FKSbzduso/physics/master/thesis.pdf · v Abstrakt Autor Bc. Tomáš Bzdušek Názov práce...

Comenius University in Bratislava

Faculty of Mathematics, Physics and Informatics

Department of Theoretical Physics and Didactics of Physics

Anomalous Spectral

Function of a Superconductor

(Master’s thesis)

Bc. Tomáš Bzdušek

Study programme: Theoretical PhysicsField of study: 4.1.1. Physics (1160)

Supervisor: Doc. RNDr. Richard Hlubina, DrSc. Bratislava, 2013

iv

Acknowledgement

I would like to thank my supervisor Richard Hlubina for the four yearsof guidance, encouragement and inspiration, and my family and friendsfor their invaluable support.

v

Abstrakt

Autor Bc. Tomáš BzdušekNázov práce Anomálna spektrálna funkcia supravodicovŠkola Univerzita Komenského v BratislaveFakulta Fakulta matematiky, fyziky a informatikyKatedra Katedra teoretickej fyziky a didaktiky fyzikyVedúci práce Doc. RNDr. Richard Hlubina, DrSc.Miesto BratislavaDátum 13. 5. 2013Pocet strán 53Druh záverecnej práce Diplomová práca

Abstrakt: Spektrum elektrónov v nesupravodivom kove je parametrizo-vané jedinou funkciou – renormalizáciou elektrónovej hmotnosti. Tentovzt’ah sa zmení pri prechode do supravodivého stavu. Vtedy sú napopis spektra elektrónov potrebné dve funkcie, pribudla tzv. funkciaenergetickej medzery. Co sa však v literatúre dostatocne nezdôraznujeje, že supravodivost’ so sebou prináša aj novú spektrálnu funkciu. Nazý-vame ju anomálnou spektrálnou funkciou. Úzko súvisí s parametromusporiadania supravodica a predpokladáme, že obsahuje informáciua párovacom mechnizme elektrónov. V tejto práci skúmame, akoanomálna spektrálna funkcia typicky vyzerá a navrhujeme možnost’ jejexperimentálneho merania.

Kl’úcové slová: supravodivost’, Gorkovov propagátor, spektrálnareprezentácia, sumacné pravidlá

vi

Abstract

Author Bc. Tomáš BzdušekTitle Anomalous Spectral Function of SuperconductorsUniversity Comenius University in BratislavaFaculty Faculty of Mathematics, Physics and InformaticsDepartment Department of Theoretical Physics and Didacits of PhysicsSupervisor Doc. RNDr. Richard Hlubina, DrSc.City BratislavaDate 13. 5. 2013Number of pages 53Type of thesis Diploma thesis

Abstract: Spectrum of electrons in a non-superconducting metal isparametrized with a single function, the renormalization of electronmass. This changes when the metal undergoes a phase transition toa superconducting state. In such a case, two functions are needed todescribe the spectrum, the energy gap function appears. However, itis not stressed much in the literature that superconductivity brings onthe scene also another spectral function. We call it the anomalous spectralfunction. It is closely related to the order parameter of the superconduc-tor and we expect it to encode information about the pairing mechanismof electrons. In this work we study how the anomalous spectral functiontypically looks like, and we propose a basic method for its experimentaldetermination.

Keywords: superconductivity, Gorkov propagator, spectral representa-tion, sum rules

Contents

Preface and motivation 1

I Introduction 3I.1 Spectral representation of Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3I.2 The Nambu-Gorkov formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6I.3 Electron-phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8I.4 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

II Properties of the anomalous spectral function 13II.1 Symmetries of the anomalous spectral function . . . . . . . . . . . . . . . . . . . . . . . . 13II.2 Sum rules and their violation in the BCS model . . . . . . . . . . . . . . . . . . . . . . . . 16II.3 Solution of the Eliashberg equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

III Experimental determination of Bk(x) 24III.1 Correlation functions ρ − ρ and s − s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24III.2 Comparison to the BCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28III.3 Solving the inversion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Conclusion 35

Appendices 36A Uniqueness of analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36B Maximum entropy method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39C Padé Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43D Calculation of sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44E Solving the Eliashberg equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

vii

Preface and motivation

In the normal state of a metal, the effect of interactions on the spectrum of electrons is usually describedby a single function – the renormalization of electron mass. The quantum field description of thenormal state leads to a single Green’s function and thereby to a single spectral function as well. Wethus have, in a sense, equal number of information in both descriptions of the problem. One canmeasure the spectral function to deduce the renormalization function and vice versa.

This changes when the metal undergoes a phase transition to a superconducting state. Thereare now two distinct functions necessary to describe the spectrum of electrons: The renormalizationfunction and the gap function. The quantum field approach leads to two coupled Green’s functionsand thus to two different spectral functions. So there are two functions on both sides. This soundsperfectly congruous but there is a fundamental complication: The second spectral function cannot beexperimentally measured. It is not possible because it is an average of a non-Hermitian operator. It isgenerally complex-valued and it changes phase under a global U(1) symmetry. We call this functionthe anomalous spectral function of the superconductor and it is the central object of this work.

What researchers in the field usually do at this point is that they ignore the anomalous spectralfunction completely on one hand, and ignore changes in mass renormalization function when goingto the superconducting state on the other one. They build a connection only between the gap functionand changes in the ordinary spectral function. This method is successful very often but not always. Forexample, for two decades it has been failing to reveal the mystery of high-Tc superconductivity [1, 2].This is not surprising because the built link is only an approximation and breaks down if the couplingof electrons becomes too strong.

In the beginning of our study there was the following set of incipient questions:

1. What does the anomalous spectral function typically look like?

2. How does it encode information about the electron interactions?

3. How does it influence the physical observables?

4. Is there an experiment that might serve us to deduce this function?

Except for the second question, we got a surprising observation for all of them. As for the firstone, that is what does the anomalous spectral function look like, we found that it has to be fundamentallydifferent from the prediction of the BCS model. To be specific, the physics laws force it to always containa non-coherent part of opposite sign. When studying its influence on the physical observables we noticed thatthe famous coherence factors can be written via the spectral functions in a bit more concise way.Moreover, this formulation survives a generalization beyond the BCS model. Finally, when trying tofigure out how it could be deduced from experimental data we found that under some assumptions theconvolutions of the anomalous spectral function are in principle observable. Furthermore, we alsofound an exact procedure of extracting this function if certain specific criteria are met.

The work is organized in the following way: Chapter I gives a resume of the theory used in thiswork. We expect the reader to be familiar with the methods of quantum field theory, especially theGreen’s functions and the Matsubara (or temperature) formalism. We discuss the spectral represen-tation of Green’s functions, and we derive basic properties of the corresponding spectral functions.We then utilize the Nambu-Gorkov formalism which is suitable to describe a superconducting state.We continue with introducing the electron-phonon Hamiltonian which is known to exhibit super-conductivity at sufficiently low temperature, and the Hubbard model which is suspected to exhibit

1

CONTENTS 2

Figure 1: The spectral functions Ak(x) and Bk(x) (the "ordinary" and the "anomalous") as Lehmann spectralrepresentations of Green's functions on one hand, gap function ∆k(ω) and renormalization of electron massZk(ω) on the other one. How to relate them correctly?

unconventional superconductivity. All these topics are widely discussed in the literature, the mainpurpose of this chapter is to have a reference of used notations and definitions.

Chapter II addresses the question "How does the anomalous spectral function typically looks like".We discuss its symmetries in both momentum and frequency argument. The anomalous spectralfunction is generally complex-valued, but we show that it can often be gauged to be real. An importantconclusion is achieved by applying the sum rules. Based on our calculation we propose that for (i) ad-wave superconducting state of the Hubbard model and for (ii) the electron-phonon Hamiltonianwithout Coulomb repulsion, the anomalous spectral function consists of a peak at the quasiparticleexcitation and of a negative non-coherent part necessary to fulfill the sum rules. We further assume,that turning on a weak Coulomb interaction preserves this feature in case (ii). In the end of the chapterwe prove these conjectures in the case of the electron-phonon Hamiltonian by numerically solving theEliashberg equations of a superconductor. The non-coherent part is, in fact, found and the sum rulesare examined.

In Chapter III we propose a method to determine the anomalous spectral function experimentally.We do so by studying the spin-spin χss(q, ω) and the density-density χρρ(q, ω) correlation functions.Keeping certain simplification, we calculate these susceptibilities for both the BCS model and theEliashberg solution. We show that the non-coherent feature of the anomalous spectral function leavesa fingerprint in the difference of these susceptibilities for frequency arguments of the order of theDebye frequency. We conclude this chapter by showing that if we know the imaginary part of bothof these susceptibilities for all arguments, it is possible to uniquely determine the anomalous spectralfunction.

To keep the work easy to read we decided to move many technical details to appendices. Theyinclude discussion of analytic continuation methods, derivation of the sum rules of the anomalousspectral function, and description of our numerical approach to solving the Eliashberg equations.

Chapter I

Introduction

The goal of this chapter is to have a reference for definitions of the used mathematical objects, but alsoto refresh the quantum field methods in condensed matter physics. Listed equations are profuselyreferenced in the later chapters. If the reader feels confident about the theoretical background, it ispossible to skip directly to the chapter II.

In section I.1 we recapitulate the spectral representation of the Green’s functions which is necessaryto introduce the notion of the anomalous spectral function. We further discuss the Lehmann represen-tation and sum rules of a general spectral function. We shall find these concepts beneficial in chapterII while studying properties of the anomalous spectral function.

In section I.2 we introduce the Nambu-Gorkov operators and formalism. It is a handy tool to keepboth the electron propagator and certain superconductive averages (that we shall call "the anomalouspropagator") concisely together. In this formalism the Green’s function is a 2× 2 matrix. We show thatits diagonal terms are related to the ordinary propagator and that the off-diagonal terms correspondto the anomalous propagator.

