schrieffer. We derived the full effective pairing Hamiltonian

in the antiferromagnetic state and separated the gap equations into spin singlet and

triplet parts. To study the instability in the singlet channel, we expanded the coherence

factors and the spin susceptibility around electron and hole pocket center in the limit

of small pocket size. We used the LAHA approach to analyze the symmetry of the

superconducting gap. The superconducting gaps of symmetries consistent with the

staggered antiferromagnetic state are also expanded for comparison with the pairing

potentials. We study the charge and longitudinal spin part and the transverse part of

the fluctuations separately. For the charge and longitudinal spin part we find that with

electron pockets, the leading superconducting instability has dx2−y2−wave symmetry,

whereas with hole pockets the leading contribution has odd parity p−wave symmetry.

For the transverse potentials, the singularities in the RPA susceptibility are avoided

because they occur in combination with the SDW coherence factors. This results in a

non-divergent contribution to the pairing from the spin waves. For the hole case, the

transverse fluctuations also support a p−wave symmetry and for the electron case

they support a dx2−y2−wave pairing. But the pairing strength from the electron pocket is

weaker to order (keF )2; therefore the longitudinal fluctuations dominate in this case.

To show that our approximations for the potentials with small pockets are justified,

we plotted the potentials along the pocket with angle dependence for the full expression

and the approximated expression. For both charge- and longitudinal spin-fluctuation

part and the transverse spin-fluctuation part, they agree with the general symmetry and

magnitudes.

On the electron doped side the superconductivity has a smooth crossover from

the pure superconducting state to the coexistence state while on the hole doped side

the superconducting gap has to change symmetry to an odd-parity singlet p−wave to

avoid nodes on the pocket, therefore less favorable. Our findings regarding the leading

superconducting gap symmetries, p−wave on the hole-doped and dx2−y2−wave electron

79

doped side suggests that the coexistence state on the cuprate phase diagram can only

exist on the electron-doped side.

80

APPENDIX AMEAN FIELD QUANTITIES IN THE PURE ANTIFERROMAGNETIC STATE

The following sections are the derivation ofW and U self-consistent equation and

filling level.

A.1 Antiferromagnetic order parameter equation: derivation

Starting with the definition of the antiferromagnetic order parameterW ,

W =U

2

∑kσ

sgn(σ)⟨c†k+Qσckσ⟩, (A–1)

we write out the spin and fold the momentum index to the reduced Brillouin zone, we

have

W =U

2

∑k

′⟨c†k+Q↑ck↑⟩ − ⟨c†k+Q↓ck↓⟩+ ⟨c†k↑ck+Q↑⟩ − ⟨c†k↓ck+Q↓⟩. (A–2)

Using the unitary transformation in Equation 2–11 to change the basis of the operators,

we get

W =U

2

∑k

′⟨(−vkα†

k↑ + ukβ†k↑)(ukαk↑ + vkβk↑)⟩ − ⟨(vkα†

k↓ − ukβ†k↓)(ukαk↓ + vkβk↓)⟩

+ ⟨(ukα†k↑ + vkβ

†k↑)(−vkαk↑ + ukβk↑)⟩ − ⟨(ukα†

k↓ + vkβ†k↓)(vkαk↓ − ukβk↓)⟩

=U∑kσ

′ukvk(−⟨α†

kσαkσ⟩+ ⟨β†kσβkσ⟩).

(A–3)

The expectation values of the number operators can be replaced by the Fermi function,

⟨γ†kσγkσ⟩ = f (E

γk ) =

1

e−Eγk /kBT + 1

=1

2− 12tanh(

E γk

2kBT) (A–4)

where γ = α, β. Plugging in the SDW coherence factors in Equation 2–14 and replacing

the expectation values of the density operators with the Fermi functions, we get the

self-consistent equation forW , and U as

W = U∑k

′ W√(εk − εk+Q)2 + 4W 2

(tanh(

Eαk

2kBT)− tanh( E

βk

2kBT)). (A–5)

81

A.2 The electron filling: derivation

The electron filling is defined by

n = 1 + x =∑k,σ

⟨c†k,σck,σ⟩. (A–6)

We reduced the sum of the momentum inside the magnetic Brillouin zone and change

the base of the operator we can get

n = 2∑k

′u2k⟨α

†k↑αk↑⟩+ v

2k ⟨α

†k↓αk↓⟩+ v

2k ⟨β

†k↑βk↑⟩+ u

2k⟨β

†k↓βk↓⟩. (A–7)

Then we replace the expectation values of the number operator with the Fermi function.

We can obtain

n = 2−∑k

′[tanh

( Eαk

2kBT

)+ tanh

( Eβk

2kBT

)]. (A–8)

82

APPENDIX BDERIVATIONS IN THE COEXISTENCE STATE OF ANTIFERROMAGNETISM AND

SUPERCONDUCTIVITY

B.1 Antiferromagnetic order parameter equation in the coexistence state withsuperconductivity: derivation

Starting with the definition and the result of the antiferromagnetic order parameter

from Appendix A.1, we have

W =U∑kσ

′ukvk(−⟨α†

kσαkσ⟩+ ⟨β†kσβkσ⟩). (B–1)

Then we change the basis to the operators in the coexistence state with the help of BCS

transformation in Equation 2–21, and obtain

W = −U∑k

′2ukvk [u

α2k ⟨γα†

k0γαk0⟩+ vα2k ⟨γα

klγα†kl ⟩ − u

β2k ⟨γβ†

k0γβk0⟩ − v

β2k ⟨γβ

klγβ†kl ⟩]. (B–2)

Now we replace the expectation values of the number operators with the Fermi function

for the quasiparticles in the coexistence state, and arrive at the following expression,

W = −U∑k

′2ukvk [u

α2k f (Ω

αk ) + v

α2k

(1− f (Ωα

k ))− uβ2k f (Ω

βk )− v

β2k

(1− f (Ωβ

k ))]. (B–3)

By evaluating both the SDW coherence factors and the BCS coherence factors, we get

the final self-consistent equation forW and U in the coexistence state as

W = U∑k

′ W

2√ε−k +W

2

[Eαk

Ωαk

tanh( Ωα

2kBT

)− E

βk

Ωβk

tanh( Ωβ

2kBT

)]. (B–4)

B.2 Filling level of electrons in the coexistence state: derivation

We start with Equation A–7, the definition of electron filling in the antiferromagnetic

state basis,

n = 2∑k

′u2k⟨α

†k↑αk↑⟩+ v

2k ⟨α

†k↓αk↓⟩+ v

2k ⟨β

†k↑βk↑⟩+ u

2k⟨β

†k↓βk↓⟩. (B–5)

83

For the derivation of the doping level in the coexistence state of superconductivity and

ferromagnetism, we use the BCS Bogoliubov transformation

αk↑ = uαk γ

αk0 + v

αk γ

α†kl ,

α†−k↓ = −vαk γα

k0 + uαk γ

α†kl

(B–6)

to change the basis of the operators from the antiferromagnetic state to the coexistence

state. We obtain

n =2∑k

′u2k⟨(uαk γ

α†k0 + v

αk γ

αkl)(u

αk γ

αk0 + v

αk γ

α†kl )⟩+ v

2k ⟨(−vαk γα

k0 + uαk γ

α†kl )(−v

αk γ

α†k0 + u

αk γ

αkl)⟩

+ v 2k ⟨(uβk γ

β†k0 + v

βk γ

βkl)(u

βk γ

βk0 + v

βk γ

β†kl )⟩+ u

2k⟨(−v

βk γ

βk0 + u

βk γ

β†kl )(−v

βk γ

β†k0 + u

βk γ

βkl)⟩.

(B–7)

With some organization, we have

n =∑k

′uα2k ⟨γα†

k0γαk0⟩+ vα2k ⟨γα

klγα†kl ⟩+ v

α2k ⟨γα

k0γα†k0 ⟩+ u

α2k ⟨γα†

kl γαkl⟩

+ uβ2k ⟨γβ†k0γ

βk0⟩+ v

β2k ⟨γβ

klγβ†kl ⟩+ v

β2k ⟨γβ

k0γβ†k0 ⟩+ u

β2k ⟨γβ†

kl γβkl⟩.

(B–8)

Here we plug in the expectation values of the number operators of the quasiparticles in

the coexistence state, ⟨γ†klγkl⟩ = f (Ω

γk). We get

n =2∑k

′uα2k f (Ω

αk ) + v

α2k [1− f (Ωα

k )] + uβ2k f (Ω

βk ) + v

β2k [1− f (Ω

βk )]

=∑k

′2− E

αk

Ωαk

tanh(Ωαk

2T)− E

βk

Ωβk

tanh(Ωβk

2T).

(B–9)

When the superconducting gaps go to zero, Ωα/βk → Eα/β

k . The equation reduces to

the result in the pure antiferromagnetic state, Equation A–8.

84

B.3 Mean field energy in the coexistence state: derivation

The Hamiltonian in the coexistence state is in the following. We separate it into

three terms,

H =H1 +H2 +H3 (B–10)

=∑kσ

ϵkc†kσckσ +

∑k,k′,σ

U

2c†kσck+Qσc

†k′+Qσck′σ +

∑k,p,q,σ

Vqc†k+qσc

†p−qσcpσckσ. (B–11)

The kinetic energy term after the sequential transformations is

⟨H1⟩ =∑kσ

ϵk⟨c†kσckσ⟩ =2∑kσ

′(εku

2k + v

2k εk+Q)(u

α2k f (Ω

αk ) + v

α2k (1− f (Ωα

k )))

+(εkv2k + u

2kεk+Q)(u

β2k f (Ω

βk ) + v

β2k (1− f (Ω

αk ))).

