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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
REFERENCES
[1] Peter Mohn. Magnetism in the Solid State: An Introduction. Springer, 2006.Available from: http://books.google.de/books?id=KaVGAAAAQBAJ.
[2] P C E Stamp. Spin fluctuation theory in condensed quantum systems. Journal ofPhysics F: Metal Physics, 15(9):1829, 1985. Available from: http://stacks.iop.org/0305-4608/15/i=9/a=005.
[3] S. Doniach and S. Engelsberg. Low-temperature properties of nearly ferromagneticFermi liquids. Phys. Rev. Lett., 17:750–753, Oct 1966. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.17.750.
[4] P. W. Anderson and W. F. Brinkman. Anisotropic superfluidity in 3He: A pos-sible interpretation of its stability as a spin-fluctuation effect. Phys. Rev. Lett.,30:1108–1111, May 1973. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.30.1108.
[5] N. F. Berk and J. R. Schrieffer. Effect of ferromagnetic spin correlations onsuperconductivity. Phys. Rev. Lett., 17:433–435, Aug 1966. Available from:http://link.aps.org/doi/10.1103/PhysRevLett.17.433.
[6] Jun Kondo. Resistance minimum in dilute magnetic alloys. Progress of TheoreticalPhysics, 32(1):37–49, 1964. Available from: http://ptp.oxfordjournals.org/content/32/1/37.abstract.
[7] A.M. Tsvelick and P.B. Wiegmann. Solution of then-channel Kondo problem (scalingand integrability). Zeitschrift fur Physik B Condensed Matter, 54(3):201–206, 1984.Available from: http://dx.doi.org/10.1007/BF01319184.
[8] N. Andrei and J. H. Lowenstein. Scales and scaling in the Kondo model. Phys. Rev.Lett., 46:356–360, Feb 1981. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.46.356.
[9] J. R. Schrieffer and P. A. Wolff. Relation between the Anderson and KondoHamiltonians. Phys. Rev., 149:491–492, Sep 1966. Available from: http://link.aps.org/doi/10.1103/PhysRev.149.491.
[10] J. Hubbard. Electron correlations in narrow energy bands. Proceedings of the RoyalSociety of London. Series A, Mathematical and Physical Sciences, 276(1365):pp.238–257, 1963. Available from: http://www.jstor.org/stable/2414761.
[11] Elliott H. Lieb and F. Y. Wu. Absence of Mott transition in an exact solution of theshort-range, one-band model in one dimension. Phys. Rev. Lett., 20:1445–1448,Jun 1968. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.20.1445.
115
[12] J. R. Schrieffer, X. G. Wen, and S. C. Zhang. Dynamic spin fluctuations and thebag mechanism of high-Tc superconductivity. Phys. Rev. B, 39:11663–11679, Jun1989. Available from: http://link.aps.org/doi/10.1103/PhysRevB.39.11663.
[13] Avinash Singh and Zlatko Tesanovic. Collective excitations in a dopedantiferromagnet. Phys. Rev. B, 41:614–631, Jan 1990. Available from:http://link.aps.org/doi/10.1103/PhysRevB.41.614.
[14] Andrey V. Chubukov and David M. Frenkel. Renormalized perturbation theory ofmagnetic instabilities in the two-dimensional Hubbard model at small doping. Phys.Rev. B, 46:11884–11901, Nov 1992. Available from: http://link.aps.org/doi/10.1103/PhysRevB.46.11884.
[15] D. J. Scalapino, E. Loh, and J. E. Hirsch. d-wave pairing near a spin-density-wave instability. Phys. Rev. B, 34:8190–8192, Dec 1986. Available from: http://link.aps.org/doi/10.1103/PhysRevB.34.8190.
[16] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer.Superconductivity in the presence of strong Pauli paramagnetism: CeCu2Si2. Phys.Rev. Lett., 43:1892–1896, Dec 1979. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.43.1892.
[17] Manfred Sigrist and Kazuo Ueda. Phenomenological theory of unconventionalsuperconductivity. Reviews of Modern physics, 63(2):239, 1991.
[18] J.G. Bednorz and K.A. Muller. Possible highTc superconductivity in the Ba-La-Cu-Osystem. Zeitschrift fur Physik B Condensed Matter, 64(2):189–193, 1986. Availablefrom: http://dx.doi.org/10.1007/BF01303701.