We then continue with a discussion of the electron-phonon Hamiltonian in section I.3. Here wederive the famous Eliashberg equations for a superconductor. We then simplify them for the case ofan isotropic s-wave superconductor with a perfect particle-hole symmetry on the Fermi surface. Theseequations are solved numerically in later chapters. We finally introduce the Hubbard Hamiltonian insection I.4. We shall study it only in section II.2 where we point out its peculiar sum rules. We alsobriefly mention the notion of unconventional superconductivity in this ultimate section.

I.1 Spectral representation of Green’s functions

What is the anomalous spectral function of a superconductor? And what do we mean by a spectral functionat all? As we show below, the definition is rather straightforward. There are however certain usefulrelations that spectral functions fulfill and that the reader may not be acquainted with. It is the aim ofthis section to refresh the definitions of Green’s functions and of their spectral representations withinthe framework of condensed matter physics. We discuss the sum rules which tell something about themoments of spectral functions, and the Lehmann representation which gives explicit expression forspectral functions via the eigenstate decomposition.

We assume the reader to be familiar with the notion of Green’s functions and of the Matsubara (ortemperature) formalism. The temperature Green’s function of operators A, B is defined as

G(τ − τ′) ≡ G(τ, τ′) = −⟨TA(−iτ)B(−iτ′)

⟩= −

⟨A(−iτ)B(−iτ′)

⟩θ(τ − τ′) + ϵ

⟨B(−iτ′)A(−iτ)

⟩θ(τ′ − τ).

In these expressions we used the following notations:

I Symbol T denotes the usual time-ordering operator. It changes sign of an expression uponswitching two neighboring fermionic operators.

I Argument (−iτ) is considered within the Heisenberg picture, i.e. A(−iτ) = eτH Ae−τH.

3

CHAPTER I. INTRODUCTION 4

I Letter ϵ equals +1 if A, B are both even functions of fermionic operators, and equals −1 other-wise.

I Angle brackets ⟨. . .⟩ = 1Z Tr [. . .], i.e. the thermal average over grand canonical ensemble.

I θ(x) is the Heaviside step function. It equals 1 for positive arguments, 0 for negative arguments,and 1

2 for x = 0.

I We put here as well as in the rest of the work h = 1.

Temperature Green’s function are studied for arguments in interval (−β, β], β = 1/kBT. Using thecyclic property of trace one can easily check that G(τ − β) = −ϵG(τ) for τ ∈ (0, β], i.e. the function is(anti)periodic.

We can perform a (discrete) temporal Fourier transformation

G(τ) =1β ∑

ℓ

G(ωℓ)e−iωℓτ, G(ωℓ) =∫ β

0dτeiωℓτG(τ) (I.1)

where ωℓ = (2ℓ+ 1)π/β for ϵ = +1 (fermionic case), and ωℓ = 2ℓπ/β for ϵ = −1 (bosonic case). Wecall the set of ωℓ as Matsubara frequencies.

A non-trivial fact is that for every Green’s function there is a unique spectral function S(x) such that

G(ωℓ) =∫ +∞

−∞

dxS(x)iωℓ − x

(I.2)

for all Matsubara frequencies. The proof of this claim is included in appendix A. One can understandequation (I.2) as a special case of

G(ω) =∫ +∞

−∞

dxS(x)ω − x

, ω ∈ C/R (I.3)

for ω = iωℓ. We often call the set of points iωℓ ∈ C as Matsubara points.According to the Sokhotski-Plemelj theorem [3], Green’s function defined in (I.3) is analytic in the

upper and lower half plane and has a cut

limε→0

(G(x + iϵ)− G(x − iϵ)) = −2πiS(x) (I.4)

on the real axis, x ∈ R. This is indeed a useful way of deducing the spectral function, especially if thespectral function is real we have

S(x) = − 1πℑ [G(x + iϵ)] .

Also, since the spectral function is uniquely determined by the infinite set of G(ωℓ) = G(iωℓ), theanalytic continuation to G(ω) of equation (I.3) is unique, too.

A spectral function can be written in so-called Lehmann representation as

S(x) =(

1 + ϵe−βx) 1

Z ∑m,n

e−βEm⟨m|A|n⟩⟨n|B|m⟩δ(x − En + Em), (I.5)

so we can write down the spectral function if we know the spectrum of the Hamiltonian. This ishowever rarely the case. Notice that the special case B = A† yields a real-valued spectral function.

A spectral function of operators A, B fulfills the set of sum rules∫ ∞

−∞dxxnS(x) =

⟨[[A,H

]n , B

]ϵ

⟩, (I.6)∫ ∞

−∞

dxxnS(x)1 + ϵe−βx =

⟨[A,H

]n B⟩

. (I.7)

for all non-negative integers n. The object[A,H

]n is an iterated commutator[

A,H]

n =[[

A,H]

n−1 ,H]

.


Relations (I.6) and (I.7) can be easily proved when one starts with the Lehmann representation (I.5).For the first one we find∫ ∞

−∞dxxnS(x) =

1Z ∑

m,n

∫ +∞

−∞

(1 + ϵeβ−x

)e−βEm xn⟨m|A|n⟩⟨n|B|m⟩δ (x − En + Em)

=1Z ∑

m,n

(e−βEm + ϵe−βEn

)(En − Em)

n ⟨m|A|n⟩⟨n|B|m⟩. (I.8)

Let us for a moment consider only the first exponential term in the first bracket. Its contribution maybe simplified as

1Z ∑

m,ne−βEm (En − Em)

n ⟨m|A|n⟩⟨n|B|m⟩ =1Z ∑


n−1 (En − Em) ⟨m|A|n⟩⟨n|B|m⟩

=1Z ∑


n−1 ⟨m|AH−HA|n⟩⟨n|B|m⟩

=1Z ∑


n−1 ⟨m|[A,H

]|n⟩⟨n|B|m⟩

=1Z ∑


n−2 ⟨m|[[

A,H]

,H]|n⟩⟨n|B|m⟩

=1Z ∑

m,ne−βEm⟨m|

[A,H

]n |n⟩⟨n|B|m⟩

=1Z ∑

me−βEm⟨m|

[A,H

]n B|m⟩ =

⟨[A,H

]n B⟩

. (I.9)

For the other term we find in complete analogy

ϵ1Z ∑

m,ne−βEn (En − Em)

n ⟨m|A|n⟩⟨n|B|m⟩ m↔n= ϵ

1Z ∑

m,ne−βEm (Em − En)

n ⟨m|B|n⟩⟨n|A|m⟩ (I.10)

= ϵ⟨

B[A,H

]n

⟩. (I.11)

Picking equations (I.8),(I.9) and (I.11) we finally find∫ ∞

−∞dxxnS(x) =

⟨[A,H

]n B⟩+⟨

B[A,H

]n

⟩=⟨[[

A,H]

n , B]

ϵ

⟩,

i.e. equation (I.6) is true.In similar fashion we write left-hand side of sum rule (I.7)∫ ∞

−∞

dxxnS(x)1 + ϵe−βx =

1Z ∑

m,n

∫ +∞

−∞e−βEm xn⟨m|A|n⟩⟨n|B|m⟩δ (x − En + Em)

=1Z ∑


n ⟨m|A|n⟩⟨n|B|m⟩ (I.9)=⟨[

A,H]

n B⟩

.

Only the set of sum rules (I.6) is exploited in the work.In this work we extensively discuss two specific Green’s functions

Gk↑(τ) = −⟨

Tck↑(−iτ)c†k↑

⟩. (I.12)

Fk↑(τ) = −⟨

Tck↑(−iτ)c−k↓

⟩. (I.13)

of electrons. We will often suppress the spin index. As the creation and annihilation operators ofelectrons are fermionic, we put ϵ = +1 in all equations. We call these Green’s functions as the normaland the anomalous propagator, respectively.


The normal propagator (I.12) describes for positive arguments propagation of an electron throughthe medium, and of the corresponding hole for negative arguments. Its spectral function

Ak↑(x) = (1 + e−βx)1Z ∑

m,ne−βEm

∣∣∣⟨m|c†k↑|n⟩

∣∣∣2 δ(x − En + Em) (I.14)

is real and it has the following probabilistic interpretation: If we create a particle with momentum kin the system, what is the probability density that we measure its energy to be x?

The anomalous propagator (I.13) is closely related to the energy gap of a superconductor. The factthat it relates the field operators annihilating a Cooper pair already means that it is non-zero only in asuperconducting state. Its spectral function is

Bk↑(x) =(

1 + e−βx) 1

Z ∑m,n

e−βEm⟨m|ck↑|n⟩⟨n|c−k↓|m⟩δ(x − En + Em). (I.15)

This function is the anomalous spectral function. It is generally complex-valued and in contrast with (I.14)it changes phase under the global U(1) transformation. We miss an easy interpretation of both theanomalous propagator and the anomalous spectral function that we have for the normal counterparts.The anomalous spectral function is non-zero too in a superconducting state only.

I.2 The Nambu-Gorkov formalism

A superconductor is a broken U(1) symmetry state. The non-zero value of the anomalous propagatormanifests itself in the equation of motion of the normal propagator – this equation is no longer inde-pendent. Equations for both propagators get coupled (see e.g. §8.3 of [4]). To handle these equationsin a handy way, Nambu [5] proposed the following bicomponent operators

αk =

(ck↑

c†−k↓

), α†

k =(

c†k↑ c−k↓

),

αk, α†

q

= δk,q1 (I.16)

where 1 is a unit 2 × 2 matrix. They are nowadays commonly called as Nambu-Gorkov operators. Withthese we define in complete analogy with (I.12) a 2 × 2 matrix of Green’s functions

Gk(τ) = −⟨

Tαk(−iτ)α†k

⟩=

−⟨

Tck↑(−iτ)c†k↑

⟩−⟨


⟩−⟨

Tc†−k↓(−iτ)c†

k↑

⟩−⟨

Tc†−k↓(−iτ)c−k↓

⟩ . (I.17)

The upper right element is exactly the anomalous propagator defined in (I.13). The lower left elementturns out to be its complex conjugate(

−⟨

Tc†−k↓(−iτ)c†

k↑

⟩)∗= −

(⟨c†−k↓(−iτ)c†

k↑

⟩)∗θ(τ) +

(⟨c†

k↑c†−k↓(−iτ)

⟩)∗θ(−τ)

= −⟨

ck↑c−k↓(+iτ)⟩

θ(τ) +⟨

c−k↓(+iτ)ck↑

⟩θ(τ)

= −⟨

ck↑(−iτ)c−k↓

⟩θ(τ) +

⟨c−k↓ck↑(−iτ)

⟩θ(τ)

= −⟨


⟩. (I.18)

In the second line we used that under Hermitian conjugation of an operator in imaginary time, wehave to flip the sign of the time argument, too. This is very easy to show as

A(−iτ) = eH0τ Ae−H0τ =⇒ [A(−iτ)]† = e−H0τ A†eH0τ = A†(+iτ).