(B–12)

For the second part of the Hamiltonian, we use the mean-field treatment to decouple the

four operators,

H2 =∑k,k′,σ

U

2c†kσck+Qσc

†k+Qσck′σ

∼=∑k,k′,σ

U

2⟨c†kσck+Qσ⟩c

†k+Qσck′σ + c

†kσck+Qσ⟨c

†k+Qσck′σ⟩ − ⟨c†kσck+Qσ⟩⟨c

†k+Qσck′σ⟩.

(B–13)

We fold the momentum into the reduced Brillouin zone. And after the unitary transforma-

tion, we obtain

H2 = −2U∑k,k′,σ

′ukvku

′kv

′k

[(⟨α†

kσαkσ⟩ − ⟨β†kσβkσ⟩)(α

†k′σαk′σ − β†

k′σβk′σ)

+ (α†kσαkσ − β†

kσβkσ)(⟨α†k′σαk′σ⟩ − ⟨β†

k′σβk′σ⟩)

− (⟨α†kσαkσ⟩ − ⟨β†

kσβkσ⟩)(⟨α†k′σαk′σ⟩ − ⟨β†

k′σβk′σ⟩)].

(B–14)

Since ⟨α†k′↑αk′↑⟩ = ⟨α†

k′↓αk′↓⟩, andW = U∑k,σ

′−ukvk(⟨α†kσαkσ⟩−⟨β†

kσβkσ⟩), we can re-write

the Hamiltonian as

H2 = −W[∑k′,σ

′− u′kv ′k(α

†k′σαk′σ − β†

k′σβk′σ) +∑k,σ

′− ukvk(α†

kσαkσ − β†kσβkσ)

]+W 2

U

=2W∑k,σ

′[ukvk(α

†kσαkσ − β†

kσβkσ)]+W 2

U.

(B–15)

85

With the BCS transformation we obtain

H2 =2W∑k,σ

′ukvk

[uα2k γα†

k0γαk0 + v

α2k γα

klγα†kl + v

α2k γα

k0γα†k0 + u

α2k γα†

kl γαkl

− uβ2k γβ†k0γ

βk0 − v

β2k γβ

klγβ†kl − v

β2k γβ

k0γβ†k0 − u

β2k γβ†

kl γβkl

]+W 2

U.

(B–16)

The expectation value of the Hamiltonian is then

⟨H2⟩ =4W∑k,σ

′ukvk

[uα2k f (Ω

αk ) + v

α2k [1− f (Ωα

k )]− uβ2k f (Ω

βk )− v

β2k [1− f (Ω

βk ])]+W 2

U.

(B–17)

The third (superconducting) part of the total Hamiltonian is

H3 =∑k,p,q,σ

Vqc†k+qσc

†p−qσcpσckσ. (B–18)

We have the same mean-field decoupling procedure,

H3 ∼=∑k,p,q,σ

Vq

[⟨c†k+qσc

†p−qσ⟩cpσckσ + c

†k+qσc

†p−qσ⟨cpσckσ⟩ − ⟨c†k+qσc

†p−qσ⟩⟨cpσckσ⟩

]. (B–19)

86

With the unitary transformation, we have

H3 =∑k,p,σ

′Vp−k[(u

2pu2k + 2upvpukvk + v

2p v2k )

(⟨α†p↑α

†−p↓⟩α−k↑αk↓ + α†

p↑α†−p↓⟨α

†−k↑α

†k↓⟩ − ⟨α†

p↑α†−p↓⟩⟨α

†−k↑α

†k↓⟩

+ ⟨β†p↑β

†−p↓⟩β

†−k↑β

†k↓ + β†

p↑β†−p↓⟨β

†−k↑β

†k↓⟩ − ⟨β†

p↑β†−p↓⟩⟨β

†−k↑β

†k↓⟩)

+(u2pv2k − 2upvpukvk + v 2p u2k)

(⟨α†p↑α

†−p↓⟩β

†−k↑β

†k↓ + α†

p↑α†−p↓⟨β

†−k↑β

†k↓⟩ − ⟨α†

p↑α†−p↓⟩⟨β

†−k↑β

†k↓⟩

+ ⟨β†p↑β

†−p↓⟩α

†−k↑α

†k↓ + β†

p↑β†−p↓⟨α

†−k↑α

†k↓⟩ − ⟨β†

p↑β†−p↓⟩⟨α

†−k↑α

†k↓⟩)]

−Vp−k+Q[(v 2p u2k + 2upvpukvk + u2pv 2k )

(⟨α†p↑α

†−p↓⟩α

†−k↑α

†k↓ + α†

p↑α†−p↓⟨α

†−k↑α

†k↓⟩ − ⟨α†

p↑α†−p↓⟩⟨α

†−k↑α

†k↓⟩

+ ⟨β†p↑β

†−p↓⟩β

†−k↑β

†k↓ + β†

p↑β†−p↓⟨β

†−k↑β

†k↓⟩ − ⟨β†

p↑β†−p↓⟩⟨β

†−k↑β

†k↓⟩)

+(v 2p v2k − 2upvpukvk + u2pu2k)

(⟨α†p↑α

†−p↓⟩β

†−k↑β

†k↓ + α†

p↑α†−p↓⟨β

†−k↑β

†k↓⟩ − ⟨α†

p↑α†−p↓⟩⟨β

†−k↑β

†k↓⟩

+ ⟨β†p↑β

†−p↓⟩α

†−k↑α

†k↓ + β†

p↑β†−p↓⟨α

†−k↑α

†k↓⟩ − ⟨β†

p↑β†−p↓⟩⟨α

†−k↑α

†k↓⟩)].

(B–20)

We can see that the terms which are generated by the folding of the Brillouin zone

overall have a different sign from the terms inside the reduced Brillouin zone. So we can

combine these two parts as

H3 =∑k,p,σ

′[Vp−k(u

2pu2k + 2upvpukvk + v

2p v2k )− Vp−k+Q(v 2p u2k + 2upvpukvk + u2pv 2k )]

(⟨α†pσα

†−pσ⟩α−kσαkσ + α†

pσα†−pσ⟨α−kσαkσ⟩ − ⟨α†

pσα†−pσ⟩⟨α−kσαkσ⟩

+ ⟨β†pσβ

†−pσ⟩β−kσβkσ + β†

pσβ†−pσ⟨β−kσβkσ⟩ − ⟨β†

pσβ†−pσ⟩⟨β−kσβkσ⟩)

+[Vp−k(u2pv2k − 2upvpukvk + v 2p u2k)− Vp−k+Q(u2pu2k − 2upvpukvk + v 2p v 2k )]

(⟨α†pσα

†−pσ⟩β−kσβkσ + α†

pσα†−pσ⟨β−kσβkσ⟩ − ⟨α†

pσα†−pσ⟩⟨β−kσβkσ⟩

+ ⟨β†pσβ

†−pσ⟩α−kσαkσ + β†

pσβ†−pσ⟨α−kσαkσ⟩ − ⟨β†

pσβ†−pσ⟩⟨α−kσαkσ⟩).

(B–21)

87

By using mean field theory and the Bogoliubov transformation, we have the Cooper pair

expectation values,

⟨γ†p↑γ

†−p↓⟩ = −⟨γp↑γ−p↓⟩ =

∆γp

2Ωγp

tanh(Ωγ

p

2T

)(B–22)

where γ = α, β. The mean field energy from the third term of the Hamiltonian produces

the following result,

⟨H3⟩ =∑k,p

′− 2[Vp−k(u2pu2k + 2upvpukvk + v 2p v 2k )− Vp−k+Q(v 2p u2k + 2upvpukvk + u2pv 2k )]

[ ∆αp

2Ωαp

tanh(Ωα

p

2T

) ∆αk

2Ωαk

tanh(Ωα

k

2T

)+∆βp

2Ωβp

tanh(Ωβ

p

2T

) ∆βk

2Ωβk

tanh(Ωβ

k

2T

)]+[Vp−k(u

2pv2k − 2upvpukvk + v 2p u2k)− Vp−k+Q(u2pu2k − 2upvpukvk + v 2p v 2k )][ ∆α

p

2Ωαp

tanh(Ωα

p

2T

) ∆βk

2Ωβk

tanh(Ωβ

k

2T

)+∆βp

2Ωβp

tanh(Ωβ

p

2T

) ∆αk

2Ωαk

tanh(Ωα

k

2T

)].

(B–23)

Now the gap equation in Equation 2–25 can replace part of this expression,

reducing the final result for the third term to the following form,

⟨H3⟩ = −2∑k

′∆αk

∆αk

2Ωαk

tanh(Ωα

k

2T

)+ ∆β

k

∆βk

2Ωβk

tanh(Ωβ

k

2T

). (B–24)

Combining the kinetic term and the antiferromagnetic term, we get

⟨H1⟩+ ⟨H2⟩ =2∑kσ

′(εku

2k + v

2k εk+Q)(u

α2k f (Ω

αk ) + v

α2k (1− f (Ωα

k )))

+(εkv2k + u

2kεk+Q)(u

β2k f (Ω

βk ) + v

β2k (1− f (Ω

αk )))

+2W 2√

ε−2k +W2

[uα2k f (Ωαk ) + v

α2k (1− f (Ωα

k ))− uβ2k f (Ω

βk )− v

β2k (1− f (Ω

αk ))]

+W 2

U.

