THERMOMAGNETIC PHENOMENA IN MESOSCOPIC AND PARAMAGNETICALLY
LIMITED SUPERCONDUCTORS
By
Mengling Hettinger

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2015

ABSTRACT
THERMOMAGNETIC PHENOMENA IN MESOSCOPIC AND
PARAMAGNETICALLY LIMITED SUPERCONDUCTORS
By
Mengling Hettinger
The superconducting fluctuation effect, due to droplets of preformed Cooper pairs above
the critical temperature Tc , governs the temperature dependence of kinetic coefficients in
superconductors at the onset of the phase transition. The transverse thermoelectric response
– Nernst effect – is particularly sensitive to the fluctuations, and the large Nernst signal found
in the various superconducting materials has raised much debate on its connection to the
origin of unconventional superconductivity. In this thesis, we present a systematic study of
the electrical and thermomagnetic transport phenomena in mesoscopic and paramagnetically
(Pauli) limited superconductors.
In the first chapter of this thesis we concentrate on the study of mesoscopic effects on
transport in superconductors. We find that long-range phase coherence developing close to
Tc triggers a great amplification of mesoscopic fluctuations due to strong pairing correlations. As a result, mesoscopic conductance fluctuations cease to be universal and exhibit
pronounced dependence on temperature. Despite the lack of universality, in the sense of random matrix theory classification, we have discovered a different kind of universality in terms
of temperature dependence of fluctuating characteristics. We find that mesoscopic fluctuations of conductivity, transversal thermoelectric coefficient and diamagnetic susceptibility
consistently display the same scaling with temperature close to Tc . We connect our results
to the existing experimental measurements of conductance fluctuations in superconducting
films. Experimental verification of the temperature scaling and the overall magnitude of

the mesoscopic fluctuations of Nernst coefficient will provide a powerful tool for a better
understanding of thermomagnetic transport phenomena in correlated systems.
In the second chapter of this thesis we examine the electrical and thermal transport
anomalies in the ultra-thin superconducting films in an external in-plane magnetic field.
We concentrate on the Clogston-Chandrasekhar phase transition, i.e., the destruction of
superconductivity by a magnetic field by virtue of the Zeeman splitting. Near the quantum
critical point of the supercooling line in the phase diagram, we discover highly non-monotonic
magnetoresistance. The most remarkable feature of this effect is that fluctuation-induced
transport is dominated by the virtual excitations rather than real preformed Cooper pairs.
We also carefully study how spin-orbit scattering and other pair-breaking effects modify the
fluctuation transport. In the strong spin-orbit scattering regime, we find that the scaling
of the thermomagnetic coefficient is the same as conductivity within the classical region
of transition, however they are drastically different near the quantum critical point. Even
though we primarily focus on the conventional superconductors our result for the Nernst
effect may have important implications to the other systems, such as iron-pnictides, and in
particular to FeSe compound, which has comparable Zeeman and superconducting gaps.

ACKNOWLEDGMENTS

First and foremost, I would like to express my sincerest gratitude to my advisor, Dr.
Alex Levchenko, who has been patient, supportive, encouraging, caring and understanding
since the day I began working with him. His vast knowledge and skills in many areas in
condensed matter physics guided me through the process of doing research and writing this
thesis. The excellent research atmosphere he provided is simply the best one can possibly
ask for.
I must thank all the physics and astronomy professors whom I have worked with and
learned from over the past five years and whom I have received the time, energy and expertise from. My thesis committee guided me through all these years. I would like to give
special thanks to professor Vladimir Zelevinsky, Carlo Piermarocchi, Norman Birge and Scott
Pratt for being excellent teachers during my courses of study, and for providing insightful
suggestions, constructive feedback and, thoughtful comments at all levels of my research
projects.
I would never have been able to finish my thesis without the help from my great and
cheerful friends and family. I owe a debt of gratitude for the constant support, concern,
trust and love to my parents He Zhang, Feng Wang and Rita Murray, and my grandparents
Rengsun Zhang and Yunfang Zhang. To my husband, Thomas Hettinger, thank you for
listening to my endless complaints and concerns, offering me precious advices, supporting
my decisions, and helping me unconditionally through this entire journey.
I want to thank all my friends who always be there cheering me up and stood by me
through the good times and bad.
The research presented in this thesis was supported in part by National Science Foundaiv

tion Grants No. DMR-1401908 and ECCS-1407875.

TABLE OF CONTENTS

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

vii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Thermomagnetic transport
2.1 Background and motivation . . . .
2.2 Foundations of UCF . . . . . . . .
2.3 Qualitative considerations . . . . .
2.4 Definitions and assumptions . . . .
2.5 Technical prerequisites . . . . . . .
2.6 Mesoscopic Nernst effect . . . . . .
2.7 Summary . . . . . . . . . . . . . .

in mesoscopic superconductors .
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33

Chapter 3 Transport anomalies in Pauli-limited superconductors .
3.1 History of the subject . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Motivation and qualitative picture . . . . . . . . . . . . . . . . . . .
3.3 Fluctuation-induced conductivity near the quantum critical point .
3.4 Effects of pair-breaking scattering . . . . . . . . . . . . . . . . . . .
3.5 Thermomagnetic phenomena . . . . . . . . . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Matsubara sums and analytical continuation . .
Appendix B: Seebeck (αxx ) and Hall (σxy ) coefficients near Tc
Appendix C: Quantum Aslamazov-Larkin terms . . . . . . . .
Appendix D: Anomalous Maki-Thompson terms . . . . . . . .

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74

BIBLIOGRAPHY

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LIST OF FIGURES

Figure 2.1:

Figure 2.2:

Figure 2.3:

Figure 2.4:

The layout of the Nernst experiment: by applying a temperature
gradient (−∇T ) in the presence of magnetic field, an electric field is
generated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Building blocks of the diagrammatic technique. Straight lines represent disorder-averaged single-particle Green’s functions. The straightdashed lines represent single impurity lines that carry an overall factor
1/2πνd τ in the diagrams. The vertex function λ(ε, ε , q) is drawn in
the ladder approximation, while diagrams with the crossed impurity
lines yield parametrically smaller contributions in 1/g
1. The polarization operator Π(ω, q) (“bubble” diagram) is also presented in
the main ladder approximation. . . . . . . . . . . . . . . . . . . . .

17

The Dyson equation for the fluctuating propagator L(ω, q) which represented graphically as the wavy line and computed in the ladder
approximation. Solid lines represent one-electron Green’s function,
greay triangle is the impurity dressed effective vertex, while each cross
between two Green’s functions is associated with the electron-electron
coupling constant −λsc . . . . . . . . . . . . . . . . . . . . . . . . . .
Feynman diagrams for the main fluctuation-induced corrections to
the conductivity. In the first row we show the Aslamazov-Larkin
diagrams (left) and the Maki-Thompson interference diagram (right).
In the second row we show two density of states diagrams. . . . . .

20

23

Figure 2.5:

Leading order diagrams for the irreducible correlator of mesoscopic
disorder-averaged two pair-propagators. This averaging contains collisions of four diffusion or Cooper modes, and involves forth and sixth
order Hikami boxes (internal impurity lines are implicit on diagrams). 28

Figure 2.6:

eh
The Aslamazov-Larkin diagrams contributing to Kxy
(Ω). The wavy
lines correspond to the fluctuation propagator L(q, ω); electric current
vertices B e and heat current vertices B h are indicated in the figure
along with running momenta and frequencies. . . . . . . . . . . . . .

vii

30

Figure 3.1:

Figure 3.2:

Figure A.3:

Above the tricritical point T ∗ the second order paramagnet to superconductor transition occurs along the (black) solid line obtained
from Eq. 3.1. At T < T ∗ this line becomes a supercooling part of the
hysteresis, and the dashed line is its superheating part. The latter is
obtained following Ref. [95]. The grey shaded area with the critical
point (0, ∆0 ) as its lowest corner bounded by the black dashed line
marks the region of quantum fluctuations (QF). . . . . . . . . . . .

37

Phase diagram of a superconducting thin film in a parallel magnetic
field parametrized by pair breaking parameter α = D(eHd)2 /6 due
to orbital mechanism. Tc0 = Tc (H = 0) is the critical temperature
in the absence of a magnetic field. At T = 0 the superconductivity
breaks down at the critical value αc = πTc0 /2γE , where ln γE ≈ 0.577
is the Euler constant. . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Integration contour in the plane of complex frequency. The lower part
of the contour corresponds to advanced-advanced products of propagators after analytical continuation. The middle section contains
mixed causality components of advanced-retarded, while the upper
third contains only retarded-retarded products of propagators. . . .

62

viii

Chapter 1
Introduction
The study of fluctuation effects at the onset of the second order phase transition, which
naturally emerged from the Landau’s theory of the phase transformations [1], was instrumental in the development of statistical mechanics and modern condensed matter theory.
The seemingly simple question – how accurately does the mean field theory describes the
second order phase transitions – lead to the ideas of scaling, universality [2–6], and eventually
to the development of the renormalization group [7]. In the context of superconductivity,
fluctuations were first studied by Ginzburg [8]. In particular, he analyzed effects of fluctuations above the critical transition temperature, Tc , on the thermodynamic properties of
superconductors and demonstrated that, in clean materials with ballistic transport, the fluctuation phenomena become important only in a very narrow temperature region near the
transition. This result explained the great level of accuracy of the Bardeen-Cooper-Schrieffer
(BCS) theory of superconductivity [9] (which in its essence is a mean field model) in application to various existing experiments of that time. Almost a decade later Aslamazov and
Larkin [10], and independently Maki [11] and Thompson [12], realized that the fluctuation
region in disordered superconducting films is determined by the resistance per unit square
1

and could be much wider than that in bulk samples. Perhaps even more importantly they
demonstrated that within microscopic BCS theory superconducting fluctuations, in fact,
play a very important role in explaining temperature dependence of kinetic coefficients, such
as conductivity. These authors discovered the phenomenon which nowadays is called the
paraconductivity effect – the decrease of the resistance of a superconductor in the normal
phase with lowering temperature towards the critical temperature. In a parallel vein this
effect was observed in experiments by Glover [13]. Little [14], Langer and Ambegaokar [15],
and McCumber and Halperin [16] showed that superconducting fluctuations associated with
the phase of the superconducting order parameter, so called phase slips, are also required
to explain the residual resistance in the superconducting phase. At very low temperatures,
thermal activation of the phase slips is severely suppressed, however resistance may appear
due to quantum phase slip tunneling [17].
Abrahams, Redi and Woo [18] demonstrated that fluctuation effects play an important
role at the level of single particle properties, namely fluctuation-induced formation of superconducting droplets in the normal state leads to the depletion in the density of states
that manifests as a pronounced zero-bias anomaly revealed by the tunneling experiments.
The delicate interplay between paraconductivity and density-of-states effects is instrumental
in explaining transport anomalies as observed in the granular superconductors. These early
studies set the stage for the new and fruitful field of research that spanned over many decades
and were recently summarized in the monograph by Larkin and Varlamov [19].
What makes superconducting fluctuations so pronounced in experiments and interesting
from the theoretical point of view is their strong dependence on temperature, magnetic field,
frequency of external drive parameter, etc. In practice these characteristic features allow one
to separate fluctuations from the other competing effects contributing to transport, and use
2

them to extract important information about the microscopic parameters of a material which
are not accessible by other means of measurement. In theory, studying fluctuation effects
advances our understanding of the underlying origin of superconductivity as a macroscopic
quantum phenomenon. This becomes especially critical in the cases when this origin is in
fact not completely known, like in the case of high-temperature cuprate superconductors or
heavy-fermion superconductors. Furthermore, many ideas of the theory of superconducting
fluctuations have been extensively used in other branches of condensed matter physics, e.g.
in developing ideas of quantum criticality. For instance, superconducting fluctuations at
zero temperature near the upper-critical field, Hc2 , give perhaps the simplest example of a
quantum critical metal.
A different kind of fluctuation effect that attracted tremendous attention over the span
of last several decades is mesoscopic fluctuations [20–23]. This phenomenon occurs due to
quantum mechanical interference between coherent electrons backscattering off random disorder potential. It manifests as completely reproducible oscillatory patterns of conductance
as a function of magnetic field or the gate voltage with universal amplitude in the units
of conductance quantum e2 /h. Since disorder substantially broadens the region of superconducting fluctuations, one may be intrigued by the question of whether there exists a
parameter range within which superconducting fluctuations coexist with mesoscopic fluctuations, and what their interplay will give for observable quantities. In particular, one may
want to investigate the ultimate fate of universality of conductance fluctuations in the presence of strong superconducting correlations. This set of questions defines the main theme
of the present dissertation. Our focus in on thermomagnetic transport in mesoscopic superconductors. Thermomagnetic effects are difficult to measure and difficult to calculate
theoretically. The mere difficulty of the problem makes it to be of fundamental importance.
3

In metals, thermomagnetic effects are usually small because of strong cancellation of currents generated by electronic excitations above and below the Fermi level – the electron-hole
asymmetry is at the heart of the nonvanishing thermomagnetic response. Because of this
compensation property, thermomagnetic effects are very sensitive to the characteristics of
the electronic spectrum, presence of impurities, and peculiarities of scattering mechanisms.
The inclusion of many-body interaction effects, such as electron-phonon renormalization,
electron-electron scattering, drag effects, etc., adds a completely new level of complexity to
the problem of calculating thermomagnetic kinetic coefficients. The observation that the
collosal Nernst signal – the electric field, Ey , generated as a response to a transverse temperature gradient, ∇x T , in the presence of a perpendicular magnetic field, H – is mediated
by superconducting fluctuations in the vicinity of transition triggered much interest to these
transport phenomena.
On the technical side there is a well defined way how one can compute response functions
based on the diagrammatic methods of many-body theory in condensed matter physics [24].
These methods are extremely powerful and, in the context of superconducting and mesoscopic
fluctuation effects, can be reduced to the summation of certain classes of diagrams. The
diagrammatic technique is especially suited for problems containing a small parameter when
the whole treatment simplifies to the summation of ladder type diagrams. Importantly for
our study, mesoscopic and superconducting fluctuations are controlled by the same parameter
– inverse dimensionless conductance – which allows us to treat these effects on equal footing
within the same formalism.
This dissertation is organized as follows. In Chapter-II we provide brief but yet sufficient
introductory discussion for the foundations of mesoscopic physics of conductance fluctuations. We devote special attention to the mesoscopic effects in supeconducting systems.
4

After reviewing existing literature we place our work in the context of existing studies. Next
we elaborate on the technical prerequisites and set the stage for the calculation of conductivity and transverse thermomagnetic power. We demonstrate that mesoscopic fluctuations
proliferate in the presence of superconducting pairing correlations and that the universality
of this phenomenon breaks down. We then relate our results to available experimental findings. In Chapter-III we concentrate on transport effects in thin superconducting films in the
in-plane magnetic field. In such systems, superconductivity is limited either by orbital effects
or by spin Zeeman effects. Depending on the film thickness and electron mean free path, one
effect dominates the other and we consider both scenarios. In either case, the phase diagram
in the field-temperature plane is interesting and we discuss fluctuation effects along the entire transition line including the most interesting quantum critical points. Despite certain
similarity of the phase diagram, the underlying nature of fluctuation effects is conceptually different. The most remarkable feature of fluctuations in the Zeeman case (so called
paramagnetically or equivalently Pauli limited superconductors) is that fluctuation-induced
transport is mediated by virtual Cooper pairs in the quantum limit because of the Zeeman
gap in the excitation spectrum at low temperatures. Chapter-III structurally is similar to
Chapter-II. We review the history of studies on the subject of Pauli limited systems, discuss
more recent tunneling results, and carry out calculation for electrical and thermomagnetic
effects. Various technical aspects of this work are delegated to multiple appendices.