In the third line of (I.18) we applied the symmetry with respect to time translation.


The diagonal elements of (I.17) are also related – the upper left element is the ordinary electronpropagator (I.12). The lower right element can be expressed as −G−k↓(−τ), as can be shown easily:

− G−k↓(−τ) =⟨

Tc−k↓(iτ)c†−k↓

⟩=

⟨c−k↓(iτ)c

†−k↓

⟩θ(−τ)−

⟨c†−k↓c−k↓(iτ)

⟩θ(τ)

=⟨

c−k↓c†−k↓(−iτ)

⟩θ(−τ)−

⟨c†−k↓(−iτ)c−k↓

⟩θ(τ)

= −⟨

Tc†−k↓(−iτ)c−k↓

⟩. (I.19)

Putting results (I.18) and (I.19) into (I.17) we have

Gk(τ) =

(Gk↑(τ) Fk↑(τ)F∗

k↑(τ) −G−k↓(−τ)

).

In analogy with (I.1) we define the temporal Fourier transformation of (I.17) as

Gk(τ) =1β ∑

ℓ

Gk(ωℓ)e−iωℓτ, Gk(ωn) =∫ β

0dτeiωnτGk(τ)

The upper row elements are defined as

Gk↑(ωn) =∫ β

0dτeiωℓτGk↑(τ) =

∫ +∞

−∞

dxAk↑(x)iωn − x

,

Fk↑(ωn) =∫ β

0dτeiωℓτ Fk↑(τ) =

∫ +∞

−∞

dxBk↑(x)iωn − x

, (I.20)

which is consistent with both (I.14) and (I.15).We are now interested in relations of the upper row elements to the elements in the lower row. For

the off-diagonal terms we find∫ β

0dτeiωnτ F∗

k↑(τ) =

[∫ β

0dτe−iωnτ Fk↑(τ)

]∗= F∗

k↑(−ωn).

Notice that if spectral function Bk↑(x) were real, then we see from (I.20) that F∗k↑(−ωn) = Fk↑(ωn).

We show in section II.2 that for s-wave superconductors we can indeed gauge the anomalous spectralfunction to be real, so in this case the off-diagonal terms are the same.

Now for the lower right element. Remembering that for the fermionic case eiωnβ = −1 and G(−τ) =−G(β − τ) and substituting β − τ 7→ τ at the second equation mark we find

−∫ β

0dτeiωnτG−k↓(−τ) =

∫ β

0dτeiωnτG−k↓(β − τ)

=∫ β

0dτeiωn βe−iωnτG−k↓(τ)

= −∫ β

0dτe−iωnτG−k↓(τ)

= −G−k↓(−ωn).

If the system is symmetric with respect to the space inversion and the spin inversion, which is thecase of singlet-pairing superconductor, then we see from Lehmann representation (I.14) that A(x,−k, ↓) = A(x, k, ↑). This can be formally proved by replacing states m, n by their space- and spin-inversedcounterparts in equation. As a consequence we find −G−k↓(−ωn) = −Gk↑(−ωn).

To summarize the analysis of the temporal Fourier transformation, in the case of singlet supercon-ductors with Bk↑(x) gauged to be real we have

Gk(ωn) =

(Gk↑(ωn) Fk↑(ωn)Fk↑(ωn) −Gk↑(−ωn)

). (I.21)


We may also introduce a matrix spectral function

Gk↑(ωℓ) =∫ +∞

−∞

dxAk(x)iωℓ − x

, Ak(x) =(

Ak↑(x) Bk↑(x)Bk↑(x) Ak↑(−x)

)(I.22)

where we kept the assumption that the anomalous spectral function is real. This matrix spectralfunction will be of use in section (III.1).

The Nambu field operators α turn out to be useful for concise manipulations with equations, e.g.to derive the Feynman rules (see chapter 10 of [6]). We will utilize the Nambu-Gorkov formalismto derive the Eliashberg equations in section I.3 and to derive the density-density and the spin-spincorrelation functions in section III.1.

I.3 Electron-phonon Hamiltonian

Phonons are for 55 years known to mediate the conventional superconductivity [7]. In this work wewill mostly study the electron-phonon Hamiltonian

H = ∑k,σ

εkc†kσckσ + ∑

q,sωqs

(a†

qsaqs +12

)+

1√Ω

∑k,σ

∑q,s

gsk,k+qc†

k+qσckσ

(aqs + a†

−qs

)+

12Ω ∑

q =0Vq ∑

p,σ∑k,σ′

c†p+qσc†

k−qσ′ckσ′cpσ (I.23)

≡ H1 +H2 +H3 +H4..

The four terms of this Hamiltonian are:

I H1 is the kinetic energy of electrons, εk is energy of a Bloch state with momentum k.

I H2 is the energy of phonons. They are labelled by branch s and momentum q. The operators a(†)qsare the (bosonic) field operators of phonons, ωqs is energy of phonon (q, s).

I H3 is the electron-phonon interaction term. Coefficients gsk,k+q can be estimated as

gsk,k′ = N

√hΩ0

2Mωk−k′,s

∫d3rψ∗

k′(r)(−ek−k′,s ·

∂U∂r

)ψk(r),

Here Ω is the volume of the sample, Ω0 is the volume of the elementary cell of the Bravais lattice,N is the number of cells in the crystal.

I H4 is the electron-electron (or Coulomb) interaction term.

This Hamiltonian exhibits superconductivity at sufficiently low temperatures. Presence of the repulsiveCoulomb interaction pushes the critical temperature down.

As the Fermi energy is usually much larger than the temperature, both states k, k′ lie close toFermi surface, i.e. magnitudes of these vectors are approximately equal to the Fermi momentum. Ifwe furthermore assume isotropic behavior, we may assume the coupling function gs

k,k′ to be a functionof the angle between the vector k, k′ only, i.e. gs(θ). In such case it turns out to be useful to definefunction

α2F(ν) = N(0)∫ dΩ′

4π ∑s|gs(θ)|2 δ(ν − ωθ,s)

which weights the spectrum of phonons by their coupling constant to electrons. This function influ-ences the solution of the Hamiltonian (I.23). In this work we utilize two simple phonon models

α2FEinstein(ν) =12 λω0δ(ω0 − ν), α2FDebye(ν) = λ

(ν

ω0

)2θ(ω0 − ν)θ(ν). (I.24)

We call ω0 as the Debye frequency. The Debye phonon model is assumed, unless otherwise stated.


Figure I.1: Renormalization of electron propagator due to electron-electron and electron-phonon interaction(expressed together schematically via the curly line). Proper ansatz of electron self-energy leads to the Eliashbergequations for the superconducting state.

The electron-phonon problem can be solved by the Eliashberg equations, i.e. by evaluating the self-energy in the self-consistent Born approximation, see figure (I.1). The exact equations for self-energyΣ(k, ωn) are1

Gk(ωn) = G 0k(ωn) + G 0

k(ωn)Σk(ωn)Gk(ωn), (I.25)

Σk(ωn) =1

βΩ ∑k′,ωm

U(k, k′, ωn, ωm)σ3Gk′(ωm)σ3, (I.26)

U(k, k′, ωn, ωm) = −Vk−k′Γk′,k(ωm, ωn) + ∑s

gsk,k′Γs

k′,k(ωm, ωn)Dsk−k′(ωn − ωm), (I.27)

where Γk′,k(ωm, ωn) is a renormalized vertex function for Coulomb interaction, Γsk′,k(ωm, ωn) is a renor-

malized vertex function for electron-phonon interaction, and

G 0k (ω) = −(iωn1 + εkσ3)/(ω2

n + ε2k) (I.28)

is a free electron Green’s function. We do not bother with renormalizing the phonon propagator sincewe work with observable phonons. In equations (I.25) to (I.28) we used the notation

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (I.29)

for the Pauli matrices, 1 is the unit matrix.We want to solve the system of equations (I.25) to (I.28). To do so we parametrize the self energy as

Σk(ωn) = (1 − Zkωn)iωn1 + Ckωn σ1 + Dkωn σ2 + χkωnσ3 (I.30)

where all introduced functions Zkωn , Ckωn , Dkωn , χkωnare real-valued, i.e. the self-energy is hermitian.

Combining (I.25) and (I.30) we find

Gk(ωn)−1 = Zkωn iωn1 − Ckωn σ1 − Dkωn σ2 − εkωn σ3 (I.31)

where we introduced εkωn = χkωn+ εk. It is straightforward to check that the inverse is

Gk(ωn) = −Zkωn iω1 + Ckωn σ1 + Dkωn σ2 + εkωn σ3

Z2kωn

ω2 + C2kωn

+ D2kωn

+ ε2kωn

, (I.32)

σ3Gk(ωn)σ3 = −Zkωn iωn1 − Ckωn σ1 − Dkωn σ2 + εkωn σ3

Z2kωn

ω2n + C2

kωn+ D2

kωn+ ε2

kωn

(I.33)

where in the second row we used that σ3σ1σ3 = −σ1 and equivalently for σ1 7→ σ2. Plugging (I.33) intoequation (I.26) and comparing coefficients standing in front of the same Pauli matrices we get a set of

1These can be written down directly if one knows the Feynman rules which we do not include in this work.


equations

(1 − Zkωn)ωn = − 1βΩ ∑

k′,ωm

U(k, k′, ωn, ωm)

Z2k′ωm

(ωm)2 + C2k′ωm

+ D2k′ωm

+ ε2k′ωm

Zk′ωm ωm,

Ckωn =1

βΩ ∑k′,ωm


Z2k′ωm

(ωm)2 + C2k′ωm

+ D2k′ωm

+ ε2k′ωm

Ck′ωm ,

Dkωn =1

βΩ ∑k′,ωm


Z2k′ωm

(ωm)2 + C2k′ωm

+ D2k′ωm

+ ε2k′ωm

Dk′ωm ,

χkωn= − 1

βΩ ∑k′,ωm


Z2k′ωm

(ωm)2 + C2k′ωm

+ D2k′ωm

+ ε2k′ωm

εk′ωm .