(B–25)

88

The total mean field energy is therefore

EMF =⟨H⟩ = ⟨H1⟩+ ⟨H2⟩+ ⟨H3⟩

=2∑kσ

′Eαk (u

α2k f (Ω

αk ) + v

α2k (1− f (Ωα

k ))) + Eβk (u

β2k f (Ω

βk ) + v

β2k (1− f (Ω

αk )))

−2∑k

′ ∆α2k

2Ωαk

tanh(Ωα

k

2T

)+∆β2k

2Ωβk

tanh(Ωβ

k

2T

)+ 2W 2

U.

(B–26)

The final result for the mean field energy is

EMF =∑kσ

′Eαk −Ωα

k + Eβk −Ωβ

k +∆α2k

2Ωαk

tanh(Ωα

k

2T

)+∆β2k

2Ωβk

tanh(Ωβ

k

2T

)+2Ωα

k f (Ωαk ) + 2Ω

βk f (Ω

βk ) +

W 2

U.

(B–27)

We can make a simple check of the final result with the limit In the limit of pure

antiferromagnetic state,

⟨HM⟩ =∑k

′2Eαk f (E

αk ) + 2E

βkf (Eβk ) +

W 2

U. (B–28)

The limit in the pure SC state,W = 0, reduces to the expectation value given in Eq.

(3.45) in Tinkham[89] which is the energy for the superconducting state,

HM =∑k

(εk − Ek +∆kbk) +∑k

Ek(γ†k0γk0 + γ†

k1γk1). (B–29)

In terms of expectation value, it would be

⟨HM⟩ =∑k

(εk − Ek +∆2k2Ektanh(

Ek2T)) +

∑k

2Ekf (Ek). (B–30)

89

APPENDIX CDERIVATTIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE PURE

ANTIFERROMAGNETIC STATE

C.1 Transverse dynamic spin susceptibility in the antiferromagnetic state:derivation

The definition of the transverse dynamic spin susceptibility is

χ+−0 (q,q′, iω) =

∫ β

0

dτ⟨TτS+q (τ)S

−−q′(0)⟩e

iωτ . (C–1)

In order to calculate the transverse part of the susceptibility, we need to use the defini-

tion of the transverse spin operators:

S+q (τ) =∑k

c†k+q↑(τ)ck↓(τ)

S−q (τ) =

∑k

c†k+q↓(τ)ck↑(τ).

(C–2)

Now the susceptibility is written in terms of raising and lowering operators as

χ+−0 (q,q′, iω) =

∫ β

0

dτ∑k,k′

⟨Tτc†k+q↑(τ)ck↓(τ)c

†k′−q′↓(0)ck′↑(0)⟩e

iωτ . (C–3)

We apply Wick theorem to decouple the four-operator, and get

χ+−0 (q,q′, iω) =

∫ β

0

dτe iωτ∑k,k′

[⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q′↓(0)⟩

+ ⟨Tτc†k+q↑(τ)c

†k′−q′↓(0)⟩⟨Tτck↑(τ)ck′↓(0)⟩].

(C–4)

Here we see two different terms for the transverse susceptibility after applying Wick

theorem. One involves the expectation values of the normal state electron operators and

the other involves the expectation value of the superconducting Cooper pair operators.

For the calculations in the pure antiferromagnetic state with no superconductivity, the

expectation values of Cooper pair operators would be zero. Therefore we can ignore the

latter for the non-superconducting calculation. We only have to evaluate:

χ+−0 (q,q′, iω) =

∫ β

0

dτe iωτ∑k,k′

⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q′↓(0)⟩. (C–5)

90

Transforming k and k′ to the reduced Brillouin zone, we have

χ+−0 (q,q′, iω) =

∫ β

0

dτe iωτ∑k,k′

′[⟨Tτc

†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q′↓(0)⟩

+ ⟨Tτc†k+q↑(τ)ck′+Q↑(0)⟩⟨Tτck↓(τ)c

†k′−q′+Q↓(0)⟩

+ ⟨Tτc†k+q+Q↑(τ)ck′↑(0)⟩⟨Tτck+Q↓(τ)c

†k′−q′↓(0)⟩

+ ⟨Tτc†k+q+Q↑(τ)ck′+Q↑(0)⟩⟨Tτck+Q↓(τ)c

†k′−q′+Q↓(0)⟩.

(C–6)

The transverse susceptibility has a diagonal term χ+−0 (q,q, iω) and the Umklapp

term χ+−0 (q,q + Q, iω) due to the breaking of translational symmetry. Here we first

calculate the diagonal parts of the transverse susceptibility and set q′ = q.

We apply the unitary transformation of Equation 2–11 to get the susceptibility in

the antiferromagnetic state. When applying the transformation, we have to consider

both case, k + q inside and outside of the reduced Brillouin zone. Due to the symmetry

of the the algebra, the two cases have the same SDW coefficients for the transverse

susceptibility. We can just assume one case that k + q is inside the reduced zone and

k+ q+Q is outside of the reduced zone and sum over k′. The susceptibility reduces to:

χ+−0 (q,q, iω) =

∫ β

0

dτe iωτ∑k

′[(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )

×⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩

+ ⟨Tτβ†k+q↑(τ)βk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩

+(u2k+qv2k + 2uk+qvk+qukvk + v

2k+qu

2k)

×⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩

+ ⟨Tτβ†k+q↑(τ)βk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩].

(C–7)

Evaluating the Fourier transform of the Matsubara Green’s function, we get

91

∫ β

0

dτe iωmτ ⟨γ†k+qσ(τ)γk+qσ(0)⟩⟨γ

′†kσ′(0)γ′kσ′(τ)⟩ =

f (E γk+q)− f (E

γ′

k )

iω − E γ′

k + Eγk+q

(C–8a)

∫ β

0

dτe iωmτ ⟨γ†k+qσ(τ)γk+qσ(0)⟩⟨γ

′kσ′(0)γ

′†kσ′(τ)⟩ =

1− f (E γ′

k )− f (Eγk+q)

iω + E γ′

k + Eγk+q

(C–8b)

∫ β

0

dτe iωmτ ⟨γk+qσ(τ)γ†k+qσ(0)⟩⟨γ

′†kσ′(0)γ

′kσ′(τ)⟩) =

f (E γ′

k ) + f (Eγk+q)− 1

iω − E γ′

k − E γk+q

(C–8c)

∫ β

0

dτe iωmτ ⟨γk+qσ(τ)γ†k+qσ(0)⟩⟨γ

′kσ′(0)γ

′†kσ′(τ)⟩) =

f (E γ′

k )− f (Eγk+q)

iω + E γ′ − E γk+q

(C–8d)

with f (E) being the Fermi function and Ek the quasiparticle energy. The final result for

the transverse dynamic spin susceptibility in the antiferromagnetic state is:

χ+−0 (q,q, iω) = −∑k,γ=γ′

′(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )

f (E γk+q)− f (E

γ′

k )

iω − E γ′

k + Eγk+q

−∑k,γ =γ′

′(u2k+qv

2k + 2uk+qvk+qukvk + v

2k+qu

2k)f (E γ

k+q)− f (Eγ′

k )

iω − E γ′

k + Eγk+q

(C–9)

where γ, γ′ are both α and β. If we plug in the values for the coherence factors, we get

the final result for the transverse off-diagonal dynamic spin susceptibility

χ+−0 (q,q, iω) = −12

∑k,γ=γ′

′

1 + ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γk )

iω − E γk+q + E

γk

−12

∑k,γ =γ′

′

1− ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γ′

k+q)− f (Eγk )

iω − E γ′

k+q + Eγk

.

(C–10)

C.2 Umklapp term for the transverse dynamic spin susceptibility

Without loss of generality, we can start with Equation C–6. For the Umklapp (off-

diagonal) term, we evaluate the spin susceptibility with q′ = q+Q and we have

92

χ+−Q (q, iω) =χ+−0 (q,q+Q, iω)

=

∫ β

0

dτe iωτ∑k,k′

′[⟨Tτc

†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q+Q↓(0)⟩

+ ⟨Tτc†k+q↑(τ)ck′+Q↑(0)⟩⟨Tτck↓(τ)c

†k′−q↓(0)⟩

+ ⟨Tτc†k+q+Q↑(τ)ck′↑(0)⟩⟨Tτck+Q↓(τ)c

†k′−q+Q↓(0)⟩

+ ⟨Tτc†k+q+Q↑(τ)ck′+Q↑(0)⟩⟨Tτck+Q↓(τ)c

†k′−q↓(0)⟩.