[19] N. P. Armitage, P. Fournier, and R. L. Greene. Progress and perspectives onelectron-doped cuprates. Rev. Mod. Phys., 82(3):2421–2487, Sep 2010.
[20] Masaki Fujita, Haruhiro Hiraka, Masaaki Matsuda, Masato Matsuura, John M.Tranquada, Shuichi Wakimoto, Guangyong Xu, and Kazuyoshi Yamada. Progressin neutron scattering studies of spin excitations in high-Tc cuprates. Journalof the Physical Society of Japan, 81(1):011007, 2012. Available from: http://jpsj.ipap.jp/link?JPSJ/81/011007/.
[21] K. Yamada, K. Kurahashi, T. Uefuji, M. Fujita, S. Park, S.-H. Lee, and Y. Endoh.Commensurate spin dynamics in the superconducting state of an electron-dopedcuprate superconductor. Phys. Rev. Lett., 90:137004, Apr 2003. Available from:http://link.aps.org/doi/10.1103/PhysRevLett.90.137004.
[22] V. F. Mitrovic, M.-H. Julien, C. de Vaulx, M. Horvatic, C. Berthier, T. Suzuki, andK. Yamada. Similar glassy features in the 139La NMR response of pure anddisordered La1.88Sr0.12CuO4. Phys. Rev. B, 78:014504, Jul 2008. Available from:http://link.aps.org/doi/10.1103/PhysRevB.78.014504.
116
[23] J. J. Wagman, G. Van Gastel, K. A. Ross, Z. Yamani, Y. Zhao, Y. Qiu, J. R. D.Copley, A. B. Kallin, E. Mazurek, J. P. Carlo, H. A. Dabkowska, and B. D. Gaulin.Two-dimensional incommensurate and three-dimensional commensurate magneticorder and fluctuations in La2−xBaxCuO4. Phys. Rev. B, 88:014412, Jul 2013.Available from: http://link.aps.org/doi/10.1103/PhysRevB.88.014412.
[24] H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld. Defects in correlated metalsand superconductors. Rev. Mod. Phys., 81:45–108, Jan 2009. Available from:http://link.aps.org/doi/10.1103/RevModPhys.81.45.
[25] Tom Timusk and Bryan Statt. The pseudogap in high-temperature superconductors:an experimental survey. Reports on Progress in Physics, 62(1):61, 1999. Availablefrom: http://stacks.iop.org/0034-4885/62/i=1/a=002.
[26] D.J. and Scalapino. The case for dx2−y2 pairing in the cuprate super-conductors. Physics Reports, 250(6):329 – 365, 1995. Available from:http://www.sciencedirect.com/science/article/pii/037015739400086I.
[27] C. C. Tsuei and J. R. Kirtley. Pairing symmetry in cuprate superconductors. Rev.Mod. Phys., 72:969–1016, Oct 2000. Available from: http://link.aps.org/doi/10.1103/RevModPhys.72.969.
[28] Masahiko Inui, Sebastian Doniach, Peter J. Hirschfeld, and Andrei E. Ruckenstein.Coexistence of antiferromagnetism and superconductivity in a mean-field theory ofhigh-Tc superconductors. Phys. Rev. B, 37:2320–2323, Feb 1988. Available from:http://link.aps.org/doi/10.1103/PhysRevB.37.2320.
[29] Boris I. Shraiman and Eric D. Siggia. Excitation spectrum of the spiral state of adoped antiferromagnet. Phys. Rev. B, 46:8305–8311, Oct 1992. Available from:http://link.aps.org/doi/10.1103/PhysRevB.46.8305.
[30] V. I. Belinicher, A. L. Chernyshev, and V. A. Shubin. Two-hole problem in the t-Jmodel: A canonical transformation approach. Phys. Rev. B, 56:3381–3393, Aug1997. Available from: http://link.aps.org/doi/10.1103/PhysRevB.56.3381.
[31] Andreas Luscher, Alexander I. Milstein, and Oleg P. Sushkov. Effective actionof the weakly doped t − J model and spin-wave excitations in the spin-glassphase of La2−xSrxCuO4. Phys. Rev. B, 75:235120, Jun 2007. Available from:http://link.aps.org/doi/10.1103/PhysRevB.75.235120.