5

Chapter 2
Thermomagnetic transport in
mesoscopic superconductors

2.1

Background and motivation

Universality of conductance fluctuations (UCF) is the hallmark of mesoscopic physics [20–
23]. This phenomenon emerges from the quantum coherence of electron trajectories and
is sensitive to changes in external magnetic field or gate voltage. At temperatures below
the Thouless energy, T < ET h , which is related to the inverse dwell time for an electron
to diffuse across the sample ET h = D/L2 , variance of conductance fluctuations saturates to
the universal value of the order of conductance quantum, ∼ e2 /h, as long as characteristic
sample size L is smaller than dephasing length, L < Lφ . Interaction effects in normal metals
barely change the magnitude and universality of conductance fluctuations although they
are crucially important in determining temperature dependence of dephasing effects, and in
particular Lφ [25]. Robustness of UCF can be rooted to random matrix theory description
of Wigner-Dyson statistics of electron energy levels in disordered conductors. Indeed, in
6

the Landauer picture of transport across a mesoscopic sample, conductance is given by e2 /h
times the number of single particle levels within the energy strip of the width of the Thouless
energy. While the average number of such levels depends on the dimensionality, random
matrix theory predicts that their fluctuations are universally of the order of one [26, 27].
When superconductivity is induced at the boundary of the mesoscopic sample via the
proximity effect, UCF remain intact [28, 29]. The magnitude of oscillations changes only
by a numerical factor of the order of unity, with a value depending on the underlying symmetry class [30–32]. Things get quantitatively different if superconducting correlations are
induced at the bulk of the sample. Experimentally this is achieved by tuning superconducting systems to the vicinity of the critical temperature Tc or the superconductor-insulator
transition. There exists compelling evidence from measurements such as those in twodimensional granular arrays [33, 34], sub-micron scale superconducting cylinders [35], and
quantum wires [36, 37], that mesoscopic oscillations could become anomalously large, sometimes reaching the level of ∼ 104 × e2 /h. These observations seemingly imply that the role
of mesoscopic effects proliferate in the presence of superconducting correlations.
Theoretical studies devoted to various aspects of mesoscopic fluctuations in superconductors cover a diverse range of topics. These works include mesoscopic effects on the Josephson
current [38–42], upper critical field [43,44], critical temperature [45], condensation energy and
glassy phase transitions, [46,47] persistent currents [48–51], density of states, gap fluctuations
and level statistics [52–56], and also some transport properties [57, 58]. In the recent years,
measurements of the Nernst-Ettingshausen effect and the diamagnetic response in superconductors [59, 60], including high-Tc [61–69] and heavy-fermion systems [70–72], attracted
tremendous attention and triggered a flood of theoretical works [73–83]. Our motivation is to
study mesoscopic effects on the thermomagnetic transport in superconductors at the onset
7

of Tc where pairing correlations due to preformed Cooper pairs are profoundly important.
We find that in a parametrically wide temperature region, g −1 < (T − Tc )/Tc

1, where

g is the dimensionless conductance, amplitude of mesoscopic fluctuations in the diagonal
component of the conductivity tensor σxx and transversal component of the thermomagnetic
tensor αxy become parametrically bigger than their bare values.

2.2

Foundations of UCF

In the multiple scattering diffusive regime, the quantum mechanical interference effects associated with coherent backscattering on impurities modify the average value of electrical
conductivity. Similarly, correlation functions of the single-particle density of states are affected by the same coherence effect. In what follows, we discuss the second moment of these
physical quantities which will lead us to the universality of conductance fluctuations.
The point of departure is the Einstein formula (throughout the text we use unites

=

kB = 1)
σαβ (ε) = 2e2 νd (ε)Dαβ

(2.1)

which relates conductivity σαβ to the diffusion constant Dαβ and density of states νd . We
introduce the mean for conductivity σ
¯ and sample specific variance δσ = σ − σ
¯ . Then the
product

δσ(ε)αβ δσγδ (ε ) = σ02 δαβ δγδ

δνd (ε)δνd (ε )
δDαβ (ε)δDγδ (ε )
+
2
ν0
D2

(2.2)

contains two contributions related to the density of states fluctuations and the diffusion coefficient fluctuations, while the cross-correlation between ν and D is absent. Here ν0 , D, and

8

σ0 are bare disorder-averaged density of states, diffusion constant, and Drude conductivity
respectively, while angular brackets . . . stand for the disorder average of the irreducible
correlation function. These correlation functions can be computed by using a conventional
Diffuson-Cooperon diagrammatic technique [20–23]. The total conductivity fluctuation can
be written in the form

δσαβ (ε)δσγδ (ε ) = σ02 [δαβ δγδ Kν (ε − ε ) + (δαγ δβδ + δαδ δβγ )KD (ε − ε )] ,
Kν (ω) =

δνd (ε)δνd (ε )
δ2
=
ν02
2βπ 2

Re
q

2

KD (ω) =

δD(ε)δD(ε )
δ
=
2
D
2βπ 2

1
2
Dq − iω
1

q

[(Dq 2 )2

+ ω2]

(2.3)

2

,

(2.4)

,

(2.5)

where ω = ε − ε , δ = 1/νd Ld is the mean level spacing and coefficient β captures the symmetry class. For the time reversal invariant system β = 1 (orthogonal ensemble), while β = 2
if it is not (unitary ensemble). To translate conductivity fluctuations into the conductance
fluctuations at finite temperature one uses g = σLd−2 (assuming cubic conductor of size L)
and g(T ) =

g(ε)∂ε f dε, where f (ε) is the Fermi distribution function, so that

δgαβ (ε)δgγδ (ε ) = δαβ δγδ G2ν (T ) + (δαγ δβδ + δαδ δβγ )G2D (T ),
G2ν (T )

4
=
β

e2 ET h
π

4
=
β

e2 ET h
π

G2D (T )
F (z) =

2

2

dω
F (ω/2T )
2T
dω
F (ω/2T )
2T

Re
q

1
2
Dq − iω
1

q

(Dq 2 )2

+ ω2

(2.6)

2

,

,

z coth(z) − 1
.
sinh2 (z)

(2.7)
(2.8)
(2.9)

The sensitivity of conductance fluctuations to a variation of the Fermi energy shows up
in the temperature dependence of the correlations. At low temperatures T

9

ET h , it is

sufficient to take ω → 0 limit in the q-summations of G2ν(D) functions since F (ω/2T ) is
sharply peaked at zero frequency. In this case both contributions become equal. As an
example, suppose that the sample is connected to leads along the x-direction and isolated
in the other directions. The leads play the role of a reservoir, and the boundary conditions
in that direction correspond to an absorbing wall. The diffusion modes in this direction are
thus quantized as qx = nx π/L with nx = 1, 2, . . .. In the other directions, the boundary
conditions are those of hard walls which implies the same quantization of diffusion modes
with the contribution of the mode ny,z = 0 added. This results in the q-summation in above
expression in the form

2
nx =0,ny ,nz (nx

+ n2y + n2z )−2 . As a result, in the zero temperature

limit (restoring Planck’s constant )

4bd
δgαβ (ε)δgγδ (ε ) =
βπ 6

e2

2

(δαβ δγδ + δαγ δβδ + δαδ δβγ ),

(2.10)

with b1 = π 4 /90, b2 ≈ 1.51, and b3 ≈ 2.52. This formula shows that conductance fluctuations do not depend on the strength of disorder since the diffusion constant dropped out
from the final expression: the fluctuations are said to be universal. They depend only on
the sample geometry and time-reversal symmetry. At high temperatures T

ET h , one

finds an algebraic decay of fluctuations ∝ (ET h /T )p with a power exponent depending on
dimensionality of the system. The significant dependence of the conductance fluctuations
on space dimensionality is a consequence of the diffusive nature of electronic transport.

10

2.3

Qualitative considerations

We proceed by discussing how results of the previous section will change in superconducting
systems close to transition. It has been emphasized early on [43,46] that quantum interference
mesoscopic effects may lead to a formation of superconducting droplets that nucleate prior to
transition of the whole system. Above Tc there are also thermally induced superconducting
fluctuations [19] that are known to be crucially important in describing transport properties. One thus expects that the combined effect of two fluctuation mechanisms may have
interesting implications for the kinetic properties of superconductors. Indeed, the probability
amplitude of the fluctuations in the pairing gap ∆ is controlled by the competition of Cooper
pair condensation energy and entropy, and can be estimated from the Ginzburg-Landau functional. The condensation energy exhibits mesoscopic fluctuations with the amplitude ∝ 1/g
and the correlation radius of the order of thermal length ∼ LT =
coincides with the superconducting coherence length ξ =

D/T . Near Tc the latter

D/Tc . On the other hand, ther-

mal superconducting fluctuations are susceptible to the Ginzburg-Landau correlation length
ξT = ξ

Tc /(T − Tc )

LT so that mesoscopic fluctuations are almost local with respect to

superconducting fluctuations, and thus should be summed randomly from different blocks
of the size ξ. For the sample size L

ξ, the above consideration leads to an estimate of

the mesoscopic fluctuations of the critical temperature δTc /Tc ∝ (1/g)(ξ/L)(4−d)/2 , where d
is the dimensionality of the system. The response functions in superconductors near Tc are
governed by the dynamic pair susceptibility propagator L(ω, q) ∝ (Dq 2 + T − Tc + |ω|)−1
for a given mode at finite frequency ω and wave vector q, which acquires mesoscopic fluctuations δL ∝ L2 δTc . Even though the whole effect seems to be small, as it scales inversely
proportional to conductance, g

1, the singular nature of L at T − Tc

11

Tc as {q, ω} → 0

translates into the substantial temperature dependence of kinetic coefficients. This is the
microscopic reason why the universality of mesoscopic effects breaks down in the case of
fluctuating superconductors. Following Ref. [58] we elaborate on this point in the context
of conductivity fluctuations and then carry out microscopic verification of this result with
further extension to thermomagnetic transport.
At the qualitative level the conductivity enhancement near Tc due to superconducting
fluctuations can be estimated as

δσ
∼
σ

∆q ∆−q τq ∼

q

1
g

Tc
T − Tc

(4−d)/2

,

(2.11)

where d is dimensionality. In essence this estimate is obtained from the Drude formula for
conductivity but applied for fluctuation-induced Cooper pairs. Indeed, the average of pairing
gap fluctuations
T
1
ν Dq 2 + T − Tc

∆q ∆−q

(2.12)

measures the average concentration of preformed pairs, while their life-time

τq

Dq 2

1
+ T − Tc

(2.13)

at a given mode q is nothing but Ginzburg-Landau relaxation time. Because of its mesoscopic
origin, discussed above, fluctuations of Tc lead to a giant mesoscopic fluctuations of the order
parameter field

δ∆q δ∆−q , where double-brackets stand for the thermal and disorder

average. Near the critical point

δ∆q δ∆−q
∆q ∆−q

∝

1
g

ξ
L
12

(4−d)/2

Tc
.
Dq 2 + T − Tc

(2.14)

This yields the estimate for mesoscopic fluctuations of conductivity from squaring Eq. 2.11
in the form
δσ 2
σ2

∼

1
g4

ξT
L

4−d

Tc
T − Tc

(8−d)/2

.

(2.15)

Despite the large factor in the denominator, this expression is substantially more singular
than Eq. (2.11) in terms of the temperature dependence near Tc . For concreteness, let’s
concentrate on the two-dimensional case d = 2. Recall that perturbative treatment of the
fluctuation effect breaks down beyond the Ginzburg region, so that at most we can allow
T − Tc ∼ Tc /g when correction to conductivity from Eq. (2.11) becomes of the same order as
a bare Drude conductivity. Remarkably, at that temperature scale, and assuming samples
size of the order of coherence length L ∼ ξT , the amplitude of mesoscopic fluctuations in
conductivity, as estimated from Eq. (2.15), exceeds the bare value by a large parameter
√

g

1.
From these considerations one can infer an estimate of expected fluctuations in thermo-

magnetic response. This can be achieved by the following lines of reasoning. When subject to
crossed electric and magnetic fields, the charged carriers acquire a drift velocity v¯x = cEy /H
in the x direction (H is in z-direction). That would result in the appearance of a transverse
current Jx = en¯
vx with respect to Ey . Under open circuit conditions, no current flows,
and the drift of carriers is compensated by the spacial variation of the electric potential
∇x ϕ = −Ex = (enc/σ)(Ey /H). Because of electroneutrality, this generates the gradient
of the chemical potential ∇x µ(n, T ) + e∇x ϕ = 0, which ultimately corresponds to the appearance of the temperature gradient ∇x T = (dµ/dT )−1 ∇x µ along the x direction. Hence,
the Nernst coefficient ν = Ey /H(−∇x T ) can be expressed in terms of the full temperature
derivative of the chemical potential ν = (σ/e2 nc)(dµ/dT ). This result can be checked for a

13

Figure 2.1: The layout of the Nernst experiment: by applying a temperature gradient (−∇T )
in the presence of magnetic field, an electric field is generated.
degenerate electron gas; the chemical potential µ(T ) = µ0 − (π 2 T 2 /6)(d ln ν/dµ) reproduces
the Sondheimer formula ν = (π 2 T /3mc)(dτ /dµ), where τ is the elastic scattering time. For
the preformed Cooper pairs, dµ/dT = −1 so that ν ∝ σ and fluctuations in ν should follow that of conductivity. This conjecture will be verified with a microscopic calculation in
Sec 2.6.