Notice that the equations for Ckωn and Dkωn are identical, so we can combine them into a singleequation for a complex number Ckωn + iDkωn . This is even compatible with the C

2kωn

+ D2kωn

=∣∣Ckωn + iDkωn

∣∣2 in the denominator. If we further reparametrize Ckωn + iDkωn = Zkωn ∆kωn we geta simplified set of equations

(1 − Zkωn)ωn = − 1βΩ ∑

k′,ωm

U(k, k′, ωn, ωm)((ωm)2 +

∣∣∆k′ωm

∣∣2) Z2k′ωm

+ ε2k′ωm

Zk′ωm ωm, (I.34)

Zkωn ∆kωn =1

βΩ ∑k′,ωm

U(k, k′, ωn, ωm)((ωm)2 +

∣∣∆k′ωm

∣∣2) Z2k′ωm

+ ε2k′ωm

Zk′ωm ∆k′ωm , (I.35)

χkωn= − 1

βΩ ∑k′,ωm

U(k, k′, ωn, ωm)((ωm)2 +

∣∣∆k′ωm

∣∣2) Z2k′ωm

+ ε2k′ωm

εk′ωm , (I.36)

where ∆kωn is now complex-valued and Zkωn , εkωn are still real-valued. The last three are also calledEliashberg equations.

Until now, the approach to solve the electron-phonon Hamiltonian was exact. Now we will applycertain simplifications, namely:

I The Migdal’s theorem holds, so we can ignore the renormalization of the electron-phonon vertex,that is we put Γs

k′,k(ωm, ωn) := gsk′,k. The reason is that higher order vertex corrections are

negligible. The physical interpretation is that phonons are about hundred times slower thanelectrons.

I Renormalization of the Coulomb interaction is difficult to do. We will treat it only in the finalequations via a single coefficient of renormalized interaction strength µ∗, see equation (I.40).

I We assume a perfect particle-hole symmetry of the Fermi surface, so χkωn= 0 and as a conse-

quence εkωn = εk.

I We will restrict to an isotropic model, i.e. spherical Fermi surface, so we put Zkωn = Zωn and∆kωn = ∆ωn . By imposing isotropy we automatically restrict to s-type superconductors only.

While these assumptions are clearly stated, it takes some time to incorporate them into Eliashbergequations. We will simplify the equation for U(k, k′, ωn, ωm) defined in (I.27). This is crucial be-cause this interaction function enters all Eliashberg equations. First of all we impose Γs

k′,k(ωm, ωn) :=

gsk′,k =

(gs

k,k′

)∗= (gs(θ))∗. For Coulomb interaction we substitute without further analysis a screened

interaction −Vk−k′Γk′,k(ωm, ωn) := −V∗(θ) where the asterisk is to remind us of the screening andrenormalization.

Let us now treat the phonon propagator. Since we consider at the vertex either absorption ofphonon with momentum q or emission of phonon with momentum −q, the same ambiguity holds forthe phonon propagator. We define it as

Dsq(τ) =

⟨T(

aqs(τ) + a†−qs(τ)

) (a†

qs + a−qs

)⟩= −

⟨Taqs(τ)a†

qs

⟩−⟨

Ta−qs(τ)a†−qs

⟩(I.37)


where the second equality holds only if we neglect influence of electron-phonon interaction on phononspectrum. Keeping the same simplification we can find the spectral representation of (I.37) to be

Dsq(ωn) =

∫ +∞

−∞dz

−δ(z − ωqs) + δ(z + ωqs)

iωn − z=

2ωqs

ω2n + ω2

qs.

Inserting these into equation (I.27) we obtain

U(k, k′, ωn, ωm) = −V∗(θ) + ∑s|gs(θ)|2 2ωθs

(ωn − ωm)2 + ω2

θs

≡ U(θ, ωn − ωm), ωθs = ωk−k′s. (I.38)

We now plunge expression (I.38) to Eliashberg equations (I.34) and (I.35). Remember that thanksto the perfect particle-hole symmetry we do not need to consider equation (I.36), we have χkωn

= 0and εkωn = εk. We further change 1

Ω ∑k′ 7→ N(0)∫ dΩ′

4π

∫ +∞−∞ dε′ with N(0) the density of states on the

Fermi surface. Equation (I.34) yields

(1 − Zωn)ωn =

=1β ∑

ωm

N(0)∫ dΩ′

4π

∫ +∞

−∞dε′(−V∗(θ) + ∑

s

|gs(θ)|2 2ωθs

(ωn − ωm)2 + ω2θs

)Zωm ωm(

(ωm)2 +∣∣∆ωm

∣∣2) Z2ωm

+ (ε′)2

= − 1β ∑

ωm

(−µ∗ +

∫ +∞

−∞

2να2F(ν)dν

(ω − ω′)2 + ν2

) ∫ +∞

−∞dε′

Zωm ωm((ωm)2 +

∣∣∆ωm

∣∣2) Z2ωm

+ (ε′)2

= −π

β ∑ωm

g(ωn − ωm)ωm√

(ωm)2 +

∣∣∆ωm

∣∣2 (I.39)

where in the first step we used definition of α2F(ν) in equation (I.24) and further defined

µ∗ = N(0)∫ dΩ′

4πV∗(θ), (I.40)

and in the second step we evaluated integral over ε′ and defined

g(ωn − ωm) =∫ +∞

−∞

2να2F(ν)dν

(ωn − ωm)2 + ν2 − µ∗.

As α2F(ν) is nonzero for positive arguments only, we can shift the lower integration boundary from−∞ to 0. Notice also that function g is even. For the Einstein and the Debye phonon models (I.24) itcan be evaluated explicitly to be

gEinstein(ωn − ωm) = λω2

0

ω20 + (ωn − ωm)

2 (I.41)

gDebye(ωn − ωm) = λ

[1 − (ωn − ωm)2

ω20

log(

1 +ω2

0(ωn − ωm)2

)]. (I.42)

In complete analogy with (I.39) we rewrite equation (I.35) as

Zωn ∆ωn =π

β ∑ωm

g(ωn − ωm)∆ωm√

(ωm)2 +

∣∣∆ωm

∣∣2 . (I.43)

Equations (I.39) and (I.43) are the Eliashberg equations of a superconductor that we will solve numer-ically in chapter II. Notice that there is a gauge invariance, i.e. if a set (Zωn , ∆ωn) is a solution, then(Zωn , α∆ωn) with α ∈ U(1) is a solution with the same observable properties. Equations (I.39) and(I.43) are also usually written in a slightly different way

Zωn = 1 +π

βωn∑ωm

g(ωn − ωm)ωm√

(ωm)2 +

∣∣∆ωm

∣∣2 (I.44)

∆ωn =π

βZωn∑ωm

g(ωn − ωm)∆ωm√

(ωm)2 +

∣∣∆ωm

∣∣2 . (I.45)


Let us finally see how do variables Zωn , ∆ωn incorporate into the ordinary and anomalous propa-gator. Comparing equations (I.21) and (I.32) we read out

Gk(ωn) =Zωn iωn + εk

Z2ωn

((iωn)2 −

∣∣∆ωn

∣∣2)− ε2k

(I.46)

Fk(ωn) =Zωn ∆ωn

Z2ωn

((iωn)2 −

∣∣∆ωn

∣∣2)− ε2k

. (I.47)

Note also that if we start with a real-valued set of ∆ωn , iteration of equations (I.43) and (I.39) will keepall successive values to be real, too. In the rest of this work we will therefore assume ∆ωn to be real.

I.4 Hubbard model

Where we refer to the Hubbard model, we mean

H = ∑k,σ

εkc†kσckσ +

UN ∑

k,p,qc†

p+q↑c†k−q↓ck↓cp↑ ≡ HT +HU . (I.48)

Here N is number of lattice points again and U is the on-site electron interaction. This Hamiltonianis suspected to exhibit d-wave superconductivity at low temperatures. It is usually solved using theQuantum Monte Carlo simulations. This approach is very different from the Eiashberg equations forthe electron-phonon hamiltonian and it demands large computational time. We discuss the Hubbardmodel only in section II.2 where we calculate the sum rules for its anomalous spectral function Bk(x).

A note has to been done about the nomenclature. A superconductor is called unconventional if thereis a symmetry operation of the point symmetry group G of the lattice that changes the order parameter⟨

ck↑c−k↓⟩≡ bk. On the other hand for a conventional superconductor all symmetry operations preserve

the order parameter, this is usually called as s-wave.An unconventional superconductor selects a subgroup of G which does not change the order pa-

rameter. Categorizing unconventional superconductors corresponds to finding the irreducible repre-sentations of G . An important information to be taken is that

∑k

⟨ck↑c−k↓

⟩= 0 (I.49)

for all unconventional superconductors. We will use this information in section (II.2) while studyingsum rules of the introduced Hamiltonians. We will also take advantage of the symmetries of G insection (II.1) while discussing symmetries of the anomalous spectral function, and in section (III.3)when seeking a method to deduce the anomalous spectral function from spectroscopic data.