[32] Cedric Weber, Kristjan Haule, and Gabriel Kotliar. Strength of correlations inelectron- and hole-doped cuprates. Nat Phys, 6(8):574–578, 08 2010. Availablefrom: http://dx.doi.org/10.1038/nphys1706.
[33] Yoichi Kamihara, Hidenori Hiramatsu, Masahiro Hirano, Ryuto Kawamura, HiroshiYanagi, Toshio Kamiya, and Hideo Hosono. Iron-based layered superconductor:LaOFeP. Journal of the American Chemical Society, 128(31):10012–10013, 2006.Available from: http://pubs.acs.org/doi/abs/10.1021/ja063355c.
117
[34] G. R. Stewart. Superconductivity in iron compounds. Rev. Mod. Phys., 83:1589–1652, Dec 2011. Available from: http://link.aps.org/doi/10.1103/RevModPhys.83.1589.
[35] Igor I. Mazin. Superconductivity gets an iron boost. Nature, 464(7286):183–186, 032010. Available from: http://dx.doi.org/10.1038/nature08914.
[36] Y. Laplace, J. Bobroff, F. Rullier-Albenque, D. Colson, and A. Forget. Atomiccoexistence of superconductivity and incommensurate magnetic order in thepnictide Ba(Fe1−xCox )2As2. Phys. Rev. B, 80:140501, Oct 2009. Available from:http://link.aps.org/doi/10.1103/PhysRevB.80.140501.
[37] Y. Laplace, J. Bobroff, V. Brouet, G. Collin, F. Rullier-Albenque, D. Colson, andA. Forget. Nanoscale-textured superconductivity in ru-substituted BaFe2As2: Achallenge to a universal phase diagram for the pnictides. Phys. Rev. B, 86:020510,Jul 2012. Available from: http://link.aps.org/doi/10.1103/PhysRevB.86.020510.
[38] Y. Texier, J. Deisenhofer, V. Tsurkan, A. Loidl, D. S. Inosov, G. Friemel, and J. Bo-broff. NMR study in the iron-selenide Rb0.74Fe1.6Se2: Determination of the super-conducting phase as iron vacancy-free Rb0.3Fe2Se2. Phys. Rev. Lett., 108:237002,Jun 2012. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.108.237002.
[39] S Graser, T A Maier, P J Hirschfeld, and D J Scalapino. Near-degeneracy of severalpairing channels in multiorbital models for the Fe pnictides. New Journal of Physics,11(2):025016, 2009. Available from: http://stacks.iop.org/1367-2630/11/i=2/a=025016.
[40] M. M. Qazilbash, J. J. Hamlin, R. E. Baumbach, Lijun Zhang, D. J. Singh, M. B.Maple, and D. N. Basov. Electronic correlations in the iron pnictides. Nat Phys,5(9):647–650, 09 2009. Available from: http://dx.doi.org/10.1038/nphys1343.
[41] G. R. Stewart. Heavy-fermion systems. Rev. Mod. Phys., 56:755–787, Oct 1984.Available from: http://link.aps.org/doi/10.1103/RevModPhys.56.755.
[42] G. R. Stewart, Z. Fisk, J. O. Willis, and J. L. Smith. Possibility of coexistence of bulksuperconductivity and spin fluctuations in UPt3. Phys. Rev. Lett., 52:679–682, Feb1984. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.52.679.
[43] H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, and J. L. Smith. p−wavesuperconductivity in UBe13. Phys. Rev. Lett., 52:1915–1918, May 1984. Availablefrom: http://link.aps.org/doi/10.1103/PhysRevLett.52.1915.
[44] S. Schmitt-Rink, K. Miyake, and C. M. Varma. Transport and thermal properties ofheavy-fermion superconductors: A unified picture. Phys. Rev. Lett., 57:2575–2578,Nov 1986. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.57.2575.
118
[45] M. Jourdan, M. Huth, and H. Adrian. Superconductivity mediated by spin fluctua-tions in the heavy-fermion compound UPd2Al3. Nature, 398(6722):47–49, 03 1999.Available from: http://dx.doi.org/10.1038/17977.