(C–11)

Here the same as in the calculation of χ+−(q,q′, iω), we use Equation 2–11 for the

unitary transformation, and obtain

χ+−Q (q, iω) =

∫ β

0

dτe iωτ∑k,k ′∈R

[⟨Tτ(uk+qα

†k+q↑(τ) + vk+qβ

†k+q↑(τ))(uk ′αk ′↑(0) + vk ′βk ′↑(0))⟩

⟨Tτ(ukαk↓(τ) + vkβk↓(τ))(vk ′−qα†k ′−q↓(0)− uk ′−qβ

†k ′−q↓(0))⟩

+ ⟨Tτ(−vk+qα†k+q↑(τ) + uk+qβ

†k+q↑(τ))(uk ′αk ′↑(0) + vk ′βk ′↑(0))⟩

⟨Tτ(vkαk↓(τ)− ukβk↓(τ))(vk ′−qα†k ′−q↓(0)− uk ′−qβ

†k ′−q↓(0))⟩

+ ⟨Tτ(uk+qα†k+q↑(τ) + vk+qβ

†k+q↑(τ))(uk ′αk ′↑(0) + vk ′βk ′↑(0))⟩

⟨Tτ(ukαk↓(τ) + vkβk↓(τ))(uk ′−qα†k ′−q↓(0) + vk ′−qβ

†k ′−q↓(0))⟩

+⟨Tτ(uk+qα†k+q↑(τ) + vk+qβ

†k+q↑(τ))(vk ′αk ′↑(0)− uk ′βk ′↑(0))⟩

⟨Tτ(vkαk↓(τ)− ukβk↓(τ))(vk ′−qα†k ′−q↓(0)− uk ′−qβ

†k ′−q↓(0))⟩.

(C–12)

93

After organizing, we have the transverse Umklapp spin susceptibility reducing to

χ+−Q (q, iω) =

∫ β

0

dτe iωτ∑k,σ

[(ukvk − uk+qvk+q)

[⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)βk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩]

− (uk+qvk+q + ukvk)

[⟨Tτα†k+q↑(τ)αk+q↓(0)⟩⟨Tτβk↑(τ)β

†k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)βk+q↓(0)⟩⟨Tταk↑(τ)α

†k↓(0)⟩].

(C–13)

With the help of the Green’s function Fourier transformations in Equation D–7 and

plugging the expressions for the coherence factors, we get the final result for the

Umklapp term of the transvere spin susceptibility,

χ+−0 (q,q+Q, iω) = χ+−Q (q, iω) =

W

2

∑k

′

1√(ε−k+q

)2+W 2

− 1√(ε−k)2+W 2

( f (Eαk+q)− f (Eα

k )

iω − Eαk+q + E

αk

−f (Eβ

k+q)− f (Eβk )

iω − Eβk+q + E

βk

)

−

1√(ε−k+q

)2+W 2

+1√(

ε−k)2+W 2

( f (Eβk+q)− f (Eα

k )

iω − Eβk+q + E

αk

−f (Eα

k+q)− f (Eβk )

iω − Eαk+q + E

βk

).

(C–14)

C.3 The longitudinal dynamic spin susceptibility

The definition of the longitudinal spin susceptibility is

χzz0 (q,q′,ω) =

∫dt

[i

2N⟨TSzq (t)Sz−q(0)⟩

]e iωt (C–15)

with the spin operator

Szq (τ) =∑kσ

σc†k+qσ(τ)ckσ(τ). (C–16)

In terms of the raising and lowering operators, the susceptibility is written as

χzz0 (q,q′,ω) =

∫ β

0

dτe iωτ∑kk′σσ′

σσ′⟨Tτ(c†k+qσ(τ)ckσ(τ)c

†k′−q′σ′(0)ck′σ′(0)⟩. (C–17)

94

Writing out the spin indices and setting q′ = q for the non-Umklapp spin susceptibility,

we get:

χzz0 (q,q,ω) =

∫ β

0

dτe iωτ∑kk′

⟨Tτ(c†k+q↑(τ)ck↑(τ)c

†k′−q↑(0)ck′↑(0)

− c†k+q↑(τ)ck↑(τ)c†k′−q↓(0)ck′↓(0)− c

†k+q↓(τ)ck↓(τ)c

†k′−q↑(0)ck′↑(0)

+ c†k+q↓(τ)ck↓(τ)c†k′−q↓(0)ck′↓(0))⟩.

(C–18)

By using Wick theorem to separate the four operators and ignoring the terms which have

expectation values zero, we have

χzz0 (q,q,ω) =

∫ β

0

dτe iωτ∑kk′

⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↑(τ)c

†k′−q↑(0)⟩

+ ⟨Tτc†k+q↓(τ)ck′↓(0)⟩⟨Tτck↓(τ)c

†k′−q↓(0)⟩.

(C–19)

Reducing the sum over k to the reduced Brillouin zone and applying the unitary transfor-

mation, we get

χzz0 (q,q,ω) =

∫ β

0

dτe iωτ∑kσ

′(u2ku

2k+q + 2ukvkuk+qvk+q + v

2k v2k+q)

×(⟨Tτ(α†kσ(τ)αkσ(0)⟩⟨Tταk+qσ(τ)α

†k+qσ(0)⟩+ ⟨Tτβ

†kσ(τ)βkσ(0)⟩⟨Tτβk+qσ(τ)β

†k+qσ(0)⟩)

+(u2kv2k+q − 2ukvkuk+qvk+q + v 2k u2k+q)

×(⟨Tτα†kσ(τ)αkσ(0)⟩⟨Tτβk+qσ(τ)β

†k+qσ(0)⟩+ ⟨Tτβ

†kσ(τ)βkσ(0)⟩⟨Tταk+qσ(τ)α

†k+qσ(0)⟩).

(C–20)

Now plugging the expectation values for the operators, we obtain the final result as

95

χzz0 (q,q,ω) = −12

∑k,γ=γ′

′

1 + ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γk )

iω − E γk+q + E

γk

−12

∑k,γ =γ′

′

1− ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

f (E γk+q)− f (E

γ′

k )

iω − E γk+q + E

γ′

k

. (C–21)

C.4 The longitudinal Umklapp susceptibility

The Umklapp term for the longitudinal dynamic spin susceptibility is zero due to the

fact that there is no symmetry breaking along the x − y plane when the system goes into

antiferromagnetic state. This fact can also be calculated algebraically.

Starting with the definition of the longitudinal dynamic spin susceptibility with

q′ = q+Q in Equation C–17, we have

χzz0 (q,q+Q, iω) =

∫ β

0

dτe iωτ∑kk′σσ′

σσ′⟨Tτ(c†k+qσ(τ)ckσ(τ)c

†k′−q+Qσ′(0)ck′σ′(0)⟩. (C–22)

Using the Wick theorem, we get

χzz0 (q,q+Q, iω) =

∫ β

0

dτe iωτ∑kk′σ

⟨Tτc†k+qσ(τ)ck′σ(0)⟩⟨Tτckσ(τ)c

†k′−q+Qσ′(0)⟩. (C–23)

Transferring k and k′into the reduced Brillouin zone, we have

χzz0 (q,q+Q, iω) =

∫ β

0

dτe iωτ∑kσ

′⟨Tτc

†k+qσ(τ)ck′σ(0)⟩⟨Tτckσ(τ)c

†k′−q+Qσ(0)⟩

+ ⟨Tτc†k+q+Qσ(τ)ck′σ(0)⟩⟨Tτck+Qσ(τ)c

†k′−q+Qσ(0)⟩

+ ⟨Tτc†k+qσ(τ)ck′+Qσ(0)⟩⟨Tτckσ(τ)c

†k′−qσ(0)⟩

+ ⟨Tτc†k+q+Qσ(τ)ck′+Qσ(0)⟩⟨Tτck+Qσ(τ)c

†k′−qσ(0)⟩.

(C–24)

96

Using the unitary transformation, we have

χzz0 (q,q+Q,ω)

=

∫ β

0

dτe iωτ∑kσ

′[⟨Tτ(uk+qα

†k+qσ(τ) + vk+qβ

†k+qσ(τ))(uk+qαk+qσ(0) + vk+qβk+qσ(0))⟩

× ⟨Tτ(ukα†kσ(τ) + vkβ

†kσ(τ))(sgn(σ))(−vkαkσ(0) + ukβkσ(0))⟩

+ ⟨Tτ(sgn(σ))(−vk+qα†k+qσ(τ) + uk+qβ

†k+qσ(τ))(uk+qαk+qσ(0) + vk+qβk+qσ(0))⟩

× (sgn(σ))⟨Tτ(−vkα†kσ(τ) + ukβ

†kσ(τ))(sgn(σ))(−vkαkσ(0) + ukβkσ(0))⟩

+ ⟨Tτ(uk+qα†k+qσ(τ) + vk+qβ

†k+qσ(τ))(sgn(σ))(−vk+qαk+qσ(0) + uk+qβk+qσ(0))⟩

× ⟨Tτ(ukα†kσ(τ) + vkβ

†kσ(τ))(ukαkσ(0) + vkβkσ(0))⟩

+ ⟨Tτ(−vk+qα†k+qσ(τ) + uk+qβ

†k+qσ(τ))(−vk+qαk+qσ(0) + uk+qβk+qσ(0))⟩

× ⟨Tτ(sgn(σ))(−vkα†kσ(τ) + ukβ

†kσ(τ))(ukαkσ(0) + vkβkσ(0))⟩.

