FERMI LIQUIDS NEAR POMERANCHUK INSTABILITIES
A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy
by
Kelly Elizabeth Reidy
August 2014 Dissertation written by
Kelly Elizabeth Reidy
B.S., American University, 2006
Ph.D., Kent State University, 2014
Approved by
Dr. Khandker Quader, Chair, Doctoral Dissertation Committee
Dr. David Allender, Members, Doctoral Dissertation Committee
Dr. Almut Schroeder,
Dr. Chuck Gartland,
Dr. Kevin Bedell,
Dr. Qi-Huo Wei,
Accepted by
Dr. James Gleeson, Chair, Department of Physics
Dr. James Blank, Dean, College of Arts and Sciences ii TABLE OF CONTENTS
LIST OF FIGURES ...... vi
LIST OF TABLES ...... xv
Acknowledgements ...... xxi
Dedication ...... xxiii
1 Introduction ...... 1
2 Fermi liquid theory ...... 7
2.1 Landau Fermi liquid theory: a phenomenological perspective . . . . . 7
2.2 Landau Fermi liquid theory: a microscopic perspective ...... 9
2.3 Ferromagnetic Fermi liquid theory ...... 14
3 Pomeranchuk Instabilities ...... 17
3.1 Pomeranchuk instabilities in 3D ...... 17
3.2 Pomeranchuk instabilities in 2D ...... 22
3.3 Nematic instabilities ...... 24
3.4 Generalized Pomeranchuk instabilities ...... 27
3.5 Quantum criticality ...... 28
iii 4 Tractable Crossing-Symmetric Equation Method ...... 32
5 Model ...... 42
6 Solution Techniques ...... 45
6.1 Parameter space near GPIs ...... 45
6.2 Graphical and numerical methods ...... 46
6.3 Calculation of other quantities ...... 48
6.4 Beyond the local limit ...... 52
7 Results: 3D ...... 59
7.1 Repulsive interaction ...... 59
7.2 Attractive interaction ...... 63
7.3 q-dependence of FL parameters ...... 65
7.4 Nematic instability: approach to GPIs ...... 68
7.4.1 Approach to GPIs ...... 69
7.4.2 Approach to q = 0 PIs ...... 70
7.5 Competing quantum fluctuations ...... 71
8 Tractable crossing-symmetric equations in two dimensions ...... 84
8.1 TCSE method formulated in 2D ...... 84
8.2 2D Model ...... 86
8.3 2D Results ...... 88
iv 9 Application to ferromagnetic superconductors ...... 97
9.1 Phase transitions in FMSCs ...... 99
9.2 Transition temperatures ...... 99
10 Conclusions and future directions ...... 104
Appendices ...... 108
A Solution details ...... 109
A.1 3D Integrations ...... 109
A.2 2D Integrations ...... 117
B Tables of 3D Results ...... 121
BIBLIOGRAPHY ...... 154
v LIST OF FIGURES
1 Three channels of two-body interaction vertex Γ. Shown are the particle-
particle (S) channel, particle-hole (T) channel, and exchange particle-
hole (U) channel with their respective momentum transfers...... 13
2 Figures from Quintanilla et al [27]. (a) Unpolarized, undeformed isotropic
Fermi surface, (b) FS with ` = 2 Pomeranchuk deformation, (c) FS
with ` = 3 Pomeranchuk deformation...... 25
3 Figures from Fradkin et al [28]. A quantum nematic phase can be
formed via a melting of a stripe phase or via a distortion of the FS due
to a Pomeranchuk instability...... 26
4 Generic phase diagram involving a quantum phase transition [36]. The
associated quantum critical regime extends to T > 0, and so the effects
of the QCP can be observed experimentally...... 29
5 Three channels of the 2-body vertex function Γ: particle-particle (S),
particle-hole (T), exchange particle-hole (U). The particle-hole and
exchange particle-hole channels are topologically equivalent, and any
channel can be transformed into any other channel by way of exchang-
ing external lines. Shown also are the momentum transfers in each
channel (K, q, and q’, respectively)...... 33
vi 6 Generic examples of reducible and irreducible diagrams. Reducible di-
agrams can be broken into independent parts by drawing a line through
an intermediate state without crossing an interaction line. Irreducible
diagrams cannot be split further into independent parts...... 34
7 Shown here are the diagrammatic representations of Eqns. 67. Parquet
equations for the particle-particle (S), particle-hole (T), and exchange
particle-hole (U) channels are given. Notice that the S,T, and U chan-
nels are all coupled to each other. When describing real many-body
systems, treatment of all three coupled channels is needed...... 35
8 Schematic form of tractable crossing-symmetric equations: F (Landau
interaction function), A (scattering amplitude), D (direct term). . . . 40
9 Two types of contribution to self-energy: (a) Hartree-type, (b) Ex-
change of collective excitations ...... 41
a s 10 Regions of F0 ,F0 parameter space and generalized Pomeranchuk in-
stabilities...... 46
s 11 Sample 3D plot for U=30; solution is intersection point at F0 =
a −0.676,F0 = −9.210...... 48
s 12 Sample 2D plot for U=30; solution is intersection point at F0 =
a −0.676,F0 = −9.210...... 49
vii 13 Schematic representation of closed path used for numerical contour
integration around a singularity marked by the dot at the origin. The
path C is comprised of parts C1,C2,C3, and C4. Path C2 has a radius
R, and path C4 has a radius ...... 55
0 0 14 For 0 ≤ q ≤ 2kF , the Lindhard function is in the range 1 ≥ χ0(q ) ≥ 0.5. 57
15 Equation surface prior to evaluating divergences with a numerical con-
tour integration. Each “spike” corresponds to a value of q in the di-
a vergent range of −2 < F0 < −1 which was sampled by the Gaussian
integration...... 58
16 After treatment of poles ...... 58
17 U=12 Equation surfaces before and after numerical contour integrations. 58
18 In the multicritical branch of solutions, it is unclear which channel
drives the multicritical behavior. The multicritical point is approached
with increasing U...... 68
0 a 19 With no q -dependence, F0 in the weak FM branch approaches -2,
the q’=2 GPI; with the addition of q0-dependence parametrized by
a c = 1, α = 0.2, to F0 , it can move into this region and approach a
GPI in the middle of this original range of finite-q’divergences. With
a a c = 1, α = 1 parametrization, F0 moves all the way through this
region and can get arbitrarily close to -1, the q’=0 FM PI...... 69
viii s 20 Scaled spin-symmetric Landau parameters (F` ) upon approach to charge
density instability. Dashed line indicates position of q0 = 0 PI for any
s s F2 s channel. Note that F2 crosses its instability ( 5 = −1) before F0
reaches -1 (the CD instability)...... 71
s 21 Scaled spin-symmetric Landau parameters (F` ) upon approach to FM
instability. Dashed line indicates position of q0 = 0 PI for any channel.
s s F2 a Note that F2 crosses its instability ( 5 = −1) before F0 reaches -2
(the GPI associated with the FM instability)...... 72
a 22 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to
charge density instability. Dashed line indicates position of q0 = 0
a s PI for any channel. Note that no F` ’s reach an instability before F0
reaches -1 (the CD instability)...... 73
a 23 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to
FM instability. Dashed line indicates position of q0 = 0 PI for any
a a channel. Note that no F` ’s reach an instability before F0 reaches -2
(the GPI associated with the FM instability)...... 74
s 24 Scaled spin-symmetric Landau parameters (F` ) upon approach to charge
0 a density instability for the case of added q -dependence in F0 (c = 1, α =
1 parametrization). Dashed line indicates position of q0 = 0 PI for any
channel...... 75
ix s 25 Scaled spin-symmetric Landau parameters (F` ) upon approach to FM
0 a instability for the case of added q -dependence in F0 (c = 1, α = 1
parametrization). Dashed line indicates position of q0 = 0 PI for any
channel...... 76
s 26 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to
0 a charge density instability for the case of added q -dependence in F0
(c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0
s s F2 PI for any channel. Note that F2 crosses its instability ( 5 = −1)
s before F0 reaches -1 (the CD instability)...... 76
s 27 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to
0 a charge density instability for the case of added q -dependence in F0
(c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0
s s F2 PI for any channel. Note that F2 crosses its instability ( 5 = −1)
s before F0 reaches -1 (the CD instability)...... 77
28 Quantum fluctuations in spin channel as function of U for the case
0 a of added q -dependence (c = 1, α = 1) in F0 . Here, fluctuations are
shown including the prefactor and sign appropriate for the spin channel. 77
29 Quantum fluctuations in density channel as function of U for the case
0 a of added q -dependence (c = 1, α = 1) in F0 . Here, fluctuations are
shown including the prefactor and sign appropriate for the density
channel...... 78
x 30 Quantum fluctuations in spin channel as function of U. Here, fluctu-
ations are shown including the prefactor and sign appropriate for the
spin channel (see (117))...... 78
31 Quantum fluctuations in density channel as function of U. Here, fluc-
tuations are shown including the prefactor and sign appropriate for the
density channel (see (117))...... 79
32 Spin fluctuations and density fluctuations in the weak FM branch with
no included prefactors or signs (i.e. independent of channel) as function
of U, where increasing U corresponds to approach to PI...... 79
33 Spin fluctuations and density fluctuations in the strong FM branch
with no prefactors or signs (i.e. independent of channel) as function of
U, where increasing U corresponds to approach to PI...... 80
a 34 Contributions of ` = 0 spin and density fluctuations to F1 with a
0 a q -dependent F0 ...... 80
s 35 Contributions of ` = 0 spin and density fluctuations to F1 with a
0 a q -dependent F0 ...... 81
a 36 Contributions of ` = 0 spin and density fluctuations to F2 with a
0 a q -dependent F0 ...... 81
s 37 Contributions of ` = 0 spin and density fluctuations to F1 with a
0 a q -dependent F0 ...... 82
xi 38 Magnetism as a function of direct repulsive interaction in the weak FM
(multicritical) branch of solutions...... 82
39 Magnetism as a function of direct repulsive interaction in the strong
FM (multicritical) branch of solutions...... 83
40 Geometry of the isotropic two dimensional system. Momentum vectors
~p and p~0 make angles φ and φ0, respectively, with the x axis...... 85
41 The real part of the 2D Lindhard function shown for several values of
frequency (ν, scaled to EF ) as a function of momentum transfer (q,
scaled to kF ). Figure from [51]...... 87
a s 42 Both F0 and F0 approach the PI of -1 for increasing U for all fre-
quencies, and this multicritical behavior is shown here for the case of
ν =0...... 91
43 ν = 0.3: Spin-symmetric (charge density channel) FL interaction func-
s s tions (F1 and F2 ) plotted as a function of distance to FM PI. The
s behavior of F0 is shown in Fig. 42. The PI value (-1) plotted as a red
s dashed line. Notice that the charge density nematic instability F2 is
crossed en route to the FM PI...... 95
44 ν = 0.3: Spin-antisymmetric (spin channel) FL interaction functions
a a (F1 and F2 ) plotted as a function of distance to FM PI. The PI value (-
1) plotted as a red dashed line. Notice that the spin nematic instability
a F1 is crossed en route to the FM PI...... 96
xii 45 One possibility for the coexistence of FM and singlet superconductiv-
ity is a long-wavelength spin-flip spiral, in which the superconducting
coherence length is larger than the magnetic correlation length [56]. . 98
46 Taken from [59]. (a) Phase diagram as a function of pressure for UGe2,
with TC indicating a first-order PM-FM phase transition, and Tx indi-
cating the first-order transition within the FM phase from a strong FM
(FM2) to weak FM (FM1). (b) Shown are the transitions in magneti-
zation as a function of pressure at T=2.3K; at low pressure, the system
is in the FM2 phase, then as pressure increases, there is a jump in M
indicating a transition to the FM1 phase, and at even higher pressure,
M drops to zero, indicating a transition to the PM phase. (c) Phases in
the presence of an external magnetic field H (px and pc indicate critical
pressure where Tx and TC → 0...... 102
47 Adapted from [63] by Anne de Visser [62], shown here is the phase
diagram as a function of pressure for UIr, as determined by resistiv-
ity, magnetization, and ac-susceptibility measurements under pressure.
UIr has 3 FM phases, and the SC dome lies in the FM3 phase (the
weakest FM phase), very close to the FM QCP...... 103
48 Calculated Tc/Tx ratio as a function of U for three different parame-
terizations of the q-dependence...... 103
xiii 49 For every value of q in the range of integration (0 to 2kF ; note that q
is scaled to kF in this figure), there is a corresponding F0,pole at which
a divergence occurs...... 112
s,a 50 For every value of F0 between -1 and -2, there is a corresponding qpole
at which a divergence occurs. Note that q is scaled to kF in this figure. 113
51 Schematic representation of closed path used for numerical contour
integration around a singularity marked by the dot at the origin. The
path C is comprised of parts C1,C2,C3, and C4. Path C2 has a radius
R, and path C4 has a radius ...... 114
xiv LIST OF TABLES
1 Solution sectors as defined in parameter space...... 45
2 Large U liming results for local FL...... 60
3 Large U limiting results for repulsive interaction...... 60
4 Sample of contributions to pairing amplitude in weak FM branch of
solutions from scattering amplitudes in s-p approximation...... 63
5 Large U limiting results for attractive interaction...... 64
a 6 Summary of results for q-dependent F0 ...... 68
a a 7 Large U limiting values of F0 (F0 maximum) for different combinations
of c, α...... 70
8 Quantum fluctuation integrals in weak FM and strong FM branches
for various values of U. Note that the trend for spin fluctuations is the
same in both branches, but the trend of density fluctuations differs
between the two branches...... 83
s,a 9 Calculated values of F0 as functions of underlying interaction U for
frequencies ν = 0, 0.1, 0.3 in 2D in the ferromagnetic state...... 88
s,a 10 Calculated values of F0 as functions of underlying interaction U for
frequencies ν = 0.5, 0.8 in 2D in the ferromagnetic state...... 89
s,a 11 Calculated values of F0 as functions of underlying interaction U for
frequencies ν = 0 and ν = 0.1 in 2D in the paramagnetic state. . . . . 89 xv s,a 12 Calculated values of F0 as functions of underlying interaction U for
frequencies ν = 0.5 in 2D in the paramagnetic state...... 90
s,a 13 Calculated values of F0 as functions of underlying interaction U for
frequencies ν = 0.8 in 2D in the paramagnetic state...... 90
s,a s,a 14 Calculated values of F1 ,F2 as functions of underlying interaction U
for frequency ν = 0.1 in 2D in the ferromagnetic state...... 93
s,a s,a 15 Calculated values of F1 ,F2 as functions of underlying interaction U
for frequency ν = 0.3 in 2D in the ferromagnetic state...... 93
s,a s,a 16 Calculated values of F1 ,F2 as functions of underlying interaction U
for frequency ν = 0.5 in 2D in the ferromagnetic state...... 94
s,a s,a 17 Calculated values of F1 ,F2 as functions of underlying interaction U
for frequency ν = 0.8 in 2D in the ferromagnetic state...... 94
18 Weights (wi) and abscissa (xi) for a 10-point Gaussian integration. . . 110
19 For F0 = −1.25 and R = kF , is varied to check the sensitivity of the
contour integration to the value of . Ji’s are the values of the integrals
along a particular segment of the contour, as shown in Fig. 51. . . . . 115
xvi 20 For F0 = −1.25 and R = kF , is varied to check the sensitivity of the
contour integration to the value of . J is the sum of all integrals over
individual contour segments (Ji’s, as shown in Table 19), i.e. the inte-
gral over the full contour, which, according Cauchy’s residue theorem,
should equal zero (since no poles are enclosed by the contour). Thus,
J can be thought of as being representative of the “error” associated
with a particular choice of and R. The principal value (PV) of the
integral is the sum of J1 and J3...... 115
21 For F0 = −1.25 and = 0.0001, R (scaled to kF ) is varied to check the
sensitivity of the contour integration to the value of R. J2 is the values
of the integral along a particular contour C2, as shown in Fig. 51.
J is the sum of all integrals over individual contour segments (Ji’s,
as shown in Table 19), i.e. the integral over the full contour, which,
according Cauchy’s residue theorem, should equal zero (since no poles
are enclosed by the contour). Thus, J can be thought of as a measure
of error associated with a particular choice of and R...... 116
22 Comparison between calculation of PV using individual contour seg-
ments J1 + J3 = −(J2 + J4) for = 0.0001 and R = kF and using
NIntegrate’s “Principal Value” option to solve integrals. Shown here
are comparisons for two values of F0,pole: -1.25 and -1.75...... 117
23 Weights (wi) and abscissa (xi) for a 6-point Gaussian integration. . . 118
xvii 24 Weights (wi) and abscissa (xi) for a 25-point Gaussian integration. . . 119
25 c = 1, α = 0.2: Strong FM solution (physical branch) ...... 123
26 c = 1, α = 0.2: Strong FM solution (unphysical branch) ...... 124
27 c = 1, α = 0.2: Weak FM solution (physical branch) ...... 125
28 c = 1, α = 0.2: Weak FM solution (unphysical branch #1) ...... 126
29 c = 1, α = 0.2: Weak FM solution (unphysical branch #2) ...... 127
30 c = 1, α = 0.2: Weak FM solution (unphysical branch #3) ...... 128
s,a 31 FL interaction functions F` : strong (large moment) FM solutions (no
s,a q-dependence in F0 :) ...... 129
s,a 32 Scattering amplitudes A` : strong (large moment) FM solutions (no
s,a q-dependence in F0 :) ...... 130
s,a 33 Scattering amplitudes A` : strong (large moment) FM solutions (no
s,a q-dependence in F0 :) ...... 131
34 Pairing amplitudes and effective mass: strong (large moment) FM so-
s,a lutions (no q-dependence in F0 :) ...... 132
s,a 35 FL interaction functions F` : weak (small moment) FM solutions (no
s,a q-dependence in F0 :) ...... 133
s,a 36 Scattering amplitudes A` : weak (small moment) FM solutions (no
s,a q-dependence in F0 :) ...... 134
37 Pairing amplitudes and effective mass: weak (small moment) FM so-
s,a lutions (no q-dependence in F0 :) ...... 135
xviii s,a s,a 38 FL interaction functions F` : PM solutions (no q-dependence in F0 :) 136
s,a s,a 39 Scattering amplitudes A` : PM solutions (no q-dependence in F0 :) . 137
40 Pairing amplitudes and effective mass: PM solutions (no q-dependence
s,a in F0 :) ...... 138
s,a 41 Negative U: FL interaction functions F` : strong (large moment) so-
s,a lutions (no q-dependence in F0 :) ...... 139
s,a 42 Negative U: Scattering amplitudes A` : strong (large moment) FM
s,a solutions (no q-dependence in F0 :) ...... 140
43 Negative U: Pairing amplitudes and effective mass: “strong” phase
s,a separation solution branch (far from PI’s) (no q-dependence in F0 :) 141
s,a 44 Negative U: FL interaction functions F` : “weak” phase separation
s,a solution branch (near PI’s) (no q-dependence in F0 :) ...... 142
s,a 45 Negative U: Scattering amplitudes A` : “weak” phase separation so-
s,a lution branch (near PI’s) solutions (no q-dependence in F0 :) . . . . 143
46 Negative U: Pairing amplitudes and effective mass: “weak” phase sep-
s,a aration solution branch (near PI’s) (no q-dependence in F0 :) . . . . 144
s,a 47 Negative U: FL interaction functions F` : PM solutions (no q-dependence
s,a in F0 :) ...... 145
s,a 48 Negative U: Scattering amplitudes A` : strong (large moment) FM
s,a solutions (no q-dependence in F0 :) ...... 146
xix 49 Negative U: Pairing amplitudes and effective mass: strong (large mo-
s,a ment) solutions (no q-dependence in F0 :) ...... 147
a 50 q-dependent F0 with “optimal” parameterization c = 1, α = 1: FL
s,a interaction functions F` : weak FM solutions) ...... 148
a 51 q-dependent F0 with “optimal” parameterization c = 1, α = 1: Scat-
s,a tering amplitudes A` : weak FM solutions ...... 149
a 52 q-dependent F0 with “optimal” parameterization c = 1, α = 1: Pairing
amplitudes and effective mass: weak FM solutions:) ...... 150
a s 53 q-dependent F0 and F0 with parameterizations c = 1, α = 1: FL
s,a interaction functions F` : weak FM solutions) ...... 151
a s 54 q-dependent F0 and F0 with parameterizations c = 1, α = 1: Scatter-
s,a ing amplitudes A` : weak FM solutions ...... 152
a s 55 q-dependent F0 and F0 with parameterizations c = 1, α = 1: Pairing
amplitudes and effective mass: weak FM solutions:) ...... 153
xx Acknowledgements
“Find something difficult to do and do it,” wrote the wise Dan Bejar of rock bands
Destroyer, The New Pornographers, and Swan Lake. It is with this spirit that I have pressed forward in completing this research and this record of it. It never would have happened without the support of my family, in particular Pat Reidy, Tom Reidy, and
Nick Reidy (and I’d get in trouble if I didn’t also mention Ann and Bill Campbell,
Carol and Susan Reidy, Hedy Scherer, and James O. (“Big Cheese”) Martin). But let’s be honest. I learned everything I know about the topic at hand (condensed matter physics) from my advisor, our collaborator, and the amazing professors I’ve had along the way. So, in terms of physics, I need to first and foremost thank
Khandker Quader, the most patient person I know. He taught me everything from
BCS theory to how to smoothly pronounce “Pomeranchuk” and so graciously shared his knowledge and experience with me throughout my time working with him. I would also like to thank Kevin Bedell, whom I consider my other advisor, for many insightful discussions, much inspiration, and a few much-needed drinks along the way.
To my committee members, especially David Allender, who taught me Statistical
Mechanics (for which I was always late. . . sorry, Dr. Allender!) and Solid State
Physics, and to all the professors at KSU that helped me along the way. Also thanks to the administrative staff throughout my time at Kent, especially Loretta Hauser and
Cindy Miller. Outside of Kent State, I would be remiss if I didn’t thank some others. xxi For many enlightening and thought-provoking discussions, I thank Andrey Chubukov
and Dmitri Maslov. For so much funding, I thank ICAM-I2CAM and in particular,
Daniel Cox, for being a great ally when I needed one. I wouldn’t be where I am
today (i.e. the greatest city in the world, the city that never sleeps) if not for Vadim
Oganesyan. I also would like to thank Harsh Mathur, my first collaborator outside
of my dissertation work, who not only taught me how to “collaborate with adults”
but made me immensely more confident in myself as a physicist. Going way back, I
must also thank Nathan Harshman, the fascinating man whose endless enthusiasm for
introductory physics got me hooked in the first place. And on a more personal note,
I wish to thank several important people in my life, who, despite being physicists,
have become my friends. They have inspired me, encouraged me, and generally kept
me sane during my time spent as a Ph.D. student. Their names are Jason Ellis,
Elena Long, Ronny Thomale, Jean-Philippe Reid, Tanmoy Das, Tanya Ostapenko,
and Andrei Bernevig.
“Poets say science takes away from the beauty of the stars - mere globs of gas atoms. I, too, can see the stars on a desert night, and feel them. But do I see less or more?” -Richard P. Feynman
xxii Dedicated to June B. Reidy.
xxiii CHAPTER 1
Introduction
The study of quantum many-body systems is a major aspect of modern physics. I present here an exploration of many-body systems near a class of instabilities known as “Pomeranchuk instabilities.” These instabilities were first described by I. Pomer- anchuk in 1958 [3]. Pomeranchuk instabilities (PIs) are related to broken symmetries and deformations of the Fermi surface in Fermi liquid systems. Pomeranchuk insta- bilities are associated with collective modes of such systems and large fluctuations about the quantum ground state. In modern parlance, these instabilities are con- sidered quantum critical points (QCPs) [4] in parameter space. Quantum criticality is currently of much interest in the condensed matter community. Many correlated- electron and atomic many-body systems which are not yet fully understood exhibit quantum critical behavior in regions of their phase diagrams; particularly, supercon- ductivity is typically found near a magnetic QCP in systems like high-Tc cuprates, iron pnictides, and ferromagnetic superconductors (see, for example, [5], [6], [8], [7]). Typ- ically, QCPs are approached from a disordered state, but in this work, we approach quantum criticality from an ordered state, thus providing an alternative viewpoint on quantum criticality.
This dissertation aims to present a general study of the physics near Pomeranchuk
1 2 instabilities (PIs) [3] in isotropic Fermi liquids (FLs) [1, 2]. When working in a FL framework, this involves the calculation of the quasiparticle interaction functions (also referred to as FL interaction functions). From these FL interaction functions, many other measurable properties of the system can be calculated following standard Fermi liquid theory [25], such as effective mass, compressibility, ultrasonic attenuation, and other transport, dynamic, and thermodynamic properties.
Historically, these FL interaction functions were fitted from experiments, but in this work, they are calculated microscopically. To calculate these FL interaction func- tions, we employ the tractable crossing symmetric equation (TCSE) method [15–20].
The TCSE method is a diagrammatic many-particle method that is used to calculate the Fermi liquid interaction functions through considering the s (particle-particle), t
(particle-hole), and u (exchange particle-hole) channels in a conserving, self-consistent fashion. After partial resummation of diagrams in these channels, quasiparticle renor- malization, and enforcement of crossing symmetry, one arrives at a set of coupled non- linear integral equations from which the FL interaction functions can be calculated. A unique aspect of this method is its ability to simultaneously consider underlying inter- actions of arbitrary strength/range and competing quantum fluctuations (i.e. density, spin, current, spin current). Another aspect of TCSE is that it goes “beyond” the usual random phase approximation (RPA) technique in the sense that the interac- tions are the renormalized interaction functions of Fermi liquid theory. Thus, density and spin fluctuations, along with higher-order fluctuations (such as current or spin 3 current fluctuations), are generally coupled, leading to feedback between channels.
The TCSE method was created to study three dimensional systems, and it is used in its usual form [15–20, 67] here to study 3D systems. One goal of my dissertation work involves the formulation (and subsequent application) of the TCSE method in two dimensions; in 2D, frequency dependence can be considered.
Once FL interactions functions have been calculated using the TCSE method, the general behaviors of Fermi systems near PIs can be explored. In a Fermi liquid, if the quasiparticle interaction function (F`) becomes too negative in a particular angular momentum channel (`), the Fermi surface becomes unstable to deformations associated with that channel; specifically, in three dimensions, F` ≤ −(2` + 1), and in two dimensions, F` ≤ −1, will result in an unstable Fermi surface. A familiar
a example of a PI is the ferromagnetic (Stoner) instability (F0 → −1); here, the Fermi surface splits into a spin-up surface and a spin-down surface, magnetic susceptibility diverges, and time-reversal symmetry is broken. Also in the ` = 0 (s-wave) channel,
s F0 → −1 marks an approach to a charge density instability.
As a PI is approached, the susceptibility in the relevant channel (henceforth re- ferred to as the “critical channel”) will diverge, indicating a “softness” of the Fermi surface with respect to its deformation. One goal of this work is to determine whether the FL functions in “non-critical” channels are affected when a PI is approached in some critical channel. A typical method for examining the behavior of systems around
QCPs invokes Hertz-Millis-type effective theories [10,11]; applying these theories, one 4
a finds that when a PI is approached in one channel, the effective mass, and hence F1 , also diverge (see, for example, [12]). This is counter to the usual assumption that all non-critical F`’s are unaffected by an approach to a PI in one channel. A re- cent analysis by Maslov and Chubukov [14] of the properties of a 2D Fermi system in the paramagnetic state near a charge nematic (` = 2) PI was done in terms of
Landau FL theory. It was found that near the transition, the system enters into a new critical FL regime, in which all spin components of the FL interaction func-
a s tions (F` ) and all charge components (F` ) with ` 6= 2 diverge at the critical point,
s while the ` = 2 charge FL component, F2 → −5, the PI for this channel. However, owing to cancellation between divergent effective mass and divergent effective Lan- dau component, non-critical channels’ susceptibilities are not found to be affected.
Other work by the same authors [13] finds that a 2D FL crosses the ` = 1 instabil- ity in the spin channel en route to the FM PI and is crossed before this instability and all other possible instabilities near the FM QCP. This is found starting in the magnetically disordered (paramagnetic) state and considering finite-range interaction.
In contrast to the recent work of Maslov and Chubukov [13, 14] and previous work by Quader et al [21], in which PIs are approached starting from the paramagnetic state, the main starting point of this calculation is the ferromagnetic state, using the well-established ferromagnetic Fermi liquid theory of Abrikosov, Dzyaloshinskii, and
Kondratenko [22, 23], which is valid for weakly ferromagnetic systems. This theory was extended by Bedell and collaborators [24] to study thermodynamics, collective 5
excitations, and superconducting instabilities in these weak ferromagnetic systems.
Starting in the magnetically ordered state, we find that both ` = 0 ferromagnetic
and charge density PIs are approached simultaneously, which creates a situation of
“quantum multicriticality” in this branch of solutions. It is also found that upon
approach to these PIs, the system crosses a d-wave (` = 2) nematic PI in the charge
s channel, i.e. F2 → −5. Thus, we conclude that this nematic transition both precedes and is driven by the approach to the ferromagnetic and charge density Pomeranchuk instabilities. As will be discussed, we see this nematic transition despite the fact that only s-wave quantum fluctuations in the exchange particle-hole channel are in- corporated into the TCSE equations; due to long range interactions generated by the medium, these ` = 0 fluctuations are enough to give information about exchange of
fluctuations in any higher angular momentum channel.