Chapter II

Properties of the anomalous spectralfunction

We study the anomalous spectral function of a superconductor. As this object is not mentioned muchin the literature, we had to start our inspection at the very beginning with the following question:How does the anomalous spectral function typically look like? We start our investigation in section II.1 byshowing that the anomalous spectral function of certain class of superconductors is odd in frequencyand that it can be gauged to be real. We also discuss its parity in the momentum argument.

In section II.2 we are inspired by an article by White [8] and study the sum rules of the anomalousspectral function. We examine both Hamiltonians that are known to exhibit superconductivity: Theelectron-phonon Hamiltonian from chapter I.3 and the Hubbard Hamiltonian from chapter I.4. Wefurther show that the simplest model of superconductivity by Bardeen, Cooper and Schrieffer [7](BCS) violates the sum rule for the first moment. We conclude this section by proposing a reasonablegeneralization of the BCS result that fulfills all derived sum rules. This is achieved by including anon-coherent weight of opposite sign to the anomalous spectral function.

In section II.3 we present our numerically obtained solution of the Eliashberg equations that de-scribe superconductivity in the electron-phonon Hamiltonian. The Hubbard Hamiltonian is not treatedin this work any further because finding its ground state requires completely different techniques.Our solution reveals that the anomalous spectral function of a superconductor in the electron-phononHamiltonian really has the non-coherent part of an opposite sign. We also examine the validity of thederived sum rules. Results of this section are also used as an input for the final chapter where we tryto find a way to deduce the anomalous spectral function experimentally.

II.1 Symmetries of the anomalous spectral function

In this section we investigate symmetries of the anomalous spectral function in both the momentumand the frequency argument. We start with the momentum. Remind the spectral representation

Fk(ω) =∫ +∞

−∞

dxBk(x)ω − x

and the definition (I.13) of the anomalous propagator. Especially notice that Fk(τ = 0) correspondsto the order parameter bk discussed in section (I.4). Since the correspondence between Bk(x) and bkis linear, the anomalous spectral function transform according to the same irreducible representation.We can therefore claim that

B−k(x) = Bk(x) for s-wave and d-wave,B−k(x) = −Bk(x) for p-wave.

We will use these relations in section (III.3). Transformation of the anomalous spectral function uponapplying other symmetry operations can be deduced as well, but we do not need them.

13

CHAPTER II. PROPERTIES OF THE ANOMALOUS SPECTRAL FUNCTION 14

Discussion of the parity in the frequency argument shall take us more time. We will show thatin certain class of superconductors the anomalous spectral function is odd in the frequency argumentand that it can be gauged to be real. Knowledge of the parity may have some physical interpretation.Knowledge of the real values is crucial to perform the deduction from the Green’s function values onthe imaginary axis.

Some information can be deduced from the Eliashberg equations (I.44) and (I.45). Notice that adirect consequence of the first one is that if

∣∣∆ω

∣∣ = ∣∣∆−ω

∣∣ then Z−ω = Zω. As for the parity of ∆ω,both odd and even are compatible with the equation (I.45). In this work we assume ∆ω to be an evenfunction of the imaginary Matsubara frequency ω because such is the usual approach.1 The even parityof ∆ω in our numerical solution is ensured by the even initial input for the iteration as is explained inappendix E below (E.2).

A consequence of the chosen parity of ∆ω together with equation (I.47) is that for the Matsubarafrequencies

Fk(ωℓ) = Fk(−ωℓ). (II.1)

Writing down this identity in the spectral representation we find that the anomalous spectral functionBk(x) satisfies

0 =∫ +∞

−∞

Bk(x)dxiωℓ − x

−∫ +∞

−∞

Bk(x)dx−iωℓ − x

=∫ +∞

−∞

−2iωℓ

ω2ℓ + x2

Bk(x). (II.2)

We will show that this equality holding for all Matsubara frequencies ωℓ forces the anomalous spectralfunction to be real and odd.

The rest of this section is rather technical. If the reader is not curious about rigorous proof ofour claims, it is possible skip directly to section (II.2). An avid reader is kindly invited to revive hisknowledge of calculus and plunge into the following paragraphs.

One can check that the Fourier expansion

2iωℓ

ω2ℓ + x2

= −i∫ +∞

−∞e−ikx

(ekωℓθ(−k) + e−kωℓθ(k)

)= −2i

∫ +∞

0e−kωℓ cos (kx)dk (II.3)

holds for ωℓ > 0 i.e. for all positive fermionic Matsubara frequencies. Inserting this into (II.2) andswitching the order of integrals we find∫ +∞

0dke−kωℓ

∫ +∞

−∞dx cos (kx)Bk(x) = 0. (II.4)

where we threw away the irrelevant numerical factor.We will now need the following theorem from complex calculus:Theorem: A continuous bounded function h(u) : [0, 1] → C that satisfies∫ 1

0unh(u)du = 0 (II.5)

for all n = 0, 1, 2 . . . is a zero function, h(u) = 0.2

1We assume that there is also a solution with ∆ω = −∆−ω but that it has much lower critical temperature and is thus notrealized.

2The following proof is taken from [9]. According to the Stone-Weierstrass theorem [10], for every continuous functionh(u) and for ∀ϵ > 0 there exists a polynomial Pϵ(u) that satisfies ∀u ∈ [0, 1] : |h∗(u)− Pϵ(u)| < ϵ. The star denotes thecomplex conjugation. Since Pϵ(u) is a polynomial, the conditions of the theorem imply∫ 1

0h(u)Pϵ(u)du = 0.

Taking the limit ϵ → 0 we find ∫ 1

0|h(u)|2 = 0.

This implies that h(u) is a zero function.


The use of the theorem to our problem is that equation (II.4) can be rewritten into the form of(II.5) with some specific function h(u). Recall that Matsubara frequencies in the upper half plane areωℓ = (2ℓ− 1)πT, ℓ = 1, 2, 3 . . .. Substituting u = e−2πTk we get

12πT

∫ 1

0duuℓ−1

∫ +∞

−∞

√udx cos

(− x

2πTlog u

)Bk(x) = 0. (II.6)

The second integral is zero according to the theorem. Switching back to k = − log u/2πT this meansthat for all real and positive k ∫ +∞

−∞dx cos (kx)Bk(x) = 0.

If interpreted as a Fourier transformation, the last equation already tells us that B(x) is an odd function.Hence our first conclusion.

Another important consequence of the theorem is that as the second integral in equation (II.6)is zero, the equation remains valid for all real and positive numbers ℓ.3 If we track the previouscalculations, this means that

Fk(ω) = Fk(−ω)

holds not only for the Matsubara frequencies, but for all frequencies on the upper half of the imaginary axis.By the Sokhotski-Plemelj theorem [3] the Green’s function Fk(ω), ω ∈ C is separately analytic in

the upper and the lower half-plane. We also now that in the adopted gauge it is real-valued on theimaginary axis. According to the Schwarz reflection principle4 [11]

Fk(iω′′ + ω′) = F∗k(iω

′′ − ω′) (II.7)

We further explot that the function Fk(ω) is even on the imaginary axis. Uniqueness of the analyticcontinuation then tells us that we can obtain the values in the lower half-plane by a π rotation aroundthe origin of the values in the upper half-plane. This means that F(ω) = F(−ω) for all ω off the realaxis. Equivalently we can write

Fk(iω′′ + ω′) = Fk(−iω′′ − ω′). (II.8)

Our argumentation is illustrated in figure (II.1) too.Combining properties (II.7) and (II.8) of the anomalous propagator we find that for ω ∈ C/R

Fk(ω∗) = Fk(ω)∗. (II.9)

If we now apply equation (I.4) to the anomalous spectral function, we find

Bk(x) = − 12πi

[Fk(x + i0+)− Fk(x − i0+)

]= − 1

2πi[Fk(x + i0+)− Fk(x + i0+)∗

],

clearly a real-valued function. This implies that the cut of the anomalous propagator on the real axisis purely imaginary and that we can calculate the anomalous spectral function as

Bk(ω) = − 1πℑ[Fk(ω + i0+)

]. (II.10)

3The limitations comes from the domain of validity of the Fourier expansion (II.3).4The Schwarz reflection principle claims the following: If a function F is analytic in the upper half-plane and has a real number

boundary values on the imaginary axis, the analytic continuation to the lower half-plane can be done by demanding F(z∗) = F∗(z) [11].Because of the uniqueness of the analytic continuation we can reverse the theorem in the following manner:If a function Fis analytic in a domain D and has real values on a non-zero conjuction of D with the real axis, then for z, z∗ ∈ D : F(z∗) = F∗(z).Finally since a rotation by π/2 preserves the analytic properties of a function, analogous statement can be made about theimaginary axis, one just has to flip sign of the real part of the argument instead of complex conjugating it: If a function Fis analytic in a domain D and has real values on a nonzero conjuction of D with the imaginary axis, then for iz′′ + z′, iz′′ − z′ ∈ D :F(iz′′ + z′) = F∗(iz′′ − z′).


Figure II.1: According to the Schwarz reection principle, function values in points A and C is complex con-jugated. Further, because the function values on the imaginary axis are even, we can get function values in thelower half-plane by a π-rotation around the origin of the values in the upper half-plane, hence function values inpoints B and C are the same. Combining these two identities we see that function values in points A and B arecomplex conjugated, the equation (II.9).

II.2 Sum rules and their violation in the BCS model

In an attempt to find out something about the anomalous spectral function we will now apply the sumrules that were introduced in section I.1. We will study both the electron-phonon Hamiltonian fromsection I.3 and the Hubbard model from section I.4. Remembering the general sum rule prescription(I.6) and the definition of the anomalous propagator (I.13) we are left with evaluating∫ +∞

−∞dxxnBk(x) =

⟨[ck↑,H

]n , c−k↓

⟩. (II.11)

The calculation is tedious and is listed in appendix D. Here we only present the results.For the electron-phonon Hamiltonian we obtained∫ +∞

−∞Bk(x)dx = 0∫ +∞

−∞xBk(x)dx =

1Ω ∑

q =0ℜ[Vq] ⟨

c−(q+k)↓c(q+k)↓

⟩∫ +∞

−∞x2Bk(x)dx =

2Ω ∑

q =0ℜ[Vq]εq+k

⟨c−(q+k)↓c(q+k)↑

⟩+

1Ω2 ∑

q =0∑pℜ[Vq]ℜ[Vp+q+k

] ⟨c−p↓cp↑

⟩+terms linear in Vqgs

.,..