(C–25)

After some organization, we get:

χzz0 (q,q+Q, iω) =

∫ β

0

dτe iωτ∑kσ

′− sgn(σ)(uk+qvk+q + ukvk)[

⟨Tτ(α†k+qσ(τ)αk+qσ(0)⟩⟨Tταkσ(τ)α

†kσ(0)⟩ − ⟨Tτβ

†k+qσ(τ)βk+qσ(0)⟩⟨Tτβkσ(τ)β

†kσ(0)⟩

]−sgn(σ)(uk+qvk+q − ukvk)[

⟨Tτα†k+qσ(τ)αk+qσ(0)⟩⟨Tτβkσ(τ)β

†kσ(0)⟩ − ⟨Tτβ

†k+qσ(τ)βk+qσ(0)⟩⟨Tταkσ(τ)α

†kσ(0)⟩

].

(C–26)

The Green’s function value does not depend on spins. But the coherence factors result

in different signs for opposite spins. Therefore the evaluation of the Umklapp term of the

longitudinal dynamic spin susceptibility is zero.

97

C.5 Analytic proof for the formation of the Goldstone mode

The Goldstone mode only appears in the transverse part of the spin susceptibility,

and occurs when the real part of the susceptibility satisfies the condition 1 − Uχ+−0

at k equals to the ordering momentum Q = (π, π) and ω = 0. This is when the RPA

susceptibility has a pole in the denominator.

Here we know that the self-consistent equation for U andW , Equation 2–16 is

always satisfied. With q = Q and ω = 0, the bare transverse Pauli susceptibility reduces

to

χ+−0 (Q, 0) = 2∑k

′ f (Eαk )− f (E

βk )

Eαk − Eβ

k

. (C–27)

Here we have used the relation E γk+Q = E

γk . For the Umklapp transverse susceptibility

equation (14), the periodic condition (ε−k+Q)2 = (ε−k )

2 makes the coherence factor

always to be zero, therefore the Umklapp term is zero and we only have to consider

the diagonal part of the transverse susceptibility matrix in Equation 3–4. Comparing

Equation C–27 with the self-consistent equation for U andW , Equation 2–16, We have

simply

χ+−0 (Q, 0) =1

U. (C–28)

This guarantees that the Goldstone mode always happens at q = Q.

98

APPENDIX DDERIVATIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE COEXISTENCE STATE

OF ANTIFERROMAGNETIC AND SUPERCONDUCTIVITY

D.1 Derivations of transverse dynamic spin susceptibility in the coexistencestate of antiferromagnetism and superconductivity

We can start the derivation from the intermediate steps of the susceptibility in the

pure antiferromagnetic state. For the non-Umklapp (diagonal) term of the transverse

dynamic spin susceptibility, we start with Equation C–4 which is

χ+−0 (q,q′, iω) =

∫ β

0

dτe iωτ∑k,k′

[⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q′↓(0)⟩

+ ⟨Tτc†k+q↑(τ)c

†k′−q′↓(0)⟩⟨Tτck↑(τ)ck′↓(0)⟩].

(D–1)

Note that in the superconducting state, we have to include the Cooper pairing operators

which were omitted in the case of non-superconducting state. We define that

χ+−0 (q,q′, iω) =χ+−nor (q,q

′, iω) + χ+−SC (q,q′, iω) (D–2)

with

χ+−nor (q,q′, iω) =

∫ β

0

dτe iωτ∑k,k′

⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q′↓(0)⟩ (D–3)

and

χ+−SC (q,q′, iω) =

∫ β

0

dτe iωnτ∑k,k′

⟨Tτc†k+q↑(τ)c

†k′−q′↓(0)⟩⟨Tτck↑(τ)ck′↓(0)⟩. (D–4)

For the normal state part of the transverse spin susceptibility χ+−nor (q,q,ω), we

already have the result from the pure antiferromagnetic state calculation which is

99

Equation C–7,

χ+−nor (q,q, iω) =

∫ β

0

dτe iωτ∑k

′(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )[

⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩+ ⟨Tτβ

†k+q↑(τ)βk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩

]+(u2k+qv

2k + 2uk+qvk+qukvk + v

2k+qu

2k)[

⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩+ ⟨Tτβ

†k+q↑(τ)βk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩

].

(D–5)

In the superconducting state, we have to continue with the BCS transformation in

Equation 2–21. The quasiparticle operators in the pure antiferromagnetic state, αk

and βk will be replaced by γαk0, γ

αkl and γβ

k0, γβkl . But the SDW coherence factors remain

unchanged. We can just evaluate the Matsubara Fourier transformation of Green’s func-

tion,∫ β

0dτe iωτ ⟨Tτα

†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩. Later we replace the integration of

the expectation value with its general expression.

To get the general expression of the expectation value, we perform the BCS

transformation and get∫ β

0

dτe iωτ ⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩

=

∫ β

0

dτe iωmτ[uα2k+qv

α2k ⟨γ†

k+q0(τ)γk+q0(0)⟩α⟨γ†k0(τ)γk0(0)⟩α

+ uα2k+quα2k ⟨γ†

k+q0(τ)γk+q0(0)⟩α⟨γkl(τ)γ†kl(0)⟩α

+ vα2k+qvα2k ⟨γk+ql(τ)γ†

k+ql(0)⟩α⟨γ†k0(τ)γk0(0)⟩α

+ vα2k+quα2k ⟨γk+ql(τ)γ†

k+ql(0)⟩α⟨γkl(τ)γ†kl(0)⟩α

](D–6)

where ⟨γγ⟩α = ⟨γαγα⟩.

100

Next we use the the following relations for the Matsubara integration of the Green’s

function,∫ β

0

dτe iωτ ⟨γ†k+q0(τ)γk+q0(0)⟩⟨γ

′†k0(0)γ

′k0(τ)⟩ =

f (Ωγk+q)− f (Ω

γ′

k )

iω −Ωγ′

k +Ωγk+q

(D–7a)

∫ β

0

dτe iωτ ⟨γ†k+q0(τ)γk+q0(0)⟩⟨γ

′kl(0)γ

′†kl (τ)⟩ =

1− f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k +Ωγk+q

(D–7b)

∫ β

0

dτe iωτ ⟨γk+ql(τ)γ†k+ql(0)⟩⟨γ

′†k0(0)γ

′k0(τ)⟩) =

f (Ωγ′

k ) + f (Ωγk+q)− 1

iω −Ωγ′

k −Ωγk+q

(D–7c)

∫ β

0

dτe iωτ ⟨γk+ql(τ)γ†k+ql(0)⟩⟨γ

′kl(0)γ

′†kl (τ)⟩) =

f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k −Ωγk+q

(D–7d)

where γ and γ′ are α and β. And we get∫ β

0

dτe iωτ ⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩

=uα2k+qvα2k

f (Ωαk ) + f (Ω

αk+q)− 1

iω +Ωαk +Ω

αk+q

+ uα2k+quα2k

f (Ωαk )− f (Ωα

k+q)

iω −Ωαk +Ω

αk+q

+vα2k+qvα2k

f (Ωαk+q)− f (Ωα

k )

iω +Ωαk −Ωα

k+q

+ vα2k+quα2k

1− f (Ωαk )− f (Ωα

k+q)

iω −Ωαk −Ωα

k+q

.

(D–8)

For the same calculation for β, we just replace all αs in the above equation with β.

Following the same calculations we can also get the interband terms which are∫ β

0

dτe iωτ ⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩

=uα2k+qvβ2k

f (Ωβk ) + f (Ω

αk+q)− 1

iω +Ωβk +Ω

αk+q

+ uα2k+quβ2k

f (Ωβk )− f (Ωα

k+q)

iω −Ωβk +Ω

αk+q

+vα2k+qvβ2k

f (Ωαk+q)− f (Ω

βk )

iω +Ωβk −Ωα

k+q

+ vα2k+quβ2k

1− f (Ωβk )− f (Ωα

k+q)

iω −Ωβk −Ωα

k+q

.

(D–9)

101

The other interband term can be obtained by interchange α and β in the above equation.

Then we get the expression for the normal part of the transverse spin susceptibility:

χ+−nor (q,q, iω) =∑kγ

′(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )

×[uγ2k+qv

γ2k

f (Ωγk) + f (Ω

γk+q)− 1

iω +Ωγk +Ω

γk+q

+ uγ2k+quγ2k

f (Ωγk)− f (Ω

γk+q)

iω −Ωγk +Ω

γk+q

+v γ2k+qvγ2k

f (Ωγk+q)− f (Ω

γk)

iω +Ωγk −Ω

γk+q

+ v γ2k+quγ2k

1− f (Ωγk)− f (Ω

γk+q)

iω −Ωγk −Ω

γk+q

]+∑kγ =γ′

′(u2k+qv

2k + 2uk+qvk+qukvk + v

2k+qu

2k)

×[uγ2k+qv

γ′2k

f (Ωγ′

k ) + f (Ωγk+q)− 1

iω +Ωγ′

k +Ωγk+q

+ uγ2k+quγ′2k

f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k +Ωγk+q

+v γ2k+qvγ′2k

f (Ωγk+q)− f (Ω

γ′

k )

iω +Ωγ′

k −Ωγk+q

+ v γ2k+quγ′2k

1− f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k −Ωγk+q

].