Another goal of this work is to compare results with experimental data on ferro-
magnetic superconductors. With the addition of a dependence on momentum transfer
in the FL parameters, we are able to find a correspondence with measured transi-
tion temperatures in these compounds; further properties may be calculated once the
proper q-dependence is matched to such results. The superconducting instability is
also explored near PIs. In the paramagnetic state, we see very clearly that triplet
pairing is the only type allowed. However, we find in the ferromagnetic state that
both singlet and triplet pairing are possible when starting with a repulsive driving
interaction, though singlet pairing is favored. This raises the intriguing possibility of 6 switching between singlet and triplet pairing via some symmetry-breaking mechanism.
The remainder of this dissertation will be organized as follows: Chapter 2 will present background theory on Landau Fermi liquid theory from both a phenomeno- logical and microscopic points of view, followed by a discussion of the ferromagnetic
Fermi liquid theory of Abrikosov, Dzyaloshinskii, and Kondratenko. In Chapter 3, I introduce Pomeranchuk instabilities with extra attention paid to nematic instabili- ties and the relation of Pomeranchuk instabilities to quantum criticality. The TCSE method is derived in 3D and discussed in detail in Chapter 4, followed by a presen- tation of our 3D model in Chapter 5. Chapter 6 highlights solution techniques of the TCSEs including details or graphical and numerical techniques employed in this work. In Chapter 7, 3D results are presented for repulsive and attractive underlying interaction along with results for the case where an explicit q-dependence is added to the FL interaction functions. Multicriticality and quantum fluctuations are also discussed in Chapter 7. The method is formulated for the first time in 2D in Chapter
8, and results of 2D calculations along with our 2D model are also presented here.
In Chapter 9, application to ferromagnetic superconductors is discussed. Chapter 10 presents a summary, discussion, and ideas for future directions. Details of calculation techniques can be found in Appendix A, and detailed results not given in the body of the work can be found in Appendix B. CHAPTER 2
Fermi liquid theory
2.1 Landau Fermi liquid theory: a phenomenological perspective
One of the greatest challenges of modern condensed matter physics is the descrip- tion of interacting many-body systems. In 1956, Landau developed his phenomeno- logical theory of interacting Fermi systems [1,2,25]. In Landau’s Fermi liquid theory
(FLT), we start with a non-interacting Fermi gas. The ground state of such a system is an isotropic distribution in which energy levels are occupied up to a cutoff energy, called the Fermi energy (EF ), which also corresponds to a Fermi momentum (pF ),
and states above this level are unoccupied; the occupation obeys the Pauli exclusion
0 principle. In other words, at T=0, the distribution function npσ = 1 for p < pF and
0 npσ = 0 for p > pF .
This boundary between occupied and unoccupied states, called the “Fermi level”, forms a surface called the “Fermi surface” (FS) in momentum space; in three di- mensional momentum space, the FS is a sphere (for such a non-interacting isotropic
Fermi system), and in two dimensions, it is a circle. Then, if interactions between these fermions are turned on adiabatically, Landau described a new interacting system resulting from this; states of the interacting system are assumed to have a one-to-one correspondence with states of the non-interacting system; the number of states does
7 8 not change between the interacting and non-interacting systems, however, spacing of the energy levels may change as interactions are turned on. This is a key idea of Landau’s FLT. Excitations of the interacting system are known as quasiparticles
(p > pF ) and quasiholes (p < pF ). These excitations correspond to taking a particle from inside the Fermi surface (which becomes the quasiparticle of the interacting sys- tem) and putting it outside the FS, leaving behind a hole in its place (which becomes the quasihole of the interacting system). Once interactions have been turned on, the energy thus becomes a functional of quasiparticle distribution. The energy of the interacting system is given by:
X 1 X E = E + δn + f 0 0 δn δn 0 0 , (1) 0 p pσ 2 pσ,p σ pσ p σ p,σ p,p0,σ,σ0
where E0 is the ground state energy, p is the kinetic energy of a quasiparticle,
δnpσ is the change of quasiparticle distribution function, and fpσ,p0σ0 is the “Landau interaction function” or “FL interaction function”. It is defined as the second func- tional derivative of E. This last term in the energy is the interaction between two quasiparticles pσ and p0σ0, where p, p0, σ, σ0 are relative momenta and spins in relative and center of mass frame.
In isotropic real or momentum space and in spin-rotationally-invariant space, the FL interaction function can be expressed in terms of a spin-symmetric and - antisymmetric part as
σσ0 s a 0 fpp0 = fpp0 + fpp0 σ · σ (2)
and are generally expressed in this work as scaled to density of states at the Fermi 9
surface:
s,a s,a N(0)fpp0 = Fpp0 . (3)
In 3D, these can be expanded in Legendre polynomials in order to separate out various components of angular momentum, as
s,a X s,a Fpp0 = F` P`(cos θ). (4) ` In 2D, they can be expanded in, for example, cosines:
s,a X s,a Fpp0 = F` cos(θ`). (5) ` where, in both cases, θ is the angle between ~p and ~p0 and is also referred to as
s,a s the Landau angle (θL), and F` are the spin-symmetric (F` ) and spin-antisymmetric
a (F` ) FL interaction functions in a particular angular momentum channel `.
Landau envisioned that the first few FL interaction functions could be obtained by fits to various experiments and then used to predict transport, dynamic, thermo- dynamic, and superconducting properties. However, with modern developments in microscopic approaches, such as the tractable crossing-symmetric equation method, these Landau interaction functions can be calculated, as will be shown shortly. The
FL interaction function in this form is the starting point for calculating properties of
Fermi liquids.
2.2 Landau Fermi liquid theory: a microscopic perspective
Landau justified his FL theory via a microscopic derivation [2] (see also [26]), which will be sketched briefly here. Consider a single-quasiparticle Green’s function, 10
which has a singular piece and a “regular” or ”incoherent” piece, given in Eqn. 6 as
Ginc(p). The singular piece (the first term in Eqn. 6) has a pole at the quasiparticle energy (p) and gives the renormalization zp at this quasiparticle pole. The imaginary
part of the singular piece gives a measure of the quasiparticle lifetime; here, is an arbitrary energy variable, and µ is chemical potential.
z G(~p,) = p + Ginc(p) (6) − p + iδsgn(p − µ)
Now consider the intermediate state GpGh of a quasiparticle-quasihole process near the Fermi surface (physically, this comes from making a virtual particle-hole excitation in the many-body medium). Recall that momentum below pF corresponds to a quasihole, and momentum above pF corresponds to a quasiparticle. If the quasiparticle-quasihole pair has momenta p + q/2 and p − q/2, either momentum can correspond to quasiparticle or quasihole, depending on its location relative to the
Fermi surface. Thus, the intermediate state is given by z z G(p + q/2)G(p − q/2) = p+q/2 p−q/2 [ + ω/2 − p+q/2 + iδ][ − ω/2 − p−q/2 − iδ] z z + p+q/2 p−q/2 (7) [ + ω/2 − p+q/2 − iδ][ − ω/2 − p−q/2 + iδ] z z +Ginc(p) p+q/2 + p−q/2 + [Ginc(p)]2, + ω/2 − p+q/2 + iδ − ω/2 − p−q/2 − iδ
where ω is the energy transfer accompanying the momentum transfer q in a scat- tering or interaction process.
The first two terms above are highly singular, having two poles each. The last two terms can be considered the “regular” pieces, as they have at most one pole. We shall 11
be concerned with the (q, ω) → 0 limit, as quasiparticles/holes are only defined in a small range around the Fermi surface. This limit ensures that the quasiparticles/holes will be in this range. The highly singular terms have the behavior of approaching the poles from opposite sides of the Fermi surface as this limit is taken, making them non-analytic in this relevant limit. In other words, the (q, ω) → 0 limit is not unique
(note: the “regular” terms have no such problem regarding this limit). In the limit
q → 0, we have
zp±q/2 → zp (8)
p±q/2 → p ± q · vp/2,
where vp is defined as the velocity corresponding to momentum p. Using the relation
z p = − ω/2 − p−q/2 − iδ
zp + 2πizpδ( − ω/2 − p + q · vp/2), (9) − ω/2 − p + q · vp/2 + iδ
we can express the first term of eq. 7 as
z2 1 p × + ( + ω/2 − − q · v /2 + iδ) ( − ω/2 − + q · v /2 + iδ) p p p p (10) 2 2πizpδ( − ω/2 − p + q · vp/2)
+ ω/2 − p − q · vp/2 + iδ
In the first term, the poles are on the same side of the real axis and thus do not
cause difficulty. However, careful attention must be given when taking the FS limit of 12 the second term. In the end, we find that the singular piece of GpGh can be written as
2 q · vp R(p, q) = 2πizp δ(p − µ)δ( − p) (11) ω − q · vp + i|δ|
If we define the ratio s = q/ω, then we can write
s 2 s · vp R = 2πizp (12) 1 − s · vp + iδ
Here, we are interested in the limit where both q and ω go to 0. Clearly, the order in which these two limits are taken matters in the ratio s = q/ω.
Case 1: q → 0 first, thus s → 0.
Case 2: ω → 0 first, thus s → ∞.
Here, I would like to introduce the two-body vertex function Γ, which sums all two- body quantum processes, which can be viewed in three ways: the particle-particle (S) channel, the particle-hole (T) channel, and the exchange particle-hole (U) channel. If the total momentum of the two-particle process is K = p1 + p2, then the momentum transfer in the particle-hole channel is q = p1 − p3 and the momentum transfer in the
0 exchange particle-hole channel is q = p1 − p4. In this way we can define the channels, as is shown in Figure 1. The 2-body vertex function Γ in a particle-hole channel is
s related to the product GpGh (and thus the singular part R ) as
X Γs(p, p0) = I(p, p0) + I(p, p00)[G(p00)]2 + Rs(p00)Γs(p00, p0), (13) p00,σ00
where I(p, p0) represents irreducible processes in the particle-hole channel and is well-behaved in the limit q, ω → 0. Thus, the singularities in Γ(p, p0) arise only due 13
Figure 1: Three channels of two-body interaction vertex Γ. Shown are the particle- particle (S) channel, particle-hole (T) channel, and exchange particle-hole (U) channel with their respective momentum transfers.
to the singularities in GpGh which have been discussed in detail above. Taking the
0 s → 0 or ∞ limit of GpGh is equivalent to taking these limits of Γ(p, p ) in the p-h channel.
We have two limiting cases:
s 0 0 lim 2πizpzp0 Γσ,σ0 (p, p ) = fσ,σ0 (p, p ) (14) s→0
s 0 0 lim 2πizpzp0 Γσ,σ0 (p, p ) = aσ,σ0 (p, p ) (15) s→∞
Thus, the two limits s → 0 and s → ∞ of the vertex function respectively yield the
0 0 Landau interaction function fσ,σ0 (p, p ) and the scattering amplitude aσ,σ0 (p, p ). This 14 is valid for p, p0 on the Fermi surface. To extrapolate away from the Fermi surface, we will use the tractable crossing-symmetric equation method discussed later in the text.
2.3 Ferromagnetic Fermi liquid theory
A key assumption in Landau’s FLT was starting with a paramagnetic ground state and adiabatically reaching an interacting state, while maintaining a one-to-one corre- spondence between all such states. In the 1950s and ’70s, Abrikosov, Dzyaloshinskii, and Kondratenko [22, 23] developed a similar theory for weakly ferromagnetic Fermi liquids (FFLT), which was further developed and extended by Bedell and Blagoev [24].
In the case of a FFL, due to interactions, an internal field (magnetic moment) is spon- taneously generated. In this theory, the ground state is a Weiss mean-field state, and the quasiparticles are built from this internal mean-field, which gives rise to internal magnetism.
The energy in a ferromagnetic FL can be expanded in terms of the magnetization
(m) as: 1 + F a E = E + 0 m2 + bm4 + ... (16) 0 2N(0)
The FFLT only holds for weak equilibrium magnetization (m0), which, in this theory, is given by
a 1/2 m0 ∼ |1 + F0 | , (17)
and in general is m0 = n↑ −n↓, where nσ is the occupation number of particles with spin σ. In this limit of weak magnetization, the quasiparticle distribution function 15 and energy function are given in terms of magnetization (mp) and effective magnetic
field (hp) as
n~pαα0 (~r, t) = n~p(~r, t)δαα0 + mp(~r, t) · ~σαα0 (18)
~pαα0 (~r, t) = ~p(~r, t)δαα0 + hp(~r, t) · ~σαα0 . (19)
This effective magnetic field hp includes the external magnetic field B and the
a internal magnetic field generated by interactions between quasiparticles (fpp0 ):
X a 0 hp = −B + 2 fpp0 mp (20) p0
In the limit of weak magnetization, the quasiparticle interaction can be treated as spin-rotationally invariant (while spins do point in a particular direction in this ferromagnetic state, any direction is equally likely), and can be expressed (similar to in Eq. 2 for regular Fermi liquids) as
σσ0 s a 0 2 fpp0 = fpp0 + fpp0 σ · σ + O(m0). (21)
2 The higher order terms O(m0) are irrelevant in the limit of weak m0, and in 3D
σσ0 we can expand fpp0 in Legendre polynomials as
σσ0 X s a 0 N(0)fpp0 = (F` + F` σ · σ )P`(cos θ), (22) ` where θ is the angle between ~p and ~p0.
Using this FFLT in conjunction with microscopic approaches (such as the one
s,a discussed in this work), these FL interaction functions F` can be calculated for 16 ferromagnetic systems. Thus, the magnetism can be readily obtained from micro-
a scopic calculation of (F0 ) (Eq. 17), and with a formulation of a Fermi liquid theory
s,a in place for FM systems, once F` are calculated, FL concepts can be applied to cal- culate other properties of such systems (scattering amplitudes, transport, dynamic, and thermodynamic properties) as discussed in the previous section on Landau FLT. CHAPTER 3
Pomeranchuk Instabilities
3.1 Pomeranchuk instabilities in 3D
In a Fermi liquid theory, there exists a requirement for thermodynamic stability,
first noted by Pomeranchuk [3]. If the FL parameters are large enough (exact val- ues depend on dimensionality and will be discussed shortly) and negative, then the stabilizing effects of the Fermi pressure are overcome, and the Fermi surface becomes
“soft” with respect to a particular deformation, indicating a thermodynamic insta- bility [12]. At this point, a corresponding susceptibility diverges. By requiring that the ground state energy is a minimum, the Pomeranchuk result can be derived. The presentation here closely follows that in Reference [25].
Consider an isotropic Fermi surface that is distorted in some way. The distribution function can be written as
np,σ = Θ(pf (θ, σ) − p), (23)
where the distortion is characterized by a varying Fermi momentum δpf (θ, σ) =
0 pf (θ, σ) − pf .(θ = polar angle of Fermi momentum, σ = spin orientation), and Θ(x) is simply a Heaviside theta function (x > 0 : Θ(x) = 1, x < 0 : Θ(x) = 0). To have thermodynamic stability, the ground state energy must be a minimum and not just stationary. Thus E − µN, the energy in the grand canonical ensemble,where µ is 17 18
chemical potential and N is particle number (conserved in grand canonical ensemble),
must be a minimum in the undistorted case. The distortion will cause some change
in E − µN, which, to second order, can be written as
X 0 1 X (E − µN) − (E − µN) = (ε − µ)δn + f 0 0 δn δn 0 0 , (24) 0 p pσ 2 pσ,p σ pσ p σ ~pσ ~pσ,p~0σ0 where
1 ∂ δn = n − n0 = (δp (θ, σ))δ(p − p) − (δp (θ, σ))2 δ(p − p). (25) pσ pσ p f f 2 F ∂p f
Here, as discussed in the chapter on FL theory, the first term on the right side of
Eq. 24 is the energy of adding a single quasiparticle (pσ), and the second term is the
0 0 interaction between quasiparticles (pσ and p σ ) with fpσ,p0σ0 being the quasiparticle
interaction function of Landau FL theory.
Following Baym-Pethick [25], Eq. 24 can be calculated to second order in δpf . In
three dimensions, first consider the first term on the right side of Eq. 24, using the
0 fact that εp = µ − vF (p − pF ), where vF is the Fermi velocity.
X 0 X (εp − µ)δnpσ = vF (p − pF )(δpF (θ, σ))δ(p − pF ) ~pσ ~pσ 1 ∂ − (p − p )(δp (θ, σ))2 δ(p − p ). (26) 2 F F ∂p F
Converting the sum over ~p to an integral in three dimensions, we have
2π 1 ∞ Z Z Z vF X 2 dφ d cos θ dpp (p − pF )(δpF (θ, σ))δ(p − pF ) (2π~)3 σ 0 −1 0 1 ∂ − (p − p )(δp (θ, σ))2 δ(p − p ). (27) 2 F F ∂p F 19
Upon integrating over p, the first term vanishes (due to the delta function δ(p−pF )
causing the (p − pF ) in this term to go to zero). The second is integrated by parts to
yield 2π 1 Z Z vF X 2 2 dφ d cos θpF (δpF (θ, σ)) . (28) 2(2π~)3 σ 0 −1 Nothing that 2 pF N(0) = 2 3 , (29) vF π ~
and integrating over φ, we find
1 v2 N(0) X Z d cos θ F (δp (θ, σ))2. (30) 4 2 F σ −1
δpF (θ, σ) can be expanded in Legendre polynomials as
X νF δpF (θ, σ) = ν`σP`(cos θ). (31) `
So Eq. 30 becomes
1 N(0) X Z d cos θ X [ν P (cos θ)]2. (32) 4 2 `σ ` σ −1 `
Upon summing over spins and integrating over cos θ, we find, for the first term on the right side of Eq. 24,
N(0) X 1 [(ν + ν )2 + (ν − ν )2]. (33) 8 2` + 1 `↑ `↓ `↑ `↓ `
Next, consider the second term on the right side of Eq. 24:
1 X f 0 0 δn δn 0 0 . (34) 2 pσ,p σ pσ p σ ~pσ,p~0σ0 20
Using the definition of δnpσ from Eq. 25, this term becomes
1 X 1 2 ∂ f 0 0 (δp (θ, σ))δ(p − p) − (δp (θ, σ)) δ(p − p) 2 pσ,p σ F F 2 F ∂p F ~pσ,p~0σ0 (35) 1 ∂ ×(δp0 (θ0, σ0))δ(p − p0) − (δp0 (θ0, σ0))2 δ(p − p0), F F 2 F ∂p0 F
2 which, to second order in (δpF ) , is
1 X 0 0 0 0 f 0 0 [(δp (θ, σ))(δp (θ , σ ))δ(p − p)δ(p − p )]. (36) 2 pσ,p σ F F F F ~pσ,p~0σ0
Converting the sums over ~p and p~0 to integrals,
2π 2π 1 1 ∞ ∞ Z Z Z Z Z Z 1 X 0 0 2 0 02 6 dφ dφ d cos θ d cos θ dpp dp p 2(2π~) 0 σ,σ 0 0 −1 −1 0 0 (37)
0 0 0 0 ×fpσ,p0σ0 [(δpF (θ, σ))(δpF (θ , σ ))δ(pF − p)δ(pF − p )].
The integrals over p and p’ yield
2π 2π 1 1 4 Z Z Z Z pF X 0 0 0 0 0 0 0 6 dφ dφ d cos θ d cos θ fpσ,p σ (δpF (θ, σ))(δpF (θ , σ )) (38) 2(2π~) 0 σ,σ 0 0 −1 −1 Integrating over φ and φ0 and recalling Eq. 29, this becomes
1 1 2 Z Z 0 [N(0)vF ] X d cos θ d cos θ X 0 0 0 fpσ,p0σ0 (δpF (θ, σ))(δpF (θ , σ )) (39) 8 0 2 2 σ,σ −1 −1 ` 0 0 Expanding δpF (θ, σ) and δpF (θ , σ ) in Legendre polynomials as in Eq. 31, we have
1 1 2 Z Z 0 [N(0)] X d cos θ d cos θ X 0 fpσ,p0σ0 ν`σP`(cos θ)ν`σ0 P`(cos θ ) (40) 8 0 2 2 σ,σ −1 −1 ` Taking the sums over spins,
1 1 2 Z Z 0 [N(0)] d cos θ d cos θ X 0 2 [P (cos θ)P (cos θ )] × [(f 0 ν 8 2 2 ` ` p↑p ↑ `↑ −1 −1 ` (41)
2 +fp↓p0↓ν`↓ + fp↑p0↓ν`↑ν`↓ + fp↓p0↑ν`↓ν`↑]. 21
The quasiparticle interaction fpσ,p0σ0 can be expressed in terms of its spin-symmetric and spin-antisymmetric components as
s a fp↑,p0↑ = fp↓,p0↓ = fpp0 + fpp0 (42) s a fp↑,p0↓ = fp↓,p0↑ = fpp0 − fpp0 .
Then Eq. 41 becomes
1 1 2 Z Z 0 [N(0)] d cos θ d cos θ X 0 s 2 [P (cos θ)P (cos θ )] × f 0 (ν + ν ) 8 2 2 ` ` pp `↑ `↓ −1 −1 ` (43)
a 2 +fpp0 (ν`↑ − ν`↓)
By the addition theorem for spherical harmonics,
Z Z Z 2 d cos θ d cos θ0P (cos θ00)P (cos θ0)P (cos θ) = d cos θ P (cos θ). (44) ` ` ` 2` + 1 `
So Eq. 43 becomes, after integrating over cos θ and cos θ0,
[N(0)]2 X 1 f s f a ` (ν + ν )2 + ` (ν − ν )2 (45) 8 2` + 1 2` + 1 `↑ `↓ 2` + 1 `↑ `↓ `
Then recalling that f` = N(0)F`, we have
N(0) X 1 F s F a ` (ν + ν )2 + ` (ν − ν )2 (46) 8 2` + 1 2` + 1 `↑ `↓ 2` + 1 `↑ `↓ `
Finally, combining this with the first term (Eq. 33) on the right side of Eq. 24,
X N(0) F s F a (ν + ν )2 1 + ` + (ν − ν )2 1 + ` (47) 8(2` + 1) `↑ `↓ 2` + 1 `↑ `↓ 2` + 1 `
Upon examination, it can be seen that this sum is positive definite (required for thermodynamic stability) only if 22
s,a F` > −(2` + 1). (48)
In three dimensions, this is Pomeranchuk’s stability criterion.
3.2 Pomeranchuk instabilities in 2D
In two dimensions, it can similarly be shown that the requirement for thermody- namic stability is
s,a F` > −1. (49)
Starting from Eq. 26 and converting the sum over ~p to an integral in two dimen- sions, we have
2π ∞ Z Z vF X dθ dpp (p − pF )(δpF (θ, σ))δ(p − pF ) (2π~)23 σ 0 0 1 ∂ − (p − p )(δp (θ, σ))2 δ(p − p ). (50) 2 F F ∂p F
Upon integrating over p, the first term vanishes (due to the delta function δ(p−pF ) causing the (p − pF ) in this term to go to zero). The second is integrated by parts to yield 2π Z vF X 2 dθ(δpF (θ, σ)) . (51) 2(2π~)2 σ 0
δpF (θ, σ) can be expanded in cosines as
X νF δpF (θ, σ) = ν`σ cos(`θ). (52) ` So Eq. 51 becomes 2π Z pF X 2 2 dθν`σν`σ cos(`θ) . (53) 2vF (2π~) σ 0 23
Upon summing over spins and integrating over θ, we find, for the first term on the right side of Eq. 24,
pF X 2 2 2 (1 + δ`0)[(ν`↑ + ν`↓) − (ν`↑ − ν`↓) ]. (54) 8π vf ~ ` Next, consider the second term on the right side of Eq. 24:
1 X f 0 0 δn δn 0 0 . (55) 2 pσ,p σ pσ p σ ~pσ,p~0σ0
Using the definition of δnpσ from Eq. 25, this term becomes
1 X 1 2 ∂ f 0 0 (δp (θ, σ))δ(p − p) − (δp (θ, σ)) δ(p − p) 2 pσ,p σ F F 2 F ∂p F ~pσ,p~0σ0 (56) 1 ∂ ×(δp0 (θ0, σ0))δ(p − p0) − (δp0 (θ0, σ0))2 δ(p − p0), F F 2 F ∂p0 F 2 which, to second order in (δpF ) , is
1 X 0 0 0 0 f 0 0 [(δp (θ, σ))(δp (θ , σ ))δ(p − p)δ(p − p )]. (57) 2 pσ,p σ F F F F ~pσ,p~0σ0 Converting the sums over ~p and p~0 to integrals, 2π 2π ∞ ∞ Z Z Z Z 1 X 0 0 0 4 dθ dθ dpp dp p 2(2π~) 0 σ,σ 0 0 0 0 (58)
0 0 0 0 ×fpσ,p0σ0 [(δpF (θ, σ))(δpF (θ , σ ))δ(pF − p)δ(pF − p )]. The integrals over p and p’ yield (again, using Eq. 52)
2π 2π 2 Z Z pF X X 0 0 0 0 0 4 dθ dθ cos(`θ) cos(`θ )fpσ,p σ ν`σν`σ (59) 2(2π~) 0 σ,σ ` 0 0
The angular integrals give π(1 + δ`0), and taking the sums over spins and using
Eq. 42, we have
2 pF π X s 2 a 2 (1 + δ )[f 0 (ν + ν ) − f 0 (ν − ν ) ] (60) 2(2π )4 `0 p,p `↑ `↓ p,p `↑ `↓ ~ ` 24
Finally, combining the first and second terms (Eqs. 54 and 60), we find
X 2 s 2 a δ(E − µN) ∝ (ν`↑ + ν`↓) [1 + F` ] + (ν`↑ − ν`↓) [1 + F` ]. (61) ` Thus, for thermodynamic stability (which requires that this sum be positive definite), it must be true that
s,a F` > −1 (62) for all ` in 2D.
3.3 Nematic instabilities
If these conditions are violated, the system is unstable to deformations of the Fermi surface. In general, if this condition is violated, the symmetry of the Fermi liquid is
a lowered and a corresponding susceptibility diverges [28]. For example, F0 ≤ −1 is a ferromagnetic instability. At this point, the system will undergo a phase transition from a paramagnetic phase to a ferromagnetic phase, in which the Fermi surface splits into spin up and spin down Fermi surfaces; here, magnetic susceptibility diverges and
s time-reversal symmetry is broken. F0 ≤ −1 corresponds to a density PI. Higher
` instabilities in two and three dimensions correspond to density or spin nematic
s phases of the Fermi liquid, with one exception. There is no PI for F1 ; referring back
s to Eq. 48, the criterion for thermodynamic stability for F1 is
F s m∗p F s N(0) 1 + 1 = f 1 + 1 > 0. (63) 3 π2~3 3
And since effective mass m∗ is defined as
m∗ F s = 1 + 1 , (64) m 3 25
it is clear that Eq. 63 will always be satisfied in Galilean-invariant systems. Note
s that in order for effective mass to be positive at the Fermi surface, F1 > −3, but violation of this criterion does not indicate a softness of the FS with respect to any particular deformation.
Otherwise, as mentioned, higher ` instabilities correspond to so-called “nematic” instabilities of a Fermi system. With the exception of the ` = 1 case just mentioned, these nematic instabilities can occur in the spin or charge channel and involve rota- tional symmetry breaking. The deformations resulting from such instabilities can be seen generically for ` = 0, 1, and 2 in Fig. 2 [27].
Figure 2: Figures from Quintanilla et al [27]. (a) Unpolarized, undeformed isotropic Fermi surface, (b) FS with ` = 2 Pomeranchuk deformation, (c) FS with ` = 3 Pomeranchuk deformation.
At these points in parameter space, a system is unstable to the formation of a
quantum (electronic) nematic phase. This results in an elliptical distortion of the
Fermi surface, analogous with the classical nematic phase of liquid crystal physics.
In the classical liquid crystal nematic phase, rod-shaped molecules form a liquid with
long-range directional order that breaks rotational symmetry but not translational 26
Figure 3: Figures from Fradkin et al [28]. A quantum nematic phase can be formed via a melting of a stripe phase or via a distortion of the FS due to a Pomeranchuk instability. symmetry, owing to their shape and the anisotropic interactions between them. How- ever, since electrons are point-like, these “quantum liquid crystal phases” (another example is the electronic smectic phase) possibly arise from local electronic phase separation, in which there exist locally Mott insulating regions amongst metallic re- gions [29]. One way that a quantum nematic phase can be reached is through the melting of the electronic smectic phase. This phase is analogous to the classical liquid crystal smectic phase, in which molecules are arranged such that they have long-range directional order along one direction; different smectic phases have differ- ent types degrees of ordering. A quantum/electronic smectic phase, also referred to as a “stripe phase” breaks both translational and rotational symmetry. When consid- ering a strong-coupling perspective on the origin of the electronic nematic phase, it is 27 theoretically understood that thermal melting of a stripe phase generates a nematic phase [30].