2.4

Definitions and assumptions

We start with the definition of kinetic coefficients concentrating on the linear response analysis. The electric Jetr and heat Jhtr transport currents are related to the electric field E and
temperature gradient ∇T by the following matrix




Jetr
Jhtr









  σ α  E 
=

,
α κ
−∇T
14

(2.16)

where σ is the electric conductivity tensor , α and α are the thermoelectric tensors (α = T α
due to Onsager relations), and κ is the thermal conductivity tensor. Applying the open
circuit conditions to Eq. (2.16) (see Figure 2.1 for the layout of the experiment), we have
E/∇T = σ −1 /α where the inverse of σ is


σ −1 =



1  σyy −σxy 

.
det |σ|
−σyx σxx

Multiplying it by α, taking the corresponding element of the matrix, the Nernst coefficient
is expressed in terms of the conductivity and thermoelectric tensors

ν=

Ey
1 αxy σxx − αxx σxy
=
.
2 + σ2
(−∇x T )H
H
σxx
xy

(2.17)

An important aspect of the calculation of the transverse thermoelectric response αxy , discussed in detail in Ref. [84], is the need to account for bulk magnetization currents. This
issue arises because the microscopic electric and heat currents, as calculated below by the
Kubo formula, are composed of transport and magnetization currents

Je = Jetr + Jemag ,

Jh = Jhtr + Jhmag .

(2.18)

The magnetization currents are currents that circulate in the sample and do not contribute
to the net currents which are measured in a transport experiment. On the other hand,
they do contribute to the total microscopic currents, and it is thus necessary to subtract
them from the total currents to obtain the transport current response. In the presence of an

15

applied electric field, it was shown in Ref. [84] that the magnetization current is given by

Jhmag = cM × E,

(2.19)

where M is the equilibrium magnetization (in the absence of the electric field). It then
follows that the transverse thermoelectric response is given by

αxy = −

Jh
cMz
cMz
+
=α
¯ xy +
.
Ex T
T
T

(2.20)

It is apparent from the above expression that αxy is obtained by subtracting the result of
two independent calculations: the response of the total current to the applied electric and
magnetic fields, and the magnetization currents as derived from the equilibrium magnetization. Therefore, we need to know magnetic susceptibility M = χH, which will be computed
diagrammatically along with α
¯ xy . Importance of the magnetization contribution to αxy in
the context of superconducting fluctuations was elaborated by Ussishkin [74].
The calculations will be carried out assuming particle-hole symmetry, i.e., neglecting any
contributions which arise due to asymmetry around the Fermi surface in properties such as
the density of states or scattering time. Particle-hole symmetry implies that σxy = αxx =
κxy = 0, and therefore the general expression for the Nernst coefficient Eq. (2.17) greatly
simplifies. The conventional result for αxy in the normal metallic state (so-called quasiparticle
contribution) also vanishes in this limit. However, it has been emphasized [74] that this result
is not required by the symmetry, and will not necessarily hold when additional processes, such
as superconducting or mesoscopic fluctuations, are taken into account without breaking the
particle-hole symmetry. Another simplification of our analysis comes from assuming s-wave

16

G	
  

1/2πντ	
  

λ	
  

Π	
  

Figure 2.2: Building blocks of the diagrammatic technique. Straight lines represent disorderaveraged single-particle Green’s functions. The straight-dashed lines represent single impurity lines that carry an overall factor 1/2πνd τ in the diagrams. The vertex function λ(ε, ε , q)
is drawn in the ladder approximation, while diagrams with the crossed impurity lines yield
parametrically smaller contributions in 1/g
1. The polarization operator Π(ω, q) (“bubble” diagram) is also presented in the main ladder approximation.
symmetry of the superconducting order parameter. In the context of the high-temperature
superconductors, it is of interest to consider also the case of d-wave symmetry in a similar
approach.

2.5

Technical prerequisites

In our calculations we closely follow Ref. [19] for all the basic definitions and notations of
the diagrammatic technique. Graphical rules for constructing Feynman diagrams in the
context of transport in the disordered systems are depicted in Fig. 2.2. We start from the

17

disorder averaged single particle Green’s function, which in the energy and momentum (ε, p)
representation reads

G(ε, p) =

1
,
ε − ξp + 2τi sgnε

ξp = p2 /2m − εF ,

(2.21)

where τ is the elastic scattering time on impurities. In the parameter T τ one should distinguish three different transport regimes in fluctuating superconductors: the diffusive scattering T τ
Tτ

1, the ballistic limit 1

Tτ

Tc /(T − Tc ), and the ultra-clean limit

Tc /(T − Tc ). We will concentrate on the diffusive case exclusively, which is also

mathematically simpler. In the ballistic case, fluctuation effects become strongly non-local
in space, while diffusive impurity scattering makes response functions isotropic and local.
The basic building block of the Cooper ladder is two Green’s functions connected by one
impurity line, which we denote by Γ. Each impurity line brings a factor 1/2πνd τ , where νd
is the density of states. In two-dimensional case ν2 = m/2π, so that

Γ(ω, q) =

1
2πνd τ

dd p R
G (ε+ , p+ )G A (ε− , p− ),
(2π)d

(2.22)

where superscript R(A) stands for the retarded (advanced) component of Eq. (2.21). We
used shorthand notations ε± = ε ± ω/2 and p± = p ± q/2. Integrating above over ξp and
expanding to the first order in ωτ, Dq 2 τ

1,

Γ(ω, q) = 1 + τ (iω − Dq 2 ),

(2.23)

where D = vF2 τ /2 is the diffusion coefficient (here and in the remaining parts of the thesis
we concentrate of the two-dimensional case unless otherwise explicitly mentioned). Having
18

Γ one can sum up the whole series of ladder diagrams to obtain the Cooperon propagator

C(ω, q) =

1
1
1
1
1
(1 + Γ + Γ2 + . . .) =
=
.
2
2
2πνd τ
2πνd τ 1 − Γ
2πνd τ Dq − iω

(2.24)

In practice it is also useful to have the same expression but rewritten in the Matsubara
representation with two different energies ε1,2 running through each of the Green’s function
lines of the Copper ladder

C(ε1 , ε2 , q) =

1
θ(−ε1 ε2 )
,
2
2
2πνd τ Dq + |ε1 − ε2 |

(2.25)

where θ(x) is the unity step-function. The vertex function dressed with impurities is represented as follows (see Fig. 2.2)

λ(ε1 , ε2 , q) = 2πνd τ C(ε1 , ε2 , q).

(2.26)

The propagator of superconducting fluctuations reads

L−1 (ω, q) = −λ−1
sc + Π(ω, q),

(2.27)

where λsc is the coupling constant in the particle-particle (Cooper) channel and

Π(ω, q) = 4πνd T
εn

1
2εn + |ω| + Dq 2
>0

(2.28)

is the polarization operator (summation goes over the fermionic Matsubara frequency εn =
(2n + 1)πT , while ωm = 2πmT corresponds to bosonic frequency). With these ingredients

19

q,

k

=

+

=

+

+
p,
p q,

+

n

q,
k

k

n

Figure 2.3: The Dyson equation for the fluctuating propagator L(ω, q) which represented
graphically as the wavy line and computed in the ladder approximation. Solid lines represent
one-electron Green’s function, greay triangle is the impurity dressed effective vertex, while
each cross between two Green’s functions is associated with the electron-electron coupling
constant −λsc .
at hand one can explicitly calculate the pair propagator which takes the form (see Fig. 2.3)

L−1 (ω, q) = −νd ln

T
+ψ
Tc

1 Dq 2 + |ω|
+
2
4πT

−ψ

1
2

,

(2.29)

where ψ(x) is the digamma function, and the critical temperature is expressed through the
coupling constant Tc = (2γE ωD /π)e−1/νd λsc with γE = 1.78 being the Euler constant and
ωD being the Debye frequency, which cuts logarithmically divergent summation at nmax =
ωD /2πT in the polarization operator. At small frequencies and momenta (the most relevant
limit for most of the further applications) one can expand digamma functions at small
argument {Dq 2 , ω}/T

1 and use ψ (1/2) = π 2 /2 to obtain a simpler expression

L(ω, q) = −

1
1
.
νd ln(T /Tc ) + πDq 2 /8T + π|ω|/8T

(2.30)

For the purpose of calculating kinetic coefficients, the crucial objects are current vertices.
In particular, the electric current vertex function consists of three Green’s functions with
20

two impurity ladders

Bie (ωm , Ωk , q) = 2eT
×

p

εn

λ(εn + Ωk , ωm − εn , q)λ(εn , ωm − εn , q)

vi (p)G(εn + Ωk , p)G(εn , q)G(ωm − εn , q − p),

(2.31)

where Ωk is the external frequency (which will be eventually set to zero in the Kubo formula
to get dc-transport coefficient). Being a function of three frequencies and momentum, this
vertex is fairly complicated, however in the classical region of fluctuations near Tc its evaluation is straightforward. The essential simplification comes from the separation of energy
scales. Bosonic modes are pinned to the energy set by the pole structure of superconducting
propagator Dq 2 ∼ |ωm | ∼ T − Tc , which can be readily seen from Eq. (2.30). At the same
time all fermionic modes are governed by the temperature |εn | ∼ T . Near the transition
T − Tc

T we can evaluate the vertex function by setting all the bosonic frequencies to

zero. As stated above, within the linear response Kubo analysis the external frequency is
also set to zero. As a result, we approximate

Bie (0, 0, q) = 2eT

εn

λ2 (εn , −εn , q)

p

vi (p)G 2 (εn , p)G(−εn , q − p).

Transforming now momentum summation into the integral

p

. . . → νd

(2.32)

dξp . . . ϑ , where

averaging goes along the Fermi surface, and using Eq. (2.21) we get
+∞

Bie (0, 0, q) = −2eνd T

εn

λ2 (q, εn , −εn )

−∞

dξp
(i¯
εn − ξp )2

vi (p)
i¯
εn + ξp − vj (p)qj

,

(2.33)

ϑ

where in addition we used ξq−p ≈ ξp − v · q and abbreviated ε¯n = εn + 1/2τ . From here we
need only the leading small-q part of the vertex. Expanding the denominator to the linear
21

order in q and using Eq. (2.26), the above equation transforms into

Bie (0, 0, q) = −2eνd T vi vj qj

|2¯
εn |2
|εn |2

ϑ
εn

+∞
−∞

(ξ 2

dξ
.
+ ε¯2n )2

(2.34)

The remaining ξ-integration, followed by a εn -summation, can be completed in the closed
form in terms of the digamma function

Bie = 2eBi (q),

Bi (q) = νd Dτ qi ψ

Focusing on the diffusive case only T τ

1
1
+
2 4πT τ

−ψ

1
2

−

1
ψ
4πT τ

1
2

.

(2.35)

1, the above expression simplifies even further

Bi (q) = −2νηqi ,

η = πD/8T .

(2.36)

We will also need the heat current vertex function

Bih (ωm , Ωk , q) = T

εn

i(2εn + Ωk )
λ(εn + Ωk , ωm − εn , q)λ(εn , ωm − εn , q)
2
×

p

vi (p)G(εn + Ωk , p)G(εn , p)G(ωm − εn , q − p).

(2.37)

Under the same approximations as specified above one finds

Bih = 2iνωm ηqi = −iωm Bi (q) .

(2.38)

It should be stressed that such simple expressions for the electrical and heat current vertices
are only possible to obtain near Tc . We will see later in the text that in the quantum low
temperature regime the above calculation has to be revisited, and frequency dependence of

22

Figure 2.4: Feynman diagrams for the main fluctuation-induced corrections to the conductivity. In the first row we show the Aslamazov-Larkin diagrams (left) and the Maki-Thompson
interference diagram (right). In the second row we show two density of states diagrams.
both B e and B h (omitted here) will play a crucially important role. The above results however
are sufficient to make further progress in addressing the classical region of superconducting
fluctuations.
As has been discussed in the introductory chapter, the superconducting fluctuations enhance the conductivity above Tc due to so called Aslamazov-Larkin [10] and Maki-Thompson
[11,12] contributions as well as density of states effects [18], which are less important for the
conductivity unless one studies granular systems. A similar identification of the microscopic
contributions applies to other transport coefficients. In the case of the transverse thermomagnetic response, the leading order contribution to αxy is due to the Aslamazov-Larkin
(AL) diagrams alone. The contributions of the Maki-Thompson (MT) and density of states
(DOS) diagrams are less divergent as T → Tc ; see Fig. 2.4 for the identification of diagrams.
Within the linear response analysis, diagonal Aslamazov-Larkin conductivity is deter-

23

ee
mined from the following current-current Kubo kernel Kxx
:

1
ee
(Ω)]R ,
Im[Kxx
Ω→0 Ω

σxx = lim

ee
(Ωm ) = 4e2 T
Kxx
qωk

Bx2 (q)L(ωk , q)L(ωk + Ωm , q) ,

(2.39)

ee
ee R
(Ωm ) as it is analytically continued
] indicates the retarded component of Kxx
where [Kxx

from the discrete Matsubara frequencies into the entire complex plane iΩm → Ω + i0. The
Aslamazov-Larkin contribution to the transversal thermoelectric coefficient is found from
eh
:
the mixed electric current-heat current Kubo response function Kxy

α
¯ xy =

H
1
eh
lim
Re[Kxy
(Q, Ωm )]R ,
cT Ω,Q→0 ΩQ

(2.40)

where

eh
Kxy
(Ωm , Q) = −4e2 T

q

Bx (q)By2 (q)

(iωn + iΩm /2)

(2.41)

ωn

× [L(ωn , q − Qx )L(ωn , q)L(ωn + Ωm , q) + L(ωn , q)L(ωn + Ωm , q)L(ωn + Ωm , q + Qx )]

We finally define magnetic susceptibility from the equilibrium magnetization M = χH.
Diagrammatically, it may be calculated to linear order in H by considering the current
response to a magnetic field at a finite wavevector Q [85]:

χµν = −

where

αβγ

4e2
c2

αγµ βκν T
ω,q

xˆγ xˆκ L2 (ω, q)Πα (ω, q)Πβ (ω, q),

(2.42)

is the anti-symmetric Levi-Civita unity tensor, and xˆ is thecoordinate operator in

momentum representation. Below we will consider only the isotropic case χµν = χδµν . With
these ingredients at hand we proceed with the calculation of fluctuation-induced corrections

24

2
2
to σxx , αxy , and χ, as well as their mesoscopic correlation functions δσxx
, δαxy
, and δχ2

along with the possible cross-correlators.