[46] K. Miyake, S. Schmitt-Rink, and C. M. Varma. Spin-fluctuation-mediated even-paritypairing in heavy-fermion superconductors. Phys. Rev. B, 34:6554–6556, Nov 1986.Available from: http://link.aps.org/doi/10.1103/PhysRevB.34.6554.
[47] Shinya Nishiyama, K. Miyake, and C. M. Varma. Superconducting transitiontemperatures for spin-fluctuation superconductivity: Application to heavy-fermioncompounds. Phys. Rev. B, 88:014510, Jul 2013. Available from: http://link.aps.org/doi/10.1103/PhysRevB.88.014510.
[48] Tomislav Vuletic, Pascale Auban-Senzier, Claude Pasquier, Silvia Tomic, DenisJerome, Michel Heritier, and Klaus Bechgaard. Coexistence of superconductivityand spin density wave orderings in the organic superconductor (TMTSF)2PF6.The European Physical Journal B-Condensed Matter and Complex Systems,25(3):319–331, 2002.
[49] P Langevin. A fundamental formula of kinetic theory. In Annales de Chimie et dePhysique, volume 5, pages 245–288, 1905.
[50] P Weiss. Hypothesis of the molecular field and ferromagnetic properties. J. Phys,6(4):661–690, 1907.
[51] Werner Heisenberg. Zur theorie des ferromagnetismus. Zeitschrift fur Physik,49(9-10):619–636, 1928.
[52] Clifford Glenwood Shull and J Samuel Smart. Detection of antiferromagnetismby neutron diffraction. Physical Review (US) Superseded in part by Phys. Rev. A,Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D, 76, 1949.
[53] Kentaro Kitagawa, Naoyuki Katayama, Kenya Ohgushi, Makoto Yoshida, andMasashi Takigawa. Commensurate itinerant antiferromagnetism in BaFe2As2: 75As-NMR studies on a self-flux grown single crystal. Journal of the Physical Society ofJapan, 77(11):114709, 2008. Available from: http://jpsj.ipap.jp/link?JPSJ/77/114709/.
[54] Felix Bloch. Uber die quantenmechanik der elektronen in kristallgittern. Zeitschriftfur physik, 52(7-8):555–600, 1929.
[55] N F Mott. The basis of the electron theory of metals, with special reference to thetransition metals. Proceedings of the Physical Society. Section A, 62(7):416, 1949.Available from: http://stacks.iop.org/0370-1298/62/i=7/a=303.
[56] J.-P. Ismer, Ilya Eremin, Enrico Rossi, Dirk K. Morr, and G. Blumberg. Theoryof multiband superconductivity in spin-density-wave metals. Phys. Rev. Lett.,105(3):037003, Jul 2010.
119
[57] G C Psaltakis and E W Fenton. Superconductivity and spin-density waves: organicsuperconductors. Journal of Physics C: Solid State Physics, 16(20):3913, 1983.Available from: http://stacks.iop.org/0022-3719/16/i=20/a=015.
[58] Bumsoo Kyung. Mean-field study of the interplay between antiferromagnetism andd-wave superconductivity. Phys. Rev. B, pages 9083–9088, Oct.
[59] A Aperis, G Varelogiannis, P B Littlewood, and B D Simons. Coexistence of spindensity wave, d-wave singlet and staggered π-triplet superconductivity. Journal ofPhysics: Condensed Matter, (43):434235.
[60] W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld. Spin excitations in layeredantiferromagnetic metals and superconductors. Phys. Rev. B, 86:134513, Oct 2012.Available from: http://link.aps.org/doi/10.1103/PhysRevB.86.134513.
[61] Yasuyuki Kato, C. D. Batista, and I. Vekhter. Antiferromagnetic order in Pauli-limited unconventional superconductors. Phys. Rev. Lett., 107:096401, Aug 2011.Available from: http://link.aps.org/doi/10.1103/PhysRevLett.107.096401.
[62] I. I. Mazin and David J. Singh. Ferromagnetic spin fluctuation induced supercon-ductivity in Sr2RuO4. Phys. Rev. Lett., 79:733–736, Jul 1997. Available from:http://link.aps.org/doi/10.1103/PhysRevLett.79.733.