If instead, a weak-coupling perspective is considered, another way to reach the electronic nematic phase is by starting with an isotropic Fermi liquid and approaching the ` ≥ 1 Pomeranchuk instabilities (see Fig 3) [28]. Nematic states can be generalized as states in which a transport anisotropy is present due to the anisotropy of the ground state [28]. These quantum nematic states are currently of much interest, as they have been experimentally observed in strontium ruthenates [31], quantum Hall states [30,32], heavy fermion systems, and iron pnictides [33], and has been discussed in the context of the the pseudogap phase of the cuprates [34]; studying such nematic phases may give clues to behavior and properties of the cuprates and pnictides in particular. These instabilities can be detected experimentally as a divergence in the relevant (particular channel) susceptibility or scattering amplitude. For example, spin-current measurements can be used to detect a divergent susceptibility in the
a a p-wave spin channel (i.e. F1 → −3 in 3D or F1 → −1 in 2D).
3.4 Generalized Pomeranchuk instabilities
The PI’s discussed until now have been related to a divergence of a q = 0 sus- ceptibility. However, susceptibilities can diverge for other values of q provided that the FL parameters have some momentum dependence, which is discussed in great detail later. For now, note that these finite-q divergences of the susceptibility can be characterized as “generalized Pomeranchuk instabilities,” (GPI’s) in contrast to 28
Pomeranchuk’s original stability criterion for q = 0 processes. These GPI’s (including
a possible spin density wave instability) are a substantial part of the work presented
here and will be explored further in upcoming chapters. It is well-known that an
approach to a PI is accompanied by a divergence in the corresponding channel’s
susceptibility, e.g. 1 1 s,a → ∞, (65) 1 + (2`+1) F` (q)χ`(q) and to consider a specific example, the ` = 0 spin susceptibility diverges when
a 0 a F0 χ0(q ) = −1. The traditional q = 0 PI occurs when F0 = −1, but a q = 2kF GPI
a occurs at F0 = −2, which, as will be discussed, potentially corresponds to a spin density wave transition.
3.5 Quantum criticality
Another reason Pomeranchuk instabilities are of recent interest is that they can be viewed as quantum critical points (QCP’s) in parameter space [4], thus providing an alternate viewpoint on quantum criticality. QCP’s are points at which quantum phase transitions (QPT’s) occur. Classical phase transition are thermally-driven, occurring at some critical temperature, with fluctuations due to thermal energy leading to a change of phase; it is possible to tune this critical temperature to T=0 via some external parameter such as chemical doping or the application of pressure. Pushing the critical temperature of a phase transition to zero creates a QCP at which the
QPT occurs [9]. QPT’s are second-order phase transitions which occur in the ground state of a system, thus at zero temperature, and are related to quantum fluctuations 29
(as classical phase transitions are related to thermal fluctuations). According to
Sachdev and Keimer [35], at a QCP, there is “a qualitative change in the ground- state wavefunction of a large many-body system on smoothly changing one of more coupling constants in its Hamiltonian.” Though quantum phase transitions occur at
T=0, their effects are still experimentally accessible due to the fact that for every
QCP, there is an associated “quantum critical regime” of the phase diagram which extends to finite temperature in which quantum critical fluctuations still play a major role. A generic phase diagram for a system undergoing a QPT is shown in Fig. 4. One reason quantum criticality is of current interest in the condensed matter community is that superconductivity appears to be found in the area of a QCP, and thus creating
QCPs through external tuning of parameters could help guide the discovery of new superconductors [9].
Figure 4: Generic phase diagram involving a quantum phase transition [36]. The associated quantum critical regime extends to T > 0, and so the effects of the QCP can be observed experimentally.
Quantum critical behavior is traditionally described using a Hertz-Millis approach
[10,11]. However, this traditional approach to quantum criticality has been known to 30
break down in certain situations, including the ferromagnetic transition in 3D [37].
The Hertz-Millis approach to quantum criticality involves integrating electrons out
of an order parameter field φ, resulting in an effective action S[φ], which keeps terms
up to quartic order in φ. Whether the electronic excitations can be integrated out is
often an issue in metals (gapless electronic excitations), as it is unclear as to whether
the (potentially singular) interactions between fluctuations of the order parameter
can be approximated by such a quartic term. Thus, an alternate viewpoint, such as
that of the Pomeranchuk instability of a Fermi liquid, can be useful.
Upon applying Hertz-Millis theories to an isotropic FL, one finds that when a PI
a is approached in one channel, the effective mass, and hence F1 , also diverge (see,
for example, [12]). This is counter to the usual assumption that all non-critical F`’s
are unaffected by an approach to a PI in one channel. A recent analysis by Maslov
and Chubukov [14] of the properties of a 2D Fermi system in the paramagnetic state
near a charge nematic (` = 2) PI was done in terms of Landau FL theory. It was found that near the transition, the system enters into a new “critical FL” regime,
a in which all spin components of the FL interaction functions (F` ) and all charge
s components (F` ) with ` 6= 2 diverge at the critical point, while the ` = 2 charge
s FL component, F2 → −5, the PI for this channel. However, owing to cancellation between divergent effective mass and divergent effective Landau component, non- critical channels’ susceptibilities are found to be not affected. Other work by the same authors [13] finds that a 2D FL crosses the ` = 1 instability in the spin channel 31 en route to the FM PI and is crossed before this instability and all other possible instabilities near the FM QCP. This is found starting in the magnetically disordered
(paramagnetic) state and considering finite-range interaction. CHAPTER 4
Tractable Crossing-Symmetric Equation Method
Strong interactions are encountered in many recently-discovered physical systems of interest; perturbation theory using these underlying bare potentials may diverge at short-range, long-range, or both. To avoid these problems involved in using bare
(potentially strong) interactions, renormalized interactions and full vertices must be considered. One way this can be done is via the parquet approach (see, for exam- ple, [38] and references therein), which is non-perturbative in that it sums all dia- grams to infinite order in the particle-partlcle (p-p), particle-hole (p-h), and exchange particle-hole (ex-p-h) channels. This approach to studying many-body systems in- volves considering interactions in all three channels, due to the exchange symmetry present in Fermi systems.
As in the microscopic derivation of Landau’s FLT, we start with a general two- body vertex. This vertex function contains all possible two-body quantum processes and can be viewed as three different channels. The particle-particle (S) channel, particle-hole (T) channel, and exchange particle-hole (U) channel. These types of diagrams can be transformed into each other by exchanging external lines of the
4-point vertex function, as shown in Fig. 5.
The fermion vertices possess crossing symmetry, which is related to exchanging
32 33
Figure 5: Three channels of the 2-body vertex function Γ: particle-particle (S), particle-hole (T), exchange particle-hole (U). The particle-hole and exchange particle- hole channels are topologically equivalent, and any channel can be transformed into any other channel by way of exchanging external lines. Shown also are the momentum transfers in each channel (K, q, and q’, respectively).
external lines:
Γ(1324) = −Γ(1423)
Γ(1324) = −Γ(2413) (66)
Γ(1324) = −Γ(1234)
The first step in the parquet formalism is the summation of reducible diagrams in each channel. Reducible diagrams are diagrams which can be “cut” into smaller, irreducible blocks; if a line can be drawn through an intermediate state of a diagram without crossing an interaction line, a diagram is said to be reducible in a particular channel. See Figure 6 for examples.
If we group all completely irreducible diagrams into one term called I and call the summed reducible diagrams by the name of their respective channel (S,T,U), then this leads to the following set of equations, where Gi’s are Green’s function in each 34
Figure 6: Generic examples of reducible and irreducible diagrams. Reducible dia- grams can be broken into independent parts by drawing a line through an interme- diate state without crossing an interaction line. Irreducible diagrams cannot be split further into independent parts.
channel:
S = (I + T + U)GsGsΓ
T = (I + S + U)GtGtΓ (67)
U = (I + S + T )GuGuΓ
Γ = I + S + T + U.
S,T, and U above are equations for the individual channels, and Γ is again, the full
2-body vertex. It contains all fully irreducible diagrams and the diagrams which are
reducible in each channel. These equations can be combined into one master equation
for the full interaction vertex, known as the parquet equation.
X Γ Γ = I + G G Γ (68) 1 + G G Γ i i i=s,t,u i i
These equations are represented diagrammatically in Figure 7. It is clear from the diagrams that all three channels are coupled. Thus, finding a self-consistent solution to the full parquet equation presents a formidable challenge; all three channels are treated completely microscopically with vertex functions’ topology taken into account, and this full treatment is an arduous task. 35
Figure 7: Shown here are the diagrammatic representations of Eqns. 67. Parquet equations for the particle-particle (S), particle-hole (T), and exchange particle-hole (U) channels are given. Notice that the S,T, and U channels are all coupled to each other. When describing real many-body systems, treatment of all three coupled channels is needed.
However, microscopic treatment of the p-p channel (Brueckner theory) prevents only short-range divergences, and microscopic treatment of the p-h channels (RPA) prevents only long-range divergences; so the completely reducible two-body vertex must indeed include both p-p and p-h channels. This implies that a consistent Fermi liquid theory cannot be formulated in terms of short-range effective interactions alone; collective excitations generated by these interactions must be exchanged between quasiparticles. This underscores the physical basis for TCSE, which can be considered a “minimal” or “tractable” parquet [18] and [15–17,19,20]. The TCSE method utilizes 36 the idea that a large part of the renormalization of quasiparticle interaction comes through the p-h processes near the Fermi surface. This suggests the regrouping of diagrams into p-h reducible and irreducible terms, as is shown next.
Recall the non-analytic limit (q, ω → 0) encountered in the particle-hole channels in the earlier section on microscopic derivation of Fermi liquid theory. The particle- hole Green’s function can be written as
r reg 2 GpGh = S + [G ] , (69)
where Sr is the highly singular piece which depends on the ratio r = q/ω and the order in which the limits are taken.
Γ can be expanded in the particle-hole channels as
Γ = (I + S + U)(1 + GtGtΓ) = (I + S + T )(1 + GuGuΓ). (70)
Recall also from FLT that the renormalized interaction f r (where r refers to the ratio q/ω) is related to the renormalization at the quasiparticle pole and the limit of the singular term:
r 2 r f = 2πizpΓ . (71)
This can be written in terms of the particle-hole (T) channel as
r r r f = fT + fT S f (72)
and equivalently in the exchange particle-hole (U) channel.
r r r f = fU + fU S f , (73) 37
where fT and fU represent diagrams which are irreducible in the particle-hole and exchange particle-hole channels, respectively.
Then to get the Landau interaction function f, we again take the limit r → 0 as
in the section on FLT.
r r r f = lim f = lim[fU + fU S f ] (74) r→0 r→0
When this limit is taken in the exchange particle-hole channel, the first term fU
becomes irreducible in both particle-hole channels. fU contains particle-particle dia-
grams, T-matrix, non-local interactions and physically direct short-range interactions
in the r → 0 limit, and is renamed ”d” to denote “driving” or “direct” interaction.
Thus, it becomes simple to regroup terms into those that are reducible/irreducible
in the particle-hole channels. We can identify the diagrams which are particle-hole
irreducible, along with particle-particle diagrams, as the “direct” or “driving” term
(d). The other diagrams can be identified as the “induced interaction” (find). This results in a Landau interaction function which, when proper momentum and spin labels are included for the U channel, is given by
0 0 X 00 r 0 s 0 fσ,σ0 (p, p ) = d + find = dσ,σ0 (p, p ) + fσ,σ0 (p, p )Sσ00 (q )fσ,σ0 (p, p ). (75) σ00
Here we can extrapolate away from the Fermi surface, following Babu and Brown
[16], and associate with Sr the phase space given by the Lindhard function χ(q0). As discussed previously, taking the other limit (r → ∞) yields the scattering amplitude, which is related to the FL interaction function. After inserting the Lindhard function 38
and splitting the FL interaction function into spin-symmetric and spin-antisymmetric
components, we arrive at the tractable crossing-symmetric equations (TCSE).
As we have seen, the tractable crossing-symmetric equations are obtained via appli- cation of regrouping along with partial resummation of certain diagrams, quasiparticle renormalization, and careful preservation of crossing symmetry. In order to obtain the appropriate phase space, Lindhard (RPA-like) functions must be included. In the
TCSE scheme, these phase space functions are made of the renormalized interaction functions and scattering amplitudes themselves, rather than just bare interactions, as in RPA. TCSE can be viewed as RPA albeit with renormalized interactions in multiple coupled particle-hole channels, in which these coupled channels then drive each other and feed back upon each other. In these ways, TCSE goes beyond RPA using bare interactions.
The tractable crossing-symmetric equations which result are a set of coupled non- linear integral equations for Landau interaction functions F (q), which are composed of quantum fluctuation terms (in which the Landau interaction functions themselves are summed to infinite order in the exchange particle-hole channel) and a term which is fully particle-hole irreducible: d(q) (the driving term d discussed above). The re-
sulting feature of the TCSE method is that it splits the renormalized FL interaction
functions into two parts: a driving term and a quantum fluctuation term. The driv-
ing term, henceforth referred to as D(q) or D (which is just d(q) scaled to Fermi
surface density of states: D(q) = N(0)d(q)), contains diagrams that are particle-hole 39
irreducible in both the direct (t) and exchange (u) particle-hole channels, such as all
p-p terms (i.e. t-matrix and non-local interactions). It is model-dependent and re-
flects the symmetry of the underlying Hamiltonian; the choice of an antisymmetrized
direct interaction is necessary to preserve the required crossing symmetry. Physi-
cally, it reflects the possibility for direct scattering via some effective potential. The
quantum fluctuation (QF) term contains diagrams that are particle-hole reducible.
The diagrams in the QF term account for medium effects and exchange of collective
excitations, such as density, spin-density, and higher-order fluctuations. One unique
aspect of the TCSE method is its ability to treat an arbitrary underlying interac-
tion (D) and these competing quantum fluctuations on the same footing. The set
of equations obtained is shown schematically in Figure 8. It should also be noted
that another way to understand and develop this method is to obtain the TCSE’s
is through functional differentiation of two types of self-energy, the Hartree-Fock-
type (short-range pseudopotential) and the type arising from exchange of collective
excitations (Figure 9).
In isotropic, spin-rotationally-invariant systems, the FL interaction functions can
be expressed as a combination of spin-symmetric and spin-antisymmetric terms:
σσ0 s a ~0 Fpp0 = Fpp0 + Fpp0~σ · σ (76)
These FL interaction functions in the symmetric and antisymmetric channels are ex- panded in Legendre polynomials for the 3D isotropic FL case; they are then calculated 40
Figure 8: Schematic form of tractable crossing-symmetric equations: F (Landau interaction function), A (scattering amplitude), D (direct term). in the TCSE scheme, given in their most general form in Equation (3). The FL func-
s,a s,a tions here are scaled to the density of states at the Fermi surface, i.e. F` = N(0)f`
s,a s,a 0 and Fpp0 = N(0)fpp0 . q is the momentum transfer in the exchange p-h channel. Near
02 0 the Fermi surface, q = 2kF (1 − cos θL), where θL =p ˆ · pˆ.
s 0 s 0 a 0 a 0 s s 1 F0 (q )χ0(q)F0 (q ) 3 F0 (q )χ0(q)F0 (q ) Fpp0 = Dpp0 + s 0 0 + a 0 0 2 1 + F0 (q )χ0(q ) 2 1 + F0 (q )χ0(q ) 02 s 0 s a 0 a 1 q F1 χ1(q )F1 F1 χ1(q )F1 + 1 − 2 s 0 + 3 a 0 + ` = 2, 3 ... (77) 2 4kF 1 + F1 χ1(q ) 1 + F1 χ1(q )
s 0 0 s 0 a 0 0 a 0 a a 1 F0 (q )χ0(q )F0 (q ) 1 F0 (q )χ0(q )F0 (q ) Fpp0 = Dpp0 + s 0 0 − a 0 2 1 + F0 (q )χ0(q ) 2 1 + F0 (q )χ0(q) 02 s 0 s a 0 a 1 q F1 χ1(q )F1 F1 χ1(q )F1 + 1 − 2 s 0 − a 0 + ` = 2, 3 ... (78) 2 4kF 1 + F1 χ1(q ) 1 + F1 χ1(q )
0 0 where χ0(q ) and χ1(q ) are the Lindhard function (density-density correlation function) and current-current correlation function, respectively, which are expressed 41
Figure 9: Two types of contribution to self-energy: (a) Hartree-type, (b) Exchange of collective excitations in 3D as:
1 q0 1 1 − 0.5q0 χ (q0) = 1 + − ln (79) 0 2 4 q0 1 + 0.5q0 1 3 1 1 1 3q0 1 − 0.5q0 χ (q0) = − − + − ln (80) 1 2 8 2q02 2q03 4q0 32 1 + 0.5q0
Note that q0 is defined as the momentum transfer in the exchange particle-hole channel, whereas q is the momentum transfer in the direct particle-hole channel.
A key input to the theory is the driving interaction (D) which must be properly antisymmetrized. For a given D, the FL interaction functions can be calculated in any angular momentum channel, along with the corresponding scattering amplitudes and effective mass (related to self-energy). From these basic quantities, various transport, thermodynamic, and pairing properties can then be calculated.
In the past, calculations have been done using this method on the PM side [21,40] and in the local FL limit (q → 0, ω → 0) [40]. In this limit, χ0 = 1, which effectively removes all q0-dependence from the problem by way of limiting the phase space. CHAPTER 5
Model
In all cases discussed here, we incorporate the momentum transfer q0 through the
Lindhard function, and in that sense it differs from a local FL (q → 0, ω → 0), so
we shall henceforth refer to this as a non-local FL. This q0-dependence of the phase space is sufficient for moving away from the local limit.
It is natural to consider predominantly ` = 0 fluctuations, as they are energetically easy to excite, and many physical systems exist which exhibit FM instabilities (e.g. layered ruthenates [52]) and CDW instabilities (e.g. cuprates [53,54]). For this reason, only the ` = 0 fluctuations are considered here, but the TCSE method allows for the possible inclusion of higher order fluctuations such as current (` = 1) and nematic
(` = 2).
For the case in which the Lindhard function contains momentum dependence but the FL interaction functions themselves have no explicit q0-dependence, the FL interaction functions and direct interactions can be expanded in Legendre polynomials as
s,a X s,a Fpp0 = F` P`(cos θL) ` (81) s,a X s,a Dpp0 = F` P`(cos θL) `
42 43
Using the orthogonality of Legendre polynomials, these functions can projected out in various angular momentum channels in the TCSE equations as given in the
“Method” section. We begin by projecting out the ` = 0 (s-wave) interaction func- tions:
1 Z 2kF F sχ (q0)F s 3 Z 2kF F aχ (q0)F a F s = Ds + 0 0 0 dq0 + 0 0 0 dq0 0 0 2 1 + F sχ (q0) 2 1 + F aχ (q0) 0 0 0 0 0 0 (82) Z 2kF s 0 s Z 2kF a 0 a a a 1 F0 χ0(q )F0 0 1 F0 χ0(q )F0 0 F0 = D0 + s 0 dq − a 0 dq . 2 0 1 + F0 χ0(q ) 2 0 1 + F0 χ0(q )
Next, an underlying interaction is chosen. Given a particular underlying interac-
s,a tion, we are led to the direct term (D0 ), which must be properly antisymmetrized.
The direct term is model-dependent; its choice is guided by the underlying interaction
of the system being studied and the degree of simplification desired.
In the work presented here, we start with a zero-range interaction as in the single-
band Hubbard model for a 3D lattice. In second quantized form, the Hubbard model
Hamiltonian is given by
N X † X H = −t (ci,σcj,σ + h.c.) + U ni↑ni↓ (83) hi,ji,σ i=1
where the first term is the kinetic energy of nearest-neighbor hopping between sites
i and j (c† and c are fermion creation and annihilation operators), and the second
term is an on-site zero-range repulsive interaction. ni↑ and ni↓ represent the num-
ber of particles (either 0 or 1 by the Pauli exclusion principle) with a given spin on
site i. This model is one of the most-studied in correlated electron systems and is
believed to be relevant for many superconducting systems, including ferromagnetic 44
superconductors. For a continuum system, the equivalent of the Hubbard model in-
volves a kinetic energy of p2/(2m∗) and a contact interaction determined by the Pauli
exclusion principal. Motivated by these Hamiltonians, we use a contact interaction
to study our continuum system. Then, by the Pauli principle, U ↑↑ = 0; U ↑↓ = U.
This gives, for the antisymmetrized ` = 0 direct interaction in TCSE scheme,
U Ds,a = ± (84) 0 2 where U is scaled to density of states at the Fermi surface surface and is related to the Hubbard U shown above.
After projections and choice of direct interaction, the model TCSE’s become
U 1 Z 2kF F sχ (q0)F s 3 Z 2kF F aχ (q0)F a F s = + 0 0 0 dq0 + 0 0 0 dq0 0 2 2 1 + F sχ (q0) 2 1 + F aχ (q0) 0 0 0 0 0 0 (85) Z 2kF s 0 s Z 2kF a 0 a a −U 1 F0 χ0(q )F0 0 1 F0 χ0(q )F0 0 F0 = + s 0 dq − a 0 dq 2 2 0 1 + F0 χ0(q ) 2 0 1 + F0 χ0(q )
0 0 where χ0(q ) is the Lindhard function, defined for q = 0 → 2kF , as in equation
0 (4) [41]. Here, and in the rest of the work unless otherwise noted, q is scaled to kF
0 0 (i.e. q → q /kF ). CHAPTER 6
Solution Techniques
6.1 Parameter space near GPIs
The two starting points of the calculation are the paramagnetic FL and the weak ferromagnetic FL, based on the well-established theory of Abrikosov, Dzyaloshinskii,
a s and Kondratenko [22–24]. We seek solutions to the TCSE’s in F0 ,F0 space. This
parameter space can be broken into four major regions: ferromagnetic, paramag-
netic, phase separation, and mixed phase (meaning both charge and ferromagnetic
PI thresholds have been crossed, and so both types of instabilities are present). See
Table 1 for details. As will be discussed shortly, the presence of solutions in various
regions depends largely on whether an attractive or repulsive interaction is considered.
a s This F0 ,F0 parameter space is shown in Fig. 10. The thatched sections of the plot are the regions of finite-q divergences, bounded on one edge by the q = 0 PI
s,a s,a (F0 → −1)and on the other edge by a q = 2kF instability (F0 → −2), with all
a s Ferromagnetic F0 < −1,F0 > −1
a s Paramagnetic F0 > −1,F0 > −1
a s Mixed (FM/CDW) F0 < −1,F0 < −1
a s Phase separation F0 > −1,F0 < −1 Table 1: Solution sectors as defined in parameter space.
45 46 other generalized Pomeranchuk instabilities between these boundaries.
4 s F0
-2 -1 2 FM PM a F0 0
-1 -1 -2 -2 -2
-4 Mixed -2 -1 PS
-6 -6 -4 -2 0 2 4
a s Figure 10: Regions of F0 ,F0 parameter space and generalized Pomeranchuk instabil- ities.
6.2 Graphical and numerical methods
For a given value of underlying interaction U, there may exist one or more so-
a s lutions (F0 ,F0 pairs) which satisfy the set of coupled non-linear crossing-symmetric equations. Due to the highly non-linear nature of these equations, it is difficult to
find solutions using numerics alone. So to find solutions to the TCSEs, both graphical and numerical techniques are employed.
In 3D, quantum fluctuation integrals are conveniently done using Mathematica’s
“NIntegrate,” [46, 47] an adaptive algorithm that numerically computes integrals.
Integrals performed using NIntegrate were rigorously compared to results from various numerical quadrature rules and found to be in agreement. In regions of GPIs, however, 47
“NIntegrate” is not sufficient for solving integrals due to the continuum of divergences in each channel which is present in these regions. These singularities are not removable and can only be evaluated in a “principal value” sense using Cauchy’s principal value theorem [71], which is a commonly-used technique in physics (see, for example, the
Anderson impurity model [39]). In 2D, integrations are done using a Gauss-Lobatto quadrature with Kronrod extension for error estimation. Numerical issues in both dimensions are discussed extensively in Appendix A, along with details of numerical quadratures.
In conjunction with numerical evaluation of integrals, graphical techniques are utilized to find self-consistent solutions to the set of coupled non-linear TCSEs:
U 1 Z 2kF F sχ (q0)F s 3 Z 2kF F aχ (q0)F a F s = + 0 0 0 dq0 + 0 0 0 dq0 0 2 2 1 + F sχ (q0) 2 1 + F aχ (q0) 0 0 0 0 0 0 (86) Z 2kF s 0 s Z 2kF a 0 a a U 1 F0 χ0(q )F0 0 1 F0 χ0(q )F0 0 F0 = − + s 0 dq − a 0 dq . 2 2 0 1 + F0 χ0(q ) 2 0 1 + F0 χ0(q ) To satisfy these coupled equations, it is convenient to define expressions in the spin-symmetric (S) and spin-antisymmetric (A) channels as
U 1 Z 2kF F sχ (q0)F s 3 Z 2kF F aχ (q0)F a S[F a,F s] = F s − − 0 0 0 dq0 − 0 0 0 dq0 0 0 0 2 2 1 + F sχ (q0) 2 1 + F aχ (q0) 0 0 0 0 0 0 (87) Z 2kF s 0 s Z 2kF a 0 a a s a U 1 F0 χ0(q )F0 0 1 F0 χ0(q )F0 0 A[F0 ,F0 ] = F0 + − s 0 dq + a 0 dq 2 2 0 1 + F0 χ0(q ) 2 0 1 + F0 χ0(q ) When both equations are satisfied, that is to say,
a s a s S[F0 ,F0 ] = A[F0 ,F0 ] = 0, (88)
a s the corresponding intersection point [F0 ,F0 ] in parameter space is considered a solu- tion to the TCSEs. This intersection point can be found graphically as the three-way 48
a s intersection of the symmetric channel equation S[F0 ,F0 ], the antisymmetric chan-
a s a s nel equation A[F0 ,F0 ], and the zero plane in a three-dimensional space (3D: F0 ,F0 , and the value of the S or A equation). Shown in Fig. 11 is an example of a graphical solution plotted in these three dimensions. This is perhaps seen more clearly in a two- dimensional plot generated by looking at a “zero plane slice” of the three-dimensional
a s space discussed (the two dimensions of these plots are then F0 , F0 ). In the 2D plot, surfaces S and A appear as lines, and their intersection represents a solution. Fig. 12 shows the same solution as Fig. 11, but here in a two-dimensional contour plot.
2 0 -2
-11 -0.4
-10 -0.6
-9 -0.8
-8
s Figure 11: Sample 3D plot for U=30; solution is intersection point at F0 = a −0.676,F0 = −9.210.