2.6

Mesoscopic Nernst effect

The first step in the derivation of desired kinetic coefficients within the Kubo formalism
requires consideration of the discrete sums over Matsubara frequencies, such as sums in the
response kernels of Eqs. (2.39) and (2.41). Such summations over bosonic frequencies can be
conveniently done with the help of closed contour integration in the complex plane by using
the following formula

f (ωm ) =

T
ωm

1
4πi

dωf (−iω) coth

ω
.
2T

(2.43)

Applying this to Eq. (2.39) one finds

ee
Kxx
(Ωm ) = 4e2
q

Bx2 (q)

1
4πi

dωL(−iω, q)L(−iω + Ωm , q) coth

ω
.
2T

(2.44)

The propagators under the integral have breaks of analyticity in the complex plane of ω at
Im(ω) = 0 and Im(ω) = −Ωm , so that the integration contour has two branch cuts along
these lines. We delegate details of this integration, followed by an analytical continuation,
to Appendix-A and present here only the result for the conductivity correction

σxx =

e2
πT

+∞
q

Bx2 (q)

−∞

dω
[ImLR (ω, q)]2 .
sinh (ω/2T )
2

25

(2.45)

For the further integrations we define the following dimensionless units: x = ηq 2 , y = πω/8T ,
and

= ln(T /Tc ) ≈ (T − Tc )/Tc . In these units, the interaction propagator and vertex

function become

ImLR (x, y) = −

y
1
,
ν ( + x)2 + y 2

Bx2 (x) = 4ν 2 ηx cos2 φ ,

(2.46)

and integrations transform into
∞

2π
q

→

dφ
0

0

+∞

dx
,
8π 2 η

−∞

dω
πT
→
2
2
sinh (ω/2T )

+∞
−∞

dy
,
y2

(2.47)

where we expanded sinh y ≈ y since major contribution comes from the range of parameters
{x, y} ∼

1. Combining these definitions together, rescaling y → ( + x)y first and then

x → x, the latter expression transforms into

σxx

e2
= 2
4π

∞

2π
2

dφ cos φ
0

0

xdx
(x + 1)3

+∞
−∞

dy
,
(1 + y 2 )2

(2.48)

with the three integrals equal to π, 1/2, and π/2, respectively, and thus with the final result

σxx =

e2 Tc
.
16 T − Tc

(2.49)

So far we have only reproduced the celebrated result of Aslamazov and Larkin [10]. Our
immediate task is to generalize this result for mesoscopic effects associated with fluctuations
of pair propagator. For that matter we return to Eq. (2.45), take its variation, square the

26

result, and average it over the realization of disorder potential. We thus find

2
=
δσxx

4e4
π2T 2

+∞
q1 q2

Bx2 (q1 )Bx2 (q2 )

R

−∞

dω1 dω2
sinh (ω1 /2T ) sinh2 (ω2 /2T )
2

R

ImL (ω1 , q1 )ImL (ω2 , q2 ) ImδLR (ω1 , q1 )ImδLR (ω2 , q2 ) .

(2.50)

In order to calculate the irreducible correlation function of the pairing susceptibility,
one has to draw two diagrams for L and connect their diffusive parts by impurity lines.
Such construction involves four colliding Diffuson-Cooperon ladders and, on a technical level,
requires computation of four- and six-order Hikami boxes [86], see Fig. 2.5 for the illustration.
Some of these diagrams have been studied before [43,45–47,58] and we invoke that knowledge
for our purposes. In particular we use

δL

R(A)

R(A)

(ω1 , q1 )δL

Aν 2
(ω2 , q2 ) = 2d
g

LT
L

2

[LR(A) (ω1 , q1 )]2 [LR(A) (ω2 , q2 )]2 .

(2.51)

Precise calculation of the numerical factor A ∼ 1 is not of principal importance in a view
of strong dependence of the whole expression on temperature in the low momentum and
frequency limit (ω, q) → 0, which in a way defines T -dependence of transport coefficients. It
is then straightforward to show that

ImδLR (ω1 , q1 )ImδLR (ω2 , q2 ) =
4Aνd2
g2

LT
L

2

ImLR (ω1 , q1 )ReLR (ω1 , q1 )ImLR (ω2 , q2 )ReLR (ω2 , q2 ).

(2.52)

2
We take this back into the equation for δσxx
, and introduce dimensionless x, y, variables,

27

Figure 2.5: Leading order diagrams for the irreducible correlator of mesoscopic disorderaveraged two pair-propagators. This averaging contains collisions of four diffusion or Cooper
modes, and involves forth and sixth order Hikami boxes (internal impurity lines are implicit
on diagrams).

28

as in the case considered above, to arrive at

2
=
δσxx

2π

Ae4 L2T
π 4 g 2 L2

∞

dφ1 dφ2 cos2 φ1 cos2 φ2
0

+∞

dx1 dx2

dy1 dy2
−∞

0

x1 x2 (x1 + )(x2 + )
,
[(x1 + )2 + y12 ]3 [(x2 + )2 + y22 ]3

(2.53)

that after rescaling of integration variables gives the final result

2
δσxx
=

Ae4
32πg 2

ξT
L

2

Tc
T − Tc

3

.

(2.54)

Thus by microscopic analysis we have confirmed our earlier result Eq. (2.15) which was based
on a qualitative considerations. We have already argued that interplay of superconducting
and mesoscopic effects trigger giant fluctuations for samples L

ξT at the temperatures T −

2
∼ (e2 /g)2 (Tc /ET h )3 .
ξT , fluctuations saturate to δσxx

Tc ∼ Tc /g. For larger samples L

We can build on this result to consider emergent mesoscopic fluctuations in the transversal
thermoelectric coefficient. We start from Eq. 2.41 where we need only contributions linear
in Q, which can be easily extracted by expanding the pair propagator and noticing that

∂L(ω, q)
= −Bx (q)L2 (ω, q).
∂qx

(2.55)

eh
Next we have to sum the resulting expression for Kxy
(Ωm , Q) over the Matsubara frequency,

as in the case of the conductivity calculation by contour integration in the complex plane.
We find

eh
Kxy
(Ωm , Q) = −4e2 Q

q

Bx2 (q)By2 (q)

1
4πi

dω coth

ω
(ω + iΩm /2)
2T

×[L3 (ω, q)L(ω + iΩm , q) − L(ω, q)L3 (ω + iΩm , q)].
29

(2.56)

Qˆ
x

Bye

q + Qˆ
x, ω

q, ω
Qˆ
x, Ω

Bxe

Byh

Ω

q + Qˆ
x, ω + Ω

q, ω
Qˆ
x, Ω

Byh

q + Qˆ
x, ω + Ω

q, ω + Ω

Bxe

Ω

Bye

Qˆ
x

eh
Figure 2.6: The Aslamazov-Larkin diagrams contributing to Kxy
(Ω). The wavy lines correspond to the fluctuation propagator L(q, ω); electric current vertices B e and heat current
vertices B h are indicated in the figure along with running momenta and frequencies.

1

30

After further calculation (see analytical continuation that is done carefully in the AppendixA) we arrive at

α
¯ xy =

4e2 H
cπT
R

+∞
q

Bx2 (q)By2 (q)
3

dω coth
−∞

ω
2T

R

× [ReL (ω, q)] ImL (ω, q) + ReLR (ω, q)[ImLR (ω, q)]3 .

(2.57)

In the dimensionless units x = ηq 2 and t = πω/2T , and after the rescaling y → (x + )y and
x → x, one finds at the intermediate step

α
¯ xy =

16e2 H η
cπ 3

∞

2π

dφ cos2 φ sin2 φ
0

0

x2 dx
(x + 1)4

+∞

dy
−∞

1
.
(1 + y 2 )3

(2.58)

After remaining integration, where each of the three integrals yields coefficient π/4, 1/3, and
3π/8 respectively, one finds
α
¯ xy =
Here

H

=

Tc
e ξT2
∝
2
2π H
T − Tc

(2.59)

c/eH is the magnetic length and α
¯ xy has the same scaling with temperature as

the conductivity. As shown by Ussishkin [74] the magnetization contribution has the same
structural form but comes with the coefficient −1/3 instead of 1/2 so that αxy = α
¯ xy +cMz /T
has an overall coefficient of 1/6.
We can address now the mesoscopic fluctuations of αxy by taking the variation of Eq. (2.57),
squaring the result, and averaging over the disorder realization with the help of the correlation function Eq. (2.51). In doing so we encounter quite a cumbersome expression with
2
several contributions to δαxy
, but we make an observation that all the emergent terms have

exactly the same scaling with temperature and differ from each other only by a numerical

31

coefficient of the order of unity. For brevity we present here one particular such term

2
δαxy

=A

e2 νd HLT
cgT L

+∞

dω1 dω2 coth
−∞

2

q1 q2

Bx2 (q1 )Bx2 (q2 )By2 (q1 )By2 (q2 )

(2.60)

ω1
ω2
coth
[ReLR (ω1 , q1 )]4 [ReLR (ω2 , q2 )]4 ImLR (ω1 , q1 )ImLR (ω2 , q2 )
2T
2T

and carry out the calculation up to a factor modulo one (we will absorb all the numerical
factors into the redefinition of coefficient A). Since most relevant frequencies ω ∼ T − Tc are
small compared to temperature we can approximate coth(ω/2T ) ≈ 2T /ω. Transforming the
above into dimensionless variables

2
δαxy
=

e2 A
g2

ξ 2 LT
2
HL

2

∞

+∞

dx1 dx2

dy1 dy2
−∞

0

x21 x22 (x1 + )4 (x2 + )4
, (2.61)
((x1 + )2 + y12 )5 ((x2 + )2 + y22 )5

followed by rescaling and integration, one finds

2
δαxy
=

e2 A
g2

ξT

4

H

LT
L

2

Tc
T − Tc

2

.

(2.62)

2
2
, namely ∝ (T −Tc )−4 ,
has exactly the same temperature scaling as δσxx
Interestingly δαxy

that we already anticipated based on qualitative considerations in Sec. 2.3, and the above
diagrammatic calculations provide the microscopic justification for our results.
It remains to consider fluctuation-induced corrections to magnetic susceptibility and its
mesoscopic fluctuations. From Eq. (2.42) we get for the Aslamazov-Larkin contribution [85]

χ=−

16e2
T
3c2

ωm ,q

Πx L3 (ωm , q) Πx Πyy − Πy Πxy ,

32

(2.63)

where derivatives of the polarization operator can be easily computed from Eq. (2.28)

Πx,y = −

πνD
qx,y ,
4T

Πyy = −

πνD
,
4T

Πxy = 0.

(2.64)

Already at this level, by simple power counting of integration variables, one can deduce that
χ ∝ Tc /(T − Tc ). Consequently one expects that δχ2 will also scale with T − Tc in the
same way as the conductivity and thermomagnetic coefficients. Indeed,

2

δχ

=A

e2 νd2 ηLT
c2 gL

2

q1 q2

Bx2 (q1 )Bx2 (q2 )

dω1 dω2 coth

ω1
ω2
coth
2T
2T

×Im[LR (ω1 , q1 )]4 Im[LR (ω2 , q2 )]4 ,

(2.65)

which, as in the previous examples, reduces with standard steps to

δχ

2

=A

e2 D
c2 g

2

LT
L

2

Tc
.
T − Tc

4

(2.66)

Finally, we will not delve into detailed calculation of the possible cross-correlation functions
between different kinetic coefficients, and merely state here that all such correlations are of
the same order and yield the same temperature dependence.