(D–10)

Next we evaluate the other part of the transvase susceptibility that has Cooper pair

operators. After applying the unitary transformation to Equation D–4, we get

χ+−SC (q, q, iω) =1

V

∫ β

0

dτe iωτ∑k,σ

′[(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )

⟨Tτα†k+q↑(τ)α

†−k−q↓(0)⟩⟨Tταk↑(τ)α−k↓(0)⟩+ ⟨Tτβ

†k+q↑(τ)β

†−k−q↓(0)⟩⟨Tτβk↑(τ)β−k↓(0)⟩

+ (u2k+qv2k + 2uk+qvk+qukvk + v

2k+qu

2k)

⟨Tτα†k+q↑(τ)α

†−k−q↓(0)⟩⟨Tτβk↑(τ)β−k↓(0)⟩+ ⟨Tτβ

†k+q↑(τ)β

†−k−q↓(0)⟩⟨Tταk↑(τ)α−k↓(0)⟩].

(D–11)

We can see that this term has the same SDW coherence factors as χ+−nor (q,q, iω). The

same as for the normal part of the transverse susceptibility, we perform BCS Bogoliubov

transformation to get new quasiparticle operators in the coexistence state. For each

102

four-operator term we have,∫ β

0

dτe iωτ ⟨α†k+qσ(τ)α

†−k−qσ(0)⟩⟨α−kσ(0)αkσ(τ)⟩

=

∫ β

0

dτe iωτ uαk+qvαk+qu

αk v

αk (⟨γ

†k+q0(τ)γk+q0(0)⟩α⟨γ

†k0(0)γk0(τ)⟩α

− ⟨γ†k+q0(τ)γk+q0(0)⟩α⟨γkl(0)γ

†kl(τ)⟩α − ⟨γk+ql(τ)γ†

k+ql(0)⟩α⟨γ†k0(0)γk0(τ)⟩α)

+ ⟨γk+ql(τ)γ†k+ql(0)⟩α⟨γkl(0)γ

†kl(τ)⟩α).

(D–12)

We evaluate the Matsubara integration of the quasiparticle operators of the coexistence

state. Each individual terms follows the relations in Equations D–7. We get∫ β

0

dτe iωτ ⟨α†k+qσ(τ)α

†−k−qσ(0)⟩⟨α−kσ(0)αkσ(τ)⟩

=uαk+qvαk+qu

αk v

αk

[ f (Ωαk+q)− f (Ωα

k )

iω −Ωαk +Ω

αk+q

−1− f (Ωα

k )− f (Ωαk+q)

iω +Ωαk +Ω

αk+q

−f (Ωα

k ) + f (Ωαk+q)− 1

iω −Ωαk −Ωα

k+q

+f (Ωα

k )− f (Ωαk+q)

iω +Ωαk −Ωα

k+q

].

(D–13)

We can get the β intraband values by replacing all the αs in the above equation with βs.

For the interband values, we have∫ β

0

dτe iωτ ⟨α†k+qσ(τ)α

†−k−qσ(0)⟩⟨β−kσ(0)βkσ(τ)⟩

=uαk+qvαk+qu

βk v

βk

[ f (Ωαk+q)− f (Ω

βk )

iω −Ωβk +Ω

αk+q

−1− f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk +Ω

αk+q

−f (Ωβ

k ) + f (Ωαk+q)− 1

iω −Ωβk −Ωα

k+q

+f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk −Ωα

k+q

].

(D–14)

103

The result for the SC part of the transverse susceptibility is

χ+−SC (q, q, iω) =∑k,γ

′(u2k+qu

2k − 2uk+qvk+qukvk + v 2k+qv 2k )

uγk+qvγk+qu

γk v

γk

[ f (Ωγk+q)− f (Ω

γk)

iω −Ωγk +Ω

γk+q

−1− f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk +Ω

γk+q

−f (Ωγ

k) + f (Ωγk+q)− 1

iω −Ωγk −Ω

γk+q

+f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk −Ω

γk+q

]+∑k,γ =γ′

′[(u2k+qv

2k + 2uk+qvk+qukvk + v

2k+qu

2k)

uαk+qvαk+qu

βk v

βk

[ f (Ωαk+q)− f (Ω

βk )

iω −Ωβk +Ω

αk+q

−1− f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk +Ω

αk+q

−f (Ωβ

k ) + f (Ωαk+q)− 1

iω −Ωβk −Ωα

k+q

+f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk −Ωα

k+q

].

(D–15)

We clearly see that the coherence factor ukvk is proportional to the superconducting gap.

In the non-superconducting limit, this term would disappear. And we would get the result

the same as in the pure antiferromagnetic state. Combining the two terms, we get the

final result for the non-Umklapp transverse spin susceptibility as

χ+−0 (q,q,ω) =∑k,γ

′1

4

1 + ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +E γk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

]f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q +Ω

γk

+1

2

[1−E γk E

γk+q +∆

γk∆

γk+q

ΩγkΩ

γk+q

](f (Ωγ

k+q) + f (Ωγk)− 1

ω + iδ +Ωγk+q +Ω

γk

+1− f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q −Ω

γk

)

+∑k,γ =γ′

′1

4

1− ε−k ε−k+q −W 2√(

ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +E γk E

γ′

k+q +∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

]f (Ωγ′

k+q)− f (Ωγk)

ω + iδ +Ωγ′

k+q −Ωγk

+1

2

[1−E γk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

](f (Ωγ′

k+q) + f (Ωγk)− 1

ω + iδ +Ωγ′

k+q +Ωγk

+1− f (Ωγ′

k+q)− f (Ωγk)

ω + iδ −Ωγ′

k+q −Ωγk

).

(D–16)

104

For the purpose of coding, we further simplify the expression by shifting k and

change ω → −ω in some terms. We get the susceptibility to be the sum of the following

six terms:

χ+−1 (q,q,ω) = −∑k

1

4

(1 +

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1 +Eαk+qE

αk + ∆

αk+q∆

αk

ΩαkΩ

αk+q

)f (Ωα

k+q)− f (Ωαk )

iω −Ωαk +Ω

αk+q

,

(D–17)

χ+−2 (q,q,ω) = −∑k

1

8

(1 +

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1−Eαk+qE

αk + ∆

αk+q∆

αk

ΩαkΩ

αk+q

)( f (Ωα

k+q) + f (Ωαk )− 1

iω +Ωαk +Ω

αk+q

+1− f (Ωα

k+q)− f (Ωαk )

iω −Ωαk −Ωα

k+q

),

(D–18)

χ+−3 (q,q,ω) = −∑k

1

4

(1−

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1 +Eαk+qE

βk +∆

αk+q∆

βk

ΩβkΩ

αk+q

)( f (Ωα

k+q)− f (Ωβk )

iω −Ωβk +Ω

αk+q

+f (Ωβ

k )− f (Ωαk+q)

iω −Ωαk+q +Ω

βk

),

(D–19)

χ+−4 (q,q,ω) =−∑k

1

4

(1−

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1−Eαk+qE

βk + ∆

αk+q∆

βk

ΩβkΩ

αk+q

)( f (Ωα

k+q) + f (Ωβk )− 1

iω +Ωβk +Ω

αk+q

+1− f (Ωα

k+q)− f (Ωβk )

iω −Ωβk −Ωα

k+q

),

(D–20)

χ+−5 (q,q,ω) = −∑k

1

8

(1+

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1−Eβk+qE

βk +∆

βk+q∆

βk

ΩβkΩ

βk+q

)( f (Ωβ

k+q) + f (Ωβk )− 1

iω +Ωβk +Ω

βk+q

+1− f (Ωβ

k+q)− f (Ωβk )

iω −Ωβk −Ω

βk+q

),

(D–21)

105

χ+−6 (q,q,ω) = −∑k

1

4

(1+

ε−k ε−k+q −W 2√

ε−2k +W2

√ε−2k+q +W

2

)(1 +Eβk+qE

βk +∆

βk+q∆

βk

ΩβkΩ

βk+q

)f (Ωβ

k+q)− f (Ωβk )

iω −Ωβk +Ω

βk+q

.

(D–22)

D.2 The Umklapp term for the transverse dynamic spin susceptibility

For the Umklapp (off-diagonal) transverse susceptibility, we evaluate the spin

susceptibility at q′ = q + Q. The Umklapp transverse susceptibility in the coexistence

state can also be separated in to normal term and superconducting term as

χ+−Q (q, iω) =χ+−0 (q,q+Q, iω)

=χ+−Qnor(q, iω) + χ+−QSC(q, iω)

(D–23)

with

χ+−Qnor(q, iω) =

∫ β

0

dτe iωτ∑k,k′

⟨Tτc†k+q↑(τ)ck′↑(0)⟩⟨Tτck↓(τ)c

†k′−q+Q↓(0)⟩ (D–24)

and

χ+−QSC(q, iω) =

∫ β

0

dτe iωnτ∑k,k′

⟨Tτc†k+q↑(τ)c

†k′−q+Q↓(0)⟩⟨Tτck↑(τ)ck′↓(0)⟩. (D–25)

To evaluate the value of χ+−Qnor(q, iω), without loss of generality, we can start with result in

the pure antiferromagnetic state. From Equation C–6. , we have

χ+−Qnor(q, iω) =

∫ β

0

dτe iωτ∑k

′[(ukvk − uk+qvk+q)

[⟨Tτα†k+q↑(τ)αk+q↑(0)⟩⟨Tταk↓(τ)α

†k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)βk+q↑(0)⟩⟨Tτβk↓(τ)β

†k↓(0)⟩]

− (uk+qvk+q + ukvk)

[⟨Tτα†k+q↑(τ)αk+q↓(0)⟩⟨Tτβk↑(τ)β

†k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)βk+q↓(0)⟩⟨Tταk↑(τ)α

†k↓(0)⟩].