6.3 Calculation of other quantities
Once the ` = 0 interaction functions have been calculated using methods dis- cussed, other properties can be calculated from these. It is useful to first obtain the
FL interaction functions in higher angular momentum channels. The orthogonality 49
s F0 -0.3
-0.4
-0.5
-0.6
-0.7
-0.8
a -10.5 -10.0 -9.5 -9.0 -8.5 -8.0 F0
s Figure 12: Sample 2D plot for U=30; solution is intersection point at F0 = a −0.676,F0 = −9.210. of the Legendre polynomials is used to project out the desired channel’s interaction function. As examples, see the ` = 1, 2 projections:
3 Z 2kF q02 F s(q0)χ (q0)F s(q0) F s = q0 1 − 0 0 0 dq0 1 4 2 1 + F s(q0)χ (q0) 0 0 0 (89) Z 2kF 02 a 0 0 a 0 9 0 q F0 (q )χ0(q )F0 (q ) 0 + q 1 − a 0 0 dq 4 0 2 1 + F0 (q )χ0(q )
3 Z 2kF q02 F s(q0)χ (q0)F s(q0) F a = q0 1 − 0 0 0 dq0 1 4 2 1 + F s(q0)χ (q0) 0 0 0 (90) Z 2kF 02 a 0 0 a 0 3 0 q F0 (q )χ0(q )F0 (q ) 0 − q 1 − a 0 0 dq 4 0 2 1 + F0 (q )χ0(q )
5 Z 2kF 3q02 q02 F s(q0)χ (q0)F s(q0) F s = 1 − 1 − 0 0 0 dq0 2 4 2 4 1 + F s(q0)χ (q0) 0 0 0 (91) Z 2kF 02 02 a 0 0 a 0 15 3q q F0 (q )χ0(q )F0 (q ) 0 + 1 − 1 − a 0 0 dq 4 0 2 4 1 + F0 (q )χ0(q ) 50
5 Z 2kF 3q02 q02 F s(q0)χ (q0)F s(q0) F a = 1 − 1 − 0 0 0 dq0 2 4 2 4 1 + F s(q0)χ (q0) 0 0 0 (92) Z 2kF 02 02 a 0 0 a 0 5 3q q F0 (q )χ0(q )F0 (q ) 0 − 1 − 1 − a 0 0 dq 4 0 2 4 1 + F0 (q )χ0(q )
The scattering amplitudes are calculated from the F`’s as
s,a s,a F` A` = s,a (93) 1 + F` /(2` + 1)
One can use sum rules to check the convergence of these scattering amplitudes; the
X s a simplest sum rule to test is the forward scattering sum rule, (A` + A` ) = 0. ` However, the Sj¨oberg sum rule [42] may be used to compensate for the inclusion of
only lower-order angular momentum scattering amplitudes:
X F s F s ( 0 + 0 + F s + F a) = 0 (94) 1 + F s 1 + F s ` ` `6=0 0 0
From here, standard Fermi liquid theory [25] can be applied to calculate properties
of the system, such as effective mass (for a Galilean-invariant system):
m∗ F s = 1 + 1 (95) m 3
and compressibility m∗ κ m = s (96) κ0 1 + F0
Superconducting pairing amplitudes gs,t (singlet and triplet pairing amplitudes) can be calculated using the Patton-Zaringhalam approximation [43]. To understand the Patton-Zaringhalam scheme, we start with scattering amplitudes from Fermi liq- uid theory in the singlet and triplet channels. They can be expanded in Legendre 51 polynomials as:
X s a As(θ, φ) = (A` − 3A` )P`(cos θ)Rs(φ) ` (97) X s a At(θ, φ) = (A` + A` )P`(cos θ)Rt(φ), ` where θ is the angle between incoming particles, and φ is the angle between the plane of incoming particles and outgoing particles. Rs,t(φ) are functions that parameterize
s,a the scattering as a function of φ in singlet and triplet channels. A` are the scattering amplitudes in the spin-symmetric and spin-antisymmetric channels, respectively, for a particular angular momentum channel `, and as described before, as related to the
FL interaction functions as
s,a s,a F` A` = s,a . (98) 1 + F` /(2` + 1)
Returning to Rs,t(φ), the Pauli exclusion principle requires that
Rs(φ) = Rs(φ + π) (99)
Rt(φ) = −Rt(φ + π), and from the definition of the FL interaction functions,
Rs = (0) = Rt(0) = 1. (100)
If, as following Patton-Zaringhalam [43] and Dy-Pethick [44], we work in the s-p approximation (in other words, keep only ` = 0, 1 angular momentum channels), then a simple form can be used as an approximation for Rs and Rt:
Rs = 1 (101)
Rt = cos φ, 52
which clearly obeys the Pauli condition (Eq. 99). Additionally, this simple form introduces no new parameters to the problem. Working in the s-p approximation, singlet and triplet pairing amplitudes can be defined as
gs = As(π, φ) (102) A (π, φ) g = t , t 3 cos φ Summing only the ` < 2 channels, we find
s a s a gs = ((A0 − 3A0)(1) + (A1 − 3A1)(cos π))/4 (103) s a s a gt = ((A0 + A0)(1) + (A1 + A1)(cos π))/12,
which can be more compactly written as
X ` s a Singlet: gs = (−) (A` − 3A` )/4 ` (104) X ` s a Triplet: gt = (−) (A` + A` )/12 ` Calculation of pairing amplitudes including up to ` = 3 scattering amplitudes were also considered, but no quantitative difference was found with the inclusion of these extra parameters. Detailed results can be found in Appendix B.
6.4 Beyond the local limit
In the past, calculations have been done in the limit of a local Fermi liquid (see, for example, [ref]). This involves the constraint of a momentum-independent self- energy (i.e. ∂Σ/∂p = 0). In this limit, q0 = 0, ω = 0 and the shape and size of the
Fermi surface are not changed by interactions [40]. This leads to a local full vertex function (from which F`’s and A`’s are derived, as discussed previously). A local FL 53
s,a is described by only two FL interaction functions, F0 and therefore two scattering
s,a amplitudes A0 . Thus, in the local limit,
s,a ` > 0 : F` = 0, (105) s,a ` > 0 : A` = 0
Considering then the forward scattering sum rule (which is essentially the Pauli exclusion principle),
X s a (A` + A` ) = 0, (106) ` we see that in the local limit,
a s A0 = −A0. (107)
a s This gives a simple relation between F0 and F0 :
a,s s,a −F0 F0 = a,s (108) 1 + 2F0
Additionally, in this local FL limit, the relationship between effective mass and quasiparticle residue is m∗ z−1 = . (109) m
s Since F1 = 0 in the local limit (Eq. 105), we have
m∗ = 1, (110) m and thus
z = 1. (111) 54
Superconducting pairing amplitudes can be calculated from scattering amplitudes, as discussed in the previous section. It follows from Eqs. 104 and 107 that the triplet pairing amplitude is always zero in the local limit :
X ` s a local s a gt = (−) (A` + A` )/12 −−→ (A0 + A0)/12 = 0 (112) `
However, in the local limit, s-wave singlet pairing may be possible:
X ` s a local s a gs = (−) (A` − 3A` )/4 −−→ (A0 − 3A0)/4 ` (113) a s a a F0 = (A0 − 3A0)/4 = −A0 = − a . 1 + F0
a On the ferromagnetic side (F0 < −1), gs is always negative. Thus, for the special case of a ferromagnetic local FL, the triplet pairing amplitude is zero, and singlet
a pairing is always attractive. In general for a LFL (arbtitary F0 ), triplet pairing amplitude is always zero, and singlet pairing may or may not be attractive, depending
a on the specific value of F0 .
When the calculation is extended from the local limit (q0 = 0, ω = 0) to include momentum transfer up to 2kF , the phase space becomes momentum-dependent. This introduces finite q divergences to the problem in addition to the q=0 PIs. These
finite q divergences were described in the introduction as “generalized Pomeranchuk instabilities” or GPIs and are shown in the parameter space of Fig. 10. The quantum
s,a 0 0 fluctuation terms diverge when 1 + F0 χ0(q ) = 0. Since 0.5 ≤ χ0(q ) ≤ 1 (for
s,a 0 ≤ q ≤ 2kF - see fig. 14), when −2 ≤ F0 ≤ −1, there are two divergences for
0 a s every value of q : one in the F0 integral and one in the F0 integral. So, in theory, 55 there exist two uncountably infinite sets of divergences. Numerically, for every value of q0 sampled in the integration, one divergence is present in each of the two (s, a) channels. These sets of divergences are bounded in parameter space for ` = 0 by
F0 = −2 on the negative edge and F0 = −1 on the positive edge (see shaded regions in Fig. 10). We treat these sets of divergences using a numerical contour integration.
This involves integrating around a divergence using a closed contour around the in the complex plane as shown schematically in Figure 51. Using the residue theorem [70] and Cauchy’s principal value (PV) technique [71], which are discussed at length in
Appendix A, these divergent integrals can be evaluated.
Figure 13: Schematic representation of closed path used for numerical contour inte- gration around a singularity marked by the dot at the origin. The path C is comprised of parts C1,C2,C3, and C4. Path C2 has a radius R, and path C4 has a radius .
The integrals along these segments of the contour Ci are given here as equations 56
Ji, where R is the radius of the larger arc, and is the radius of the smaller arc:
Z 2 2 F0 χ0(q) J1 = dq qp+ 1 + F0χ0(q) Z π iReiθF 2χ (1 + Reiθ) J = dθ 0 0 2 1 + F χ (1k + Reiθ) 0 0 0 F (114) Z qp− 2 F0 χ0(q) J3 = dq 0 1 + F0χ0(q) Z 0 iθ 2 iθ ie F0 χ0(qp + e ) J4 = dθ iθ . π 1 + F0χ0(qp + e )
According to the residue theorem, the sum of these integrals should equal zero since no pole is enclosed by the contour. The original divergent integral can then be evaluated as the PV (J1 + J3) of the contour. It should be noted that if qp is located at 0 or 2kF , the integration requires special attention.The type of contour integration discussed here involves integrating over a small symmetric range about a singularity.
The only physically relevant values of q for this calculation are q = 0 → 2kF .A problem occurs when the singularity lies at exactly 0 or 2kF , as the consequences of extending the integration outside of the range (to either (0 − ) or (2kF + )) are unclear. Thus, these points are problematic in the PV scheme. However, the exclusion of these points does not make a qualitative difference in results, and in fact, these
“endpoint singularities” may be signaling strong instabilities of the physical system; it is well-known that q = 0 corresponds to the FM instability, and perhaps q = 2kF also corresponds to a strong physical instability (e.g. a spin-density wave instability).
The validity of using the numerical contour integration to make the divergences integrable was checked with an additional calculation using Mathematica’s “Principal 57
Value” option; again, refer to Appendix A for details. Thus, the regions of GPIs can be evaluated, and these regions become, in a “principal value” sense, integrable. The surfaces defined by the equations (see Eqn. 87) are shown in Figs. 15 and 17 before and after the treatment of poles via numerical contour integration. The surfaces shown
s,a are the TCSE’s as functions of F0 when U = 12. In Fig. 15 (when no considerations were given to the region of divergences/before contour integration), the spikes along
a a the F0 axis are poles at Gauss points sampled in the integration of F0 in the quantum
fluctuation terms of the TCSE’s. The more Gauss points used, the more spikes appear in the surfaces, since each Gauss point corresponds to a different value of q. Recall
s that for each value of q, there is a divergence in each channel. If F0 had also been
s plotted in the range of finite q divergences (F0 between -1 and -2), similar spikes would be present along its axis as well. It can be seen from Fig. 17 that the principal value integrals, obtained by numerical contour integrations, provide a way to treat the finite-q divergences in these regions.
Χ0 1.0
0.9
0.8
0.7
0.6 q¢ 0.5 1.0 1.5 2.0 kF 0 0 Figure 14: For 0 ≤ q ≤ 2kF , the Lindhard function is in the range 1 ≥ χ0(q ) ≥ 0.5. 58
Figure 15: Equation surface prior to evaluating divergences with a numerical contour integration. Each “spike” corresponds to a value of q in the divergent range of −2 < a F0 < −1 which was sampled by the Gaussian integration.
Figure 16: After treatment of poles Figure 17: U=12 Equation surfaces before and after numerical contour integrations. CHAPTER 7
Results: 3D
After extension to finite q and treatment of related divergences, solutions are found for both attractive and repulsive interactions. From these solutions, higher-order FL parameters can be projected out, along with corresponding scattering amplitudes, effective masses, and pairing amplitudes. Refer to earlier sections for details of calcu- lation methods. We start with the simplest non-local case, in which the FL parameters do not have explicit q−dependence, but the phase space does, via the Lindhard func- tion (some results for this case are presented in [48]). In each case presented below, we
find three primary branches of solutions, corresponding to ground states of different physical systems. The solution sectors in which these three solutions are found vary with the sign of the interaction, as will be discussed shortly.
7.1 Repulsive interaction
To study systems with dominant repulsive interactions between the fermions, we choose a repulsive contact interaction (i.e. positive value of U, the adjustable inter- action parameter) as the underlying interaction. As an example, this may be used to model ferromagnetic superconducting systems, as will be shown.
The three types of solutions found for the case of repulsive interactions include one paramagnetic and two ferromagnetic branches: one near the FM PI (weak FM) 59 60
a s Solution Type F0 F0 PM −0.5(+) U FM (strong) −U/3 −0.5(−) FM (weak) −1(−) −1(+)
Table 2: Large U liming results for local FL.
a s Solution Type F0 F0 (m ∗ /m) Pairing Amplitudes
PM −0.63 U Large gt < 0, gs > 0
FM (strong) −U/3 −0.63 Modest gt < 0, gs < 0, but|gs| > |gt|
(−) (+) FM (weak) −2 −1 Large gt < 0, gs < 0, but|gs| > |gt| Table 3: Large U limiting results for repulsive interaction. and one far beyond the FM PI (strong FM). At least one other branch of solutions is found, but guided by the behavior of solutions in the local FL limit and previous work [40] (see Table 2 for large U limiting results in LFL case), we deem these so- lutions unphysical and focus on the three branches mentioned initially; these extra solutions may be considered purely mathematical in natural, arising as a result of the nonlinearity of the TCSE equations. Considering the large-U limit of the solu- tions gives insight into the general behavior and properties of these solutions. See
Table 3 for these limiting results in the general (non-local) case of a repulsive contact interaction.
In the paramagnetic branch, in agreement with previous work [21, 40], we find
a that F0 approaches −0.63 from the positive side, meaning it moves in the direction
s of the FM instability with increasing U, and F0 approaches U. The effective mass
s (related to F1 ) is large and on the order of U. Using the Patton-Zaringhalam scheme 61
(Eq. 104) to calculate singlet and triplet effective pairing amplitudes, we find that
only triplet pairing is found to have an attractive amplitude in this branch.
a The next solution branch is the strongly ferromagnetic solution. Here, F0 ap-
s proaches −U/3 and F0 approaches −0.63 from the positive side (moving in the direc- tion of the density instability with increasing U). Note that as the underlying repulsive interaction increases, this solution moves deeper into the ferromagnetic regime. While weakly FM systems are well described by FM Fermi liquid theory [22] [23], [24], no rigorous Fermi liquid theory exists at present for describing strongly FM systems.
One may argue that this branch of solutions is therefore only ”academic” in nature; however, the merit of this large moment solution branch is that these results will apply if such a theory is developed in the future. In this branch, the effective mass is modest but greater than one. In contrast to the PM solution, both singlet and triplet pairing amplitudes are attractive (and thus both types of pairing are possible), but singlet is favored due to its larger negative magnitude.
a Finally, in the weakly ferromagnetic branch that is near the FM PI, F0 approaches
s −2 from the negative side and F0 approaches −1 from the positive side. Due to the
a region of GPI’s in the antisymmetric channel, F0 cannot move all the way to the FM instability, but instead moves in the direction of the q0 = 0 PI (-1) and gets “stuck” at the q0 = 2 GPI of -2 (possible SDW instability). It will be shown in a later section
0 a 0 that with added q -dependence, F0 will truly approach the FM (q = 0) PI of -1 as U increases. Also of note in this branch is its “multicritical” nature, meaning that both 62
channels move towards instabilities together as U increases; the weak FM branch of
solutions is the only branch found to exhibit this quantum multicritical behavior.
The effective mass in this weak FM branch is found to be large and of the order of U.
As in the other FM solution branch, both singlet and triplet pairing are attractive, but singlet pairing is preferred.
For both strong and weak FM branches, we find this to be the case. Recall that a negative pairing amplitude indicates the possibility of pairing in a given channel, and a more negative amplitude is favored over a less negative amplitude. These ampli- tudes are calculated in the Patton-Zaringhalam scheme (Eq. 104) and give the same qualitative results for either an s-p approximation (` = 0, 1 scattering amplitudes only) or an s-p-d approximation (` = 0, 1, 2). For s-p approximation, the singlet and triplet pairing amplitudes reduce to
1 s a s a gs = [(A0 − 3A0) − (A1 − 3A1)] 4 (115) 1 g = [(As + Aa) − (As + Aa)] , t 12 0 0 1 1
a Therefore, a positive A0 increases attraction in the singlet channel while decreasing
a attraction in the triplet channel; A1 being negative has the same effect. Recalling
s,a s,a s,a a a that A` = F` /[1 + F` /(2` + 1)], one can see that on the FM side (F0 < −1) A0
a a is always positive. Also, when F1 is between -3 and 0, A1 will be negative; in this
a branch of solutions, we always find F1 to be in this range. These conditions help favor singlet pairing over triplet, and in the FM branches, these factors conspire to generate a singlet amplitude which is significantly more negative than the corresponding triplet 63
s a s a U A0 A0 A1 A1 gs gt gs/gt 13 -7.51 1.77 1.99 -0.27 -3.91 -0.62 6.3 15 -14.2 1.94 2.36 -0.76 -6.15 -1.15 5.3 18 -27.4 1.99 2.58 -1.57 -10.17 -2.20 4.6
Table 4: Sample of contributions to pairing amplitude in weak FM branch of solutions from scattering amplitudes in s-p approximation. amplitude. Shown in Table 4 are the contributions to singlet and triplet pairing from the four scattering channels considered in the s-p approximation. These are shown for three representative values of repulsive interaction in the weak FM branch. The
final column presents the ratio of singlet to triplet pairing amplitude, which shows that for this branch, singlet-favoring becomes less predominant for stronger repulsive interaction.
7.2 Attractive interaction
Next, we choose an attractive underlying interaction (negative values of U) as the underlying interaction. Physically, an attractive interaction such as this is not repre- sentative of a bare attractive potential, but of an effective interaction which may be generated through collective behavior (such as phonons in conventional superconduc- tors) or exchange correlations, and so many physical systems of recent interest can be described using a negative U Hubbard model. For the case of an attractive inter- action, the pairing instability will always be present (at least in the s-wave channel); one question of interest is what happens to other instabilities in this situation.
As in the repulsive interaction case, three major solution branches are found, but 64
a s Solution Type F0 F0 (m ∗ /m) Pairing Amplitudes PM −U/3 −0.63 Modest Both possible, singlet preferred PS (strong) −0.63 U Modest Triplet PS (weak) −1(+) −2(+) Large Triplet
Table 5: Large U limiting results for attractive interaction. in different regions of parameter space. One branch is paramagnetic, and the other two lie in the phase separation region: one near the PS PI (weak PS) and one far from it (strong PS). This is the region which lacks magnetic order but in which solutions are beyond the charge-density instability. As in the case of repulsive interaction, at least one additional branch is found. Guided by the local FL results, we deem these solutions “unphysical” and focus on the other solution branches. These extra solutions may, again, be considered purely mathematical in nature (see Appendix B for detailed results in these “unphysical” branches).
Now considering these paramagnetic and phase separation solutions, large U lim- iting values of calculated quantities (Table 5) can be examined for insight into of the behavior of these solutions.
a s In the paramagnetic branch, F0 approaches −U/3 and F0 approaches −0.63 from the negative side, meaning it moves in the direction of the PS instability with in- creasing U. The effective mass is modest but greater than one. Using the Patton-
Zaringhalam scheme (Eq. 104) to calculate singlet and triplet effective pairing am- plitudes, we find that both singlet and triplet pairing amplitudes are attractive, but singlet is favored due to its larger negative value. 65
a In the phase separation branch where solutions are found away from PI’s, F0 approaches −0.63 from the positive side (moving in the direction of the FM instability)
s with increasing U, and F0 approaches U. The effective mass is also modest and greater than one in this branch. Only triplet pairing is attractive.
a In the phase separation branch where solutions are found near PI’s, F0 approaches
s −1 from the positive side, and F0 approaches −2 from the positive side (see Table
44 in Appendix B).These are the system’s instability points; F0 → −1 is the ` = 0
0 PI, and F0 → −2 marks the edge of the region of GPI’s/finite-q divergences. These
a s points are instabilities for both F0 and F0 , meaning that again, the system is in a
“multicritical” regime. As in the repulsive case, solutions never push beyond these instabilities without the effect of added q−dependence in the FL parameters (see discussion in repulsive attractive section). In this solution branch of phase separation near PI’s, the effective mass is large - on the order of U - and only triplet pairing is attractive. Thus, for both phase separation solutions (near and away from PS PI), only triplet pairing is possible.
7.3 q-dependence of FL parameters
In the next phase of the calculation, q0-dependence is added to the FL parameters
s,a s,a so that F` → F` (q). The major goal of this is to push the weak FM solution as close as possible to the FM instability. With no added q0-dependence, recall from the repulsive interaction section that the weak FM solution moves in the direction of the FM instability, approaching from the negative side, but never moves past the 66
0 a q = 2kF GPI at F0 = −2. This particular GPI is significant because it marks the negative edge of the region of GPI’s/finite-q0 divergences. This region of divergences
s,a 0 is controlled by the denominator of the quantum fluctuation terms, 1 + F0 χ0(q ).
0 Because (for q between 0 and 2kF ) 0.5 ≤ χ0 ≤ 1, the range of finite q’ divergences is
s,a 0 −2 ≤ F0 ≤ −1. Thus, an added q -dependence can change the location and size of
s,a the range of divergences, thereby changing the large U limiting value of F0 in the weak FM branch of solutions. The range of GPI’s becomes subject to the condition
s,a 0 1 + F0 (q)χ0(q ) = 0.
We chose a q0-dependence with two adjustable parameters, c and α:
αq02 F s,a(q0) = F s,a c 1 + (116) 0 0 4
The motivation behind this particular form of momentum dependence lies in a calculation of Quader and Ainsworth [45] [49], which involved studying the PM side in the TCSE scheme using a full q-dependence of FL parameters. From this self- consistent treatment in both p-h momentum channel, a similar q-dependence emerged.
The chosen form of q-dependence also satisfies one of our original goals of adding q- dependence, which was to find solutions near the FM PI; this form of q-dependence can drive the solutions arbitrarily close to the FM PI. With a suitable choice of
a parameters c and α, F0 in the weak FM branch of solutions is moved through the region of GPIs and truly approaches the FM PI. This “optimal” parameterization
(c = 1, α = 1) thus allows the system to approach the multicritical FM/charge-
0 s a density q = 0 PI point (both F0 and F0 → −1). A parameterization using certain 67 values of c, α can also be chosen to move the system continuously through the FM instability and into the PM region. These are two things that cannot be accomplished without this q0-dependence. As shown in Table 3, with no q0-dependence in the FL
s 0 parameters, F0 naturally approaches the q = 0 PI, since it approaches the region of GPI’s (-1 → -2) from the positive side. In the context of adding q0-dependence in order to push the solutions are close as possible to the q0 = 0 PI, it is therefore not necessary or desirable to add q-dependence to the charge channel’s FL function, and no qualitative difference is seen between cases where q0-dependence is present or absent in this channel. The more interesting case is the addition of q0-dependence to the spin channel, as discussed above. The addition of q0-dependence to both channels simultaneously was explored and found to be qualitatively similar to the addition of
0 a 0 q -dependence in only the F0 channel. A summary of results for added q -dependence in the spin channel only is given in Table 6 and shown graphically in Figure 19.
0 a With the c = 1, α = 1 q -dependence added to F0 (hence, in effect reducing the q- dependence to having only one parameter), the system can approach the multicritical q0 = 0 PI. One may then ask which channel is driving this multicritical behavior.
s a The answer to this is unclear, because while F0 approaches its instability first, F0 approaches its instability more quickly, especially in the range of smaller U values.
This is shown in Figure 18.
In order to connect with experiments, it is useful to consider several combinations of these parameters, because within this model, different systems could correspond 68
F0
-0.95 s F0 -1.00
-1.05 a F0 -1.10
-1.15
-1.20
U 15 20 25 30 Figure 18: In the multicritical branch of solutions, it is unclear which channel drives the multicritical behavior. The multicritical point is approached with increasing U.
a s Solution Type Region F0 F0 (m ∗ /m) Pairing Amplitudes Large FM −U/3 > −2 Modest (> 1) Singlet
Small (<∼ 20) FM −1 −1 Large (∼ O(U)) Singlet Arbitrary PM -2/3 U Large (∼ O(U)) Triplet
a Table 6: Summary of results for q-dependent F0
to a different set of c, α parameters. For a given value of underlying interaction (U),
a s and by fitting F0 to spin susceptibility measurements and F1 (related to effective
mass) to specific heat experiments, the values of c and α in the parametrization can be fixed to model a particular system. A few examples of large U limiting values of
a a F0 (F0 maximum) for different combinations of c, α are shown in Table 7.
7.4 Nematic instability: approach to GPIs
The weak FM branch of solutions (repulsive interaction) exhibits another fasci-
nating behavior in addition to its multicriticality. Upon approach to either the FM or 69
4 s F0 -2 -1 No q'-dependence 3
2 With c=1, Α=0.2 q'-dependence
1 a With c=1, Α=1 q'-dependence F0 0
-1 -1
-2 -2
-3 -6 -5 -4 -3 -2 -1 0
0 a Figure 19: With no q -dependence, F0 in the weak FM branch approaches -2, the 0 a q’=2 GPI; with the addition of q -dependence parametrized by c = 1, α = 0.2, to F0 , it can move into this region and approach a GPI in the middle of this original range a of finite-q’divergences. With a c = 1, α = 1 parametrization, F0 moves all the way through this region and can get arbitrarily close to -1, the q’=0 FM PI.
0 CD instability, χ2(q ) diverges in the spin-symmetric channel, leading to a charge ne-
s matic instability. In FL language, F2 → −5 in the weak FM branch of solutions. We
can therefore make the statement that this charge nematic transition both precedes
and is driven by the approach to the s-wave instabilities.
7.4.1 Approach to GPIs
The system only truly approaches the FM instability with added q0-dependence,
but without this dependence (when the system instead approaches a GPI near the
FM instability in the spin channel, see Fig. 19), the same behavior discussed above
s related to nematic instabilities occurs in F2 . Figures 20 and 21 show the higher 70
a c α F0 Max 0.9 0.5 -1.39 0.9 0.2 -1.85 1.2 0.2 -1.39 1.5 0.2 -1.11 1.65 0.2 -1.01 1 0 -2 1 1 -1
a a Table 7: Large U limiting values of F0 (F0 maximum) for different combinations of c, α. angular momentum harmonics of the FL interaction function in the charge channel upon approach to the ` = 0 density and spin instabilities, respectively, with no
0 a q -dependence added to F0 . Figures 22, 23 show the higher angular momentum harmonics of the FL interaction function in the spin channel upon approach to the
` = 0 spin and density instabilities, respectively, also with no q0-dependence added to
a F0 .
7.4.2 Approach to q = 0 PIs
0 a When explicity q -dependence is added to F0 (optimal parameterization c = 1, α =
1), the system approaches the q0 = 0 FM PI. The corresponding figures for added
0 a q -dependence in F0 (c = 1, α = 1 parametrization) are shown (Figures 24, 25, 26,
s and 27). It can be seen that in the charge channel (F` ), the nematic transition occurs en route to the FM and CD transitions (Figs. 20 and 21) as discussed, and in the
a spin channel (F` ), the system passes through no other instabilities as the FM or CD
PIs are approached (Figs. 22 and 23) . 71
s Fl 2 l + 1 6 CD PI
4 s F1 2 s F3 s 0.04 0.06 0.08 0.10 F0 + 1 s -2 F2
s Figure 20: Scaled spin-symmetric Landau parameters (F` ) upon approach to charge density instability. Dashed line indicates position of q0 = 0 PI for any channel. Note s s F2 s that F2 crosses its instability ( 5 = −1) before F0 reaches -1 (the CD instability).
7.5 Competing quantum fluctuations
In the TCSE method, an arbitrary underlying interaction is treated on the same footing as quantum fluctuations. In our model, only ` = 0 fluctuations are taken into
s,a account. F0 can then be thought of as a sum of the direct interaction and quantum
fluctuations. Schematically, this can be expressed as
s 1 1 3 F0 = (DI) + (QF )density + (QF )spin 2 2 2 (117) 1 1 1 F a = − (DI) + (QF ) − (QF ) , 0 2 2 density 2 spin
where DI is the direct interaction, and (QF )density,spin are quantum fluctuations
in the density and spin channels, respectively. QFs defined as in Equation (117) do 72
s Fl 2 l + 1 6 FM PI
4 s F1 2 s F3 a -1.25 -1.20 -1.15 -1.10 -1.05 F0 + 1
-2 s F2
s Figure 21: Scaled spin-symmetric Landau parameters (F` ) upon approach to FM 0 s instability. Dashed line indicates position of q = 0 PI for any channel. Note that F2 s F2 a crosses its instability ( 5 = −1) before F0 reaches -2 (the GPI associated with the FM instability). not include sign or prefactor, and hence are given as follows:
Z 2kF F sχ (q0)F s (QF ) = 0 0 0 dq0 density 1 + F sχ (q0) 0 0 0 (118) Z 2kF a 0 a F0 χ0(q )F0 0 (QF )spin = a 0 dq 0 1 + F0 χ0(q ) Upon approach to PIs’s (increasing U), quantum fluctuations are enhanced, as
evident from Figures 32 and 33 in both strong and weak (multicritical) FM branches
of solutions. Shown in these plots are values of the fluctuations as given by the in-
tegrals in Equation (118) in the weak and strong FM branches, here also including
the prefactor and sign appropriate for the chosen channel (see Eq. (117) for signs
and prefactors). The interplay between the short-range direct interaction and these
enhanced fluctuations is shown for the weak FM/multicritical branch in Figures 31
and 30 (density and spin channels, respectively). In the spin channel, fluctuations
work together to cancel most of the contribution from the driving interaction, which, 73
a Fl 2 l + 1 1.5 CD PI
1.0 a F2 0.5 Fs + 1 0.04 a 0.06 0.08 0.10 0 F1 -0.5 a F3 -1.0
a Figure 22: Scaled spin-antisymmetric Landau parameters (F` ) upon approach to charge density instability. Dashed line indicates position of q0 = 0 PI for any channel. a s Note that no F` ’s reach an instability before F0 reaches -1 (the CD instability).
a as can be seen by equation (117), leads to a small F0 , where by “small”, we mean
a “close to the F0 = −1 PI” or, equivalently for the case of q-independent Landau
a parameters, “close to the F0 = −2 GPI”. In the density channel, the density fluc- tuations, together with the driving interaction, compete against spin fluctuations to
s s give a small F0 , where by “small”, we mean “close to the F0 = −1 PI”. Thus, this
enhancement, interplay, and feedback of quantum fluctuations results in multicritical
behavior in which both channels simultaneously approach a PI or GPI. With added
0 a a 0 a q -dependence in F0 , which allows F0 to approach its q = 0 PI (F0 = −1), the
fluctuations in the density channel do not change qualitatively. However, the spin
fluctuations in the spin channel are stronger and more competitive with density fluc- tuations than they are in the case of no added q0-dependence. See Figures 28 and 29
for these fluctuations as a function of U. 74
a Fl 2 l + 1 1.5 FM PI 1.0 a F2 0.5 a -1.25 -1.20 -1.15 -1.10 -1.05 a F0 + 1 a F1 -0.5 F3 -1.0
a Figure 23: Scaled spin-antisymmetric Landau parameters (F` ) upon approach to FM instability. Dashed line indicates position of q0 = 0 PI for any channel. Note that a a no F` ’s reach an instability before F0 reaches -2 (the GPI associated with the FM instability).