2.7

Summary

The main results of this Chapter are expressions Eqs. (2.54), (2.62), and (2.66) for variances
of different kinetic coefficients in mesoscopic superconductors. Because of the long-range
phase coherence developing close to Tc , sample-specific mesoscopic fluctuations should be
observable at large length scales. Similarly to normal samples, these fluctuations are sensi-

33

tive to magnetic field strength, impurity configuration, and gate voltage. However, in sharp
contrast to the normal case, where such fluctuations are universal, interaction effects in
the Cooper channel trigger a great amplification of fluctuations due to pairing correlations.
This interplay of coherent impurity scattering and interactions leads to a spectacular example of quantum mesoscopic phenomena occurring at a macroscopic scale. Despite the fact
that mesoscopic fluctuations are no longer universal, in the sense of random matrix theory
classification, we have discovered a different kind of universality in the sense of temperature dependence, which was found to be consistently the same for all the considered kinetic
coefficients.
These calculations have been carried out for homogeneously disordered superconductors.
Therefore, our results cannot be directly compared to the existing experimental findings
where the samples were granular in their origin [33–36]. Granularity adds another parameter into the model – inter-grain conductance – which leads to a strong competition between
Aslamazov-Larkin, Maki-Thompson, and DOS effects [87]. Nonetheless, the main features
predicted by the theory should be present for inhomogeneously disordered superconductors
as well. Indeed, the predicted sample-specific conductance fluctuations were observed experimentally in samples of macroscopic length, and only in a narrow temperature range in
the immediate vicinity of Tc , consistent with the theory. The amplitude of the conductance
fluctuations was found to greatly exceed that of the UCF in normal samples. It should
be also emphasized that some other features accompanying giant mesoscopic effects, such
as suppression of h/2e oscillations in cylindrical samples, negative mangetoresistance, and
its asymmetry, can be also addressed within the same theoretical model. As of today we
are unaware of experimental measurements of mesoscopic effects in thermomagnetic transport of superconductors, except for the measurements of magnetic susceptibility [88]. The
34

mesoscopic Nernst effect has been studied experimentally only in the non-superconducting
systems [89]. Verification of the temperature scaling and the overall magnitude of the effect for mesoscopic fluctuations of the Nernst coefficient predicted here would provide an
important test for our understanding of thermomagnetic transport phenomena in correlated
systems.

35

Chapter 3
Transport anomalies in Pauli-limited
superconductors

3.1

History of the subject

According to the microscopic BCS theory [9] a magnetic field H extinguishes superconductivity. In the absence of spin-orbit interaction there are two basic mechanisms. The first one is
the diamagnetic effect associated with the action of the field on the orbital motion of electrons
forming a Cooper pair. The second, paramagnetic mechanism, is due to Zeeman splitting
of the states with the same spatial wave function but opposite spin. In the former case, the
estimate for the upper critical field follows from the condition Hc2 ξ 2

Φ0 , where Φ0 = hc/2e

is the flux quantum. In contrast, Zeeman splitting destroys superconductivity at a different
critical field Hz that follows from the condition Ez

∆, where Ez = gL µB H is the Zeeman

energy, µB = e /2mc is the Bohr magneton and gL is the renormalized giro factor, while ∆ is
the superconducting gap. The ratio between the two fields is Hz /Hc2 ∼ kF ∼ g
kF is Fermi momentum and

1, where

is the elastic scattering length. Thus, in bulk systems, the
36

Ez
∆0 2
√

2

1

superheating
2−nd order
QF

supercooling

T∗
Tc0

1

T
Tc0

Figure 3.1: Above the tricritical point T ∗ the second order paramagnet to superconductor
transition occurs along the (black) solid line obtained from Eq. 3.1. At T < T ∗ this line
becomes a supercooling part of the hysteresis, and the dashed line is its superheating part.
The latter is obtained following Ref. [95]. The grey shaded area with the critical point
(0, ∆0 ) as its lowest corner bounded by the black dashed line marks the region of quantum
fluctuations (QF).
suppression of superconductivity is typically governed by the first diamagnetic mechanism.
The situation changes in the case of restricted dimensionality. For example, in the case of a
thin-film superconductor in a parallel field, the above ratio becomes Hz /Hc2 ∼ (kF )(d/ξ),
which can be small provided that the film is thin enough d

ξ/kF , such that spin effects

dominate.
The scenario of paramagnetically limited superconductivity has a long history that goes
back to pioneering works by Clogston and Chandrasekhar [90, 91]. The first order phase
transition from superconductor to paramagnet was predicted at the critical field approaching
√
Ez = 2∆ at low temperatures. In practice, the measured film resistance follows a hysteresis
loop [92–96] instead of a sharp first order transition, and the experimental phase diagram
is qualitatively as in Fig. 3.1. At low temperatures, the system remains superconducting as

37

the field increases up to the superheating field; above the critical field, the film is trapped in
a metastable state. At fields exceeding the superheating threshold the film becomes normal.
When the field is reduced, the film stays normal down to the supercooling field Ezsc (T ),
which corresponds to the zero binding energy of a Cooper pair. At T = 0, the normal state
√
is metastable in the interval ∆ < Ez < 2∆ [97, 98]. One should ote that these papers also
√
predicted spatially inhomogeneous state for 2 < Ez /∆ < 1.52. We neglect such possibility
in this work.
In this Chapter, we study the transport properties near the supercooling field, which is
determined by the equation [99]

ln

Tc
Tc0

=ψ

1
2

− Re ψ

1
iE sc
+ z
2 4πTc

(3.1)

similar to that in the theory of paramagnetic impurities [100]. Here ψ is the digamma
function and Tc0 = Tc (H = 0) is the critical temperature in the absence of a magnetic field.
The zero temperature solution of Eq. 3.1, Ezsc (0) = ∆, defines the quantum critical point
(QCP), which is the premier interest of our study. Since in this case the critical parameter
depends on H, it allows for a well controlled exploration of the QCP and its vicinity by
varying the applied magnetic field.
In the scenario when orbital effects dominate the phase diagram is determined by a
similar equation
ln

Tc
Tc0

=ψ

1
2

−ψ

1
α
+
2 4πTc

(3.2)

where pair-breaking parameter α ∼ τd−1 can be deduced from the time-scale of the loss
of phase coherence of the Cooper pairs in the presence of the finite magnetic field τd−1 ∼
DH 2 d2 /Φ20 . This transition is of the second order throughout the entire H − T line, see
38

1.0

α/Tc0	
  

0.8

Normal	
  Phase	
  

0.6
Superconduc/ng	
  
Phase	
  

0.4
0.2
0.0

0.0 0.2 0.4 0.6 0.8 1.0
T Tc0
Figure 3.2: Phase diagram of a superconducting thin film in a parallel magnetic field
parametrized by pair breaking parameter α = D(eHd)2 /6 due to orbital mechanism.
Tc0 = Tc (H = 0) is the critical temperature in the absence of a magnetic field. At T = 0 the
superconductivity breaks down at the critical value αc = πTc0 /2γE , where ln γE ≈ 0.577 is
the Euler constant.
Fig. 3.2. Despite apparent similarity between the two cases at the level of the phase diagram,
we will demonstrate that the microscopic nature of the fluctuation-induced transport near
respective QCPs is conceptually different.
Equation (3.2) implicitly defines the critical temperature Tc as a function of the pairbreaking parameter α. A full analytic solution of this equation in terms of Tc (α) is not
possible, but asymptotic expressions can be easily extracted. For α

Tc0 we can expand

the digamma function ψ(x + 1/2) to first order in x and thus obtain

ln

Tc0
Tc (α)

−

π2
α
= 0.
2 4πTc (α)

(3.3)

To first order in α this yields Tc (α) ≈ Tc0 − πα/8. The expansion for large values of α is a

39

bit more involved, because there exists a critical value of the pair-breaking parameter αc at
which the critical temperature vanishes nonanalytically as a function of α. In order to see
this, we rewrite (3.2) in the following form

Tc0
= exp ψ
Tc (α)

1
α
+
2 4πTc (α)

exp −ψ

1
2

.

(3.4)

Since we are in the regime where α is finite but Tc goes to zero, we make use of the asymptotic
expansion exp[ψ(x + 1/2)] ≈ x + 1/(4!x) + . . . being valid for large x. This expansion yields
Tc0
≈ 4γE
Tc

α
1
+
4πTc 4!(α/4πTc )

.

(3.5)

The critical pair-breaking parameter is defined as the value at which the critical temperature
vanishes Tc (αc ) = 0. In this case we can neglect the second term on the right hand side of
the above equation and obtain αc = πTc0 /2γE ≈ 0.88Tc0 . Expressed in terms of this quantity
√
the critical temperature becomes Tc ≈ ( 6/π) α(αc − α).

3.2

Motivation and qualitative picture

The renewed interest in the physics of paramagnetically limited superconductors is motivated
by the rapid growth of its experimental realizations. Recent parallel magnetic field studies of
two-dimensional superconducting systems were extended to much lower temperatures thus
making it feasible to approach the limit of QCP. Tunneling spectroscopy of ultrathin Al
and Be films revealed field-induced spin mixing and anomalous resonances in the density of
states (DOS) [94, 101, 102]. The latter was successfully explained in theory [103–105], which
emphasized the crucial role of superconducting pairing correlations in the paramagnetic
40

state even far from the transition region. A surprising enhancement of superconductivity
by a parallel magnetic field, deduced from the transport measurements, was observed in
ultrathin, homogeneously disordered amorphous Pb films and in the two-dimensional electron
gas realized at the interface of oxide insulators LaAlO3 and SrTiO3 [106]. In addition,
pronounced negative magnetoresistance (NMR), concomitant with the enhanced Tc , was
reported. Although we do not dwell onto the issue of Tc enhancement in these systems (see
Ref. [107] for the recent theoretical proposals), we show that transport anomalies, such as
NMR, can be successfully addressed within BCS theory.
The issue of NMR in superconductors, either near the QCP or near the parallel field-tuned
superconductor-insulator transition, was previously discussed in the literature experimentally [108, 109] and attributed theoretically [110–112] to the proliferation of superconductive
fluctuations. These studies emphasized mainly the orbital effect of a magnetic field on the
preformed Cooper pairs. In this Chapter we develop transport theory of paramagnetically
limited ultrathin superconductors focusing on the quantum regime of zero temperature near
the critical Zeeman field. The regime of classical fluctuations was partially discussed in the
early papers [113–116].
The conceptual difference of our analysis from the problem of fluctuation-induced transport close to Tc is that unpaired particles, have finite excitation energy Ez . As a result,
the activation probability of such pairs is suppressed exponentially ∝ exp(−Ez /T ) at low
temperature with the statistical Boltzmann factor. We argue that, while in the standard
case the real gapless pairs are only important in the paramagnetically limited case, such
pairs are always virtual.
Let us illustrate this point by taking the Aslamazov-Larkin correction to the conductivity as an example. Consider first the standard case of near–Tc . In the AL diagram the
41

triangular vertex can be estimated as B(ω, Ω, q) ∝ Dqx ∂Π(ω, q)/∂ω. At small momenta
we can take Π(ω, 0) in a clean system. The imaginary part of the polarization operator
ImΠ ≈

dξ[n(−ξp + ω)n(ξp ) − f˜(−ξp + ω)f˜(ξp )]δ(ω − 2ξ) = ν(ω/2) tanh(ω/2T ), where the

particle and hole occupation numbers are f (ε) = (1 + eε/T )−1 , f˜(ε) = 1 − f (ε). The real
2
part, due to virtual pairs ReΠ(ω, q) ≈ ln |(ω 2 − T 2 )/ωD
|, is the familiar Cooper logarithm.

The imaginary part contribution B(ω, Ω, q) ∝ Dqx /T . In contrast, the real part contribution vanishes at ω = 0 due to the particle-hole symmetry, ν(ω) = ν. The expansion in
ω ∼ T − Tc

T yields a correction small in the parameter (T − Tc )/Tc

1.

In the presence of a Zeeman field the situation is very different. The pair activation
rate, ImΠ(ω, q) ≈ ν(ω)[f (ω/2 − Ez /2) − f (ω/2 + Ez /2)], gives an exponentially suppressed
contribution ∝ Dqx exp(−Ez /T )/T . The real part, due to virtual pair excitation, can be
2
|. Its contribution
obtained by the Kramers-Kronig relation, Re(ω, q)Π ≈ ln |(ω 2 − Ez2 )/ωD

to B(ω, Ω, q) is suppressed only algebraically ∝ Dqx T /Ez2 . Unlike the standard case the
virtual quasiparticles make a dominant contribution to the triangular vertex excitations. The
algebraic suppression of vertexes is most pronounced in the case of the AL and is manifested
in additional factors of Dq 2 and ω in σ AL , which makes it logarithmic in Ez /(Ez − ∆0 ).
Note that in the case of the near–Hc2 problem [110] the AL contribution is also suppressed
due to the current matrix elements connecting adjacent Landau levels. The regular MT and
DOS contributions are proportional to a second derivative of the real part of the polarization
operator ReΠ(ω, q). Since the latter is finite at ω = 0, these contributions are as singular as
AL terms. These technicalities will be explained in great details in the following section.

42

3.3

Fluctuation-induced conductivity near the quantum critical point

We approach the problem of transport in the in-plane magnetic field close to the superconducting transition based on the diagrammatic perturbation theory. Note that the technique
based on the time-dependent Ginzburg-Landau formalism applied for studying transport
near QCP [117, 118] accounts correctly only for the classical part of AL-type contribution
to the conductivity, but it misses completely the quantum zero-temperature corrections. A
microscopic approach takes care of all the contributions including the DOS part, resulting
from the depletion of the normal state density of states by superconducting fluctuations,
and also the MT interference term [110–112]. In fact, at T = 0, where the corrections
come from purely quantum fluctuations, these effects turn out to be of the dominant nature. In calculations T

{Ez , ∆}

τ −1

εF , these conditions are satisfied in many

experiments [94, 101].
Our starting point is the current-current response kernel which can be conveniently presented as a sum of three contributions K = K AL + K M T + K DOS . Within this section we
will be discussing only longitudinal electrical transport so the subscript Kxx is suppressed in
all the expressions for brevity. The general expression for the AL term reads

K AL (Ωn ) = −e2 T

q,ωk

B 2 (ωk , Ωn , q)L(ωk , q)L(ωk + Ωn , q),

(3.6)

where ωk = 2πkT . Notice here the factor of 4 difference compared to the earlier expression
Eq. (2.39). This is due to the fact that, in the finite in-plane magnetic field, different spin
projections σ = ± contribute unevenly to the current vertex B. It reads explicitly now as
43

follows

B(ωk , Ωn , q) = T
σ
=
JAL
p

σ
,
λσ (εm+n , ωk − εm , q)λσ (εn , ωk − εn , q)JAL

(3.7)

vp G σ (εn+m , p)G σ (εn , p)G −σ (−εn + ωk , −p + q),

(3.8)

σ,εm

with the impurity ladders

λσ (εn , εm , q) =

θ(−εn εm )
,
τ [Dq 2 + |εn − εm | − iσEz sgn(εn − εm )]

(3.9)

and an integral over the block of three Green’s functions with Gεσn ,p = (iεn − ξp + σEz /2 +
isgn(εn )/2τ )−1 . Here we used notations: εm = 2πT (m + 1/2), ξp = p2 /2m − εF , vp = ∂p ξp ,
θ(ε) is the step-function, and sgn(ε) is the sign-function. The propagator of fluctuating
Cooper pairs in Eq. (3.6) is given by

L−1 (ωk , q) = −ν ln

1
2

T
−ψ
Tc0

Ψσ (ωk , q) = ψ

+

1
Ψσ (ωk , q) ,
2 σ=±

1 Dq 2 + |ωk | + iσEz
+
2
4πT

.