(D–26)

106

We replace the Fourier transform of the four-operators with Equations D–6, D–9. We

have

χ+−Qnor(q, iω) =∑k,γ

′(ukvk − uk+qvk+q)

[uγ2k+qv

γ2k

f (Ωγk) + f (Ω

γk+q)− 1

iω +Ωγk +Ω

γk+q

+ uγ2k+quγ2k

f (Ωγk)− f (Ω

γk+q)

iω −Ωγk +Ω

γk+q

+v γ2k+qvγ2k

f (Ωγk+q)− f (Ω

γk)

iω +Ωγk −Ω

γk+q

+ v γ2k+quγ2k

1− f (Ωγk)− f (Ω

γk+q)

iω −Ωγk −Ω

γk+q

]−∑k,γ =γ′

′(uk+qvk+q + ukvk)

[uγ2k+qv

γ′2k

f (Ωγ′

k ) + f (Ωγk+q)− 1

iω +Ωγ′

k +Ωγk+q

+ uγ2k+quγ′2k

f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k +Ωγk+q

+v γ2k+qvγ′2k

f (Ωγk+q)− f (Ω

γ′

k )

iω +Ωγ′

k −Ωγk+q

+ v γ2k+quγ′2k

1− f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k −Ωγk+q

].

(D–27)

We apply the unitary transformation, Equation 2–11 to the superconducting part of

the Umklapp transverse susceptibility Equation C–24 and get

χ+−QSC(q, iω) =1

V

∫ β

0

dτe iωτ∑k,σ

′(uk+qvk+q + ukvk)

[⟨Tτα†k+q↑(τ)α

†−k−q↓(0)⟩⟨Tταk↑(τ)α−k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)β

†−k−q↓(0)⟩⟨Tτβk↑(τ)β−k↓(0)⟩]

+ (uk+qvk+q − ukvk)

⟨Tτα†k+q↑(τ)α

†−k−q↓(0)⟩⟨Tτβk↑(τ)β−k↓(0)⟩ − ⟨Tτβ

†k+q↑(τ)β

†−k−q↓(0)⟩⟨Tταk↑(τ)α−k↓(0)⟩].

(D–28)

107

After replacing the Fourier transform of the four-operator by using Equations D–13 and

D–14, we get

χ+−QSC(q, iω) =∑k,γ

′(uk+qvk+q + ukvk)

uγk+qvγk+qu

γk v

γk

[ f (Ωγk+q)− f (Ω

γk)

iω −Ωγk +Ω

γk+q

−1− f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk +Ω

γk+q

−f (Ωγ

k) + f (Ωγk+q)− 1

iω −Ωγk −Ω

γk+q

+f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk −Ω

γk+q

]+∑k,γ =γ′

′(uk+qvk+q − ukvk)

uγk+qvγk+qu

γ′

k vγ′

k

[ f (Ωγk+q)− f (Ω

γ′

k )

iω −Ωγ′

k +Ωγk+q

−1− f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k +Ωγk+q

−f (Ωγ′

k ) + f (Ωγk+q)− 1

iω −Ωγ′

k −Ωγk+q

+f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k −Ωγk+q

].

(D–29)

The Umklapp transverse spin susceptibility is the sum of the following terms:

χ+−(a)Q =−∑k

1

4

[ W√(εk+q − εk+q+Q)2 + 4W 2

− W√(εk − εk+Q)2 + 4W 2

][(Eαk+q

Ωαk+q

− Eαk

Ωαk

)f (Ωα

k ) + f (Ωαk+q)− 1

iω +Ωαk +Ω

αk+q

+ (Eαk

Ωαk

−Eαk+q

Ωαk+q

)1− f (Ωα

k )− f (Ωαk+q)

iω −Ωαk −Ωα

k+q

− (−Eβk

Ωβk

+Eβk+q

Ωβk+q

)f (Ωβ

k ) + f (Ωβk+q)− 1

iω +Ωβk +Ω

βk+q

− (Eβk

Ωβk

−Eβk+q

Ωβk+q

)1− f (Ωβ

k )− f (Ωβk+q)

iω −Ωβk −Ω

βk+q

],

(D–30)

108

χ+−(b)Q =−∑k

1

4

[ W√(εk+q − εk+q+Q)2 + 4W 2

+W√

(εk − εk+Q)2 + 4W 2

][2(−E

βk

Ωβk

+Eαk+q

Ωαk+q

)f (Ωβ

k ) + f (Ωαk+q)− 1

iω +Ωβk +Ω

αk+q

+ 2(Eβk

Ωβk

+Eαk+q

Ωαk+q

)f (Ωβ

k )− f (Ωαk+q)

iω −Ωβk +Ω

αk+q

+ 2(−Eβk

Ωβk

−Eαk+q

Ωαk+q

)f (Ωα

k+q)− f (Ωβk )

iω +Ωβk −Ωα

k+q

+ 2(Eβk

Ωβk

−Eαk+q

Ωαk+q

)1− f (Ωβ

k )− f (Ωαk+q)

iω −Ωβk −Ωα

k+q

],

(D–31)

χ+−(c)Q =∑k

[ W√(εk+q − εk+q+Q)2 + 4W 2

+W√

(εk − εk+Q)2 + 4W 2

][ ∆α

k+q∆βk

4Ωαk+qΩ

βk

( f (Ωαk+q)− f (Ωα

k )

iω −Ωαk +Ω

αk+q

+1− f (Ωα

k )− f (Ωαk+q)

iω +Ωαk +Ω

αk+q

+f (Ωα

k ) + f (Ωαk+q)− 1

iω −Ωαk −Ωα

k+q

+f (Ωα

k )− f (Ωαk+q)

iω +Ωαk −Ωα

k+q

)−∆βk+q∆

αk

4Ωβk+qΩ

αk

( f (Ωβk+q)− f (Ω

βk )

iω −Ωβk +Ω

βk+q

+1− f (Ωβ

k )− f (Ωβk+q)

iω +Ωβk +Ω

βk+q

+f (Ωβ

k ) + f (Ωβk+q)− 1

iω −Ωβk −Ω

βk+q

+f (Ωβ

k )− f (Ωβk+q)

iω +Ωβk −Ω

βk+q

)],

(D–32)

χ+−(d)Q =∑k

[ W√(εk+q − εk+q+Q)2 + 4W 2

− W√(εk − εk+Q)2 + 4W 2

][ ∆α

k+q∆βk

4Ωαk+qΩ

βk

( f (Ωαk+q)− f (Ω

βk )

iω −Ωβk +Ω

αk+q

+1− f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk +Ω

αk+q

+f (Ωβ

k ) + f (Ωαk+q)− 1

iω −Ωβk −Ωα

k+q

+f (Ωβ

k )− f (Ωαk+q)

iω +Ωβk −Ωα

k+q

)−∆βk+q∆

αk

4Ωβk+qΩ

αk

( f (Ωβk+q)− f (Ωα

k )

iω −Ωαk +Ω

βk+q

+1− f (Ωα

k )− f (Ωβk+q)

iω +Ωαk +Ω

βk+q

+f (Ωα

k ) + f (Ωβk+q)− 1

iω −Ωαk −Ω

βk+q

+f (Ωα

k )− f (Ωβk+q)

iω +Ωαk −Ω

βk+q

)].

(D–33)

109

After organizing, we have the Umklapp transverse spin susceptibility reducing to

χ+−0 (q,q+Q,ω) =W

4

∑k,γ

′

1√(ε−k+q

)2+W 2

− 1√(ε−k)2+W 2

±(E γk+q

Ωγk+q

+E γk

Ωγk

)f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q +Ω

γk

±(E γk+q

Ωγk+q

− Eγk

Ωγk

)(1− f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q −Ω

γk

+f (Ωγ

k+q) + f (Ωγk)− 1

ω + iδ +Ωγk+q +Ω

γk

)

+W

4

∑k,γ =γ′

′

1√(ε−k+q

)2+W 2

+1√(

ε−k)2+W 2

±

(E γk+q

Ωγk+q

+E γ′

k

Ωγ′

k

)f (Ωγ

k+q)− f (Ωγ′

k )

ω + iδ −Ωγk+q +Ω

γ′

k

±

(E γk+q

Ωγk+q

− Eγ′

k

Ωγ′

k

)(1− f (Ωγ

k+q)− f (Ωγ′

k )

ω + iδ −Ωγk+q −Ω

γ′

k

+f (Ωγ

k+q) + f (Ωγ′

k )− 1ω + iδ +Ωγ

k+q +Ωγ′

k

).

(D–34)

D.3 The longitudinal dynamic spin susceptibility

The longitudinal dynamic spin susceptibility, in terms of the raising and lowering

operators, is written as

χzz0 (q,q,ω) =

∫ β

0

dτe iωτ∑kk′

⟨Tτ(c†k+q↑(τ)ck↑(τ)c

†k′−q↑(0)ck′↑(0)

− c†k+q↑(τ)ck↑(τ)c†k′−q↓(0)ck′↓(0)− c

†k+q↓(τ)ck↓(τ)c

†k′−q↑(0)ck′↑(0)

+ c†k+q↓(τ)ck↓(τ)c†k′−q↓(0)ck′↓(0))⟩.