Fluctuations in spin and density channels also affect the higher ` FL parameters, as evidenced by Equations 89, 90, 91, and 92. Fluctuation contributions to higher `
0 a parameters for the case of added q -dependence (c = 1, α = 1) in F0 , which allows for a “true” q0 = 0 approach to PIs in both channels (multicriticality), can be examined in Figures 34 through 37.
Several features can be attributed to the interplay shown in Figures 30 and 31:
a a 1/2 1. In the FM FL theory, the ferromagnetism is related to F0 as m ∼ [1+F0 ] . in
the multicritical branch of solutions, this quantity becomes weaker for increasing
direct interaction, as opposed to becoming stronger, as in the strongly FM
a branch (see Figures 38 and 39). This counterintuitive behavior of F0 is due
to the large fluctuations which are opposite in sign to the direct interaction,
a leading to a small F0 , which gets even smaller for increasing U (i.e. approach 75
s Fl 2 l + 1 4 Density PI
s F1 2 s F3 s 0.01 0.02 0.03 0.04 0.05 0.06 F0 + 1 PI s -2 F2
-4
s Figure 24: Scaled spin-symmetric Landau parameters (F` ) upon approach to charge 0 a density instability for the case of added q -dependence in F0 (c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel.
to PI) due to enhancement of fluctuations in this area.
2. As discussed, pairing amplitudes for the ferromagnetic solutions are found to be
attractive for both singlet and triplet, but singlet is found to be more attractive.
This is due to the interplay and competition between quantum fluctuations
and direct interaction. This result raises the intriguing possibility of switching
between singlet and triplet via some symmetry-breaking effect. 76
s Fl 2 l + 1 4 FM PI s F1 2 s F3 a -0.20 -0.15 -0.10 -0.05 0.00 F0 + 1 PI
-2 s F2
-4
s Figure 25: Scaled spin-symmetric Landau parameters (F` ) upon approach to FM 0 a instability for the case of added q -dependence in F0 (c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel.
a Fl 2 l + 1
a 6 F2
a Density PI 4 F1
2 a F3 s 0.01 0.02 0.03 0.04 0.05 0.06 F0 + 1 PI
s Figure 26: Scaled spin-antisymmetric Landau parameters (F` ) upon approach to 0 a charge density instability for the case of added q -dependence in F0 (c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel. Note s s F2 s that F2 crosses its instability ( 5 = −1) before F0 reaches -1 (the CD instability). 77
a Fl 2 l + 1 a F2 6
a 4 FM PI F1
2
a F3 a -0.20 -0.15 -0.10 -0.05 0.00 F0 + 1 PI
s Figure 27: Scaled spin-antisymmetric Landau parameters (F` ) upon approach to 0 a charge density instability for the case of added q -dependence in F0 (c = 1, α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel. Note s s F2 s that F2 crosses its instability ( 5 = −1) before F0 reaches -1 (the CD instability).
5 Spin Fluctuations Density Fluctuations
15 20 25 30 U
FM PI -5
-10 Driving Term
-15
Figure 28: Quantum fluctuations in spin channel as function of U for the case of 0 a added q -dependence (c = 1, α = 1) in F0 . Here, fluctuations are shown including the prefactor and sign appropriate for the spin channel. 78
Driving Term 10 Density Fluctuations
15 20 25 30 U
Density PI
-10
Spin Fluctuations -20
Figure 29: Quantum fluctuations in density channel as function of U for the case of 0 a added q -dependence (c = 1, α = 1) in F0 . Here, fluctuations are shown including the prefactor and sign appropriate for the density channel.
Spin Channel Fluctuations vs. U: Weak FM Branch
4
Spin Flucts 2
U 14 15 16 17 18 Density Flucts -2
-4 Driving Int.
-6
-8
Figure 30: Quantum fluctuations in spin channel as function of U. Here, fluctua- tions are shown including the prefactor and sign appropriate for the spin channel (see (117)). 79
Density Channel Fluctuations vs. U: Weak FM Branch
Spin Flucts 5
Density Flucts U 14 15 16 17 18
Driving Int. -5
Figure 31: Quantum fluctuations in density channel as function of U. Here, fluctua- tions are shown including the prefactor and sign appropriate for the density channel (see (117)).
Fluctuation Integrals vs. U: Weak FM
20
15
Spin Flucts 10
5 Charge Flucts
U 14 15 16 17 18
Figure 32: Spin fluctuations and density fluctuations in the weak FM branch with no included prefactors or signs (i.e. independent of channel) as function of U, where increasing U corresponds to approach to PI. 80
Fluctuation Integrals vs. U: Strong FM
6
4 Spin Flucts
2 Charge Flucts
U 14 15 16 17 18
Figure 33: Spin fluctuations and density fluctuations in the strong FM branch with no prefactors or signs (i.e. independent of channel) as function of U, where increasing U corresponds to approach to PI.
15 a Contributions to F1
10
Density Fluctuations
5
Spin Fluctuations
15 20 25 30 U
a 0 Figure 34: Contributions of ` = 0 spin and density fluctuations to F1 with a q - a dependent F0 . 81
15 s Contributions to F1
10
Density Fluctuations
5
Spin Fluctuations
15 20 25 30 U
s 0 Figure 35: Contributions of ` = 0 spin and density fluctuations to F1 with a q - a dependent F0 .
a Contributions to F2
15
10 Density Fluctuations
5 Spin Fluctuations
15 20 25 30 U
a 0 Figure 36: Contributions of ` = 0 spin and density fluctuations to F2 with a q - a dependent F0 . 82
20 s F2 = -5 HPIL 10 Density Fluctuations
15 20 25 30 U
-10
-20 Spin Fluctuations
-30
-40 s Contributions to F2
s 0 Figure 37: Contributions of ` = 0 spin and density fluctuations to F1 with a q - a dependent F0 .
Magnetism mHUL: WFM m
1.12
1.10
1.08
1.06
1.04
1.02
U 14 15 16 17 18 Figure 38: Magnetism as a function of direct repulsive interaction in the weak FM (multicritical) branch of solutions. 83
Magnetism mHUL: SFM m
2.0
1.8
1.6
U 14 15 16 17 18 Figure 39: Magnetism as a function of direct repulsive interaction in the strong FM (multicritical) branch of solutions.
Branch U Spin Fluct Density Fluct WFM 13 -10.2 7.5 WFM 14 -10.8 10.0 WFM 15 -11.3 15.1 WFM 16 -11.9 15.1 WFM 18 -12.8 20.2 SFM 13 -10.1 4.3 SFM 14 -10.7 3.5 SFM 15 -11.3 3.2 SFM 16 -12.0 2.0 SFM 18 -13.3 2.6
Table 8: Quantum fluctuation integrals in weak FM and strong FM branches for various values of U. Note that the trend for spin fluctuations is the same in both branches, but the trend of density fluctuations differs between the two branches. CHAPTER 8
Tractable crossing-symmetric equations in two dimensions
8.1 TCSE method formulated in 2D
In this chapter we formulate the TCSE method two-dimensions for the first time, and then apply this 2D TCSE method to an isotropic 2D Fermi liquid. The formu- lation and results may find applications in 2D and quasi-2D or layered systems with large inter-planar separation. One example is graphene, in which quantum fluctua- tions may play an important role. We start by formulating the TCSE equations in
2D. In carrying out the formal development, we start from the two-body vertex func- tions in the particle-particle, particle-hole, and exchange particle-hole channels and arrive at expressions for F, A, as in 3D case (see sections on microscopic FL theory and TCSE method in 3D). The TCSE equations in general form are then
s 0 s 0 a 0 a 0 s s 1 F0 (q )χ0(q)F0 (q ) 3 F0 (q )χ0(q)F0 (q ) Fpp0 = Dpp0 + s 0 0 + a 0 0 2 1 + F0 (q )χ0(q ) 2 1 + F0 (q )χ0(q ) 02 s 0 s a 0 a 1 q F1 χ1(q )F1 F1 χ1(q )F1 + 1 − 2 s 0 + 3 a 0 + ` = 2, 3 ... (119) 2 4kF 1 + F1 χ1(q ) 1 + F1 χ1(q )
s 0 0 s 0 a 0 0 a 0 a a 1 F0 (q )χ0(q )F0 (q ) 1 F0 (q )χ0(q )F0 (q ) Fpp0 = Dpp0 + s 0 0 − a 0 2 1 + F0 (q )χ0(q ) 2 1 + F0 (q )χ0(q) 02 s 0 s a 0 a 1 q F1 χ1(q )F1 F1 χ1(q )F1 + 1 − 2 s 0 − a 0 + ` = 2, 3 .... (120) 2 4kF 1 + F1 χ1(q ) 1 + F1 χ1(q )
84 85
In 3D, they are projected in Legendre polynomials, and in 2D, where only one relevant angle is present (see Figure 40 for the 2D geometry), they can be projected out in cosines, for example, following [50], as
s,a X s,a 0 Fpp0 = F` g` cos `(φ − φ ) (121) `
where g` is a normalization defined as
g` = δ`,0 + 2(1 − δ`,0) (122)
and φ, φ0 are the angles p and p’ make with the x-axis of Figure 40, respectively.
Figure 40: Geometry of the isotropic two dimensional system. Momentum vectors ~p and p~0 make angles φ and φ0, respectively, with the x axis.
This yields, for the FL interaction functions,
Z 2π s,a dφ s,a 0 F` = Fp,p0 cos `(φ − φ ). (123) 0 2π 86
Recall that in 2D, all Pomeranchuk instabilities occur at -1,
s,a F` → −1. (124)
Also in 2D, effective mass [50] is given by
m∗ = 1 + F s, (125) m 1
In 2D, scattering amplitudes [50] are related to the FL interaction functions as
s,a s,a F` A` = s,a (126) 1 + F`
8.2 2D Model
For a model calculation in 2D, we choose (as in 3D) a contact interaction of
strength U (U = U ↑↓, note also that U ↑↑ = 0) as the underlying interaction. This
s,a leads to an antisymmetrized driving term given by D0 = ±U/2. we keep only the
` = 0 quantum fluctuations and choose the driving term to be a contact interaction
(±U/2), then the 2D TCSEs take the form:
U 1 Z 2kF dq0 F sχ (q)F s F s = + 0 0 0 0 p 02 0 s 2 2π 0 1 − 0.25q 1 + χ0(q )F0 3 Z 2kF dq0 F aχ (q0)F a + 0 0 0 (127) p 02 0 a 2π 0 1 − 0.25q 1 + χ0(q )F0
U 1 Z 2kF dq0 F sχ (q0)F s F a = − + 0 0 0 0 p 02 0 s 2 2π 0 1 − 0.25q 1 + χ0(q )F0 1 Z 2kF dq0 F aχ (q0)F a − 0 0 0 (128) p 02 0 a 2π 0 1 − 0.25q 1 + χ0(q )F0 87
As in 3D, the momentum transfer in the exchange particle-hole channel is given
02 0 by q = 2kF (1 − cos θL), where θL =p ˆ· pˆ. One major difference between 3D and 2D is the form of the Lindhard function. In 2D, the real part of the Lindhard function is given by the following expression [51]:
s 2 2 2 2 |ν − q | 1 ν − q χ0(q) = 1 + Sgn[ν − q ]Θ − 1 − 1 2q 2 2q s 2 2 2 2 |ν + q | 1 ν + q + Sgn[ν + q ]Θ − 1 − 1 (129) 2q 2 2q
where Sgn[x] gives the algebraic sign of the argument x, and Θ[x] is the Heaviside
theta function (x > 0 : Θ[x] = 1; x < 0 : Θ[x] = 0). The 2D Lindhard function depends on momentum transfer q (scaled to kF ), but it also depends on frequency
ν (scaled to Fermi energy EF ), as shown in Figure 41. The square root singularities
present in the Lindhard function pose additional computational challenges that did
not appear in the 3D calculation.
Figure 41: The real part of the 2D Lindhard function shown for several values of frequency (ν, scaled to EF ) as a function of momentum transfer (q, scaled to kF ). Figure from [51]. 88
a s a s a s ν U F0 F0 ν U F0 F0 ν U F0 F0 0 10 -1.424 -0.7749 0.1 10 -1.458 -0.7753 0.3 10 -1.576 -0.775 0 11 -1.33 -0.8061 0.1 11 -1.362 -0.8064 0.3 11 -1.463 -0.8183 0 12 -1.274 -0.8186 0.1 12 -1.302 -0.8251 0.3 12 -1.402 -0.8338 0 13 -1.231 -0.8373 0.1 13 -1.265 -0.8438 0.3 13 -1.367 -0.8523 0 14 -1.206 -0.856 0.1 14 -1.237 -0.8563 0.3 14 -1.342 -0.8709 0 15 -1.19 -0.8625 0.1 15 -1.218 -0.8688 0.3 15 -1.321 -0.8844 0 20 -1.125 -0.8996 0.1 20 -1.162 -0.9031 0.3 20 -1.279 -0.9203 0 25 -1.093 -0.9196 0.1 25 -1.142 -0.9251 0.3 25 -1.265 -0.9322 0 30 -1.076 -0.9343 0.1 30 -1.134 -0.9377 0.3 30 -1.257 -0.9467 0 35 -1.063 -0.9437 0.1 35 -1.131 -0.9473 0.3 35 -1.253 -0.9543 0 40 -1.055 -0.9507 0.1 40 -1.127 -0.9543 0.3 40 -1.251 -0.9607 0 45 -1.048 -0.9561 0.1 45 -1.126 -0.9601 0.3 45 -1.249 -0.9665 0 50 -1.042 -0.9606 0.1 50 -1.125 -0.9639 0.3 50 -1.248 -0.9697
s,a Table 9: Calculated values of F0 as functions of underlying interaction U for fre- quencies ν = 0, 0.1, 0.3 in 2D in the ferromagnetic state.
8.3 2D Results
Calculations of ` = 0 FL interaction functions were carried out for 5 different
choices of frequency (ν = 0, 0.1, 0.3, 0.5, 0.8). As in 3D, the integrals are done numer-
ically. The ` = 0 integrals are done using a 25-point Gaussian integration. Once the
values of the quantum fluctuation integrals have been obtained, we again search for
solutions graphically in parameter space. For all frequencies calculated, two solutions
were found - one paramagnetic and one ferromagnetic. Results for ferromagnetic
` = 0 FL interaction functions are given in Tables 9 and 10 and for paramagnetic
` = 0 FL interaction functions in Tables 11, 12 and 13.
For all frequencies shown above, the system approaches multicriticality in the
` = 0 spin-symmetric and spin-antisymmetric channels. In other words, F0a and F0s
a both approach -1 simultaneously and from opposite sides (F0 from the negative side,
s F0 from the positive side) as in 3D. This multicritical behavior (which occurs for all 89
a s a s ν U F0 F0 ν U F0 F0 0.5 10 -1.739 -0.7846 0.8 10 N/S N/S 0.5 11 -1.589 -0.8251 0.8 11 -2.016 -0.8407 0.5 12 -1.53 -0.8501 0.8 12 -1.879 -0.875 0.5 13 -1.489 -0.8719 0.8 13 -1.817 -0.8937 0.5 14 -1.466 -0.8797 0.8 14 -1.772 -0.9125 0.5 15 -1.443 -0.8906 0.8 15 -1.754 -0.9218 0.5 20 -1.396 -0.928 0.8 20 -1.706 -0.9505 0.5 25 -1.374 -0.9436 0.8 25 -1.698 -0.9649 0.5 30 -1.364 -0.9553 0.8 30 -1.692 -0.9726 0.5 35 -1.359 -0.9617 0.8 35 -1.689 -0.9768 0.5 40 -1.356 -0.9677 0.8 40 -1.688 -0.9806 0.5 45 -1.353 -0.9721 0.8 45 -1.687 -0.9839 0.5 50 -1.351 -0.9751 0.8 50 -1.686 -0.9855
s,a Table 10: Calculated values of F0 as functions of underlying interaction U for fre- quencies ν = 0.5, 0.8 in 2D in the ferromagnetic state.
a s a s ν U F0 F0 ν U F0 F0 0 1 -0.3477 1.015 0.1 1 -0.3443 1.011 0 2 -0.4412 2.219 0.1 2 -0.4346 2.186 0 3 -0.4813 3.376 0.1 3 -0.4798 3.36 0 4 -0.5125 4.505 0.1 4 -0.5069 4.474 0 5 -0.5398 5.64 0.1 5 -0.534 5.619 0 6 -0.5568 6.732 0.1 6 -0.543 6.703 0 7 -0.5713 7.879 0.1 7 -0.5611 7.787 0 8 -0.5883 9.007 0.1 8 -0.5792 8.996 0 9 -0.6004 10.09 0.1 9 -0.6007 10.09 0 10 -0.6102 11.17 0.1 10 -0.6128 11.21 0 15 -0.6587 16.79 0.1 15 -0.661 16.76 0 20 -0.7002 22.38 0.1 20 -0.6971 22.33 0 25 -0.729 27.95 0.1 25 -0.7242 27.95 0 35 -0.7715 39.16 0.1 35 -0.7724 39.21 0 45 -0.8049 50.46 0.1 45 -0.805 50.49 0 100 -0.8903 112.8 0.1 100 -0.887 112.6 0 500 -0.9738 567.4 0.1 500 -0.9732 567.3
s,a Table 11: Calculated values of F0 as functions of underlying interaction U for fre- quencies ν = 0 and ν = 0.1 in 2D in the paramagnetic state. 90
a s ν U F0 F0 0.5 1 -0.376 0.968 0.5 2 -0.5059 2.309 0.5 3 -0.5693 3.614 0.5 4 -0.6127 4.937 0.5 5 -0.6512 6.287 0.5 6 -0.6851 7.689 0.5 7 -0.7212 9.18 0.5 8 -0.7513 10.78 0.5 9 -0.782 12.56 0.5 10 -0.8233 14.97
s,a Table 12: Calculated values of F0 as functions of underlying interaction U for fre- quencies ν = 0.5 in 2D in the paramagnetic state.
a s ν U F0 F0 0.8 1 -0.005772 0.9941 0.8 2 -0.4375 1.932 0.8 3 -0.5473 3.167 0.8 4 -0.6127 4.398 0.8 5 -0.6572 5.656 0.8 6 -0.6958 6.939 0.8 7 -0.7212 8.199 0.8 8 -0.7483 9.516 0.8 9 -0.7694 10.84 0.8 10 -0.7905 12.15 0.8 15 -0.8637 19.16 0.8 25 -0.944 37.04 0.8 26 -0.9521 40.17
s,a Table 13: Calculated values of F0 as functions of underlying interaction U for fre- quencies ν = 0.8 in 2D in the paramagnetic state. 91
frequencies discussed here) is shown for the case of ν = 0 in Figure 42.
-0.8 s -0.9 F0
-1.0 PI
-1.1 a F0
-1.2
-1.3
-1.4 20 30 40 50 U
a s Figure 42: Both F0 and F0 approach the PI of -1 for increasing U for all frequencies, and this multicritical behavior is shown here for the case of ν = 0.
Additionally, higher angular momentum component of the FL interaction func- tions can be projected out as, for example,
1 Z 2kF [1 − q02/2] F sχ (q0)F s F s = dq0 0 0 0 1 p 02 s 0 2π 0 1 − q /4 1 + F0 χ0(q ) 3 Z 2kF [1 − q02/2] F aχ (q0)F a + dq0 0 0 0 (130) p 02 a 0 2π 0 1 − q /4 1 + F0 χ0(q )
1 Z 2kF [1 − q02/2] F sχ (q0)F s F a = dq0 0 0 0 1 p 02 s 0 2π 0 1 − q /4 1 + F0 χ0(q ) 1 Z 2kF [1 − q02/2] F aχ (q0)F a − dq0 0 0 0 (131) p 02 a 0 2π 0 1 − q /4 1 + F0 χ0(q ) 92
1 Z 2kF [2(1 − q02/2)2 − 1] F sχ (q0)F s F s = dq0 0 0 0 2 p 02 s 0 2π 0 1 − q /4 1 + F0 χ0(q ) 3 Z 2kF [2(1 − q02/2)2 − 1] F aχ (q0)F a + dq0 0 0 0 (132) p 02 a 0 2π 0 1 − q /4 1 + F0 χ0(q )
1 Z 2kF [2(1 − q02/2)2 − 1] F sχ (q0)F s F a = dq0 0 0 0 2 p 02 s 0 2π 0 1 − q /4 1 + F0 χ0(q ) 1 Z 2kF [2(1 − q02/2)2 − 1] F aχ (q0)F a − dq0 0 0 0 (133) p 02 a 0 2π 0 1 − q /4 1 + F0 χ0(q )
Integrals in the higher angular momentum channel equations were computed nu- merically using a 20-point Gauss-Lobatto quadrature with Kronrod Extension (see
Appendix A for integration details). Results for higher ` FL interaction functions can be found in Tables 14 through 17 for frequencies in the range ν = 0.1 − 0.8. For
0 ν = 0, the Lindhard function χ0(q ) becomes simply the 2D density of states, which is a constant. Thus, the integrand does not have q’-dependence, and hence gives zero- projections for higher ` FL interaction functions. This was confirmed numerically for a range of underlying interaction (U) between 10 and 50.
The case of ν = 0 corresponds to the limit of a local Fermi liquid (LFL), which was
s,a discussed in an earlier section. Thus, F` = 0 for all ` > 0. For frequencies 0.1, 0.3,
s a and 0.5, two instabilities are crossed en route to the multicritical point (F0 ,F0 → −1):
a s s F1 and F2 . F2 also precedes and is driven by the ` − 0 multicriticality in 3D. For
a s the highest frequency used, 0.8, F2 and F2 cross their instabilities en route to the 93
a s a s ν U F1 F1 F2 F2 0.1 10 -0.0936493 0.41263 -0.0237388 -0.226881 0.1 11 -0.130141 0.586123 -0.0084333 -0.374906 0.1 12 -0.181937 0.797429 0.0238334 -0.556694 0.1 13 -0.234167 1.03035 0.0558704 -0.764612 0.1 14 -0.304357 1.30801 0.107047 -1.01381 0.1 15 -0.372535 1.59774 0.156173 -1.27999 0.1 20 -1.03784 3.98867 0.727933 -3.52296 0.1 25 -2.19973 7.98626 1.79847 -7.38904 0.1 30 -3.92973 13.6741 3.45226 -12.9708 0.1 35 -5.68473 19.5099 5.14623 -18.7528 0.1 40 -32.3629 100.136 31.7016 -99.1362 0.1 45 -6.75388 23.9778 6.00408 -22.8473 0.1 50 -7.69893 27.3808 6.87734 -26.1451
s,a s,a Table 14: Calculated values of F1 ,F2 as functions of underlying interaction U for frequency ν = 0.1 in 2D in the ferromagnetic state.
a s a s ν U F1 F1 F2 F2 0.3 10 -0.302303 1.17551 -0.0609047 -0.520911 0.3 11 -0.416077 1.71134 -0.0259138 -0.951439 0.3 12 -0.572291 2.28587 0.0712978 -1.4095 0.3 13 -0.713751 2.87693 0.150964 -1.90095 0.3 14 -0.857388 3.54007 0.232341 -2.48348 0.3 15 -1.06357 4.38756 0.36648 -3.2091 0.3 20 -2.06309 8.47517 1.12572 -6.94729 0.3 25 -3.38231 13.0974 1.94451 -10.2603 0.3 30 -4.64797 18.1595 3.37647 -16.1501 0.3 35 -6.27013 24.0424 4.86704 -21.8803 0.3 40 -7.6586 29.392 6.14102 -27.1528 0.3 45 -10.3665 39.0041 8.70588 -36.6539 0.3 50 -13.0144 48.0247 11.2558 -45.6028
s,a s,a Table 15: Calculated values of F1 ,F2 as functions of underlying interaction U for frequency ν = 0.3 in 2D in the ferromagnetic state. 94
a s a s ν U F1 F1 F2 F2 0.5 10 -0.465783 1.74786 -0.17081 -0.501518 0.5 11 -0.648686 2.53805 -0.114516 -1.08092 0.5 12 -0.782904 3.18227 -0.0664703 -1.60269 0.5 13 -0.939343 3.96388 -0.0234161 -2.19258 0.5 14 -1.10802 4.61599 0.0507231 -2.62329 0.5 15 -1.3375 5.55092 0.14836 -3.25999 0.5 20 -2.08277 9.31842 0.616106 -6.67644 0.5 25 -3.5822 15.1411 1.31645 -10.431 0.5 30 -4.56874 19.7525 2.54954 -16.1424 0.5 35 -5.80215 24.8084 3.62392 -20.9903 0.5 40 -6.9402 30.0026 4.56132 -25.9135 0.5 45 -9.16841 38.4955 6.61325 -34.1964 0.5 50 -12.1502 49.0496 9.42959 -44.5274
s,a s,a Table 16: Calculated values of F1 ,F2 as functions of underlying interaction U for frequency ν = 0.5 in 2D in the ferromagnetic state.
a s a s ν U F1 F1 F2 F2 0.8 11 -1.27616 4.36657 -0.888965 1.05461 0.8 12 -0.767047 3.22489 -0.625274 -0.330291 0.8 13 -0.751073 3.51878 -0.654006 -0.725778 0.8 14 -0.664356 3.7763 -0.727184 -1.19221 0.8 15 -0.690385 4.21476 -0.846297 -1.2926 0.8 20 -0.384344 5.35453 -1.31619 -2.34294 0.8 25 0.226352 5.90646 -1.9372 -3.18442 0.8 30 0.389654 7.7528 -2.78307 -3.23513 0.8 35 1.23694 7.15123 -3.09123 -4.43588 0.8 40 1.7579 8.07716 -3.81801 -4.96199 0.8 45 2.64455 8.54065 -4.59611 -6.00473 0.8 50 3.06086 9.32129 -5.18707 -6.41845
s,a s,a Table 17: Calculated values of F1 ,F2 as functions of underlying interaction U for frequency ν = 0.8 in 2D in the ferromagnetic state. 95
multicritical point. This is the only case in which we find a spin-nematic PI crossed
in either 3D or 2D. See Figures 43 and 44 for higher ` FL interaction functions as
a a function of 1 + F0 , which can be interpreted as distance to the multicritical point
s a where both F0 ,F0 → −1.
F s 10 { Ν=0.3 FM PI
5 s F1
a -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 1+F0 PI
s -5 F2
-10 Figure 43: ν = 0.3: Spin-symmetric (charge density channel) FL interaction functions s s s (F1 and F2 ) plotted as a function of distance to FM PI. The behavior of F0 is shown in Fig. 42. The PI value (-1) plotted as a red dashed line. Notice that the charge s density nematic instability F2 is crossed en route to the FM PI.
Using the Patton-Zaringhalam pairing amplitudes in the s-p approximation (Eq. 104), we find that on the FM side, both singlet and triplet pairing are attractive, and singlet pairing is preferred. On the PM side, we find that only triplet pairing is attractive.
These pairing results are qualitatively the same as in the 3D isotropic Fermi liq- uid. Detailed tables of results for scattering amplitudes and pairing can be found in
Appendix B. 96
F a 10 { Ν=0.3 FM PI 5 a F2
a -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 1+F0 PI a -5 F1
-10
a Figure 44: ν = 0.3: Spin-antisymmetric (spin channel) FL interaction functions (F1 a and F2 ) plotted as a function of distance to FM PI. The PI value (-1) plotted as a a red dashed line. Notice that the spin nematic instability F1 is crossed en route to the FM PI. CHAPTER 9
Application to ferromagnetic superconductors
One hallmark of superconductivity is the Meissner effect, which is the expulsion of a magnetic field from a superconductor below its superconducting transition temper- ature. Another well-known effect is that strong enough magnetic fields can destroy conventional superconductivity by breaking Cooper pairs in one of two ways:
1. The paramagnetic effect, in which FM coupling forces all electron spins to point
in the same direction, or
2. The orbital effect, in which the Lorentz force (q~v × B~ ) acts in opposite direc-
tions on the two electrons in Cooper pairs, due to the fact that have opposite
momenta.
As stated by Flouquet and Buzdin [55], “superconductivity and magnetism usually try to avoid each other.” However, in recent years (since 2000), several materials have been observed to simultaneously exhibit ferromagnetism and superconductivity at low temperatures; the list of these materials is short and primarily includes ura- nium compounds which exhibit band (itinerant) magnetism, such as UGe2 and UIr.