(3.10)
(3.11)

When calculating the B-vertex one should follow a few basic steps which we already discussed
in the previous Chapter: i ) In the leading order in the transferred momentum q one can
approximate G −σ (−εn + ωk , −p + q) ≈ G −σ (−εn + ωk , p) + v · q[G σ (−εn + ωk , p)]2 . ii )
Furthermore, one can neglect Zeeman energy as compared to the inverse scattering time in
the Green’s functions (provided the condition T

{Ez , ∆}

τ −1

εF is satisfied) and

then complete p integration in a standard way with density of states and angular averaging
over the Fermi surface. iii ) Next is the fermionic Matsubara εm sum in Eq. (3.7), which can

44

be found in the closed form with the result

B(ωk , Ωn , q) =

νd q x D
Ωn

σ
σ

[Ψσ (|ωk | + Ωn , q) − Ψσ (|ωk |, q)

+Ψ (|ωk+n | + Ωn , q) − Ψσ (|ωk+n |, q)] .

(3.12)

iv ) The remaining step of the calculation is a bosonic ωk sum followed by an analytical
continuation iωn → ω. The latter is accomplished via the contour integration over the circle
with two-branch cuts at Im(ω) = 0, −Ωn where the product of propagators in Eq. (3.6) has
R
breaks of analyticity. After the Ω-expansion of KAL
(Ω) to the linear order, one finds for the
AL
AL
AL conductivity correction σ AL = σclAL + σq1
+ σq2
, where

σclAL =

+∞

e2
4πT

−∞

q

∞

2

AL
σq1

e
=
4π

AL
σq2
=−

e2
4π

dω
[B RA (ω, q)]2 [ImLR (ω, q)]2 ,
sinh (ω/2T )
2

dω coth
0

q

ω
Re [B RA (ω, q)]2 − [B RR (ω, q)]2 ]∂ω [LR (ω, q)]2 , (3.14)
2T

+∞

dω coth
q

−∞

(3.13)

ω
2T

× ∂Ω [B RR (ω, Ω, q)]2 [LR (ω, q)]2 − ∂Ω [B AA (ω − Ω, Ω, q)]2 [LA (ω, q)]2
+∂Ω [[B RA (ω − Ω, Ω), q]2 − [B RA (ω, Ω, q)]2 ]|LR (ω, q)|2 .

(3.15)

The superscript R(A) in the vertex functions and propagators stands for the retarded (advanced) component while subscript cl(q) refers to classical (quantum). This convention comes
form the observation that as T → 0 the classical contribution vanishes while the quantum
contribution remains finite.

45

We turn now to the derivation of the MT contribution whose response kernel is given by

K M T (Ωn ) = e2 T
ωk ,q

L(ωk , q)ΣM T (ωk , Ωn , q),

(3.16)

where

ΣM T (ωk , Ωn , q) = T
σ,εm

λσ (εm+n , ωk−n − εm , q)λσsgn[ m

m+n ]

(εm , ωk − εm , q)JM T

(3.17)

vp vq−p G (εm+n , q)G −σ (ωk−n − εm , q − p)G σ (εm , p)G −σ (−εm + ωk , q − p). (3.18)
σ

JM T =
p

Momentum integration in the block of Green’s functions JM T is done under the same approximations as in the case of the AL term described above. According to the standard
convention [19] we now split the MT term into the so-called regular and anomalous contributions:

ΣM T (reg) (ωk , Ωn , q) = −
ΣM T (an) (ωk , Ωn , q) = −

νd D
Ωn

νd D
2(Dq 2 + Ωn )

[Ψσ (|ωk | + 2Ωn , q) − Ψσ (|ωk |, q)] ,

(3.19)

[Ψσ (−|ωk | + 2Ωn , q) − Ψσ (|ωk |, q)] .

(3.20)

σ

σ

After the analytical continuation these translate into the conductivity correction σ M T =
MT
MT
σreg
+ σan
, where

MT
σreg

MT
σan
=

e 2 νd D
=− 3 2
8π T

e 2 νd D
8πT

σq

∞

ω
Im[LR (ω, q)[Ψσ (−iω, q)] ],
2T

(3.21)

LR (ω, q)[Ψσ (iω, q) − Ψσ (−iω, q)]
dω
.
Dq 2 + Γ
sinh2 (ω/2T )

(3.22)

dω coth
σq
+∞

−∞

0

In order to regularize the logarithmically divergent momentum integral in the case of the

46

anomalous contribution we have introduced a pair-breaking cutoff parameter Γ. The microscopic origin of the latter (e.g. spin-orbit scattering) will be discussed in the next section.
We finally discuss the DOS contribution to the conductivity. It is given by the similar to
Eq. (3.16) expression with

K DOS (Ωn ) = e2 T
ωk ,q

L(ωk , q)ΣDOS (ωk , Ωn , q),

(3.23)

where

ΣDOS (ωk , Ωn , q) = 2T
σ,εm

JDOS =
p

[λσ (εm , ωk − εm , q)]2 JDOS ,

(3.24)

vp2 [G σ (εm , p)]2 G σ (εm + Ωn , p)[G −σ (ωk − εm , q − p)
+

1
2πνd τ

p

[G σ (εm , p )]2 G −σ (ωk − εm , q − p )].

(3.25)

After completing the standard steps outlined above one arrives at the conductivity correction
σ DOS = σclDOS + σqDOS in the form

σclDOS = −

e 2 νd D
16π 2 T 2

+∞
σq

−∞

dω[[Ψσ (iω, q)] − [Ψσ (−iω, q)] ] R
L (ω, q),
sinh2 (ω/2T )

MT
σqDOS = σreg
.

(3.26)
(3.27)

The equality between the two contributions in Eq. (3.27) has parallels with the original
fluctuation transport considerations at T − Tc

T . In the original near–Tc problem, the

typical energy of diffusing pairs Dq 2 ∼ T − Tc is smaller than the thermal energy of the
quasiparticle ∼ T . In our case, Ez adds to the energy of pairs making it bigger than T .
Correspondingly, unlike the near–Tc case, the off-shell energy of a pair, 2ε ∼ T , falls below
47

the pair excitation energy set by Ez . This causes a sign inversion of the energy denominator
associated with the unbound intermediate state, and the correction Eq. (3.27) turns out to
be positive. In general, the derived above conductivity corrections are applicable at any field
H and temperature T above the transition. In the following, we discuss limiting cases of
interest.
It is convenient to regroup all contributions and present the total conductivity correction
as the a sum of zero-temperature (δσq ) and finite-temperature (δσT ) terms, namely

δσ(H, T ) = δσq (H) + δσT (H, T ).

(3.28)

The first term here is determined by the quantum AL [Eqs. (3.14)-(3.15) and DOS [see
Eq. (3.27)] contributions, and also the regular part of the MT conductivity [see Eq. (3.21)].
The remaining terms define δσT . The magnitude of δσq decreases monotonically with an
increasing field. This leads to a pronounced magnetoresistance at zero temperature. At
finite temperature, based on how the quantum critical point is approached, there are several
regimes that show different T and H dependencies, which should be experimentally accessible. Below we focus on QCP only and extract the leading singularity in δσq as the function of
the Zeeman field. Thermal contribution δσT and various crossover regimes will be discussed
in the next section.
At zero temperature Ψσ (±iω, q) → ln[(Dq 2 ± iω + iσEz )/4πT ] and the pair propagator
can be taken in the leading pole approximation

LR(A) (ω, q) ≈ −

2∆20 /νd
,
Ec2 − (ω ± iDq 2 )2

48

(3.29)

which is obtained from Eq. (3.10) under the conditions Dq 2

∆0 and |Ec ± Ω|

∆0 = πTc0 /2γE where ln γE ≈ 0.57 is the Euler constant, and Ec =

∆0 . Here

Ez2 − ∆20 . The branch

cut of the propagator (due to the logarithmic structure) also contributes to δσq , but gives
the subleading singularity. Within the same accuracy we compute the vertex functions:

[B RR(AA) (ω(−Ω), Ω, q)]2 =

8νd2 D 2
Dq (Dq 2 ∓ iω)(Dq 2 ∓ iω − 2iΩ),
4
Ez

[B RA(AR) (ω, Ω, q)]2 =

8νd2 D
(Dq 2 )2 (Dq 2 − 2iΩ).
4
Ez

(3.30)

(3.31)

Alltogether this leads to the conductivity correction near the Zeeman field-induced quantum
critical point
δσq (H) =

2e2
ln
π2

Ez
Ez − ∆0

,

(3.32)

which is obtained within the logarithmic accuracy. Equation (3.32) is the main result of this
section. We have checked explicitly that other contributions, such as the diffusion coefficient
renormalization or the contribution with only one or no Cooperon vertex, are either small
or nonsingular. Since the temperature can be set to zero in integrations over fast-fermion
degrees of freedom, the additional factors of τ results in small prefactors τ Ez , τ DQ2 or τ Ω.

3.4

Effects of pair-breaking scattering

In the preceding calculations we assumed that impurity scattering of electrons does not cause
spin flips. There are two sources of spin relaxation of conduction electrons: localized spins
(magnetic impurities) and spin-orbit (SO) scattering of electrons by nonmagnetic disorder.
The latter is characterized by the scattering amplitude ivso ([p×p ]·σ)/p2F , where p and p are
the initial and final momenta of an electron, and σ is the spin operator whose components
49

are the Pauli matrices. Let us discuss the effect of SO scattering first starting with the
qualitative physical picture [103]. In the absence of both, SO interactions and magnetic field
two-spin states, which belong to a given orbital, have the same energy. Magnetic field split
this degeneracy. It is important that the splitting energy Ez is exactly the same for all of
the orbital states, which is no longer the case for finite SO interaction. Without an external
magnetic field the states are still doubly degenerate due to time-reversal invariance (Kramers
doublets). A magnetic field splits the Kramers doublets similar to how it splits pure spin
states in the absence of SO interactions. The main difference is that this splitting is not
exactly uniform anymore [119].
The spin-orbit scattering and finite thickness effects modify the fluctuation transport, due
to the finite spectral weight in the particle-particle channel at zero frequency. The addition
of a finite spin-orbit scattering introduces a finite lifetime Γ−1 to the Cooperon. At lowest
temperatures the superconductivity survives if this scattering is not too strong, Γ

Ez with

a somewhat lower critical field. While Ez approaches the supercooling transition from above
the results obtained in the previous section are expected to crossover to a different regime
at Γ ≈ Ec . The finite film thickness affects the crossover in a similar way. Inclusion of these
effects was shown to be necessary for quantitative analysis of measurements of the density
of states [102].
As was discussed early in Sec. 2.5 [see Eq. (2.25)] the Cooperon is formed by two electron
Green’s functions. In the absence of an external magnetic field it is convenient to classify
Cooper poles by the total spin of the two electrons S+ = (σ 1 + σ 2 )/2. Spin-orbit scattering
does not affect the spin singlet part of the Cooperon (S2+ = 0), however, this scattering
leads to total spin relaxation, i.e., the triplet (S2+ = 2) component of the Cooperon decays
and, consequently, the pole in the ω plane is shifted from the real axis even for q = 0. An
50

external magnetic field is coupled with the difference S+ = (σ 1 + σ 2 )/2 of two electron
spins, and as a result we classified the Cooperon by the eigenvalue of the operator S− · Ez .
These eigenvalues for S2− = 2 are 0, ±Ez , corresponding to S−z = 0, ±1 and 0 for S2− = 0.
Neither of those two classifications is exact when both a magnetic field and SO scattering
take place simultaneously. We assume that the SO effect is weak that allows us to evaluate
the Cooperon perturbatively

Π(ω, q) = 4πνd T
εn

1
,
2+Γ
2ε
+
|ω|
+
iσE
+
Dq
n
z
so
>0 σ=±

(3.33)

−1
2
where Γso = 2/3τso , and τso
= 2πνd vso
is the time of the spin relaxation by SO scattering

[compare Eq. (3.33) to Eq. (2.28)]. This consideration suggests that any physical mechanism
of violation of either time-reversal invariance or conservation of spin will have a similar effect
on a Cooperon field. In the following discussion we assume that Γ = Γso + Γs + ΓH is the
total scattering rate that include spin-orbital, spin-flip and finite film thickness effects.