Notice that for Vq = 0, i.e. the electron-phonon Hamiltonian without the Coulomb interactions all 0th ,1st and 2nd moments must vanish. Remind that the BCS solution [7] neglects the Coulomb interactiontoo so it should satisfy these constraints.

Our results for the Hubbard Hamiltonian are∫ +∞

−∞Bk(x)dx = 0∫ +∞

−∞xBk(x)dx = −U

N ∑q

⟨c−q↓cq↑

⟩∫ +∞

−∞x2Bk(x)dx = −U

N ∑q(2εk + U)

⟨cp↑c−p↓

⟩.

Notice again that for a d-wave superconducting state of the Hubbard model all 0th , 1st and 2nd momentsmust vanish. This is because the average value of order parameter in a d-wave superconductor si zeroas was mentioned in equation (I.49).


Figure II.2: (left) In the BCS model, the anomalous spectral function consists of two δ-peaks at energies ±Ekwith weight ±∆/2Ek. This function annihilates the zeroth and the second moment, but not the rst one as isdemonstrated in equation (II.15).

Figure II.3: (right) A small non-coherent part of opposite sign is necessary to annihilate the rst moment of theanomalous spectral function (i) of an s-wave superconductor in the electron-phonon Hamiltonian (I.23) and (ii)of a d-wave superconductor in the Hubbard model (I.48).

To summarize these important results, for the Hubbard model and for the electron-phonon Hamil-tonian wih Vq = 0 we have ∫ +∞

−∞Bk(x)dx = 0 (II.12a)∫ +∞

−∞xBk(x)dx = 0 (II.12b)∫ +∞

−∞x2Bk(x)dx = 0 (II.12c)

for all k. These are strong and clearly stated limitations on Bk(x).Before we immerse the reader into the involved numerical treatment of the Eliashberg equations,

let us test the validity of the sum rules in the BCS solution [7]. In this case the spectral funtions are(clarify equations (8.19) and (8.57) in §8.3 of [4])

ABCSk (x) =

12

(1 +

εk

Ek

)δ (x − Ek) +

12

(1 − εk

Ek

)δ (x + Ek) (II.13)

BBCSk (x) =

∆2Ek

[δ (x − Ek)− δ (x + Ek)] (II.14)

with Ek =√|∆|2 + ε2

k the quasiparticle excitation energy. Variable ∆ is complex, its absolute value isthe energy gap on the Fermi surface. Notice that function (II.14) is in accordance with the previoussection odd and can be gauged to be real by taking real-valued ∆. We do so and we plot the function(II.14) in figure (II.2).

Now for the sum rules. As the anomalous spectral function (II.14) is odd, it satisfies the results forthe zeroth and the second moment trivially. However the first moment∫ +∞

−∞xBBCS

k (x)dx =∫ +∞

−∞x

∆2Ek

[δ(x − Ek)− δ(x + Ek)]dx = ∆ = 0 (II.15)

is clearly nonzero. This means that the BCS solution is in a sense non-physical. To ensure the firstmoment to be zero, we expect that the fuller treatment solution must contain a small non-coherentweight of opposite sign. A sketch of our proposal is shown in figure (II.3). We also hypothesize thatturning on a small Coulomb interaction in the electron-phonon Hamiltonian should lead to a smoothchange in the anomalous spectral function thus preserving the non-coherent feature of opposite sign.These conjectures are to be numerically tested in the following section.


II.3 Solution of the Eliashberg equations

Here we put the conjectures of the previous section to a numerical verification. We keep all simplifi-cations mentioned in section I.3 that is an isotropic s-wave superconductor with a perfect particle-holesymmetry. We also neglect the vertex corrections, which is according to the Migdal’s theorem feasibleif the Coulomb repulsion is weak enough.

Both the ordinary and the anomalous spectral function are obtained in the following three steps:

1. Equations (I.44) and (I.45) are solved iteratively on the imaginary axis. The procedure, precisionof the calculation and further subtleties are discussed in appendix E.

2. The solution on the imaginary axis is analytically continued to the real axis. This is performedusing the method of Padé approximants. Discussion of the method is included in appendix C.

3. Since we know that anomalous spectral function is gauged to be real, we can obtain it fromequation (II.10). For the ordinary spectral function we proceed in complete analogy.

As for the second step, i.e. the analytic continuation, this is in the literature usually done for the or-dinary particle propagator using the maximum entropy principle. This is unfortunately not suitablefor our problem because it requires the corresponding spectral function to have a probabilistic inter-pretation5 which is not the case of the anomalous spectral function. For purely punctilious reasons weinclude a brief review of the maximum entropy reasoning in appendix B. For illustration of the analyticcontinuation we also include graphs of variables Zω and ∆ω on both the imaginary and the real axisin figure (II.4).

Unless otherwise stated, we work with parameters listed in table (II.1). We included there also thecritical temperature and the energy gap size for those parameters.

We now proceed by presenting our results. Most of the figures in this subsection are composed ofa pair of pictures. The left one always correspond to the Einstein phonon model, the right one alwaysto the Debye phonon model, clarify equations (I.41) and (I.42).

We start with the normal spectral function Ak(x) because its properties are well-known and it mayserve as a quick check of the veracity of our calculations. We present it in figure (II.5). Notice (i) the

Notation Value Meaningλ 0.5 coupling constant of electrons to phononsµ∗ 0 renormalized Coulomb interaction parameter

kBT 0.015ω0 temperatureTc 0.039ω0 (Einstein) temperature of the phase transition

0.024ω0 (Debye)2∆ 2 × 0.073ω0 (Einstein) energy gap on the Fermi surface

2 × 0.037ω0 (Debye) at temperature kBTN 1500 number of considered Matsubara pointsF 40 number of digits of variables

used in the numerical computationp 30 accuracy of the results

according to definition (E.3)2r 50 number of Matsubara points used

to obtain the analytic continuationspread of Matsubara points used

s 1 to obtain the analytic continuation,s = n means taking every nth point

Table II.1: Overview of parameter values in our numerical calculations.

5Especially it has to be positive real valued.


Matsubara points

1.1

1.2

1.3

1.4

1.5Z_Ω

Matsubara points

0.01

0.02

0.03

0.04

0.05D_Ω Ω0

1 2 3 4 5ΩΩ0

0.5

1.0

1.5

Z

Im@ZD

Re@ZD

1 2 3 4 5ΩΩ0

-0.04

-0.02

0.02

0.04

0.06

0.08

0.10

DΩ0

Im@DΩ0 D

Re@DΩ0 D

Figure II.4: Demonstration of the Padé approximation. The graphs in the upper row show a quite boringstructure of Zω and ∆ω on the imaginary axis obtained by solving the Eliashberg equation. The peak centrescorrespond to ω = 0, compare also to gure (E.2) and (E.2). The pictures in the lower row show the analyticcontinuation of the data in the upper row to the real axis.

Figure II.5: The normal spectral function Ak(x) in the Einstein (left) and in the Debye (right) phonon model.The horizontal axis expresses momentum by means of εk/ω0, the horizontal axis is the observed energy of theelectrons x/ω0. Energies on both axes are thus counted as multiples of the Debye frequency. Red color correspondto positive values, white corresponds to zero. Saturation means absolute value. Several well-known features canbe observed in these pictures: The renormalization of electron spectrum in x ∈ (−ω0, ω0) energy shell aroundthe Fermi surface, the blurred spectrum away from the Fermi surface (not visible in the Einstein phonon modelbecause all phonons have energy ω0), and the energy gap 2∆ at the Fermi surface.

renormalization of the electron mass close to the Fermi surface, (ii) blurred spectrum away from theFermi surface, and (iii) the energy gap at the Fermi surface.

We further calculated several fixed momentum cuts of Ak(x). These correspond to vertical line cutsof the graphs in figure (II.5). Such cuts are functions of the energy x only and have the aforementioned


probabilistic interpretation. We show these cuts in figure (II.6). Broadening of the quasiparticle peaksand increasing weight of the non-coherent part of the spectral function can be observed for the mo-mentum argument going away from the Fermi surface. Details of the quasiparticle peaks close to theFermi surface are shown for the Debye phonon model in figure (II.7).

We have checked the validity of the numerical results for the normal spectral function by calculatingthe integral

I0A(k) =

∫ +∞

−∞Ak(x)dx = − 1

πℑ∫ +∞

−∞Gk(z)dz

(II.16)

which according to the theory should be equal to 1 for all momenta k. We checked this sum rule usingthe NIntegrate command in Mathematica 8.0.4. To obtain better convergence we shifted the integralpath of (II.16) from the real axis to a horizontal line in the upper half-plane,6 and we introduced acutoff instead of the infinite boundaries. We find that the sum rule for I0

A(k) is satisfied with precisionbetter than 2 × 10−3.

Let us now summarize the anomalous spectral function Bk(x) at last. Our results are shown infigure (II.8). The fixed momentum cuts are shown in figures (II.9) and (II.10). The key observation isthat the non-coherent part is in fact present. It is very small compared to the peaks that are located atquasiparticle energies. However, as the non-coherent feature goes to very high energies, the first sumrule turns out to be fulfilled very precisely as is demonstrated by calculating

I1B(k) =

∫ +∞

−∞Bk(x)xdx = − 1

πℑ∫ +∞

−∞Gk(z)zdz

which gives

∣∣I1B(k)

∣∣ < 10−7. As the anomalous spectral function we found is manifestly odd in fre-quency argument, it is no surprise that the zeroth and the second moment turn out to be of the orderof the machine epsilon, i.e. zero. All sum rules listed in (II.12) are thus fulfilled.