In ferromagnetic superconductors (FMSCs), the coexistence of these two orders can either be uniform or non-uniform. Uniform superconductivity and ferromagnetism in- volves the same electrons being involved in SC pairing and magnetic ordering. In this 97 98 situation, s-wave singlet pairing is not possible due to strong exchange interactions which result in the FM alignment of spins, thus, pairing must be of a more exotic symmetry. The other case of coexistence involves non-uniform phases. In these situa- tions, SC/FM order parameters are intertwined in a spatially-modulated way (spiral, stripe, etc.). If a FMSC exhibits singlet pairing, it is most likely of this second type, in which interplay between SC and magnetism plays a large role in the physics of the system. For example, a scenario in which one can imagine FM and singlet SC pairing in shown in Fig. 45; this involves a long-wavelength spiral in which the super- conducting coherence length is larger than the magnetic correlation length [56]. In either case (uniform or non-uniform coexistence), strong local repulsion is favorable, as it can lead to long-range FM order or long-wavelength magnetic correlations [55].
As previously discussed, an underlying repulsive contact interaction (U) was chosen for this calculation with application to FMSCs in mind.
Figure 45: One possibility for the coexistence of FM and singlet superconductivity is a long-wavelength spin-flip spiral, in which the superconducting coherence length is larger than the magnetic correlation length [56]. 99
9.1 Phase transitions in FMSCs
UGe2 is a material known to exhibit itinerant 5f ferromagnetism. It has a SC
phase, but only at high pressures ( 8 kbar-16 kbar) and only near the FM instability.
In UGe2, in addition to the first-order PM→FM transition at the Curie temperature
TC , there exists a first-order transition from a large moment FM phase (“FM2”) to a small moment FM phase (“FM1”) at a temperature Tx. Interestingly, the pressure
(px) at which Tx → 0 is the pressure at which Tsc is highest (as shown in single-crystal measurements) [59]. Figure 46 summarizes these results.
In UIr, another FMSC, pressure-induced magnetic transitions have also been ob- served. UIr is of much current interest because it exhibits SC despite a lack of inversion symmetry (inversion symmetry is typically thought of as favorable for SC).
Additionally, and of more relevance here, UIr has been observed to have three FM phases, with the weakest FM phase (F3) corresponding to the highest pressure 47.
Note that in both examples discussed here, the SC transition is in close proximity to a FM QCP, which suggests that SC is perhaps related to enhanced quantum
fluctuations near the QCPs.
9.2 Transition temperatures
Magnetic phase transitions are accompanied by a divergence in magnetic suscepti-
a bility [1/(1+F0 χ0(q))], which is related to FL parameters. Until now, all calculations have been done at T = 0, but the Lindhard function also has a temperature-dependent 100 form, so the susceptibility can be written as a function of temperature as
1 χ(q, T ) ∼ a , (134) 1 + F0 (q)χ0(q, TC )
0 where χ0(q ,T ) is the temperature-dependent Lindhard function [41, 58]. When
T/TF is small, a valid approximation is to expand the Lindhard function to quadratic order in q and T [41], which gives " # q 2 1 T 2 T 2 χ0(q, T ) ≈ 1 − + γ1 − γ2 , (135) kF 12 TF TF where TF is the Fermi temperature and
2 γ1 = π /96 (136) 2 γ2 = π /24.
The ferromagnetic transition (Curie) temperature, TC , is given by the temperature at which the q = 0 magnetic susceptibility diverges. Here Tc is scaled to TF .
s a 1 + F0 TC = a (137) γ2F0
. As discussed, several experiments on FMSCs [60] [63] show more than one magnetic transition as a function of doping or pressure (see Figs. ?? and 47). In our approach, one extra benefit of having a q-dependence in the spin-antisymmetric FL parameter
a (F0 (q)) is that a second magnetic transition temperature can be calculated. So for
a a F0 → F0 (q) there exists another situation in which magnetic susceptibility χ(q, T ) diverges; this occurs at some critical value of q (qx). The temperature at which this occurs, Tx, is again scaled to TF . s 2 qx 1 2 Tx = 1 − + a / (γ1qx + γ2) (138) 12 F0 (qx) 101
.
Different parameterizations of the q-dependence (i.e. different combinations of c and α, see Eq. 116) result in different Tc/Tx ratios which can be compared to different sets of experimental results. For example, in UGe2, TC /Tx (where Tx is the FM2-FM1 transition temperature) is found experimentally to be between 2.4 and 2.9 [60], [61]
(Figure 46). With the parametrization c=0.9, α=0.2, we find a similar ratio, as is shown in Fig. 48. In UIr, a ratio of 1-1.4 is found for TC2/TC1 (Figure 47) [63]; with parametrization of c ≈ 1.25, α = 0.2, we obtain ratios in this range. By adjusting the q-dependence to match future experimental results via spin susceptibility and specific heat measurements, we hope to have the ability to predict properties of other FMSC materials. 102
Figure 46: Taken from [59]. (a) Phase diagram as a function of pressure for UGe2, with TC indicating a first-order PM-FM phase transition, and Tx indicating the first- order transition within the FM phase from a strong FM (FM2) to weak FM (FM1). (b) Shown are the transitions in magnetization as a function of pressure at T=2.3K; at low pressure, the system is in the FM2 phase, then as pressure increases, there is a jump in M indicating a transition to the FM1 phase, and at even higher pressure, M drops to zero, indicating a transition to the PM phase. (c) Phases in the presence of an external magnetic field H (px and pc indicate critical pressure where Tx and TC → 0. 103
Figure 47: Adapted from [63] by Anne de Visser [62], shown here is the phase diagram as a function of pressure for UIr, as determined by resistivity, magnetization, and ac- susceptibility measurements under pressure. UIr has 3 FM phases, and the SC dome lies in the FM3 phase (the weakest FM phase), very close to the FM QCP.
TcTx vs. U TcTx 3.0
2.5 c=0.9, Α=0.2
2.0
c=1.2, Α=0.2 1.5 c=1.3, Α=0.2
1.0
0.5
0.0 U 10 12 14 16 18 20
Figure 48: Calculated Tc/Tx ratio as a function of U for three different parameteri- zations of the q-dependence. CHAPTER 10
Conclusions and future directions
In summary, the TCSE method allows for exploration of Fermi liquids near gener- alized Pomeranchuk instabilities, taking into account competing quantum fluctuations and an arbitrary underlying interaction. For the first time, the TCSE method has been applied starting in the ordered state. Pomeranchuk instabilities provide a dif- ferent viewpoint on quantum criticality, which is of current interest due to the usual proximity of quantum criticality to novel superconducting phases. In this way, ap- proaching PIs from the ordered state is a way to approach quantum criticality from the ordered state. In addition to finding results in 3D, the TCSE method has been formulated and applied in two dimensions. This allows for the inclusion of frequency dependence, and is also relevant for many 2D and quasi-2D correlated electron sys- tems.
As discussed at length in the introduction, the major goals of this dissertation were:
1. Development of a theory (TCSE) for describing physics near PI and GPI starting
from ordered side, for the first time, alongside PI from the disordered side
in 3D, using a short-range interaction and truncation at a particular order of
fluctuations 104 105
2. Formulation of TCSE in 2D for the first time, and also for the first time pro-
viding a way to explore frequency dependence
3. Constructing suitable and robust numerical/graphical methods of solutions that
can be utilized in future work
4. Exploring possible applications, such as to ferromagnetic superconductors
Looking more specifically at the work presented here, a contact interaction has been chosen as the underlying interaction, and truncate the quantum fluctuations beyond the s-wave angular momentum channel. We explore both a repulsive and an attractive driving interaction. With a repulsive interaction, we find PM and FM so- lutions. With an attractive interaction (physically, this can be thought as an effective interaction generated by collective modes or exchange correlations), we find PM and
PS (phase separation) solutions. For the case of repulsive interaction, we start on the FM side and find quantum multicriticality in the weakly ferromagnetic branch of solutions, meaning that both spin and density Fermi surfaces go soft simultane- ously. For this type of weak FM ground state, when an instability is approached in one channel (spin-symmetric or spin-antisymmetric), it is thus also approached in the other. Additionally, upon approach to either spin or charge-density ` = 0 PI, the sys-
s tem passes through a charge nematic (` = 2) PI, i.e. F2 → −5. Thus, this nematic transition both precedes and is driven by the ferromagnetic and density instabili- ties. With regards to superconducting pairing, model calculations find an attractive amplitude in both singlet and triplet pairing channels in the ferromagnetic region, 106 indicating that both singlet and triplet pairing are possible in this region. However, singlet pairing is preferred. This may be due to the strong competition and interplay between quantum fluctuations and the underlying interaction. The singlet-to-triplet amplitude ratio decreases significantly (but never becomes < 1) upon approach to the multicritical regime. The fact that both singlet and triplet pairing amplitudes are attractive raises the intriguing possibility of switching between singlet and triplet pairing states via some symmetry-breaking effect.
As discussed, numerical and graphical techniques were developed to explore these systems near instabilities (further details on numerics can be found in Appendix A).
Additionally, the TCSE method was formulated in two dimensions and presented here for the first time. This involves a frequency dependence which was not investigated in the 3D case.
When a q0 (exchange particle-hole momentum transfer) dependence is added to
a to F0 , we find potential for application to ferromagnetic superconducting systems
0 (such as UGe2 and UIr). Adjustable parameters c and α of the q -dependence may correspond to particular physical systems; to explore this further, we await exper- imental results (i.e. specific heat and spin susceptibility measurements) on more
FMSC compounds. Additionally, this method has the potential for application to other correlated-electron systems, such as strontium ruthenate compounds, quantum
Hall systems, and in general, materials that exhibit phases involving magnetic tex- tures (spiral magnetism, for example). 107
In the future, in addition to application to more materials as discussed above, sev- eral major ideas may be explored. One is the investigation of polarized systems with higher ` fluctuations included explicitly. Another is the possibility of a longer-range underlying interaction, which we expect to change the pairing results in particular.
This would change the structure of the induced (quantum fluctuation) terms, and thus would require a fresh consideration of the numerical integration strategies used for the contact interaction in this work.
Also of interest is a self-consistent solution with full q, q’-dependence of the Lan- dau interaction functions in the direct and exchange particle-hole channels. A q, q’-dependence would thus be generated self-consistently as opposed to the form used by ansatz discussed in this work. Now that the TCSE method has been formulated in 2D for isotropic systems, a natural extension is to apply the method to 2D lattice systems. We hope to explore a link with dynamical mean-field theory (DMFT) in this case, and in general, a possible connection between TCSE and functional renor- malization group (fRG) [64].
As discussed throughout, this work provides FL interaction functions and scat- tering amplitudes, which can now be used as input for determining thermodynamic, transport, dynamic and collective mode properties [56, 57] (serving as input into so- lutions kinetic equations [65], and ongoing work on collective modes of general spin- orbit coupled systems [66]. The FL interaction functions can also be used as input to calculations of the Higgs amplitude mode in weak ferromagnetic metals [68]. Appendices
108 CHAPTER A
Solution details
In this appendix, further details of graphical and numerical methods involved in
finding solutions are discussed. First, I will discuss how the quantum fluctuation integrals are solved, then I will elaborate further on the graphical methods used to
find solutions to this set of equations.
A.1 3D Integrations
The set of coupled non-linear integral equations to be solved in 3D is
U 1 Z 2kF F sχ (q0)F s 3 Z 2kF F aχ (q0)F a F s = + 0 0 0 dq0 + 0 0 0 dq0 0 2 2 1 + F sχ (q0) 2 1 + F aχ (q0) 0 0 0 0 0 0 (139) Z 2kF s 0 s Z 2kF a 0 a a U 1 F0 χ0(q )F0 0 1 F0 χ0(q )F0 0 F0 = − + s 0 dq − a 0 dq 2 2 0 1 + F0 χ0(q ) 2 0 1 + F0 χ0(q ) As mentioned, in 3D, the integrations were initially solved using Mathematica’s “NIn- tegrate” routine, which is a globally adaptive algorithm for numerically computing integrals. Integrals were also solved by hand using a 10-point Gaussian quadrature and were found to agree with results generated by NIntegrate. The Gaussian quadra- ture is a simple quadrature which can be used to approximate an integral over [-1,1] as: ∞ n Z 1 X X f(x)dx = wif(xi) ' wif(xi). (140) −1 i=1 i=1
109 110
i Weight (wi) Abscissa (xi) 1 0.2955242247147529 -0.1488743389816312 2 0.2955242247147529 0.1488743389816312 3 0.2692667193099963 -0.4333953941292472 4 0.2692667193099963 0.4333953941292472 5 0.2190863625159820 -0.6794095682990244 6 0.2190863625159820 0.6794095682990244 7 0.1494513491505806 -0.8650633666889845 8 0.1494513491505806 0.8650633666889845 9 0.0666713443086881 -0.9739065285171717 10 0.0666713443086881 0.9739065285171717
Table 18: Weights (wi) and abscissa (xi) for a 10-point Gaussian integration.
Gaussian abscissa (xi) are defined as roots of the nth Legendre polynomial, and weights (wi) are given by [69]
2 wi = − 2 2 . (141) (1 − xi )(Pn(xi))
However, Eq. 140 can be generalized to integrals over an arbitrary interval [a,b] with a change of limits:
Z 1 b − a Z 1 b − a b + a f(x)dx = f xi + dx −1 2 −1 2 2 n b − a X b − a b + a ' w f x + (142) 2 i 2 i 2 i=1
This form can be used to integrate from q = 0 to q = 2kF , as is needed if using a
Gaussian quadrature to compute the integrals in Eqn. 139. See Table 18 for weights and abscissas of the n = 10 Gaussian quadrature used.
These methods (NIntegrate, Gaussian quadrature) work well until a divergence in 111
s,a 0 the quantum fluctuation integrals occurs, i.e. when 1 + F0 χ0(q ) = 0, so
s,a 1 F0,pole = − (143) χ0(q)
0 where χ0(q ) is the Lindhard function, given in 3D by
1 q 1 1 − 0.5q χ (q) = 1 + − ln . (144) 0 2 4 q 1 + 0.5q
As can be seen from Eq. 143, the Lindhard function alone dictates the form and
s,a range of the two sets of divergences (F0 ). We are considering the range of momentum
0 transfer q between 0 and 2kF . In this range, the Lindhard function χ0(q ) is a smooth, well-behaved function which decreases monotonically from 1 to 0.5 with increasing q (see Fig. 14). Thus, it is instructive to consider the two “endpoint” scenarios
χ0(0) = 1 and χ0(2kf ) = 0.5 of the range of integration to determine the range of
s,a F0 for which divergences will occur. At the q = 0 end, a divergence occurs when
s,a s,a 1 + F0 = 0 → F0 = −1 (145)
And at the q = 2kF end, a divergence occurs when
F s,a 1 + 0 = 0 → F s,a = −2 (146) 2 0
In light of this, there are two ways to view the divergent region: (1) For every value of q in the range of integration (0 to 2kF ), there is a corresponding F0,pole at
s,a which a divergence occurs, or (2) for every value of F0 between -1 and -2, there is a corresponding qpole at which a divergence occurs. These two viewpoints are reflected in Figures 49 and 50, respectively. 112
F0,pole -1.0
-1.2
-1.4
-1.6
-1.8
0.5 1.0 1.5 2.0 q
Figure 49: For every value of q in the range of integration (0 to 2kF ; note that q is scaled to kF in this figure), there is a corresponding F0,pole at which a divergence occurs.
s,a In Mathematica, qp is found for a given F0 using the FindRoot function, which
s,a 0 is set to find the root of 1 + F0 χ0(q ) = 0. Once the q value of the divergence is found (henceforth referred to as qp), a numerical contour integration can be used to evaluate integrals in this divergent region of parameter space. The singularities involved in the quantum fluctuation integrals are not removable or integrable in the usual sense, but they can be evaluated using the principal value technique, which will be discussed shortly.
Since the pole qp lies on the real axis, it is natural to use a contour of the type shown in Fig 51 so that the singularity does not lie on the path of integration. In
Fig. 51, qp is shown as a blue point at the origin; the closed contour C starts a distance of + from the pole. Initially, the integration takes a path along the real axis (C1) to some distance R, follows a large arc of radius R around a path Reiθ, where θ : 0 → π
((C2), follows a path along the real axis (C3) from -R to qp − , then follows a small 113
qpole 2.0
1.5
1.0
0.5
-1.8 -1.6 -1.4 -1.2 -1.0 F0 s,a Figure 50: For every value of F0 between -1 and -2, there is a corresponding qpole at which a divergence occurs. Note that q is scaled to kF in this figure.
iθ arc of radius around a counterclockwise path e , where θ : π → 0 (C4).
Define Ji as the integral corresponding to path Ci, where the value of the integral over the full closed contour is J = J1 + J2 + J3 + J4. These integrals, which follow the paths shown and described above, are given by
Z 2 2 F0 χ0(q) J1 = dq qp+ 1 + F0χ0(q) Z π iReiθF 2χ (1 + Reiθ) J = dθ 0 0 2 1 + F χ (1k + Reiθ) 0 0 0 F (147) Z qp− 2 F0 χ0(q) J3 = dq 0 1 + F0χ0(q) Z 0 iθ 2 iθ ie F0 χ0(qp + e ) J4 = dθ iθ . π 1 + F0χ0(qp + e ) For a simple closed curve that does not intersect itself, the residue theorem states that the value of the integral around this contour is equal to 2πi multiplied by the sum of the residues of any poles enclosed by the contour [70]:
I X f(z)dz = 2πi Res(f, z0), (148) 114
Figure 51: Schematic representation of closed path used for numerical contour inte- gration around a singularity marked by the dot at the origin. The path C is comprised of parts C1,C2,C3, and C4. Path C2 has a radius R, and path C4 has a radius .
where Res(f, z0) is the residue of function f at singular point z0 which is enclosed by the contour of integration. Since the contour used in this calculation (Fig. 51) does not enclose any singular points, the residue theorem indicates that
J = J1 + J2 + J3 + J4 = 0, (149)
which is useful for computing the Cauchy principal value (PV) [71] of the divergent
quantum fluctuation integrals. One way the principal value can be defined (the way
that is relevant to this calculation) is in the context of an integral over a range (a to
b) which contains a singular point (c):
Z b Z c− Z b PV f(x)dx = lim f(x)dx + f(x)dx . (150) + a →0 a c+ 115
J3 J2 J1 J4 0.0001 -34.8462 9.87507 + i9.76182 24.9718 −0.000637292 − i9.76182 0.001 -27.6886 9.87507 + i9.76182 17.8199 −0.00637292 − i9.76182 0.01 -20.5051 9.87507 + i9.76182 10.6938 −0.0637295 − i9.76182 0.1 -13.063 9.87507 + i9.76182 3.82557 −0.637589 − i9.76182
Table 19: For F0 = −1.25 and R = kF , is varied to check the sensitivity of the contour integration to the value of . Ji’s are the values of the integrals along a particular segment of the contour, as shown in Fig. 51.
J PV 0.0001 (2.71709 × 10−7) − i(1.51863 × 10−9) -9.87443 0.001 (2.70825 × 10−7) − i(1.51505 × 10−9) -9.86869 0.01 (6.53638 × 10−8) − i(1.51418 × 10−9) -9.81134 0.1 (4.52492 × 10−8) − i(1.51419 × 10−9) -9.23748
Table 20: For F0 = −1.25 and R = kF , is varied to check the sensitivity of the contour integration to the value of . J is the sum of all integrals over individual contour segments (Ji’s, as shown in Table 19), i.e. the integral over the full contour, which, according Cauchy’s residue theorem, should equal zero (since no poles are enclosed by the contour). Thus, J can be thought of as being representative of the “error” associated with a particular choice of and R. The principal value (PV) of the integral is the sum of J1 and J3.
Thus, the principal value (PV) of our integral is (see Eqns. 147 and 149)
PV = J1 + J3 = −J2 − J4. (151)
The two parameters involved in this integration are R (large radius of contour) and (small radius of contour). Multiple values of R and were used in integrations to check for sensitivity to these parameters, and to find the parameters which min- imized error (“error” here meaning the distance of J from its expected value of 0).
Tables 19, 20, and 21 present the results of those checks. The value of F0 used in these cases is -1.25 (which corresponds to a pole of qp = 1.4511kF ). 116
R J2 J 0.1 2.05951 − i(1.12688 × 10−13) −7.81556 − i9.76182 0.3 6.96367 + i(3.60822 × 10−14) −2.9114 − i9.76182 0.5 12.5441 + i9.76182 2.66908 + i(1.24981 × 10−10) 0.8 9.35949 + i9.76182 −0.515576 + i(1.04681 × 10−11) 1.0 9.87507 + i9.76182 (2.71709 × 10−7) − i(1.51863 × 10−9) 1.5 12.7326 + i10.0851 2.85753 + i0.323264 2.0 16.8896 + i10.4948 7.01451 + i0.732975
Table 21: For F0 = −1.25 and = 0.0001, R (scaled to kF ) is varied to check the sensitivity of the contour integration to the value of R. J2 is the values of the integral along a particular contour C2, as shown in Fig. 51. J is the sum of all integrals over individual contour segments (Ji’s, as shown in Table 19), i.e. the integral over the full contour, which, according Cauchy’s residue theorem, should equal zero (since no poles are enclosed by the contour). Thus, J can be thought of as a measure of error associated with a particular choice of and R.
When varying the value of R, the only integral affected is J2. Shown in Table 21
is the sensitivity of J2 and J to changing R. Notice that J is closest to its expected
value of zero when R = kF .
Solving these integrals allows for the calculation of the PV, and so the divergent
intervals can be evaluated. However, Mathematica has an PV option in its numerical
integration routine (NIntegrate), which automatically computes the PV of a given
integral. As a check, we compared the results of this routine to the PV calculated
from the numerical contours discussed above (J1 + J + 3, or equivalently, −(J2 + J4)),
with parameters = 0.0001 and R = kF . This comparison is shown for two different values of F0,pole in Table 22.
The quantum fluctuation integrals, whether in the range of divergences or not, are then used as input to find graphical solutions, as discussed in the body of the text. 117
F0,pole J1 + J3 NIntegrate -1.25 -9.87443 -9.87507 -1.75 -11.2354 -11.2363
Table 22: Comparison between calculation of PV using individual contour segments J1 + J3 = −(J2 + J4) for = 0.0001 and R = kF and using NIntegrate’s “Principal Value” option to solve integrals. Shown here are comparisons for two values of F0,pole: -1.25 and -1.75.
Note: The type of contour integration discussed here involves integrating over a small symmetric range about some singularity. However, in this calculation, the only physically relevant values of q lie between 0 and 2kF . A problem occurs when the singularity lies at exactly 0 or 2kF , as the consequences of extending the integration outside of the range (to either (0 − ) or (2kF + )) are unclear. Thus, these points are problematic in the PV scheme. However, the exclusion of these points does not make a qualitative difference in results, and in fact, these “endpoint singularities” may be signaling strong instabilities of the physical system; it is well-known that q = 0 corresponds to the FM instability, and perhaps q = 2kF also corresponds to a strong physical instability (e.g. a spin-density wave instability).
A.2 2D Integrations
In 2D, we are interested in solving the set of coupled non-linear integral equations given by
U 1 Z 2kF dq0 F sχ (q)F s F s = + 0 0 0 0 p 02 0 s 2 2π 0 1 − 0.25q 1 + χ0(q )F0 3 Z 2kF dq0 F aχ (q0)F a + 0 0 0 (152) p 02 0 a 2π 0 1 − 0.25q 1 + χ0(q )F0 118
i Weight (wi) Abscissa (xi) 1 0.3607615730481386 0.6612093864662645 2 0.3607615730481386 -0.6612093864662645 3 0.4679139345726910 -0.2386191860831969 4 0.4679139345726910 0.2386191860831969 5 0.1713244923791704 -0.9324695142031521 6 0.1713244923791704 0.9324695142031521
Table 23: Weights (wi) and abscissa (xi) for a 6-point Gaussian integration.
U 1 Z 2kF dq0 F sχ (q0)F s F a = − + 0 0 0 0 p 02 0 s 2 2π 0 1 − 0.25q 1 + χ0(q )F0 1 Z 2kF dq0 F aχ (q0)F a − 0 0 0 , (153) p 02 0 a 2π 0 1 − 0.25q 1 + χ0(q )F0
where the Lindhard function in 2D is
s 2 2 2 2 |ν − q | 1 ν − q χ0(q) = 1 + Sgn[ν − q ]Θ − 1 − 1 2q 2 2q s 2 2 2 2 |ν + q | 1 ν + q + Sgn[ν + q ]Θ − 1 − 1 (154) 2q 2 2q and is shown in Fig. 41. To solve integrals in 2D, we used several different quadratures.
Solutions were found in the PM phase using a 6-point Gaussian quadrature, the weights and abscissas of which are given by Table 23 [69]. Discussion of the Gaussian quadrature can be found in the previous section of this appendix.
Solutions on the FM side were found using a 25-point Gaussian quadrature, the weights and abscissas of which are given by Table 24 [69].
s,a Lastly, for the higher ` projections (F` for ` > 0), a 20-point Gauss-Lobatto 119
i Weight (wi) Abscissa (xi) 1 0.1231760537267154 0.0000000000000000 2 0.1222424429903100 -0.1228646926107104 3 0.1222424429903100 0.1228646926107104 4 0.1194557635357848 -0.2438668837209884 5 0.1194557635357848 0.2438668837209884 6 0.1148582591457116 -0.3611723058093879 7 0.1148582591457116 0.3611723058093879 8 0.1085196244742637 -0.4730027314457150 9 0.1085196244742637 0.4730027314457150 10 0.1005359490670506 -0.5776629302412229 11 0.1005359490670506 0.5776629302412229 12 0.0910282619829637 -0.6735663684734684 13 0.0910282619829637 0.6735663684734684 14 0.0801407003350010 -0.7592592630373576 15 0.0801407003350010 0.7592592630373576 16 0.0680383338123569 -0.8334426287608340 17 0.0680383338123569 0.8334426287608340 18 0.0549046959758352 -0.8949919978782753 19 0.0549046959758352 0.8949919978782753 20 0.0409391567013063 -0.9429745712289743 21 0.0409391567013063 0.9429745712289743 22 0.0263549866150321 -0.9766639214595175 23 0.0263549866150321 0.9766639214595175 24 0.0113937985010263 -0.9955569697904981 25 0.0113937985010263 0.9955569697904981
Table 24: Weights (wi) and abscissa (xi) for a 25-point Gaussian integration. 120 quadrature with Kronrod extension (for error estimation) was used, henceforth re- ferred to as GL+K. Several other integration routines were checked (6-Point Gauss,
25-Point Gauss, Monte Carlo, Clenshaw-Curtis) before deciding on GL+K and were all found to be in general agreement, with GL+K yielding the smallest error. Gauss-
Lobatto integration is a type of closed Gaussian quadrature (meaning the endpoints are included), and the Kronrod extension adds n − 1 points to an n-point Gauss-
Lobatto quadrature.
a s In 3D, a different integration strategy was used in the “pole region” F0 − F0 pa- rameter space than outside this region. However, due to the complexity and numerical difficulty involved in the 2D Lindhard function (for example, square root divergences and frequency dependence), a contour integration was not attempted. In the range
s,a of divergent F0 , multiple quadratures gave similar results, so the methods discussed continued to be used in these regions. CHAPTER B
Tables of 3D Results
Tables of results in this appendix are grouped as follows to give a sample of results for each type of solution found:
• q-dependence with extra solution branches (Tables 25 through 30): As
mentioned earlier, multiple solution branches to the TCSE’s were found on the
FM side in 3D, but some were deemed “unphysical” due to their limiting behav-
ior not agreeing with the limiting behavior of the local Fermi liquid. Included
here are detailed results for the two physical branches of solutions discussed ear-
lier (strong FM and weak FM) as well as for all other solutions on the FM side
for one representative set of c, α (c = 1, α = 0.2); recall that the q-dependence
a added here to F0 is a function of two adjustable parameters, c and α. These
extra branches also exist in the q-independent case.
• q-independent solutions (Tables 31 through 40): solutions for the case of no
s,a explicit q-dependence added to F0 are given here for the two FM branches of
solutions (strong/far from PI and weak/near PI) as well as the PM solution.