3.5

Thermomagnetic phenomena

Since the Cooperon is no longer a soft mode at (ω, q) → 0 and has a finite gap Γ, due to
spin-related scattering processes, it inevitably enters into the pair propagator L(ω, q), shifts
its pole and ultimately changes temperature dependence of the kinetic coefficients. Thus
with Eq. (3.33) at hand we have to recompute all the basic ingredients of the diagrammatic
technique. Most importantly we find the generalized form of the fluctuation propagator

LR (ω, q) = −

2∆20 /νd
,
Ec2 − 2iΓ(ω + iDq 2 ) − (ω + iDq 2 )2

51

(3.34)

which is different from the expression that we used before [Eq. (3.29)] in two important
aspects. First is the presence of the new term in the denominator which introduces an
additional scale Ec2 /Γ to the problem. Second is shifted value of the critical point Ec =
Ez2 − ∆2Γ with ∆2Γ = ∆20 + Γ2 . Another important building block of the theory is the
vertex function B. We present only its mixed retarded-advanced component
[B RA(AR) (ω, Ω, q)]2 =

8νd2 D
(Dq 2 + Γ)2 (Dq 2 − 2iΩ),
4
Ez

(3.35)

which is the most relevant for the transport regime that we will specify next. The subsequent
calculations will be carried out assuming Γ
critical line Ec2 /Γ

T

Γ

Ec and for temperatures not too close to the

Ez , which is quite relevant for the actual experimental

realization. Under these conditions the third term in the denominator of Eq. (3.34) can be
neglected so that one finds approximately

ImLR (ω, q) = −

2Γω
2∆20
.
2
νd (Ec + 2ΓDq 2 )2 + 4Γ2 ω 2

(3.36)

In order to determine δσT (H, T ) we have to reexamine all the contributions to fluctuationinduced conductivity. In the course of this analysis we found that the most singular term
originates from the classical part of the AL contribution Eq. (3.13), so that in large δσT (H, T )
is governed by σclAL . To calculate this term explicitly in the above discussed regime, we
introduce dimensionless variables x = Dq 2 /2T , y = ω/2T , γ = Γ/2T , and

= Ec /2T and

obtain from Eq. (3.13)

δσT (H, T ) =

16e2
π2

∞

+∞

dx
0

−∞

dy
sinh2 y [(

52

γ 2 y 2 x(x + γ)2
,
2 + 2γx)2 + 4γ 2 y 2 ]2

(3.37)

where, in addition, we set ∆0 /Ez → 1 assuming that H is tuned sufficiently close to the
transition line. The remaining integrations can be done with having small parameter

2

/γ

1, which leads to the result

δσT (H, T ) =

e2
2π

TΓ
Ez2

Ez
Ez − ∆Γ

.

(3.38)

It is worth mentioning here that, unlike the quantum regime Eq. 3.32, which is logarithmically
singular in Ez −∆, the magnetoresistance in the classical region is more pronounced. We have
further checked that the other terms remain smaller and scale logarithmically ∝ ln(Ez Γ/Ec2 )
(see Appendices C and D for further details).
In a similar spirit we can calculate now αxy . We have verified that, in the regime of
classical fluctuations, the relation between current B e and heat B h vertices remains the same
B h = (−iω/2e)B e , so that we can proceed immediately to Eq. (2.45) with the vertex and
propagator taken from Eqs. (3.35) and (3.34) respectively. At the intermediate step one finds
for transverse thermoelectric coefficient

αxy

512e L2T
=
π 2H

∞

+∞
2 4

dxx γ
0

dy coth(y)
−∞

γy( 2 + 2γx)
[( 2 + 2γx)2 + 4γ 2 y 2 ]3

(3.39)

and after final integrations
αxy = 2e
where Lz =

L2z

Γ
Ez − ∆Γ

2
H

,

D/Ez . We reiterate that this result is valid provided Ec2 /Γ

(3.40)
T

Γ

Ez

and observe that the singularity in αxy is the same as in the conductivity δσT . This resembles
similar situation as in the case of near Tc transport at zero field, namely identical scaling
of the α and σ with the critical parameter, while the underlying physics is very different.

53

It should be stressed, however, that the field dependence of αxy is extremely sensitive to
the pair-breaking scattering. For completeness, we also analyzed behavior of αxy in the
limit of negligible Γ

max{T, Ec }

Ez and find dependence different than that give by

Eq. (3.40). For that purpose, we use Eqs. (3.29) and (3.30) in the expression for αxy defined
by Eq. (2.57), which leads us to expression

αxy =

256e L2T
π 2 2H

∞

+∞

dx

dy coth(y)
−∞

0

x7 y( 2 + x2 − y 2 )
,
[( 2 + x2 − y 2 )2 + 4x2 y 2 ]3

(3.41)

which is valid in the thermal region of fluctuations T > Ec . The double integral gives as
factor of 9π 2 /256 . This implies that αxy has square-toot singularity
√ L2
αxy = 18 2e 2z
H

Ez
.
Ez − ∆0

(3.42)

We conclude this section by briefly discussing behavior of αxy in the regime when orbital
effects of pair-breaking dominate over the spin-related effects. The phase diagram has been
already discussed above based on Eq. (3.2), where we analyzed limiting case of classical
and quantum fluctuation regimes. At a given pair-breaking strength α, superconductivity is
destroyed at T = Tc (α) and at a given temperature T , at α = αc (T ), obtained by solving
Eq. (3.2) for T and for α, respectively. In the neighborhood of this classical transition, for
α

Tc0 we can define the quantity

T (α, T )

= [T − Tc (α)]/Tc (α)/ that measures the relative

distance from the critical temperature Tc (α). Conversely, in the vicinity of the the pairbreaking quantum phase transition on can define the quantity

α (α, T )

= [α − αc (T )]/αc (T ),

which can be interpreted as the relative distance from the critical pair-breaking strength
αc (T ) at a given T

αc0 . In a parametrically broad temperature regime near the transition

54

line (but not too close to the quantum region) it is legitimate to approximate pair-propagator
and vertex function by the following simple expressions

LR (ω, q) = −

1
1
,
νd ln(α/αc (T )) + (Dq 2 − iω)/2αc (T )

Bi = −νd Dqi /αc (T ).

(3.43)

When combined with Eq. (2.57) this yields

αxy =

2e L2T
π 2 2H

∞

+∞

dy

dx
0

−∞

x2 y coth(αc y/T )( α + x)
,
[( α + x)2 + y 2 ]3

(3.44)

where we introduced dimensionless variables for the momentum x = Dq 2 /2αc and frequency
y = ω/2αc . This result is valid for temperatures away from the critical region T

αc so

that it is legitimate to expand the cotangent at small argument. The double integral gives
a factor πT /8αc

α

so that one arrives at

αxy =

e L2α αc (T )
,
4π 2H α − αc (T )

where we introduced new length-scale Lα =

(3.45)

D/αc (T ), and also expanded the logarithm

ln(α/αc ) ≈ (α − αc )/αc assuming close vicinity to the transition line.
Given the plethora of the different regimes and behaviors, it is desirable to have a systematic experimental study aimed specifically at exploring the physics of a pair-breaking phase
transition in superconducting films from the measurements of the conductivity δσ(H, T ) and
thermoelectric response αxy . There have been few works using a parallel field as a control parameter to scan across the phase diagram, but they have had other goals mainly focussing on
the physics of superconductor-insulator transition. To begin with, it will be useful to observe
the finite temperature classical transition and verify the predictions of the fluctuation the55

ory presented in this section in its vicinity. By slowly increasing the pair-breaking strength
and lowering the temperature, one could approach the quantum phase transition. Having
identified the right films, measurements of the temperature and pair-breaking parameter
dependence of the conductivity, would afford an exciting possibility of discovering different
regimes in the vicinity of the pair-breaking quantum phase transition. The non-monotonic
magnetoresistance due to the presence of superconducting fluctuations that we find, is in
stark contrast to the intuitive expectation and is a purely quantum effect. A clear experimental signature of such a characteristically quantum behavior would be an important step
forward in the study of quantum phase transitions and low temperature superconductivity.

3.6

Summary

In this Chapter we studied electrical and thermal transport anomalies of low dimensional superconducting films in an external in-plane magnetic field. We concentrated on the ClogstonChandrasekhar (CC) phase transition, i.e., the destruction of superconductivity by a magnetic field by virtue of the Zeeman splitting. As a result, a normal paramagnetic state of
electrons is formed. The main conclusion we can draw from this study of the CC state
is, that despite this state being normal (namely with the mean-field superconducting order
parameter vanishing), it is drastically different from a usual normal metal with some attractive interaction. The latter state appears, e.g., in a superconductor at temperatures higher
than the transition temperature Tc . The difference becomes apparent when one studies excited states rather than those close to the ground state. In particular, fluctuation-induced
transport is dominated by the virtual excitations rather than real preformed Cooper pairs.
This leads to a nontrivial magnetoresistance near QCP. The reason why the effects of su-

56

perconducting fluctuations in a CC metal are dramatically enhanced, in comparison with
the usual case, is the presence of the pole-like singularity in the correlation function of these
fluctuations. This pole at a finite frequency appears due to the fact that the CC transition
is of the first order. In contrast, the temperature-driven transition from superconductor to
normal metal is of the second order, and in a usual normal state the correlator of the superconducting fluctuations is a smooth function of the frequency, i.e., any superconducting
excitations decay very rapidly.
Near the QCP of the supercooling line of the phase diagram, magnetoresistance is governed by density of states and regular Maki-Thompson terms. We found complete cancellation of quantum Aslamazov-Larkin corrections, while anomalous Maki-Thomspon and
classical Aslamazov-Larkin terms vanish in the zero-temperature limit. Nevertheless, the
latter terms are crucially important in describing the crossover regimes at finite temperature
where the sign of the magnetoresistance essentially depends on the direction at which the
transition line is approached in the field-temperature plane. Surprisingly, we find that, near
the transition line, scaling of the conductivity corrections is the same as the scaling of the
transversal thermoelectric coefficient, which is analogous to usual transport anomalies near
Tc and in the absence of the field. A priori this result is difficult to foresee since the mere
mechanism of fluctuation corrections is different in the CC phase. The apparent universality between the singular field dependences of σxx and αxy near the transition is another
important observation of this study.
We close this Chapter by briefly discussing outstanding problems that remain largely
unsolved, where our microscopic approach may give an opportunity to systematically study
thermomagnetic phenomena in other various superconducting systems. Since the Nernst
effect is highly sensitive to fluctuations, its measurements may shed light on the intimate
57

connection between quantum criticality and unconventional superconductivity with competing or coexisting orders. Perhaps, the most interesting systems from that perspective are
heavy-fermion superconductors (e.g. URu2 Si2 ) and iron-pnictide superconductors (e.g. FeSe
compounds).
It was recently reported [120] that the Nernst coefficient in URu2 Si2 is anomalously
enhanced as compared to the theoretically expected value of the standard Gaussian fluctuations. Moreover, contrary to the conventional wisdom, the enhancement is more significant
with the reduction of the impurity scattering rate. This unconventional Nernst effect intimately reflects the highly unusual superconducting state embedded in the so-called hiddenorder phase of URu2 Si2 that appears to point to a new type of superconducting fluctuations
generated by a degree of freedom which has not been hitherto taken into account. It is
tempting to consider that such a degree of freedom is intimately related to the superconducting state with broken time-reversal symmetry. To properly address the data one has
to seriously consider possible chiral or Berry-phase fluctuations associated with the broken
time-reversal symmetry of the superconducting order parameter [121].
Our theory of Pauli limited superconductivity may be highly relevant for the study of
FeSe superconductors. This is a unique solid state system that offers the possibility to enter
the previously unexplored realm where the three energies, Fermi energy, superconducting
gap, and Zeeman energy, become comparable, and thus access the crossover regime between
the weakly coupled Bardeen-Cooper-Schrieffer (BCS) limit and the strongly coupled BoseEinstein-condensate (BEC) limit. The results of the transport properties of FeSe near the
BCS-BES crossover reveal intriguing features [122]. What is remarkable is that, for the
Nernst effect, the Seebeck coefficient, the temperature derivative of the resistivity, and the
Hall coefficient, all show peaks at around 20 K, which is twice as large as the critical tem58

perature (Tc ∼ 10 K). One of the most important subject in the BCS-BEC crossover is the
debate concerning the mechanism — preformed pair or pseudogap. It is useful to recall
that at the pseudogap temperatures of YBCO and CeCoIn5 , the Nernst effect exhibits its
peak. Whether one can draw any conclusion from this similarity for FeSe systems remain to
be seen. What is clear, is that theoretical input is urgently needed and so thermomagnetic
transport in superconductors will continue to attract tremendous attention from researchers.

59

APPENDICES

60

Appendix A:
Matsubara sums and analytical
continuation
As outlined in the main text we convert the bosonic Matsubara sum into the integral

S(Ων ) =

1
4πi

dz coth
C

z
L(−iz)L(−iz + Ων ),
2T

(A.46)

where the contour of integration is a circle which contains two branch-cuts at Imz = 0
and Imz = −Ων , [see Fig. (A.3)], where functions L(−iz) and L(−iz + Ων ) have breaks of
analyticity respectively (q-dependence of propagators is suppressed here for brevity). We
thus have explicitly
+∞
1
z R
S(Ων ) =
dz coth
L (−iz + Ων )[LR (−iz) − LA (−iz)] +
4πi −∞
2T
+∞−iΩν
1
z A
dz coth
L (−iz)[LR (−iz + Ων ) − LA (−iz + Ων )] .
4πi −∞−iΩν
2T

61

(A.47)

Figure A.3: Integration contour in the plane of complex frequency. The lower part of the
contour corresponds to advanced-advanced products of propagators after analytical continuation. The middle section contains mixed causality components of advanced-retarded, while
the upper third contains only retarded-retarded products of propagators.
In the second integral we shift variable z + iΩν = z which gives us
+∞
z R
1
dz coth
L (−iz + Ων )[LR (−iz) − LA (−iz)] +
S(Ων ) =
4πi −∞
2T
+∞
1
z − iΩν A
dz coth
L (−iz − Ων )[LR (−iz ) − LA (−iz )] .
4πi −∞
2T

(A.48)

z
ν
Taking into the account that iΩν is the periodic of cotangent coth z −iΩ
= coth 2T
,
2T

changing back z → z in the second integral, and taking the analytic continuation step
Ων → −iΩ, we get
1
S (Ω) =
2π

+∞

R

dz coth
−∞

z R
[L (−iz − iΩ, q) + LA (−iz + iΩ, q)]ImLR (−iz, q) .
2T

62

(A.49)

Since LR (−iz) = LA (iz) we get for the response kernel
2e2
π

ee
(Ω)]R =
[Kxx

+∞

Bx2 (q)

q

dω coth
−∞

ω A
[L (ω+Ω, q)+LR (ω−Ω, q)]ImLA (ω, q) . (A.50)
2T

We do not write imaginary i in the frequency argument of LR(A) (iz), which is implicit in the
definition. To the linear order in external frequency

LA (ω + Ω, q) + LR (ω − Ω, q) ≈ Ω∂ω [LA (ω, q) − LR (ω, q)] = 2iΩ∂ω ImLA (ω, q) .