[63] Toru Moriya and Kazuo Ueda. Spin fluctuation spectra and high temperaturesuperconductivity. Journal of the Physical Society of Japan, 63(5):1871–1880,1994. Available from: http://journals.jps.jp/doi/abs/10.1143/JPSJ.63.1871.
[64] Valentin Stanev, Jian Kang, and Zlatko Tesanovic. Spin fluctuation dynamics andmultiband superconductivity in iron pnictides. Phys. Rev. B, 78:184509, Nov 2008.Available from: http://link.aps.org/doi/10.1103/PhysRevB.78.184509.
[65] T. A. Maier, S. Graser, D. J. Scalapino, and P. J. Hirschfeld. Origin of gap anisotropyin spin fluctuation models of the iron pnictides. Phys. Rev. B, 79:224510, Jun 2009.Available from: http://link.aps.org/doi/10.1103/PhysRevB.79.224510.
[66] T. A. Maier and D. J. Scalapino. Theory of neutron scattering as a probe of thesuperconducting gap in the iron pnictides. Phys. Rev. B, page 020514, Jul.
[67] H. A. Mook, M. Yethiraj, G. Aeppli, T. E. Mason, and T. Armstrong. Polarizedneutron determination of the magnetic excitations in YBa2Cu3O7. Phys. Rev. Lett.,70:3490–3493, May 1993. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.70.3490.
[68] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, A. Ivanov, G. D. Gu, N. Koshizuka,and B. Keimer. Neutron scattering from magnetic excitations in Bi2Sr2CaCu2O8+δ.Nature, 398(6728):588–591, 04 1999. Available from: http://dx.doi.org/10.1038/19255.
120
[69] Stephen D. Wilson, Pengcheng Dai, Shiliang Li, Songxue Chi, H. J. Kang, and J. W.Lynn. Resonance in the electron-doped high-transition-temperature superconductorPr0.88LaCe0.12CuO4−δ. Nature, 442(7098):59–62, 07 2006. Available from:http://dx.doi.org/10.1038/nature04857.
[70] I. I. Mazin and Victor M. Yakovenko. Neutron scattering and superconducting orderparameter in YBa2Cu3O7. Phys. Rev. Lett., 75:4134–4137, Nov 1995. Availablefrom: http://link.aps.org/doi/10.1103/PhysRevLett.75.4134.
[71] Dirk K. Morr, Peter F. Trautman, and Matthias J. Graf. Resonance peak in Sr2RuO4:Signature of spin triplet pairing. Phys. Rev. Lett., 86:5978–5981, Jun 2001.Available from: http://link.aps.org/doi/10.1103/PhysRevLett.86.5978.
[72] J. Knolle, I. Eremin, A. V. Chubukov, and R. Moessner. Theory of itinerant magneticexcitations in the spin-density-wave phase of iron-based superconductors. Phys.Rev. B, 81:140506, Apr 2010. Available from: http://link.aps.org/doi/10.1103/PhysRevB.81.140506.
[73] S. Doniach and E. H. Sondheimer. Green’s functions for solid state physicists: a reprint volume with additional material on the physics of correlated electronsystems. Imperial College Press ; World Scientific [distributor], London; Singapore;River Edge, NJ, 1998.
[74] Fuxiang Han. A modern course in the quantum theory of solids. World ScientificPub Co Inc., Singapore; Hackensack, New Jersey, 2013.
[75] N. M. R. Peres and M. A. N. Araujo. Spin-wave dispersion in La2CuO4. Phys. Rev.B, 65:132404, Mar 2002. Available from: http://link.aps.org/doi/10.1103/PhysRevB.65.132404.
[76] Andrey V. Chubukov and Karen A. Musaelian. Magnetic phases of the two-dimensional Hubbard model at low doping. Phys. Rev. B, 51:12605–12617, May1995. Available from: http://link.aps.org/doi/10.1103/PhysRevB.51.12605.
[77] Avinash Singh, Zlatko Tesanovic, and Ju H. Kim. Doped Hubbard antiferromagnet:Instability and effective interactions. Phys. Rev. B, 44:7757–7759, Oct 1991.Available from: http://link.aps.org/doi/10.1103/PhysRevB.44.7757.
[78] P. M. R. Brydon and C. Timm. Spin excitations in the excitonic spin-density-wavestate of the iron pnictides. Phys. Rev. B, 80:174401, Nov 2009. Available from:http://link.aps.org/doi/10.1103/PhysRevB.80.174401.