• Attractive underlying interaction (Tables 41 through 49): For an attractive
underlying interaction (a negative U), three solution branches are found. One
121 122
on the PM side, and 2 in the phase separation (PS) region. One is a solution far
from PIs (labeled “strong” PS, as solutions are “deep” into the PS region) and
the other is a solution near PIs (labeled “weak” PS, as solutions are close to the
instability). Solutions in the PS region were only found for when a negative U
was used as the underlying interaction.
a • Optimal q-dependence in F0 (Tables 50 through 52): With a parameter
a choice of c = 1, α = 1 in the q-dependence of F0 , the weak FM solution truly
a approaches the FM PI, as opposed to approaching the GPI of F0 → −2 as
a in the solutions with no q-dependence in F0 . This is the multicritical branch
of solutions (the other FM solution and the PM solution are not qualitatively
affected by the added q-dependence as so are not given here).
a s • Double q-dependence (i.e. q-dependence in both F0 and F0 (Tables 53
through 55): A q-dependence with parameter choice of c = 1, α = 1 was added
a s to both F0 and F0 . Again, the other (strong) FM solution and the PM solution
are not qualitatively affected by this q-dependence and so are not given here. 123 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` s 0 A A a 0 A 2: Strong FM solution (physical branch) . = 0 s 1 F , α ), scattering amplitudes ( = 1 c s,a ` F a 1 F Table 25: s 0 F a 0 UF 50 -14.9840 -0.8442 -11.8530 -0.0887052 -0.8221 -8.67627 3.84464 -0.0954464 -0.7969 -7.72126 1.07153 -0.0915875 3.2515 -5.41849 -7.403 -0.7924 2.69703 1.09217 -0.7884 -6.754 -0.09140822 1.13028 -4.62114 -0.085085 -0.0801519 -0.7845 -3.92368 -6.11120 -0.0985829 1.6851 2.50841 -0.7814 2.5513 -0.0944717 -0.07396 -5.445 1.5603518 1.15618 -2.6481 -0.0668887 1.14879 -0.7799 1.42023 -2.43843 -4.77714 -3.7259 2.42313 -0.495054 -2.25454 2.35063 -0.415894 -3.7619 -0.7806 -0.056743 1.17379 -3.34613 -0.343264 -0.0823522 1.19566 -0.0875686 -0.0449542 -3.64037 -0.7977 -3.57457 -2.923 2.2959 1.36613 1.37876 2.27183 -0.0758295 -0.0142912 -0.8118 -0.0684141 1.22497 -2.2019 -2.21243 1.26476 1.34044 -0.00309825 2.48033 -3.54339 1.31795 -0.321125 -0.325359 -3.55789 -2.18242 2.73544 1.42626 -2.17118 -0.057837 -0.0456381 -0.310933 1.52002 -3.94315 -0.302371 1.29281 -4.3135 1.30057 -0.0143596 -2.19547 -0.00310145 -2.1731 1.35776 -0.295025 1.43081 -2.40569 -0.296763 -2.57842 -0.321692 -0.351765 300 -93.23150 -0.9919 -46.33100 -0.9572 -30.67 3.89672 -0.9206 0.828078 23.3705 0.208272 10.9414 1.01084 1.02206 -122.457 7.21016 -22.3645 1.0337 1.69503 0.648951 -11.5945 2.65871 2.35444 0.194751 -30.7657 -6.45956 -10.4833 2.11853 -2.02882 -4.05746 -1.07284 as a function of U in the strongly FM solution (physical branch in which limits are consistent with LFL limits). Shown are FL interaction functions ( 124 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` A s 0 A a 0 A 2: Strong FM solution (unphysical branch) . = 0 s 1 F , α ), scattering amplitudes ( = 1 s,a ` c F a 1 F Table 26: s 0 F a 0 UF 50 -15.0840 -1.078 -11.9230 -5.89157 -1.084 -8.77527 -1.93918 -6.04947 -1.091 1.07102 -7.81426 -2.68967 -6.22202 13.8205 -1.094 1.09158 -7.49624 -3.41703 -6.30333 12.9048 6.1125 -1.094 1.12862 -6.85322 -3.65265 5.95134 -6.29125 -5.48402 -1.095 11.989 1.1467620 -3.68888 -6.21 -26.0013 -6.30655 8.60724 11.6383 5.79328 1.15394 -5.556 -1.09518 -3.79613 13.3713 5.72452 11.6383 1.18859 -1.095 24.5814 -6.28616 1.1708514 -4.88 2.83719 16.7899 5.73452 -6.26941 0.350401 11.5263 -3.8573 -3.458 -1.09513 -3.90757 16.0646 2.14543 5.72187 -1.43809 1.19194 -1.092 -6.25887 1.21949 -3.04912 -0.810778 11.5263 14.3047 2.32886 -6.17531 -3.93919 11.5263 -1.089 -2.476 5.73876 -3.72043 1.25773 -0.750574 2.71866 5.75279 -6.09803 -1.081 1.40683 11.5263 13.4981 -3.45191 -0.610783 12.9166 -5.89353 11.8696 5.76169 1.48804 2.91717 -2.56482 3.05241 5.83437 12.5828 12.236 1.67751 -0.543215 -0.493631 15.4925 13.3457 3.11386 5.90507 2.41493 6.11039 -0.463368 22.9155 -17.6812 -0.67087 0.642878 11.0814 -1.25805 2.21617 300 -93.26150 -1.008 -46.33100 2.08275 -1.032 -30.73 -2.97208 21.5626 -1.05 7.14122 1.01084 -4.55581 1.02206 2.45796 126 32.25 1.03364 1.22931 -319.351 21 2.11253 2.63359 -232.745 31.0055 8.78476 29.2092 10.2623 1.35103 10.7256 0.991487 as a function of U in the strongly FM solution (unphysical branch in which limits are not consistent with LFL limits). Shown are FL interaction functions ( 125 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` A s 0 A a 0 A 2: Weak FM solution (physical branch) . = 0 s 1 , α F = 1 ), scattering amplitudes ( c s,a ` F a 1 F Table 27: s 0 F a 0 UF 24 -1.66722 -0.9903 -1.66720 0.561627 -0.986 -1.66818 -0.9803 30.6787 -0.755718 -1.67614 2.49925 -0.988124 -0.9707 29.3614 -1.786 -102.09313 25.4622 2.49925 -0.604686 -0.9318 -1.885 0.473065 2.49701 -70.4286 -0.0456995 19.4105 -0.9101 -49.7614 9.20064 2.73277 2.47929 -1.01019 0.0122589 2.27226 -33.1297 -1.47344 -27.726 6.86795 2.72189 -13.6628 -0.757336 2.68379 2.12994 -20.9197 -8.56661 -0.0464064 -16.0891 -10.1235 2.5984 2.26233 -5.80342 -4.03956 -5.72028 -11.3595 0.012209 -1.13387 -2.70762 2.08796 -4.64116 -0.841141 as a function ofLFL U limits). in the weak FM solution (the only physical branch of weak FM in which limits are consistent with Shown are FL interaction functions ( 126 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` s 0 A A a 0 A 2: Weak FM solution (unphysical branch #1) . s 1 = 0 F , α ), scattering amplitudes ( s,a ` = 1 F c a 1 F Table 28: s 0 F a 0 UF 24 -1.66722 -1.009 -1.66720 -1.18365 -1.012 -1.66818 -2.68693 28.9334 -1.017 -1.67614 -3.62699 27.4302 2.49925 -1.024 -1.778 112.11113 22.8233 2.49925 -3.8856 -1.046 -1.864 84.3333 -1.95512 2.49701 -4.7585 -1.054 16.1295 -25.7473 -2.101 59.8235 2.71816 -4.99844 -1.069 2.70424 2.47929 4.7813 17.3542 -5.53244 42.6667 24.0075 -0.777669 2.2322 2.65148 -0.860346 2.28535 9.15631 13.1626 9.48727 2.15741 1.90827 22.7391 25.4359 2.52952 19.5185 15.4928 8.118 3.52624 7.50352 6.55388 18.0468 -1.20629 1.27988 1.84338 2.45449 7.65897 8.56924 9.59843 1.00445 1.07438 1.25526 as a function of U inwith the LFL weak limits). FM solution (the first of 3 unphysical branches of weak FM in which limits are not consistent Shown are FL interaction functions ( 127 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` s 0 A A a 0 A 2: Weak FM solution (unphysical branch #2) . = 0 s 1 F , α ), scattering amplitudes ( s,a ` = 1 F c a 1 F Table 29: s 0 F a 0 8 -1.1294 -0.8289 -1.076 -0.3966 10.7536 11.7391 -29.1228 8.75194 -35.0626 -4.84454 14.1579 -0.657275 2.34563 2.38938 3.34453 3.2807 -6.852 -9.81088 -0.148563 0.652545 UF 27 -1.66726 -0.9941 -1.66724 2.45834 -0.993 -1.66622 -1.664 -0.99 1.79339 32.575420 -0.986 -1.651 2.4992518 -1.13876 31.9105 -0.9797 -1.615 -1.31187 -168.49214 -0.779842 2.49925 35.3308 -0.9723 -1.381 31.029813 1.35115 -141.857 0.0325049 24.4495 -0.9458 2.5015 -1.306 2.5060212 18.1608 2.74702 1.12241 -0.9367 2.5361 3.95362 -1.251 -70.428611 -43.6707 2.62602 -99 -0.9252 5.70216 -48.2611 -0.756596 -1.211 2.7422 -14.1742 10 -2.33135 -35.1011 3.62467 7.16479 -1.05377 -1.178 -37.1825 -0.91 -7.4048 0.0321565 2.73553 -17.4502 -1.83549 -0.8897 -11.9352 2.67213 -13.1889 2.57469 -21.9191 4.26797 8.29765 1.70571 2.7652 -15.4257 -11.3643 4.98406 9.2818 -5.69389 -14.7978 -17.9963 -3.94528 -1.01176 -2.92349 -12.369 -28.694 -22.3281 1.96577 -5.54883 5.73934 -8.11902 6.61798 -1.20996 2.11459 -10.1111 5.04323 -8.06618 3.88331 -6.6869 2.20337 2.26721 -6.21518 -1.46157 3.60015 -1.11524 3.46564 -6.07979 -6.14603 -0.847942 -0.598422 as a function ofconsistent with U LFL in limits). the weak FM solution (the second of 3 unphysical branches of weak FM in which limits are not Shown are FL interaction functions ( 128 ) t , g s g t g s g s 1 A a 1 A ), and singlet and triplet pairing amplitudes ( s,a ` A s 0 A a 0 A 2: Weak FM solution (unphysical branch #3) . s 1 = 0 F , α ), scattering amplitudes ( s,a ` = 1 F c a 1 F Table 30: s 0 F a 0 8 -1.1334 -1.08 -1.082 -1.139 4.7924 4.86493 -34.4166 -41.5199 8.5188 13.1951 8.19424 13.5 1.85568 3.23365 1.84503 -7.26443 3.28647 1.35834 -2.45194 1.40727 UF 27 -1.66726 -1.006 -1.66724 0.886115 -1.007 -1.66622 31.0032 0.105844 -1.009 -1.66420 2.49925 30.2229 -2.77177 -1.012 -1.651 167.66718 2.49925 -3.24308 33.6978 -1.017 0.684062 -1.615 143.85714 -3.32175 29.0986 2.5015 -1.024 2.73532 0.102237 -1.38813 2.50602 112.111 -3.41389 21.9076 -1.038 2.7291 39.8714 -1.317 84.333312 -36.4337 -0.367325 14.7144 2.5361 -1.042 13.8955 -4.36714 -1.26 40.0247 33.484311 2.75475 2.62602 59.8235 1.04617 3.57732 -1.217 -1.048 42.6667 2.7196110 -1.86228 11.9604 30.9717 27.3158 -1.055 -10.7301 -1.182 2.18078 24.7451 -0.418577 48.5424 12.3576 4.15457 2.63866 -1.061 3.05936 9.58311 -16.9158 24.8095 2.49194 3.67459 3.94892 35.6229 -22.3068 4.84615 0.775673 1.43625 5.60829 21.8333 26.633 -26.9991 4.16427 2.39577 19.1818 1.81071 6.49451 1.26281 1.50463 2.62714 17.3934 1.51469 3.64674 1.70483 3.46616 2.00201 1.85914 3.37501 0.858715 -0.0876462 1.81416 1.65077 1.56734 as a functionconsistent of with U LFL in limits). the weak FM solution (the third of 3 unphysical branches of weak FM in which limits are not Shown are FL interaction functions ( 129 :) s,a 0 F s 3 F a 3 F s 2 F a 2 F s 1 F : strong (large moment) FM solutions (no q-dependence in s,a ` F a 1 ) as a function of U in the strong FM solution with no explicit q-dependence F s,a ` F s 0 F a 0 UF 50 -15.9440 -0.6558 -12.5830 -0.019422 -0.663120 -9.21 -0.0210241 0.949228 -5.767 -0.676218 0.998434 0.110064 -0.7079 -0.0242127 -5.052 0.11863416 -0.270855 -0.0330538 1.09379 -0.7203 -0.288649 -4.309 -0.035154815 1.36609 -0.0386643 -0.0380465 -0.7391 0.133929 0.135817 -3.90814 1.49665 0.144564 0.188811 -0.0465469 -0.323944 -0.7544 -3.49113 -0.429789 1.71589 -0.0438953 0.216093 -0.0486233 -0.7711 -2.975 -0.0622133 0.166141 -0.483979 1.90553 0.264536 -0.0659659 -0.8041 0.23452 -0.0720122 -0.577546 2.18766 0.309183 -0.0915518 0.271122 -0.0894438 -0.658407 2.83808 0.378706 0.337352 -0.104876 -0.794588 0.556213 0.398547 -0.132179 -1.12105 0.501733 -0.201134 0.774603 . 300 -99.33150 -0.6338 -49.32100 -0.0154463 -0.6379 -32.64 0.816561 -0.0161841 -0.6424 0.0880203 0.839891 -0.0165994 -0.224442 0.0917812 0.865017 -0.0278152 -0.232472 0.0958861 0.106992 -0.0290639 -0.241007 0.111557 -0.0303959 0.116499 s,a 0 F Table 31: FL interaction functions added to Shown are FL interaction functions ( 130 s 3 A :) s,a 0 F a 3 A s 2 A + 1)), as a function of U in the strong ` (2 / s,a ` A a 2 A (1 + / s,a ` A = s,a ` s 1 . A A s,a 0 F : strong (large moment) FM solutions (no q-dependence in a 1 s,a ` A A ), related to F’s as s,a ` A s 0 A a 0 Table 32: Scattering amplitudes UA 50 1.06693440440 -1.905287623 1.08635578630 -0.019548557 -1.96823983420 0.721073587 1.12180268 -0.021172477 0.107693367 -2.08832612718 1.20977554 0.74911878 -0.286367832 -0.024409708 1.246791708 -2.42348510816 0.115884433 -0.035332243 0.801548199 -2.575259206 -0.033422042 -0.306333576 1.30220610515 0.13141468 0.132225311 -0.039169115 0.938659075 -0.038254421 -2.832886163 -0.346385929 1.34387895514 0.998510002 0.181940525 0.141638874 -0.047280487 -0.044172294 -3.071661238 -0.470206955 1.401445203 0.2071406713 1.091558539 -0.049424358 0.16228916 -0.062771186 -3.368719965 0.251243414 -0.535846711 1.506329114 1.165335856 0.226917612 -0.067449012 -0.652970048 -0.072760723 -4.104645227 0.291177569 1.265113751 -0.090601477 0.261012537 -0.094433657 -0.758255092 0.352041922 0.321841449 1.458397281 -0.106471182 -0.944720755 0.500532467 0.377077959 -0.134722935 -1.445043117 0.468175953 -0.20708424 0.69742738 300 1.010169836150 -1.730748225 1.020695364100 -0.015526241 -1.761668048 1.031605563 0.641856111 -0.016271882 -1.796420582 0.086497591 0.65618347 -0.016691758 -0.23499034 0.671420333 0.090126811 -0.027926167 0.094081871 -0.243807692 0.105381292 -0.253212182 -0.029185076 -0.030528463 0.109807037 0.11459188 Shown are scattering amplitudes ( FM solution with no explicit q-dependence added to 131 s 3 A :) s,a 0 F a 3 A s 2 A + 1)), as a function of U in the strong ` (2 / s,a ` F a 2 A (1 + / s,a ` F = s,a ` s 1 . A A s,a 0 F : strong (large moment) FM solutions (no q-dependence in a 1 s,a ` A A ), related to F’s as s,a ` A s 0 A a 0 Table 33: Scattering amplitudes UA 50 1.06693440440 -1.905287623 1.08635578630 -0.019548557 -1.96823983420 0.721073587 1.12180268 -0.021172477 0.107693367 -2.08832612718 1.20977554 0.74911878 -0.286367832 -0.024409708 1.246791708 -2.42348510816 0.115884433 -0.035332243 0.801548199 -2.575259206 -0.033422042 -0.306333576 1.30220610515 0.13141468 0.132225311 -0.039169115 0.938659075 -0.038254421 -2.832886163 -0.346385929 1.34387895514 0.998510002 0.181940525 0.141638874 -0.047280487 -0.044172294 -3.071661238 -0.470206955 1.401445203 0.2071406713 1.091558539 -0.049424358 0.16228916 -0.062771186 -3.368719965 0.251243414 -0.535846711 1.506329114 1.165335856 0.226917612 -0.067449012 -0.652970048 -0.072760723 -4.104645227 0.291177569 1.265113751 -0.090601477 0.261012537 -0.094433657 -0.758255092 0.352041922 0.321841449 1.458397281 -0.106471182 -0.944720755 0.500532467 0.377077959 -0.134722935 -1.445043117 0.468175953 -0.20708424 0.69742738 300 1.010169836150 -1.730748225 1.020695364100 -0.015526241 -1.761668048 1.031605563 0.641856111 -0.016271882 -1.796420582 0.086497591 0.65618347 -0.016691758 -0.23499034 0.671420333 0.090126811 -0.027926167 0.094081871 -0.243807692 0.105381292 -0.253212182 -0.029185076 -0.030528463 0.109807037 0.11459188 FM solution with no explicit q-dependence added to Shown are scattering amplitudes ( 132 :) s,a 0 F /m ∗ m (sp) t /g s g (spd) t g (sp) t g (spd) s g . s,a 0 (sp) F s g U 50 -1.47145252440 -1.412099672 -1.50998585130 -0.185966657 -1.445885317 -1.58212787220 -0.21880509 -0.190358697 -1.508426362 -1.77293423218 -0.225543531 7.91245349 -0.198680908 7.932318701 -1.6691264416 -0.238565178 -1.85791292 1.316409333 1.332811333 -0.221764961 7.963160053 -1.993226119 -1.73808924315 -0.276110584 1.364596667 -1.844814649 -0.232030983 -2.10422675814 7.994654451 -0.293946598 -0.24968777 -1.930103882 1.455363333 13 -2.26012909 8.007176003 -0.266731665 -0.325038892 1.498883333 -2.591332705 -0.354184387 7.982874443 -2.0420429 7.888927471 -2.26166268 1.571963333 -0.286466867 1.635176667 -0.338178617 -0.394530424 -0.500309916 7.889670147 7.662615482 1.946026667 1.72922 300 -1.362423142150 -1.315133193 -1.382188314100 -0.173627976 -1.332847748 -1.403183219 -0.200418637 -0.175825672 -1.351638902 7.846795055 -0.203653547 -0.178326171 7.861129156 1.272187 -0.207267342 1.279963667 7.868633133 1.288339 Table 34: Pairing amplitudes and effective mass: strong (large moment) FM solutions (no q-dependence in Shown are singlettriplet and ratio triplet (in pairingq-dependence s-p added amplitudes approximation) to and (in effective both mass, the as s-p a and function of s-p-d U approximations), in along the with strong singlet FM to solution with no explicit 133 :) s,a 0 F s 3 F a 3 F s 2 F a 2 F s 1 F : weak (small moment) FM solutions (no q-dependence in a 1 s,a ` F F ) as a function of U in the weak FM solution with no explicit q-dependence s,a ` F s 0 F a 0 UF 18 -2.00916 -0.9648 -2.03315 -1.03022 -0.9476 -2.06914 -0.841235 18.6019 -0.934 13.9463 -2.13 8.4534113 -0.609531 5.60965 -14.8433 -2.293 -0.9139 11.1699 -10.2464 -0.8825 -0.490456 -4.68781 4.09626 -0.250101 -3.04809 8.69207 20.7518 -7.48189 5.87496 2.84332 12.7831 -2.06897 -5.34028 1.6176 8.62796 -1.37548 -3.03261 -0.679825 5.55439 2.74197 . s,a 0 F Table 35: FL interaction functions Shown are FL interactionadded to functions ( 134 s 3 A :) s,a 0 F a 3 A s 2 A + 1)), as a function of U in the weak ` (2 / s,a ` F a 2 A (1 + / s,a ` F = s,a ` A . s 1 A s,a 0 F : weak (small moment) FM solutions (no q-dependence in s,a ` a 1 A A ), related to F’s as s,a ` A s 0 A a 0 Table 36: Scattering amplitudes UA 18 1.99108027816 -27.40909091 1.96805421115 -1.569038167 -18.08396947 1.935453695 2.58336998114 -1.16905036 -14.15151515 3.141735069 1.88495575213 -0.764951564 2.468910618 7.539798645 -10.61440186 1.773395205 2.364850846 2.643654597 -0.586308907 -14.19203007 -7.510638298 2.251617698 9.765172309 5.234348763 2.23024751 -0.272847475 15.07296858 -5.399067793 1.985910922 1.812574267 -2.937071971 4.523138436 1.222195358 78.46890796 3.864593971 -7.707190745 -1.711854523 -0.752949879 3.096982808 1.970216496 FM solution with no explicit q-dependence added to Shown are scattering amplitudes ( 135 :) s,a 0 F /m ∗ m (sp) t /g s g (spd) t g (sp) t g (spd) s g (sp) s g U 18 -10.1682040616 1.784405685 -7.49104844915 -2.103980253 -2.310962995 -6.15439544414 -1.737474955 -1.367838558 -2.985442973 -5.064610836 4.832841962 -0.77437874913 -1.07976387 -3.006474242 7.200633333 5.476558914 -3.908819315 -0.806900099 -0.011317963 5.648766667 -2.851552782 5.581127708 5.69976049 -0.585439592 6.276626858 -1.329555101 3.897356667 6.676725265 4.7233 2.95832 . s,a 0 F Table 37: Pairing amplitudes and effective mass: weak (small moment) FM solutions (no q-dependence in Shown are singlet and tripletratio pairing (in amplitudes s-p approximation) (in and bothadded effective the to mass, s-p as and a s-p-d function approximations), of along U with in singlet the to weak FM triplet solution with no explicit q-dependence 136 :) s,a 0 F s 3 F a 3 F s 2 F a 2 F s 1 : PM solutions (no q-dependence in F s,a ` F a 1 F s 0 F ) as a function of U in the paramagnetic solution with no explicit q-dependence s,a ` F a 0 84 -0.5862 -0.549 8.481 -0.487 4.336 0.010 0.571 2.125 0.007 -0.054 0.449 0.003 -0.042 -0.038 0.295 0.0169 -0.037 -0.020 0.0332 0.0116 -0.028 0.0244 0.0060 0.0147 UF 50 -0.62240 50.650 -0.62030 0.014 40.650 -0.61620 0.726 0.014 30.640 -0.61018 -0.078 0.715 0.014 20.600 -0.047 -0.60816 -0.077 0.699 0.0246 0.014 18.590 -0.047 -0.60615 -0.074 0.0460 0.671 0.0241 0.013 16.570 -0.047 -0.60514 -0.070 0.0451 0.663 0.0234 0.012 15.590 -0.046 -0.60413 -0.068 0.0438 0.654 0.0220 0.012 14.570 -0.046 -0.60112 -0.067 0.0414 0.649 0.0215 0.011 13.550 -0.045 -0.59911 -0.066 0.0407 0.643 0.0209 0.012 12.550 -0.045 -0.59710 -0.065 0.0399 0.633 0.0206 0.011 11.540 -0.044 -0.594 -0.064 0.0394 0.625 0.0203 0.011 10.520 -0.044 -0.062 0.0389 0.615 0.0199 0.010 -0.044 -0.061 0.0382 0.604 0.0195 -0.044 -0.059 0.0375 0.0190 -0.043 0.0366 0.0184 0.0357 300 -0.629150 300.600 -0.627100 0.014 150.700 -0.625 0.759 0.015 100.700 -0.083 0.751 0.015 -0.048 -0.082 0.742 0.0263 -0.048 -0.081 0.0489 0.0259 -0.048 0.0482 0.0256 0.0475 Table 38: FL interaction functions . s,a 0 F Shown are FL interaction functionsadded ( to 137 :) s,a 0 F s 3 A + 1)), as a function of U in the ` (2 a 3 / A s,a ` F s 2 (1 + A / s,a ` F . a 2 s,a = 0 A F s,a ` A s 1 : PM solutions (no q-dependence in A s,a ` A a 1 A s 0 A ), related to F’s as s,a ` A a 0 84 -1.413 0.8952 -1.217 0.010 0.813 -0.948 0.480 0.007 0.680 -0.055 0.390 0.003 -0.043 -0.038 0.268 0.0169 -0.037 -0.020 0.0331 0.0116 -0.028 0.0244 0.0060 0.0147 UA 50 -1.64540 0.981 -1.62830 0.014 0.976 -1.60320 0.585 0.014 0.968 -0.079 -1.56218 0.578 0.014 -0.047 0.954 -0.078 -1.55016 0.567 0.0245 0.014 -0.047 0.949 -0.075 -1.539 0.0457 15 0.548 0.0240 0.013 -0.047 0.943 -0.071 -1.530 0.0448 14 0.543 0.0233 0.012 -0.046 0.940 -0.069 -1.523 0.0435 13 0.537 0.0219 0.012 -0.046 0.936 -0.068 -1.508 0.0412 12 0.533 0.0214 0.011 -0.046 0.931 -0.067 -1.496 0.0405 11 0.529 0.0209 0.012 -0.045 0.926 -0.066 -1.480 0.0397 10 0.523 0.0206 0.011 -0.045 0.920 -0.064 -1.465 0.0392 0.517 0.0202 0.011 -0.045 0.913 -0.063 0.0386 0.510 0.0198 0.010 -0.044 -0.062 0.0380 0.503 0.0194 -0.044 -0.060 0.0373 0.0189 -0.043 0.0365 0.0183 0.0356 300 -1.694150 0.997 -1.680100 0.014 0.993 -1.667 0.606 0.015 0.990 -0.085 0.600 0.015 -0.048 -0.084 0.595 0.0262 -0.048 -0.082 0.0485 0.0258 -0.048 0.0479 0.0255 0.0472 Table 39: Scattering amplitudes paramagnetic solution with no explicit q-dependence added to Shown are scattering amplitudes ( 138 :) s,a 0 F . s,a 0 F /m ∗ m (spd) t g (sp) t g (spd) s g (sp) s g 84 1.1712 1.024 1.201 0.816 1.043 -0.083979825 -0.092115475 0.824 -0.066837064 -0.073106682 1.190356 -0.044932459 1.149557667 -0.048900528 1.098167 U 5040 1.34330 1.331 1.39120 1.313 1.378 -0.10520259118 1.283 -0.115770362 1.358 -0.10364204416 1.274 -0.114077341 1.325 1.241998 -0.10128278715 1.265 -0.111497804 1.315 1.238396 -0.09753198414 1.232873333 1.258 -0.107304565 1.304 -0.09645786613 1.252 -0.106086745 1.297 1.223679 -0.09541347212 1.220964667 1.241 -0.104857802 1.290 -0.09465351411 1.218099667 1.232 -0.103999508 1.279 -0.09397148410 1.216193333 1.221 -0.103196006 1.269 -0.092566847 1.214312333 1.209 -0.101676383 1.256 -0.091494584 1.211132333 -0.100459139 1.243 -0.09009023 1.208411 -0.088728506 -0.098892781 -0.097332699 1.204993 1.201457667 300 1.379150 1.369100 1.430 1.360 1.420 -0.109784899 1.410 -0.12085875 -0.108493284 -0.119476963 1.253147333 -0.107214771 1.250235333 -0.118109957 1.247363667 Table 40: Pairing amplitudes and effective mass: PM solutions (no q-dependence in Shown are singlet andas triplet a pairing function amplitudes of (in U both in the the s-p paramagnetic and solution s-p-d with approximations), no along explicit with q-dependence effective added mass, to 139 :) s,a 0 F s 3 F a 3 F s 2 F . s,a 0 F a 2 F : strong (large moment) solutions (no q-dependence in s 1 s,a ` F F a 1 ) as a function of an attractive (negative) U in the “strong” phase separation F s,a ` F s 0 F a 0 -6 0.7348 -4.468 0.400242 0.502103 -0.195142 -0.218671 0.0989662 0.10719 UF -50 -0.6384-35 -49.25 -0.6413-20 0.0155473 -34.23 -0.6522-10 0.809503 0.0171818 -19.18 -0.6779 -0.091892 0.826494 0.0171501 -0.0952054 -8.98 -0.0486115 0.887054 -0.0494988 0.0290425 0.0238227 -0.105303 0.0302337 0.0535099 1.05673 -0.0495965 0.0553085 0.0334233 -0.137779 0.0609873 -0.0519486 0.044482 0.0795149 -300-150 -0.63 -0.63 -299.3 0.0163837 -149.3 0.767561 0.0182784 -0.0854473 0.769456 -0.0486878 -0.0863664 0.027085 -0.049607 0.0499088 0.0275352 0.050359 Table 41: Negative U: FL interaction functions solution branch (far from PI’s) with no explicit q-dependence added to Shown are FL interaction functions ( 140 s 3 :) A s,a 0 F a 3 A s 2 A + 1)), as a function of an attractive ` (2 / s,a ` F a 2 A (1 + / s,a ` F = s,a ` s 1 A A : strong (large