What if one turns off the Coulomb interaction? Our analysis shows that even though the firstmoment of the anomalous spectral function is now non-zero, the non-coherent feature is preserved,see figure (II.11).

The last object to be studied in this section is related to integral

bk =∫ +∞

−∞

Bk(x)dx1 + e−βx = − 1

πℑ∫ +∞

−∞

Fk(z)dz1 + e−βz

. (II.17)

This integral is according to (I.7) equal to⟨

ck↑c−k↓

⟩which is the order parameter of a superconductor.

-2 2 4 6Ω Ω0

0.5

1.0

1.5Ω0 AHΩL

¶ Ω0 = 2.0

¶ Ω0 = 1.5

¶ Ω0 = 1.0

¶ Ω0 = 0.5

¶ Ω0 = 0

-2 2 4 6Ω Ω0

0.5

1.0

1.5Ω0 AHΩL

¶ Ω0 = 2.0

¶ Ω0 = 1.5

¶ Ω0 = 1.0

¶ Ω0 = 0.5

¶ Ω0 = 0

Figure II.6: Fixed momentum cuts of the normal spectral function in the Einstein (left) and in the Debye (right)phonon models. The Einstein phonon model is in a sense pathologic all phonos have energy ω0, therefore allquasiparticle peaks in (−ω0, ω0) energy shell are equally sharp, and all non-coherent features are outside thisshell. There are simply no processes aecting these electrons. The Debye phonon model is more realistic. Thepeaks broaden for energies away from the Fermi energy.

6This is feasible by a virtue of the residue theorem because the Green’s function is analytic in the upper half-plane.


-0.4 -0.2 0.2 0.4Ω Ω0

0.5

1.0

1.5

2.0

2.5Ω0 AHΩL

¶ Ω0 = 0.4

¶ Ω0 = 0.2

¶ Ω0 = 0

Figure II.7: Superconductive quasiparticle peaks in the Debye phonon model in the low energy limit. Thebroadening is easy to see. The weight of these two peaks is qualitatively well-described by the BCS model.

Figure II.8: The anomalous spectral function Bk(x) that we study in this work calculated for the Einstein(left) and for the Debye (right) phonon model. Red color correspond to positive values, blue to negative, whitecorresponds to zero. Saturation means absolute value. The peaks are located at the same place as the quasiparticleexcitations of the normal spectral function. Notice also that the peaks of Bk(x) lie within interval (−ω0, ω0),i.e. Bk(x) selects the states that mediate superconductivity. The non-coherent part is too small to be seen onthis scale. We demonstrate its presence in gure (II.10).

In the BCS model this variable can be calculated analytically. According to chapter VI of [6]⟨c−k↓, ck↑

⟩(BCS)

≡ bk = ukv∗k[1 −

⟨γ†

k↑γk↑

⟩−⟨

γ†k↓γk↓

⟩].

We use now that ukv∗k = ∆/2Ek and that bogolyubons are eigenstates with Fermi-Dirac statistics f (Ek).If we further realize that 1 − 2 f (Ek) = tanh (Ek/2T), we find that for the BCS model

bk (BCS) =∆

2√

∆2 + ε2k

tanh

√

∆2 + ε2k

2kBT

.

At the Fermi surface εk = 0 at temperature kBT = 0.015ω0 and

I for ∆ = 0.073ω0 (Einstein) this equals 0.49,


0.5 1.0 1.5 2.0 2.5 3.0Ω Ω0

-0.05

0.05

0.10

0.15Ω0 BHΩL

¶ Ω0 = 2.0

¶ Ω0 = 1.5

¶ Ω0 = 1.0

¶ Ω0 = 0.5

¶ Ω0 = 0

0.1 0.5 1.0 5.0 10.0 50.0Ω D

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05Ω0 BHΩL

Figure II.9: (left) Detail of the cuts of the anomalous spectral function in the Debye phonon model. Notice thesmall scale on the vertical axis. Here we can nally observe the non-coherent feature. It is small but it extendsto high frequencies thus annihilating the rst moment almost perfectly.

Figure II.10: (right) The red curve from the left graph shown in logarithmic scale. This corresponds toεk/ω0 = 0, i.e. exactly at the Fermi surface.

0.2 0.4 0.6 0.8 1.0 1.2 1.4Ω Ω0

-0.05

0.05

0.10

0.15Ω0 BHΩL

¶ Ω0 = 2.0

¶ Ω0 = 1.5

¶ Ω0 = 1.0

¶ Ω0 = 0.5

¶ Ω0 = 0

0.1 0.5 1.0 5.0 10.0 50.0100.0Ω D

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05Ω0 BHΩL

Figure II.11: The left picture shows the anomalous spectral function Bk(x) of the Debye phonon model inpresence of the Coulomb repulsion µ∗ = 0.12. The gap is much smaller ∆ ≈ 0.012ω0, the critical temperatureis also pushed down by the presence of the electron-electron interaction. In these calculations we decreased thetemperature to kBT = 0.003ω0.

Figure II.12: The integral (II.17) for the Einstein (left) and for the Debye (right) phonon model. Its value atthe Fermi surface is the order parameter of a superconductor. Notice that the values are smaller than the BCSprediction.


I for ∆ = 0.037ω0 (Debye) this equals 0.42.

Within the Eliashberg model the order parameter has a smaller value. Graphs of bk for both phononmodels are shown in figure (II.12).

Chapter III

Experimental determination of Bk(x)

We seek a way to experimentally measure the anomalous spectral function. Since it corresponds to theanomalous propagator which is non-Hermitian, it can hardly be directly observable. The Josephsoneffect may give information about the frozen phase of the condensate, but that is only a single parame-ter. We would really like to obtain some spectroscopic information. To do so we shall use the fact, thatthe anomalous spectral function influences certain physical characteristics of a superconductor. Thesemight serve us to deduce it experimentally. We decided to study the spin-spin and the density-densitycorrelation functions of a superconductor as these are among the simplest physical observables in thequantum field approach.

Section III.1 is dedicated to deriving the spin-spin and the density-density correlation functions.Two important simplification of our calculation have to be stressed. The first one, we calculate onlythe irreducible susceptibilities, while experimentally observable are the reducible ones. The secondsimplification is that we neglect vertex corrections. The discussion of this is the same as in sectionI.3 where we derived the Eliashberg equations: Renormalization of electron-phonon vertex is indeednegligible by virtue of the Migdal’s theorem. If there is a Coulomb repulsion of electrons present inthe system however, ignoring the vertex corrections is not a good approximation.

In section (III.2) we apply the derived formula for the correlation functions to the BCS model onone hand, and to our numerical data for the Eliashber model on the other one. We seek some inherentdistinction between these two models. We only concentrate on the imaginary part of the differenceof the correlation functions at q = 0. Motivation for this will be clear from the text. We find thatthe Eliashberg solution in fact contains fingerprints of the the non-coherent features of the anomalousspectral function. They appear for arguments of the order of the Debye frequency. This motivates usthat studying these correlation functions might be useful indeed.

We conclude this chapter by section III.3 which deals with the following problem: Suppose thatwe somehow obtained the irreducible spin-spin and density-density correlation functions for all argu-ments (q, ω). Is it then possible to deduce the anomalous spectral function? Surprisingly, the answerseems to be affirmative. There are certain subtleties though that one has to take care about, neverthe-less we still succeeded in deriving an analytic expression to deduce the anomalous spectral functionfrom the susceptibilities.

III.1 Correlation functions ρ − ρ and s − s

We decided to study the spin-spin and the density-density correlation functions in a blind belief thatthey shall reveal some information about the anomalous spectral function. As we have already spoiledin the previous paragraphs, they happily turned out to reveal much more. The following pages tellour story with a surprising conclusion.

First of all, we have to derive the expression for the correlation functions. For brevity we willoften call them only as susceptibilities and denote them χρρ and χss. Derivation of the formulas is moretractable if one employs the Nambu-Gorkov formalism from section I.2. Remind that σ3 = diag(1,−1)is the z-component Pauli matrix (I.29). We will formulate the susceptibilities in the space-time domainand then Fourier transform them into functions of momentum and frequency. The result shall be

24

CHAPTER III. EXPERIMENTAL DETERMINATION OF BK(X) 25

susceptibilities expressed using the propagators. We shall then use their spectral representation toobtain relation between the susceptibilities and the discussed spectral functions.

We will need the following Fourier transformation of the field operators

ψσ(r) =1√Ω

∑k

eik·rckσ and ψ†σ(r) =

1√Ω

∑k

e−ik·rc†kσ

with Ω being the volume of the crystal. It is further convenient to introduce the Fourier transorm ofthe Nambu operators (I.16)

α(r) =1√Ω

∑k

eik·rαk =

(1√Ω ∑k eik·rck↑

1√Ω ∑k eik·rc†

−k↓

)=

(ψ↑(r)ψ†↓(r)

)α†(r) =

(ψ†↑(r) ψ↓(r)

). (III.1)

We can use operators (III.1) to express the local density ρ(r) and the local spin density1 s(r). Fordensity operator we find

ρ(r) = ψ†↑(r)ψ↑(r) + ψ†

↓(r)ψ↓(r)

= ψ†↑(r)ψ↑(r)− ψ↓(r)ψ

†↓(r) + 1

= α†(r)σ3α(r) + 1. (III.2)

Similarly for the local spin density operator

s(r) = ψ†↑(r)ψ↑(r)− ψ†

↓(r)ψ↓(r)

= ψ†↑(r)ψ↑(r) + ψ↓(r)ψ

†↓(r)− 1

= α†(r)α(r)− 1, (III.3)

where we intentionally suppressed constant h/2.With operators (III.2) we can write down the density-density correlation function in the Matsubara

formalism as

χρρ(r, τ) = − ⟨Tρ(r, τ)ρ(0, 0)⟩ = −⟨

T[α†(r, τ+)σ3α(r, τ−) + 1

] [α†(0, 0+)σ3α(0, 0−) + 1

]⟩= −1 −

⟨Tα†(r, τ+)σ3α(r, τ−)

⟩−⟨

Tα†(0, 0+)σ3α(0, 0−)⟩

−⟨

Tα†(r, τ+)σ3α(r, τ−)α†(0, 0+)σ3α(0, 0−)⟩

. (III.4)

The superscripts +,− at time arguments are introduced to tell the time-ordering operator about theordering of equal time operators.