[79] Sadao Nakajima. Paramagnon effect on the bcs transition in He3. Progressof Theoretical Physics, 50(4):1101–1109, 1973. Available from: http://ptp.oxfordjournals.org/content/50/4/1101.abstract.
121
[80] D. Fay and J. Appel. Coexistence of p−state superconductivity and itinerantferromagnetism. Phys. Rev. B, 22:3173–3182, Oct 1980. Available from: http://link.aps.org/doi/10.1103/PhysRevB.22.3173.
[81] David M. Frenkel and W. Hanke. Spirals and spin bags: A link between the weak-and the strong-coupling limits of the Hubbard model. Phys. Rev. B, 42:6711–6714,Oct 1990. Available from: http://link.aps.org/doi/10.1103/PhysRevB.42.6711.
[82] Andrey V. Chubukov and Dirk K. Morr. Electronic structure of underdoped cuprates.Physics Reports, 288(1Aı6):355 – 387, 1997. I.M. Lifshitz and Condensed MatterTheory. Available from: http://www.sciencedirect.com/science/article/pii/S0370157397000331.
[83] S. Maiti, M. M. Korshunov, T. A. Maier, P. J. Hirschfeld, and A. V. Chubukov.Evolution of symmetry and structure of the gap in iron-based superconductorswith doping and interactions. Phys. Rev. B, 84:224505, Dec 2011. Available from:http://link.aps.org/doi/10.1103/PhysRevB.84.224505.
[84] Kiyohisa Tanaka, W. S. Lee, D. H. Lu, A. Fujimori, T. Fujii, Risdiana, I. Terasaki,D. J. Scalapino, T. P. Devereaux, Z. Hussain, and Z.-X. Shen. Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212. Science,314(5807):1910–1913, 2006. Available from: http://www.sciencemag.org/content/314/5807/1910.abstract.
[85] I. M. Vishik, M. Hashimoto, Rui-Hua He, Wei-Sheng Lee, Felix Schmitt, Donghui Lu,R. G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, Kazuhiro Fujita,S. Ishida, M. Ishikado, Yoshiyuki Yoshida, Hiroshi Eisaki, Zahid Hussain, Thomas P.Devereaux, and Zhi-Xun Shen. Phase competition in trisected superconductingdome. Proceedings of the National Academy of Sciences, 109(45):18332–18337,2012. Available from: http://www.pnas.org/content/109/45/18332.abstract.
[86] A. Ino, C. Kim, M. Nakamura, T. Yoshida, T. Mizokawa, Z.-X. Shen, A. Fujimori,T. Kakeshita, H. Eisaki, and S. Uchida. Electronic structure of La2−xSrxCuO4 in thevicinity of the superconductor-insulator transition. Phys. Rev. B, 62:4137–4141, Aug2000. Available from: http://link.aps.org/doi/10.1103/PhysRevB.62.4137.
[87] E. Razzoli, G. Drachuck, A. Keren, M. Radovic, N. C. Plumb, J. Chang, Y.-B.Huang, H. Ding, J. Mesot, and M. Shi. Evolution from a nodeless gap to dx2−y2-wave in underdoped La2−xSrxCuO4. Phys. Rev. Lett., 110:047004, Jan 2013.Available from: http://link.aps.org/doi/10.1103/PhysRevLett.110.047004.
[88] Yingying Peng, Jianqiao Meng, Daixiang Mou, Junfeng He, Lin Zhao, Yue Wu,Guodong Liu, Xiaoli Dong, Shaolong He, Jun Zhang, Xiaoyang Wang, Qinjun Peng,Zhimin Wang, Shenjin Zhang, Feng Yang, Chuangtian Chen, Zuyan Xu, T. K. Lee,and X. J. Zhou. Disappearance of nodal gap across the insulator–superconductortransition in a copper-oxide superconductor. Nat Commun, 4, 09 2013. Availablefrom: http://dx.doi.org/10.1038/ncomms3459.
122
[89] M. Tinkham. Introduction to Superconductivity: Second Edition. Dover Bookson Physics Series. Dover Publications, Incorporated, 2012. Available from:http://books.google.de/books?id=VpUk3NfwDIkC.
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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.
124
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