moment) FM solutions (no q-dependence in s,a ` A a 1 A ), related to F’s as s,a ` A s 0 A a 0 . Table 42: Negative U: Scattering amplitudes -6 0.423564676 1.288350634 0.353129571 0.430115562 -0.203067396 -0.228671777 0.097586519 0.10557337 UA s,a -50 -1.765486726-35 1.020725389 -1.787844996-20 0.015467143 1.030093289 -1.875215641-10 0.637487095 0.017083956 1.055005501 -0.093612447 -2.104625893 0.647977496 0.017052615 -0.049088756 1.125313283 -0.097053402 0.684621824 0.028922503 0.023635017 -0.049993726 -0.107568456 0.053103959 0.781464381 0.030103679 -0.050093391 -0.141683194 0.054874921 0.03326447 -0.052493998 0.044201121 0.060460539 0.078621813 0 -300 -1.702702703-150 1.00335233 -1.702702703 1.006743088 0.016294711 0.018167708 0.611186654 -0.086932937 0.61238757 -0.049166562 -0.087884453 0.026980604 -0.050104103 0.049555478 0.027427312 0.049999298 Shown are scattering amplitudes(negative) ( U in theF “strong” phase separation solution branch (far from PI’s) with no explicit q-dependence added to 141 /m ∗ m (sp) t /g s . g s,a 0 F (spd) t g (sp) t g (spd) s g (sp) s :) g s,a 0 F U -6 0.161733 0.256865042 0.0773891 0.041410917 3.161199832 1.16737 -50-35 1.43152 1.489462121-20 1.44923 -0.116476 1.509517282-10 1.5118 -0.128368065 -0.118568 1.68216 1.579950106 -0.13082169 1.775296803 -0.126824 0.861485916 -0.148701 -0.139962202 0.859791253 -0.164882433 1.26983 0.858237475 0.849148994 1.2755 1.29568 1.35224 -300 1.38729-150 1.440197541 1.38924 -0.110569 1.442629001 -0.121910936 -0.110543 -0.122041954 0.861874743 0.860295831 1.25585 1.25649 Table 43: Negative U:(no Pairing q-dependence amplitudes in and effective mass: “strong” phase separation solution branch (far from PI’s) Shown are singlet andratio triplet (in pairing s-p amplitudes approximation) (in andsolution effective both mass, branch the as (far s-p a from and function PI’s) s-p-d of n approximations) with attractive along no (negative) with U explicit singlet in q-dependence the to added “strong” triplet phase to separation 142 s 3 F a 3 F s 2 F . a 2 s,a 0 F F s 1 : “weak” phase separation solution branch (near PI’s) (no q- F s,a ` F ) as a function of an attractive (negative) U in the “weak” phase separation a 1 s,a ` F F s 0 F a 0 :) s,a 0 F 84 1.1712 1.024 1.201 -0.083979825 0.816 1.043 -0.092115475 -0.066837064 0.824 -0.073106682 1.190356 -0.044932459 1.149557667 -0.048900528 -0.037 -0.043 1.098167 0.0116 0.0169 -0.028 0.0244 0.0331 0.0060 0.0147 -9 -0.9334-8 -2.067 -0.9162-7 -2.125 0.659196-6 -0.892 -2.232 0.479408 -0.853 -2.477 9.92339 0.324396 7.90229 0.182574 -4.12223 5.9764 -2.92884 4.05253 0.622271 0.405718 2.12719 -1.92163 -1.06403 1.40698 4.50769 0.218915 2.91852 0.0687779 0.850242 0.423989 1.71079 0.817281 UF 1312 1.24111 1.232 1.27910 -0.092566847 1.221 1.269 -0.101676383 -0.091494584 1.209 1.256 1.211132333 -0.100459139 1.243 -0.09009023 -0.045 1.208411 -0.088728506 -0.098892781 -0.097332699 1.204993 0.0198 1.201457667 -0.044 -0.043 0.0380 -0.044 0.0194 0.0183 0.0189 0.0373 0.0356 0.0365 -16 -0.9824-15 -0.9795-14 -2 -0.9753-13 -2 -2.001 -0.9703-12 0.558665 -2.003 -0.964-11 1.6427 1.05737 -2.007 -0.956 1.42966-10 24.5373 -2.017 -0.95 1.42966 23.0412 21.2852 -2.035 1.05197 18.8679 -13.5208 0.67006 18.8679 -12.9217 -12.2447 6.56303 -10.3803 14.1944 6.15136 4.76588 2.73469 -10.3803 12.5422 2.16378 21.5984 6.73454 7.36813 -6.97387 6.17931 2.16378 19.8489 17.9481 -5.62456 14.5868 1.19742 6.17931 3.98078 14.5868 1.36327 8.79953 2.86776 6.80017 dependence in solution branch (near PI’s) with no explicit q-dependence added to Table 44: Negative U: FL interaction functions Shown are FL interaction functions ( 143 . s,a 0 s 3 F A a 3 A s 2 A + 1)), as a function of an attractive ` (2 / s,a ` F a 2 (1 + A / s,a ` F = s,a ` A s 1 A : “weak” phase separation solution branch (near PI’s) solutions (no s,a ` A a 1 A ), related to F’s as s,a ` A s 0 A :) s,a 0 F a 0 -9 -14.01501502-8 1.937207123 -10.93317422-7 0.54044331 1.888888889 -8.259259259-6 0.413353076 1.811688312 2.3035883 -5.802721088 2.17448536 0.292741298 1.67704807 -23.48126502 1.997370884 -7.070530524 0.553398262 0.172100319 -3.121181015 0.375267448 1.631425444 1.723862217 0.209732291 1.171509864 2.741977756 -1.351674428 0.758154207 2.059746817 0.067844657 1.374792642 0.399774703 0.731835916 UA -16 -55.81818182-15 -47.7804878-14 -39.48582996-13 2 1.999000999 -32.67003367-12 1.06147285 1.997008973 2 -26.77777778-11 0.968241355 2.629403917 1.993048659 0.470961723 -21.72727273 2.588437847-10 8.450798515 0.968241355 2.673170572 1.983284169 9.646581046 0.781814328 2.588437847 1.767808406 7.933996808 0.778858185 1.510222257 2.654393807 3.589674509 9.646581046 2.476573768 2.837936942 -19 3.282051185 5.035922575 8.155888256 1.510222257 3.274149594 17.66547442 4.730094317 2.440066845 3.282051185 5.286617433 0.966063297 3.432352303 4.730094317 2.537657616 1.966183575 5.174971787 3.898641922 0.547724015 2.4209314 45.02817984 1.071202385 2.034334033 3.449319103 Table 45: Negative U: Scattering amplitudes Shown are scattering amplitudes(negative) ( U in the “weak” phase separation solution branch (near PI’s) the with no explicit q-dependence added to q-dependence in 144 /m ∗ m (sp) t /g s g . s,a 0 (spd) F t g (sp) t g (spd) s g (sp) s g :) s,a 0 F U -9 10.82499845-8 28.57429678 8.438496357-7 -1.243486625 13.83521111 6.367579775-6 -3.154142189 -0.969343648 8.760898609 4.469412519 0.378836915 -1.527282237 -0.728140261 5.500129504 4.307796667 0.609928991 -0.970760988 -0.501802963 3.634096667 0.726818111 -0.608788777 2.992133333 0.812601324 2.350843333 -16 42.04856501-15 36.80755164 36.25812815-14 -4.746859509 30.75122867 30.25287638-13 -3.849198364 -4.101391328 24.35672959 25.08084905 1.142389622-12 -3.21839507 -3.431475477 18.22346883 20.66066705-11 -2.579924901 1.179079007 -2.852475325 13.80328683 9.1791 16.75627578-10 -1.922741716 1.24207465 -2.361784027 3.748685791 14.54710606 1.376293903 8.6804 -1.432050418 8.095066667 -1.916618376 -18.95622823 1.496793286 -0.363990233 -1.666872653 7.2893 4.469906713 2.174742532 7.2893 5.731466667 -0.767405091 5.180733333 Table 46: Negative U:q-dependence Pairing in amplitudes and effective mass: “weak” phase separation solution branch (near PI’s) (no Shown are singlet and tripletratio pairing (in amplitudes s-p (in approximation) and bothsolution effective the branch mass, s-p (near as and PI’s) a s-p-d with function approximations), of no along n explicit with attractive singlet q-dependence (negative) to added U triplet in to the “weak” phase separation 145 s 3 :) F s,a 0 F a 3 F s 2 F a 2 : PM solutions (no q-dependence in F s,a ` F s 1 F ) as a function of an attractive (negative) U in the PM solution branch with no a 1 s,a ` F F . s,a 0 F s 0 F a 0 Table 47: Negative U: FL interaction functions -5-4 2.03-3 1.663 -0.4802-2 -0.4562 1.287 -0.00332588 -0.00265448-1 0.8938 -0.4223 0.289062 -0.3685 0.244687 -0.00185604 0.4733 0.0187642 -0.00114239 0.192045 0.0146386 -0.273 -0.0644926 0.128614 -0.0531806 0.000105202 0.0101984 -0.00568128 0.00561564 0.0557086 -0.00441736 -0.0402836 -0.0256461 0.0254805 0.00155947 0.0205871 -0.0030636 -0.0101653 -0.00168 -0.000457724 0.0152028 0.00356812 0.00936507 UF -50-30 17.27 -0.6076-20 10.57 -0.0117138 -0.5945-10 7.204 -0.0100382 -0.5795 3.794 0.68355 -0.00836334 -0.5405 0.625205 0.0675821 0.56452 -0.0055204 0.0591806 -0.180171 0.0508325 -0.161457 0.43413 -0.0211211 -0.142518 -0.0183945 0.0343732 0.0825164 -0.0157077 -0.103928 0.0725757 0.0627716 -0.0105063 0.0436385 -300 100.6-200 67.31 -0.627-150 -0.6262 50.64-100 -0.0128154 -0.0112014 -0.6225 33.96 0.774936 -0.0227184 -0.6188 0.766227 0.0814226 -0.0125358 0.0800507 0.78317 -0.209665 0.735452 -0.206111 0.082831 -0.0255242 0.0753408 -0.0249633 -0.216968 0.0988482 -0.196891 0.0970227 -0.0267801 -0.0236018 0.101829 0.091677 Shown are FL interaction functionsexplicit ( q-dependence added to 146 s 3 A :) s,a 0 F a 3 A . s 2 s,a A 0 F + 1)), as a function of an attractive ` (2 / s,a ` F a 2 A (1 + / s,a ` F = s,a ` s 1 A A : strong (large moment) FM solutions (no q-dependence in s,a ` A a 1 A ), related to F’s as s,a ` A s 0 A a 0 Table 48: Negative U: Scattering amplitudes -5-4 0.669966997 -0.923816853-3 0.624483665 -0.003329571 -0.838911364-2 0.562745955 0.263657541 -0.002656831 0.018694044 -0.73100225-1 0.471961136 0.226234765 -0.065335329 -0.583531275 -0.001857189 0.014595867 0.321251612 -0.005685895 -0.001142825 0.180490877 -0.053752316 -0.375515818 0.025388086 0.123326815 0.010177641 -0.004420149 0.000105198 0.020526731 0.00560934 -0.04061079 0.054692977 -0.025778323 -0.003064941 0.001558984 -0.001680403 0.015169854 -0.010186009 0.009352558 -0.000457754 0.003566302 UA -50 0.945265463-30 -1.54841998 0.913569576-20 -1.466091245 -0.011759717-10 0.87810824 -0.010071901 0.556704809 0.791405924 -1.378121284 0.517381776 0.066680814 -1.176278564 0.058488325 -0.186906008 -0.00838672 -0.005530577 -0.166844647 -0.021185022 0.475115864 0.379248893 -0.018442964 0.081555025 0.050320912 0.03413851 0.07183096 -0.146699463 -0.015743027 -0.10613406 0.062213707 -0.010522093 0.04336814 -300 0.99015748-200 0.985360855 -1.680965147-150 -1.675227394 -0.01287038 0.980635167-100 -0.011243381 -1.649006623 0.971395881 0.61585362 0.610340534 -0.022891755 -1.623294858 0.078789273 0.080117918 0.621042671 -0.012588402 -0.214972645 -0.218841689 0.081481167 0.590653019 -0.025052642 -0.02561761 -0.226810107 0.074222405 0.095696312 -0.026882947 0.097471784 -0.204962036 0.100368933 -0.023681647 0.090491854 Shown are scattering amplitudes(negative) ( U in the PM solution branch the with no explicit q-dependence added to 147 :) s,a 0 F /m ∗ m (sp) t /g s g (spd) t g (sp) t g . s,a 0 F (spd) s g (sp) s g U -5-4 -0.801841 -0.83219539-3 -0.736642 -0.761026884-2 -0.0428481 -0.651326 -0.0365005 -0.669111568 -0.046734926-1 -0.531543 -0.039763507 -0.0289075 -0.542194079 18.7135719 -0.348412 20.18169614 -0.031443594 -0.0194795 -0.352127749 1.081562333 1.096354 22.53138459 -0.00908853 -0.021160259 -0.009807451 27.28730204 1.064015 38.33535236 1.042871333 1.018569533 -50-30 -1.24405 -1.340787195-20 -1.1886-10 -0.095675 -1.12818 -1.274176769 -0.986584 -1.202596057 -0.105693734 -0.0883193 -1.038721637 13.00287431 -0.0805619 -0.097348989 -0.0632159 -0.088593395 13.4579871 1.22785 -0.069215542 14.00389018 1.208401667 15.60657999 1.188173333 1.14471 -300 -1.32648-200 -1.441274448 -1.31885-150 -1.431680275 -0.107816 -1.30827-100 -1.437970916 -0.107414 -1.29148 -0.119376223 -1.398382494 12.3031832 -0.118762255 -0.10483 12.27819465 -0.102497 1.258312 -0.117654276 1.255409 -0.113391935 12.47991987 12.60017366 1.261056667 1.245150667 Table 49: Negative U: Pairing amplitudes and effective mass: strong (large moment) solutions (no q-dependence in Shown are singlet and tripletratio pairing (in amplitudes s-p (in bothwith approximation) the no and s-p explicit effective and q-dependence s-p-d mass, added approximations), as to along a with singlet function to of triplet n attractive (negative) U in the PM solution branch 148 : weak FM s,a ` F s 3 F a 3 F s 2 F = 1: FL interaction functions , α a 2 = 1 F c s 1 F a 1 F ) as a function of an U in the weak FM solution branch with “optimal” q- s,a ` . F a 0 F s 0 F with “optimal” parameterization a 0 a 0 F = 1) added to UF 33 -1.00432 -0.9985 -1.00531 16.7157 -0.9982 -1.006 12.455930 15.6876 -0.9979 34.1316 -1.008 12.422129 14.8397 -26.6002 -0.9975 31.8054 12.3482 -1.0128 13.8193 22.0527 -24.5141 29.8927 -1.012 12.4727 -0.99727 15.7069 -22.8499 20.004 -0.9965 -1.015 27.367 12.854526 12.0634 18.3497 -0.9958 15.4845 12.3264 -19.8044 -1.018 12.155725 11.1461 15.192 25.1466 -0.9949 16.4289 -1.021 23.338 11.934624 -17.7862 10.2379 15.1667 21.2175 -0.994 -16.2011 -1.026 11.5411 14.652223 -0.9929 -14.273 9.49701 13.2379 19.2718 -1.031 14.6375 22 8.72182 11.1958 -13.0131 -0.9915 14.0935 11.6557 -1.037 10.9068 17.688421 7.95073 10.1312 13.3923 15.8874 -11.9697 -1.044 -0.99 10.480920 12.438 -10.2762 -0.9879 14.2515 -1.053 8.9377 7.2719419 7.76478 -9.17644 6.5207 -0.9855 10.0987 -1.064 11.6238 18 10.8273 5.83332 6.63449 9.58871 12.7773 -0.9826 -1.077 9.10744 11.261117 -8.04281 9.88481 5.17541 -0.9789 9.84327 -7.15766 -1.093 8.60461 5.6992616 4.52057 -6.17643 -0.9744 4.70985 8.51483 -1.115 9.03696 8.0526315 3.90354 3.87483 -5.27472 -0.9687 8.04523 7.26472 7.49502 -1.1414 7.12479 3.30537 3.13477 -4.5417 6.12015 -1.178 -0.9618 6.9159813 6.22699 -3.87295 -0.9523 2.45359 5.02571 2.7574 -1.232 2.19984 1.87381 -3.18141 -0.9399 5.33865 6.34892 5.73301 4.51111 1.67914 1.37822 4.10001 3.18555 5.12197 -2.68453 3.71353 -2.14627 2.38504 0.975691 0.631841 -1.67086 3.02644 2.34851 0.371986 1.75569 , α = 1 c dependence ( Table 50: q-dependent Shown are FL interaction functions ( solutions) 149 s 3 A a 3 : weak FM solutions A s,a ` A s 2 A + 1)), as a function of a U in the weak ` (2 . / a 0 s,a ` F F a 2 (1 + A = 1: Scattering amplitudes / s,a ` , α F = = 1 = 1) added to c s,a ` , α A s 1 A = 1 c a 1 A ), related to F’s as s,a ` A s 0 with “optimal” parameterization A a 0 F a 0 UA 333231 251 167.666666730 201 -475.190476229 -665.6666667 2.495507211 2.54351100928 126 2.41361202 -554.5555556 2.417698096 84.33333333 2.518397226 4.28351775627 101 4.361130135 -284.7142857 2.416421888 67.66666667 6.4005680726 6.157396691 2.402525326 4.320751846 -237.0952381 56.55555556 2.406164017 5.313409769 5.06703826925 -332.3333333 2.363782244 6.28112493 -399 -195.0784314 4.117792364 4.842065628 4.791997116 48.61904762 2.432337822 2.39737254424 2.320133858 5.185454007 -165.6666667 2.412777952 4.046438448 7.23192365 39.46153846 2.381064706 4.820720941 23 2.279827735 4.170719086 7.695999137 -139.8450704 2.464900442 3.969998105 4.578800172 33.25806452 2.36600966522 6.955232986 2.232201143 4.373456906 4.677009505 2.41833035 8.119891178 -116.6470588 3.898115336 28.02702703 2.352834585 4.736950518 4.597132251 21 2.178136983 4.227608367 4.139721677 8.586954962 3.803106179 4.735413056 23.72727273 2.332388787 4.479164523 20 6.688687147 3.925528778 9.738258595 3.701399891 -81.6446281 19.86792453 4.908565916 -9919 10.98595933 -67.96551724 3.68129156 2.054691357 4.368958 4.789477008 18 1.981130538 2.285073689 3.406172875 4.251406551 16.625 13.98701299 2.256655412 3.462588632 4.097983335 17 2.123826658 -46.39336493 11.75268817 3.31573501 16.58662625 2.31290891516 1.803282198 -56.47126437 3.593712206 9.695652174 26.25073315 2.81548867 2.18571416915 1.89913778 -38.0625 13.21608973 -30.94888179 2.961633042 2.494182438 8.142857143 3.74315381 2.22444614714 1.572645221 3.141507458 -49.54942178 3.530921876 -25.17801047 1.696321018 3.150180209 6.617977528 2.092374127 3.944558071 13 1.816783888 1.436794386 2.142450419 -19.96436059 96.00174723 2.506411017 5.310344828 3.028738963 2.037321958 2.751828887 1.269177513 -8.746913818 -15.63893511 2.16515915 2.252750272 -17.18180205 1.969427494 1.151502348 1.076569626 -5.796944033 1.478132843 3.295453463 1.945837482 2.426344071 1.891894454 0.856331696 2.743242832 -3.760464375 1.614777984 2.112921436 0.57953081 -2.509446884 0.353215809 1.758523016 1.403639233 FM solution branch with “optimal” q-dependence ( Table 51: q-dependent Shown are scattering amplitudes ( 150 /m ∗ m (sp) t /g s g = 1: Pairing amplitudes and effective mass: weak , α (sp) t = 1 g c (sp) s g with “optimal” parameterization . a 0 a 0 U 33 -353.363457932 -34.96898965 -288.104196531 10.10505198 -29.87419789 -243.2793917 5.151966667 30 9.643914039 -26.03607739 -193.005907329 9.343934111 -23.1569359 -157.6122744 5.1407 5.116066667 28 -19.68153742 8.334691088 -133.228218427 8.008128179 5.157566667 -17.09913681 -108.85031626 7.791517193 5.1088 -14.51581052 -90.041440325 7.498741861 -76.76258401 5.0519 -11.9353395324 -10.14112137 7.544103798 -63.4714792323 4.9782 4.847033333 7.56943746 -8.747380639 -53.0548075522 7.256055481 4.731933333 -7.324960004 -44.7556275121 7.243016688 -6.284142379 -37.23686148 4.6356 4.493633333 20 7.121994508 -5.188093368 -30.97063866 4.366233333 19 7.177369187 -4.361281555 -25.71832429 4.196236667 18 7.101270181 -3.664154025 -21.28256787 4.035813333 17 7.018898256 -3.032945692 -17.59351297 3.868203333 16 7.017127911 -2.512381939 -14.3525691915 7.002722276 -2.076520747 3.68421 -11.8333801714 6.911835201 -1.709105806 3.49834 -9.495047032 3.305326667 13 6.923725924 -1.382082339 -7.558038793 3.116306667 6.870102282 -1.108087864 2.911003333 6.820793767 2.707323333 F F = 1) added to , α = 1 c Table 52: q-dependent FM solutions:) Shown are singlet andapproximation) triplet and pairing effective amplitudes mass,( (in as the a s-p function approximation), of along U with in singlet the to weak triplet FM ratio solution (in branch s-p with “optimal” q-dependence 151 : weak FM s,a ` F s 3 F a 3 F s 2 F = 1: FL interaction functions , α a 2 F = 1 c s 1 F ) as a function of an U in the weak FM solution branch with q-dependence a 1 . s,a F ` s 0 F F with parameterizations s 0 F and s 0 a 0 F F and a 0 F a 0 UF 35 -1.00330 -0.7833 -1.00829 -2.41769 -0.7753 -1.0128 -8.02959 -2.79035 -1.012 -0.773427 -4.137 -2.86345 -0.771 -1.015 9.190426 -0.7691 -3.39151 -2.86614 -1.01825 -2.92916 -57.2664 6.09551 -0.7664 -2.77385 -1.021 5.3530124 -2.14065 -2.91281 5.33011 -0.7639 -41.0759 -1.026 4.8673823 -1.60962 -2.88916 -37.5798 -0.7607 4.11625 -3.61332 -1.032 2.5714322 -1.19041 -2.86809 -34.6717 2.10971 -0.7577 3.65131 -1.038 -31.3742 -0.68316321 1.30922 -2.84751 1.72253 -0.7537 3.27477 -1.046 2.09497 -28.6336 -0.26036220 1.38155 -2.7721 2.7221 -0.7499 -1.056 2.57813 -26.383419 1.08508 -2.71233 0.0949219 2.18645 3.11812 -0.7449 -1.062 -23.4416 0.86362518 0.409852 -2.61851 3.39181 -0.7424 1.87509 -20.7625 -1.082 0.62633917 3.54972 0.706359 -2.57337 -0.7378 1.48498 0.455747 -18.5653 -1.09516 3.68886 0.833383 -2.50007 -0.7286 1.16537 3.72895 -16.3433 -1.114 0.30154815 -2.32022 1.05665 -0.7217 1.0001 -14.1819 -1.139 0.194625 3.64426 14 -2.20423 1.27532 -0.7122 0.0967223 -1.187 0.47936 3.51684 13 -13.1504 -2.05906 1.40648 -0.7025 3.30726 0.503378 -1.24912 -1.91951 1.53367 -10.699 0.0634 -0.6875 0.345618 -9.29874 -1.368 -1.74156 1.59828 0.0523935 -0.0438805 -0.6676 0.238408 -7.92976 3.18644 -1.55158 1.67528 2.85812 0.00640599 2.56394 -0.064924 -6.59418 -5.06319 1.73208 -0.0684878 -0.0877413 2.28303 -0.0606082 -3.83586 1.97329 1.59078 -0.14374 -0.060211 -2.65036 1.24265 -0.0437247 0.879463 = 1) added to both , α = 1 c ( Table 53: q-dependent solutions) Shown are FL interaction functions ( 152 s 3 A a 3 A : weak FM solutions s,a ` A s 2 A + 1)), as a function of a U in the weak ` . s 0 (2 F / s,a ` F and a 2 a 0 A F (1 + / = 1: Scattering amplitudes s,a ` , α F = = 1 c s,a ` s 1 A A = 1) added to both , α = 1 a 1 c A ), related to F’s as s,a ` with parameterizations A s 0 F s 0 and a A 0 F a 0 UA 35 334.333333330 -3.61467466529 -12.4556851228 126 4.78941027 84.3333333327 101 3.238245574 -3.366812227 67.6666666726 -3.450378282 -64.23442403 5.478318767 -3.330879168 56.55555556 -39.92869067 -36.7965951825 -3.413062665 -124.0468662 3.025988414 -3.280821918 10.91556728 2.466399389 48.61904762 -62.90992311 -7.47303194324 -7.468446975 -100.2228467 -3.235493435 2.746836333 25.98791857 5.842553679 2.257644316 39.46153846 -3.47305053323 -78.19812342 -3.178854994 5.692983404 2.585243325 1.382363833 2.110264226 5.94789605 -1.97350228522 -65.2283375 1.880597779 5.767346638 1.884178853 6.057815991 27.31578947 32.25 1.97876799 1.15382596321 -0.88460647 -3.060089322 1.102936256 1.621124053 0.939453908 22.73913043 2.157203117 20 6.169131195 -36.49100483 1.762538688 1.612406638 -3.127115147 2.284748278 -2.99840064 18.85714286 0.768777123 0.09201062619 -56.02026362 6.355630748 -28.28584837 -2.92003136 17.12903226 2.355326966 1.36368396618 -0.28510555 0.574898782 0.360589257 -2.881987578 -20.5917062 13.19512195 6.84293749517 1.521230928 2.415788026 1.144937995 -18.0955629 -2.813882532 0.571740892 11.52631579 0.28909431316 6.586042823 -15.00252035 7.203944179 0.652204332 -2.684598379 9.771929825 2.396579941 0.9450933215 -10.23957751 0.78142063 0.189360113 0.42788858 0.833402777 8.194244604 -2.5932447 7.722748015 0.89489441714 2.340805793 8.067334119 0.437423349 -2.474635163 2.43291748 6.347593583 -8.309800571 0.095404057 0.45733547713 -6.564903182 0.062830931 9.386734515 -2.361344538 0.957553421 5.016064257 2.246069275 10.81565761 1.01485330912 -5.329554184 0.052004259 2.18968354 0.323272258 3.717391304 -0.044157306 0.227557685 1.042746418 2.029478237 -2.008423586 1.876588519 13.53312217 -2.2 20.68204343 0.006397793 -3.213667306 -0.065531798 -0.088855051 400.6322203 1.098087944 1.721551045 1.539349559 -0.061137548 -0.147994547 -4.151711643 -5.639927819 1.296210589 1.074981605 -0.043999538 -0.069438944 0.781302101 -16.47508032 -0.060733403 1.055309882 Table 54: q-dependent FM solution branch with q-dependence ( Shown are scattering amplitudes ( 153 = 1) , α = 1 c /m ∗ m (sp) t /g s g = 1: Pairing amplitudes and effective mass: weak FM (sp) , α t g = 1 c (sp) with parameterizations s s 0 g F and . a s 0 0 U 35 -262.192785130 28.19874446 -128.038004429 -9.298030465 12.63022876 -130.282687628 -10.1374256 11.20907849 -1.67653 -103.068372327 -11.62296149 15.16646169 -142.7496114 -0.130503333 26 -6.795808697 -0.379 16.32130713 -117.5357445 0.075383333 25 -8.746211946 13.0808859 -95.4283760724 10.46293166 -8.985304618 0.28645 -79.090969123 -9.120615444 -66.91320012 8.532968953 0.603196667 22 0.46346 7.119021169 -9.26886873 -48.6431207121 -9.399213534 5.054557863 -39.10848158 0.913212667 20 -9.623615365 0.772279 3.972165742 -30.45957986 1.031640633 19 -9.845631858 2.996423067 -27.30199435 1.136617333 18 -10.1653135 2.640866937 -22.0470575217 -10.33826959 2.050194928 -17.21929317 1.235453 1.277794333 16 -10.75363967 1.515533375 -14.44899733 1.352216667 15 -11.36186999 1.210911023 -11.94173296 1.425106667 14 -11.93233611 0.939138276 -9.608883564 1.468826667 13 0.689421401 -12.715628 -7.69457732612 -13.93760558 0.491066191 1.511223333 -5.974921841 -15.66912458 0.318712257 1.53276 1.558426667 -18.74707268 1.57736 F F and a 0 F added to both Table 55: q-dependent solutions:) Shown are singlet andapproximation) and triplet effective pairing mass, amplitudes as (in a function the of s-p U approximation), in along the with weak FM singlet solution to branch triplet with q-dependence ratio ( (in s-p BIBLIOGRAPHY
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