(A.51)

As a result the retarded component of the current-current response kernel reduces to

ee
[Kxx
(Ω)]R =

2iΩe2
π

+∞
q

Bx2 (q)

dω coth
−∞

ω
∂ω ImLA (ω, q)
2T

2

,

(A.52)

which can be integrated by parts to yield Eq. (2.45).
Calculation of the Matsubara sum in the case of thermomagnetic response functions
follows the same steps as above but is more involved since it contains three propagators

S(Ωm ) =

1
4πi

dω coth
C

ω
(ω + iΩm /2) L3 (ω, q)L(ω + iΩm , q) − L3 (ω + iΩm , q)L(ω, q)
2T
(A.53)

We transform this integral into a contour with branch cuts

S(Ωm ) =

1
4πi

+∞

dωω+ coth
−∞

ω
[L3R (ω, q) − L3A (ω, q)]LR (ω + iΩ, q)
2T

−[LR (ω, q) − LA (ω, q)]L3R (ω + iΩ, q)
+

1
4πi

+∞−iΩ

dωω+ coth
−∞−iΩ

ω
[LR (ω + iΩ, q) − LA (ω + iΩ, q)]L3A (ω, q)
2T

−[L3R (ω + iΩ, q) − L3A (ω + iΩ, q)]LA (ω, q) ,
63

(A.54)

where we used notation ω± = ω ± iΩm /2. In the second integral we shift ω + iΩm → ω,
ω
m
= coth 2T
, and then perform the analytic
use the periodicity of the cotangent coth ω−iΩ
2T

continuation iΩm → Ω, which gives
+∞
ω
1
dωω+ coth
LR (ω + Ω, q)ImL3R (ω, q) − L3R (ω + Ω, q)ImLR (ω, q)
S(Ω) =
2π −∞
2T
+∞
1
ω
+
dωω− coth
L3A (ω − Ω, q)ImLR (ω, q) − LA (ω − Ω, q)ImL3R (ω, q) . (A.55)
2π −∞
2T

To the linear order in Ω, there are terms of two kind. First is the direct term proportional
to Ω due to the vertex. Second, is the linear term from the expansion of propagators. We
focus on the first possibility since it gives the most important contributions. We set Ω → 0
in propagators and get

S(Ω) =

Ω
4π

+∞

dω coth
−∞

ω
2T

(A.56)

LR (ω, q)ImL3R (ω, q) − L3R (ω, q)ImLR (ω, q) − L3A (ω, q)ImLR (ω, q) + LA (ω, q)ImL3R (ω, q) .

This can be rewritten as

S(Ω) =

Ω
2π

+∞

dω coth

ω
ReLR (ω, q)ImL3R (ω, q) − ReL3R (ω, q)ImLR (ω, q) ,
2T

dω coth

ω
[ReLR (ω, q)]3 ImLR (ω, q) + ReLR (ω, q)[ImLR (ω, q)]3 ,
2T

−∞

(A.57)

or equivalently

Ω
S(Ω) =
π

+∞
−∞

(A.58)
which eventually translates into Eq. (2.57).

64

Appendix B:
Seebeck (αxx) and Hall (σxy )
coefficients near Tc
The Aslamazov-Larkin contribution to the diagonal (Seebeck) thermoelectric coefficient is
found from the mixed electric-heat currents Kubo response function

AL
αxx
=−

1
1
eh
lim Im[Kxx
(Ω)]R ,
T Ω→0 Ω

eh
Kxx
(Ων ) = 2ieT
qω

ωn Bx2 (q)L(ωn , q)L(ωn + Ων , q).
(B.59)

Summation over the Matsubara frequency ωn and analytical continuation follows the same
way as in the case of the conductivity calculation, and we obtain

AL
αxx
=

e
2πT 2

+∞
q

Bx2 (q)

−∞

ωdω
[ImLR (ω, q)]2 .
sinh2 (ω/2T )

(B.60)

Without particle-hole asymmetry, αxx is zero. Indeed, [ImLR ]2 is even in frequency while the
rest of the integrand is odd. We have to use the generalized form of the pair propagator that
explicitly accounts for the particle-hole asymmetry factor Υω , which is essentially dictated

65

by gauge invariance [123]

L(ωm , q) = −

1
1
,
2
νd πDq /8T + + π|ωm |/8T + Υω

Υω =

iωm ∂Tc
.
2Tc ∂εF

(B.61)

Expanding LR to the leading linear in Υω order, ImLR = ImLR |Υ=0 + Υω ∂ ImLR |Υ=0 produces
AL
=
αxx

e
πT 2

+∞
q

Bx2 (q)

−∞

ωΥω dω
ImLR (q, ω)∂ ImLR (ω, q),
2
sinh (ω/2T )

(B.62)

where now both propagators are taken at Υω = 0. After introducing dimensionless variables,
at the intermediate step, one has

AL
αxx

8T e d ln Tc
= 3
π εF d ln εF

+∞

xmax

dx

dy
−∞

0

xy 2 (x + )
.
[( + x)2 + y 2 ]3

(B.63)

Logarithmically divergent momentum (x-integration) has to be regularized so that we introduced upper cut-off xmax

1/ (in the original notations this corresponds to (ξT qmax )2

1.

This choice is natural since LR , in the form we use, was obtained from the expansion of
the digamma function which works only as long as max{Dq 2 , ω} < T ). After the final
integrations one finds
AL
αxx
=

2T e d ln Tc
ln
π 2 EF d ln EF

Tc
T − Tc

.

(B.64)

In order to calculate the Hall coefficient, we need to know the transversal component
ee
of the current-current correlation function Kxy
∼

Bx By LL. In the presence of Landau

quantization the vertex in real space becomes an operator

Bˆi = −2νd η(−i∇i + 2eAi ),

66

(B.65)

where we choose the vector potential in the Landau gauge A = (0, Hx, 0). Different components of the vertex, Bˆx and Bˆy , do not commute and the matrix elements are

Bˆαnn


√

a−a
ˆ† |n
2 2νd η  i n|ˆ
=−

H
 n|ˆ
a+a
ˆ† |n

α=x
,

(B.66)

α=y

where a
ˆ, a
ˆ† are oscillator operators. Recalling that n|ˆ
a|n = n |ˆ
a† |n =

√

nδn,n +1 , we see

that only transitions between nearest Landau levels n → n ± 1 are allowed. With this at
hand we find for the Matsubara response kernel [123]

ee
Kxy
(Ω) =

(4eνd η)2
T
8π 4H

∞

(n−1)[Ln+1 (ω, q)Ln (ω−Ω, q)−Ln (ω, q)Ln+1 (ω−Ω, q)] . (B.67)
ω

n=0

After analytic continuation one gets

AL
σxy

(4eνd η)2
=−
4π 2 4H

∞

+∞

(n + 1)

n=0
R
ImLn (ω, q)∂ω ReLR
n+1 (ω, q)

ω
2T

dω coth
−∞

R
− ImLR
n+1 (ω, q)∂ω ReLn (ω, q) .

(B.68)

In the weak field limit, one needs only the first term in the expansion in powers of 1/n
and then substitute integral for the n summation (1/
∂n Ln = 2ν(η/

2
2
H )Ln

H)

2
n

→

q.

Taking into account

and, after some algebra, we find

AL
σxy
=−

(4eη)2 νd3 η
3πT 2H

+∞

q2
q

−∞

dω
[ImLR (ω, q)]3 ,
sinh (ω/2T )
2

(B.69)

where we also used integration by parts with respect to the energy variable. Since [ImLR (ω, q)]3
AL
is odd in energy without particle-hole asymmetry, σxy
vanishes in this case. Expanding to

67

the lowest non-vanishing order we get

AL
=
σxy

(4eη)2 νd4 η ∂ ln Tc
2πT 2H ∂εF

+∞

q2
q

−∞

ωdω
[ImLR (ω, q)]2 Im[LR (ω, q)]2 .
sinh2 (ω/2T )

Introducing, as usual, dimensionless variables and using integrals
and

+∞
−∞

+∞
0

(B.70)

xdx/(x + 1)4 = 1/6

y 2 dy/(y 2 + 1)4 = π/16 we finally get

AL
σxy
=

∂ ln Tc
e2
(ωc τ )
48
∂ ln εF

where ωc = eH/m is cyclotron frequency.

68

Tc
T − Tc

2

,

(B.71)

Appendix C:
Quantum Aslamazov-Larkin terms
Within this section we analyze more carefully the unconventional AL terms in the quantum
limit T → 0. We start from Eq. (3.14) and obtain, in the dimensionless variables of the main
text

AL
σq1
=−

e2
2π 2

∞

+∞

dy coth(πy)
−∞

dx (−iy)(−iy + 2x)x
0

∂
∂y

−2
2 + (x − iy)2

2

+ c.c. .
(B.72)

Here c.c. stands for the complex conjugate term. In this expression we can replace the
integrand with

+∞
−∞

dy coth(πy)[. . .] → 2

∞
0

dy coth(πy)Re[. . .]. At

1, coth πy ≈ 1, and

using the integral
∞

4

dy(−iy)(−iy + 2x)∂y
0

4
16
=
−
[ 2 + (x − iy)2 ]2
x2 +

2

,

(B.73)

we obtain
AL
=
σq1

8e2 Ez
ln
π2
Ec

8e2
ln
π2

69

Ez
.
2(Ez − ∆0 )

(B.74)

In the opposite limit,

1, the expansion of coth(πy) ≈ 1/πy gives the vanishing real part

of the integrand. We therefore conclude that this contribution is not singular in the high
temperature limit when T

Ec .

The second type of unconventional AL corrections comes from differentiating the triangular vertices B instead of the propagators. In the dimensionless notations, Eq. (3.15) can
be reduced to the form

AL
=
σq2

ie2
π2

∞

+∞

dy coth πy
−∞

0

dxx (−iy + x)[lR (x, y)]2 − (iy + x)[lA (x, y)]2 ,

(B.75)

where
lR(A) (x, y) = −
In the low-temperature, quantum limit T
∞

dy
0

2

2
.
+ (x ∓ iy)2

(B.76)

Ec , one uses the integral

4(−iy + x)
=
[ + (x − iy)2 ]2
2

2

2i
+ x2

(B.77)

to obtain
AL
σq2
=−

8e2 Ez
ln .
π
Ec

(B.78)

AL
In the opposite limit, at higher temperatures away from the critical point, σq2
= −2e2 /π .
AL
AL
Comparing σq1
and σq2
in the quantum limit T → 0, we observe acomplete cancella-

tion effect, namely AL terms have no divergent, singular correction in Ez /Ec near QCP.
As discussed in the main text, magnetoresistance δσq (H) is determined by DOS and MT
contributions.

70

Appendix D:
Anomalous Maki-Thompson terms
In the main text we introduced the pair-breaking (dephasing) parameter Γ in order to regularize the anomalous MT term. As discussed in details in Sec. (3.4), Γ naturally appears as
a result of spin-flip scattering. We thus concentrate on the Γ-dependence of the anomalous
MT term from Eq. (3.22) in various temperature regimes. In the dimensionless notations,
Eq. (3.22) reduces to

MT
σan
=

e2
π

+∞
−∞

dy
sinh2 (πy)

∞

dx
0

In the high temperature limit

MT
σan
=

1 (or equivalently in original notations T

∞

+∞

e2
π3

dy
−∞

dx
0

1
(xy)2
.
(Γ/2πT + x) [y 2 − ( − ix)2 ][y 2 − ( + ix)2 ]

(B.79)

Ec )

1
x2
,
(Γ/2πT + x) [y 2 − ( − ix)2 ][y 2 − ( + ix)2 ]

(B.80)

which eventually reduces to

MT
σan
=

e2
2π 2

∞

dx
0

1
x
2
(Γ/2πT + x) x +

2

=

71

e2 π /2 − (Γ/2πT ) ln(2πT /Γ)
.
2π 2
(Γ/2πT )2 + 2

(B.81)

From this expression we can extract various limiting cases. For Γ/2 < 2πEc < T , which is
MT
possible if Γ/2 < T , one finds σan
= e2 /4π . Alternatively, for 2πEc < Γ/2 < T , which

is possible if Γ/2 < T , or for 2πEc < T < Γ/2, which is possible if Γ/2 > T , one finds
MT
σan
= (e2 /π)(T /Γ) ln(Γ/Ec ).

We proceed with the low temperature limit

1 (or equivalently T

Ec ). In this case

we have
MT
σan

+∞

e2
=
π

−∞

∞

dyy 2
sinh2 (πy)

dx
0

1
(Γ/2πT + x) [

x2
.
2 + x2 ]2

(B.82)

At T < Ec < Γ,
MT
σan
=

e2 2πT
3π 2 Γ

∞
0

x2 dx
e2 2πT
=
.
(x2 + 2 )2
12π Γ

(B.83)

Alternatively, at T < Γ < Ec or Γ < T < Ec ,

MT
σan
=

e2
3π 2

∞
0

xdx
e2
=
,
(x2 + 2 )2
6π 2 2

(B.84)

and consequently, the anomalous MT term has no singular contributions near QCP. The
logarithmically divergent correction declared in Eq. (3.32) of the main text originates from
the regular part of the MT term. Indeed, from Eq. (3.21) we obtain

MT
σreg

4e2
= 2
π

At high temperatures T

MT
σreg
=

4e2
π3

∞

∞

dx
0

dy coth(πy)
0

xy
[(y +

)2

+

x2 ][(y

− )2 + x2 ]

.

(B.85)

Ec ,
∞

∞

dx
0

dy
0

x
e2
=
,
[(y + )2 + x2 ][(y − )2 + x2 ]
2π

72

(B.86)

while at low temperatures T

MT
=
σreg

4e2
π2

∞

∞

dx
0

Ec , near QCP ,

dy
0

2e2
xy
≈
ln(Ez /Ec ).
[(y + )2 + x2 ][(y − )2 + x2 ]
π2

73

(B.87)

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