(D–35)

By using Wick theorem to separate the four operators and ignoring the terms which have

expectation values being zero, we have

χzz0 (q,q,ω) = χzznor(q,q,ω) + χzzSC(q,q,ω) (D–36)

with

χzznor(q,q,ω) =

∫ β

0

dτe iωτ∑kk′,σ

⟨Tτc†k+qσ(τ)ck′σ(0)⟩⟨Tτckσ(τ)c

†k′−qσ(0)⟩ (D–37)

110

and

χzzSC(q,q,ω) =

∫ β

0

dτe iωτ∑kk′,σ

⟨Tτc†k+qσ(τ)c

†k′−qσ(0)⟩⟨Tτckσ(τ)ck′σ(0)⟩. (D–38)

Reducing the sum over k to the reduced Brillouin zone and applying the unitary transfor-

mation to χzznor(q,q,ω), we get

χzznor(q,q,ω) =

∫ β

0

dτe iωτ∑kσ

′(u2ku

2k+q + 2ukvkuk+qvk+q + v

2k v2k+q)

×(⟨Tτ(α†kσ(τ)αkσ(0)⟩⟨Tταk+qσ(τ)α

†k+qσ(0)⟩+ ⟨Tτβ

†kσ(τ)βkσ(0)⟩⟨Tτβk+qσ(τ)β

†k+qσ(0)⟩)

+(u2kv2k+q − 2ukvkuk+qvk+q + v 2k u2k+q)

×(⟨Tτα†kσ(τ)αkσ(0)⟩⟨Tτβk+qσ(τ)β

†k+qσ(0)⟩+ ⟨Tτβ

†kσ(τ)βkσ(0)⟩⟨Tταk+qσ(τ)α

†k+qσ(0)⟩).

(D–39)

Plugging the expectation values for the four-operators, we obtain the expression as

χzznor(q,q,ω) =∑kγ

′1

2(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)

[uγ2k+qv

γ2k

f (Ωγk) + f (Ω

γk+q)− 1

iω +Ωγk +Ω

γk+q

+ uγ2k+quγ2k

f (Ωγk)− f (Ω

γk+q)

iω −Ωγk +Ω

γk+q

+ v γ2k+qvγ2k

f (Ωγk+q)− f (Ω

γk)

iω +Ωγk −Ω

γk+q

+ v γ2k+quγ2k

1− f (Ωγk)− f (Ω

γk+q)

iω −Ωγk −Ω

γk+q

]+∑kγ =γ′

′1

2(1−

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)

[uγ2k+qv

γ′2k

f (Ωγ′

k ) + f (Ωγk+q)− 1

iω +Ωγ′

k +Ωγk+q

+ uγ2k+quγ′2k

f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k +Ωγk+q

+v γ2k+qvγ′2k

f (Ωγk+q)− f (Ω

γ′

k )

iω +Ωγ′

k −Ωγk+q

+ v γ2k+quγ′2k

1− f (Ωγ′

k )− f (Ωγk+q)

iω −Ωγ′

k −Ωγk+q

].

(D–40)

111

For the SC part of the longitudinal spin susceptibility from Equation D–38, after the

unitary transformation, we have

χzzSC(q,q,ω) =µ202

∫ β

0

dτe iωnτ∑k∈Rσ

(u2ku2k+q + 2ukvkuk+qvk+q + v

2k v2k+q)

×(⟨Tτ(α†kσ(τ)α

†−kσ(0)⟩⟨Tταk+qσ(τ)α−k−qσ(0)⟩+ ⟨Tτβ

†kσ(τ)β

†−kσ(0)⟩⟨Tτβk+qσ(τ)β−k−qσ(0)⟩)

+(u2kv2k+q − 2ukvkuk+qvk+q + v 2k u2k+q)

×(⟨Tτα†kσ(τ)α

†−kσ(0)⟩⟨Tτβk+qσ(τ)β−k−qσ(0)⟩+ ⟨Tτβ

†kσ(τ)β

†kσ(0)⟩⟨Tταk+qσ(τ)α−k−qσ(0)⟩).

(D–41)

Plugging the expectation values for the four-operators, we obtain the expression as

χzzSC(q,q,ω) = −∑kγ

′1

2(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)

[(uγk+qv

γk+qu

γk v

γk )(f (Ωγ

k+q)− f (Ωγk)

iω −Ωγk +Ω

γk+q

−1− f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk +Ω

γk+q

−f (Ωγ

k) + f (Ωγk+q)− 1

iω −Ωγk −Ω

γk+q

+f (Ωγ

k)− f (Ωγk+q)

iω +Ωγk −Ω

γk+q

)]

+∑kγ =γ′

′1

2(1−

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)(uγk+qvγk+qu

γ′

k vγ′

k )

[ f (Ωγk+q)− f (Ω

γ′

k )

iω −Ωγ′

k +Ωγk+q

−1− f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k +Ωγk+q

−f (Ωγ′

k ) + f (Ωγk+q)− 1

iω −Ωγ′

k −Ωγk+q

+f (Ωγ′

k )− f (Ωγk+q)

iω +Ωγ′

k −Ωγk+q

].

(D–42)

112

The SC term has the same SDW coefficients. The final expression for the total longitudi-

nal spin susceptibility, χzz0 (q,q,ω) is

χzz0 (q,q,ω) =∑k,γ

′1

4

1 + ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +E γk E

γk+q + ∆

γk∆

γk+q

ΩγkΩ

γk+q

]f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q +Ω

γk

+1

2

[1−E γk E

γk+q +∆

γk∆

γk+q

ΩγkΩ

γk+q

](f (Ωγ

k+q) + f (Ωγk)− 1

ω + iδ +Ωγk+q +Ω

γk

+1− f (Ωγ

k+q)− f (Ωγk)

ω + iδ −Ωγk+q −Ω

γk

)

+∑k,γ =γ′

′1

4

1− ε−k ε−k+q +W

2√(ε−k)2+W 2

√(ε−k+q

)2+W 2

[1 +E γk E

γ′

k+q +∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

]f (Ωγ′

k+q)− f (Ωγk)

ω + iδ −Ωγ′

k+q +Ωγk

+1

2

[1−E γk E

γ′

k+q + ∆γk∆

γ′

k+q

ΩγkΩ

γ′

k+q

](f (Ωγ′

k+q) + f (Ωγk)− 1

ω + iδ +Ωγ′

k+q +Ωγk

+1− f (Ωγ′

k+q)− f (Ωγk)

ω + iδ −Ωγ′

k+q −Ωγk

).

(D–43)

For the purpose of coding, we simplify the expression by shifting k and change

ω → −ω in some terms The total longitudinal spin susceptibility, χzz0 (q,q,ω) is also the

sum of the following six terms:

χzz(1)0 (q,q,ω) =−∑k∈R

1

4(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)

(1 +Eαk+qE

αk +∆

αk+q∆

αk

Ωαk+qΩ

αk

) f (Ωαk+q)− f (Ωα

k )

iω −Ωαk +Ω

αk+q

,

(D–44)

χzz(2)0 (q,q,ω) = −∑k∈R

1

8(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)(1−Eαk+qE

αk + ∆

αk+q∆

αk

Ωαk+qΩ

αk

)[ f (Ωα

k ) + f (Ωαk+q)− 1

iω +Ωαk +Ω

αk+q

+1− f (Ωα

k )− f (Ωαk+q)

iω −Ωαk −Ωα

k+q

],

(D–45)

113

χzz(3)0 (q,q,ω) = −∑k∈R

1

4(1−

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)(1 +Eαk+qE

βk + ∆

αk+q∆

βk

Ωαk+qΩ

βk

)[ f (Ωα

k+q)− f (Ωβk )

iω −Ωβk +Ω

αk+q

+f (Ωβ

k )− f (Ωαk+q)

iω −Ωαk+q +Ω

βk

],

(D–46)

χzz(4)0 (q,q,ω) = −∑k∈R

1

4(1−

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)(1−Eαk+qE

βk + ∆

αk+q∆

βk

Ωαk+qΩ

βk

)[1− f (Ωβ

k )− f (Ωαk+q)

iω −Ωβk −Ωα

k+q

+f (Ωβ

k ) + f (Ωαk+q)− 1

iω +Ωβk +Ω

αk+q

],

(D–47)

χzz(5)0 (q,q,ω) = −∑k∈R

1

8(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)(1−Eβk+qE

βk + ∆

βk+q∆

βk

Ωβk+qΩ

βk

)[ f (Ωβ

k ) + f (Ωβk+q)− 1

iω +Ωβk +Ω

βk+q

+1− f (Ωβ

k )− f (Ωβk+q)

iω −Ωβk −Ω

βk+q

] (D–48)

and

χzz(6)0 (q,q,ω) = −∑k∈R

1

4(1 +

ε−k ε−k+q +W

2√ε−2k +W

2

√ε−2k+q +W

2

)

(1 +Eβk+qE

βk +∆

βk+q∆

βk

Ωβk+qΩ

βk

) f (Ωβk+q)− f (Ω

βk )

iω −Ωβk +Ω

βk+q

.

(D–49)

114

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BIOGRAPHICAL SKETCH

Wenya Rowe was born in Taiwan in 1980. She studied in Fu-Jen Catholic University

and received her bachelor degrees in Physics and Mathematics in 2003. She received

her Master degree from Cheng-Kung University in 2005. In 2007 she entered the

University of Florida. She received her PhD degree in May 2014.

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