The second and the third term in (III.4) are the same due to the symetry with respect to the timeand space translation. They can be simplified to

−⟨

Tα†(r, τ+)σ3α(r, τ−)⟩= + [σ3]ij

⟨T[α(r, τ−)

]j

[α†(r, τ+)

]i

⟩= − [σ3]ij Gji(0, 0) = −Tr [σ3G(0, 0)].

(III.5)With the last term of (III.4) there is a bit more shuffling to be made. We have

−⟨

Tα†(r, τ+)σ3α(r, τ−)α†(0, 0+)σ3α(0, 0−)⟩

= − [σ3]ij [σ3]kℓ

⟨T[α†(r, τ+)

]i

[α(r, τ−)

]j

[α†(0, 0+)

]k

[α(0, 0−)

]ℓ

⟩= − [σ3]ij [σ3]kℓ

⟨T[α(r, τ−)

]j

[α(0, 0−)

]ℓ

[α†(0, 0+)

]k

[α†(r, τ+)

]i

⟩= . . . . (III.6)

We now have to apply the Feynman rules. For the Nambu-Gorkov formalism they are listed e.g. inparagraph §8.3 of the Rickayzen’s book [4], page 219. Especially, we are supposed to draw all Feynmandiagrams that contain the vertices (0, 0) and (r, τ), see figure (III.1).

1We mean the z-component of spin density.


Figure III.1: To evaluate the four-point correlator in equation (III.6) we are supposed to draw all diagrams thatcontain these two vertices and whatever in between. There is innitely many of them.

Figure III.2: To simplify the calculation of (III.6) we consider these two diagrams only. This is to say that wecalculate the irreducible susceptibility instead of the reducible one, and we neglect all vertex corrections.

We do here a cruel simplification. We ignore all digrams except the two shown in figure (III.2).This means that we are at the same time (i) calculating the irreducible susceptibility instead of theobservable reducible one, and (ii) we neglect the vertex corrections. After doing so equation (III.6)simplifies to

. . . = − [σ3]ij [σ3]kℓ

⟨T[α(0, 0−)

]ℓ

[α†(0, 0)

]k

⟩ ⟨T[α(r, τ−)

]j

[α†(r, τ)

]i

⟩+ [σ3]ij [σ3]kℓ

⟨T[α(r, τ−)

]j

[α†(0, 0)

]k

⟩ ⟨T[α(0, 0−)

]ℓ

[α†(r, τ)

]i

⟩= − [σ3]ij [σ3]kℓ Gℓk(0, 0−)Gji(0, 0−) + [σ3]ij [σ3]kℓ Gjk(r, τ)Gℓi(−r,−τ)

= −(Tr[σ3G(0, 0−)

])2+ Tr

[τ3G(r, τ)σ3G(−r,−τ)

]. (III.7)

Adding results (III.5) and (III.7) together, we have for the density-density correlation function

χρρ(r, τ) = −(1 + Tr

[σ3G(0, 0−)

])2+ Tr [σ3G(r, τ)σ3G(−r,−τ)]. (III.8)

The first term of (III.8) can be simplified. Taking the spatial Fourier transform of (I.17) at τ = 0−

we find that

G(0, 0−) = −⟨

Tα(r, 0−)α†(r, 0+)⟩

=

−⟨

Tc↑(r, 0−)c†↑(r, 0+)

⟩−⟨

Tc↑(r, 0−)c↓(r, 0+)⟩

−⟨

Tc†↓(r, 0−)c†

↑(r, 0+)⟩

−⟨

Tc†↓(r, 0−)c↓(r, 0+)

⟩ =

⟨c†↑(r)c↑(r)

⟩ ⟨c↓(r)c↑(r)

⟩⟨c†↑(r)c

†↓(r)

⟩ ⟨c↓(r)c

†↓(r)

⟩ =

(n↑ · · ·· · · 1 − n↓

)where n↑ and n↓ is density of up- and down-spin electrons. It is easy to check that if we introducen = n↑ + n↓, then 1+ Tr [σ3G(0, 0−)] = n. This term addresses the non-zero average density of electronand for our purposes can be omitted. We can therefore simplify (III.8) to

χρρ(r, τ) = Tr [σ3G(r, τ)σ3G(−r,−τ)]. (III.9)

For the spin-spin correlation function we can proceed in a complete analogy. Comparing (III.2)with (III.3) we see that we have to change σ3 7→ 1 and flip some signs. After doing so we find

χss(r, τ) = − ⟨Ts(r, τ)s(0, 0)⟩ = . . . = − (1 − Tr [G(0, 0)])2 + Tr [G(r, τ)G(−r,−τ)]. (III.10)


We can easily check that 1 − Tr [G(0, 0−)] = n↓ − n↑. Since superconductor is non-magnetic, this iszero. We can therefore simplify (III.10) to

χss(r, τ) = − ⟨Ts(r, τ)s(0, 0)⟩ = . . . = Tr [G(r, τ)G(−r,−τ)]. (III.11)

We further want to transform results (III.9) and (III.11) from space-time domain to momentum-frequency arguments. To do so, let us remember the Fourier transform of the Nambu propagator

G(r, τ) =1

βΩ ∑k

∑ℓ

eik·r−iωℓτGk(ωℓ).

For the density susceptibility we obtain

χρρ(q, iζ) =∫

d3r∫ β

0dτe−iq·r+iζτχρρ(r, τ)

=1

(βΩ)2 ∑k,k′

∑ℓ,m

∫d3re−iq·r+ik·r−ik′·r︸︷︷︸

Ωδk′,k−q

∫ β

0dτe+iζτ−iωℓτ+iωmτ︸︷︷︸

βδωm,ωℓ−ζ

Tr[σ3G(k, ωℓ)σ3G(k′, ωm)

].

=1

βΩ ∑k,ℓ

Tr[σ3G(k, ωℓ)σ3G(k − q, ωℓ − ζ)

]

We use the cyclic property of trace and shift the variables summed over as k 7→ k+ q and ωℓ 7→ ωℓ + ζso that the previous result modifies to a more comfortable form

χρρ(q, iζ) =1

βΩ ∑k,ℓ

Tr[σ3G(k, ωℓ)σ3G(k + q, ωℓ + ζ)

]. (III.12)

For the spin-spin correlation function (III.10) we may follow the same steps to obtain

χss(q, iζ) =1

βΩ ∑k,ℓ

Tr[G(k, ωℓ)G(k + q, ωℓ + ζ)

]. (III.13)

We will now express the propagators in the spectral representation. Using the matrix spectralfunction (I.22) we find

χρρ(q, iζ) =1

βΩ ∑k,ℓ

∫R×R

dxdy1

(iωℓ − x) (iωℓ + iζ − y)Tr[σ3Ak(x)σ3Ak+q(y)

]. (III.14)

The summation over Masubara frequencies ωℓ can be performed using the residue trick shown onpage 351 in equations (B.15) – (B.19) in Rickayzen’s book [4]. It yields

1β ∑

ℓ

1(iωℓ − x) (iωℓ + iζ − y)

=f (x)− f (y)x − y + iζ

where f is the Fermi-Dirac distribution function. We insert this into equation (III.14) and switch toretarded Green’s functions formalism by changing iζ 7→ ω + i0+. We further simplify the matrixmultiplication

Tr[(

1 00 −1

)(Ak(x) Bk(x)Bk(x) Ak(−x)

)(1 00 −1

)(Ak+q(y) Bk+q(y)Bk+q(y) Ak+q(−y)

)]= Tr

[(Ak(x) Bk(x)−Bk(x) −Ak(−x)

)(Ak+q(y) Bk+q(y)−Bk+q(y) −Ak+q(−y)

)]= Tr

(Ak(x)Ak+q(y)− Bk(x)Bk+q(y) · · ·

· · · Ak(−x)Ak+q(−y)− Bk(x)Bk+q(y)

)= Ak(x)Ak+q(y) + Ak(−x)Ak+q(−y)− 2Bk(x)Bk+q(y).


Returning to (III.14) we have for the density-density correlation function

χret.ρρ (q, ω) =

1Ω ∑

k

∫R×R

dxdyf (x)− f (y)

x − y + ω + i0+[Ak(x)Ak+q(y) + Ak(−x)Ak+q(−y)− 2Bk(x)Bk+q(y)

].

The same steps applied to the spin-spin correlation function (III.13) lead to

χret.ss (q, ω) =

1Ω ∑

k

∫R×R

dxdyf (x)− f (y)

x − y + ω + i0+[Ak(x)Ak+q(y) + Ak(−x)Ak+q(−y) + 2Bk(x)Bk+q(y)

].

A comment has to be made at this point. The last two formulas do not look remarkably pleasingbut they are. In literature the correlation functions are usually written down explicitly only for theBCS model. This is done using the so-called coherence factors. This standard approach has twodisadvantages. First, the coherence factors are rather clumsy long expressions. Second, they can’tbe generalized beyond the BCS model. Our approach using the spectral functions overcomes both ofthese. Our notation is relatively concise and can be applied for any superconductor!

But the greatest feature of these equations is yet another one. We can combine them to obtain onlythe contribution of each of the spectral functions. If we define χret.

± = χret.ss ± χret.

ρρ , then

χret.+ (q, ω) =

2Ω ∑

k

∫R×R

dxdyf (x)− f (y)

x − y + ω + i0+[Ak(x)Ak+q(y) + Ak(−x)Ak+q(−y)

], (III.15)

χret.− (q, ω) =

4Ω ∑

k

∫R×R

dxdyf (x)− f (y)

x − y + ω + i0+Bk(x)Bk+q(y). (III.16)

So convolutions of the anomalous spectral function incorporate into the difference of the spin-spin andthe density-density correlation functions. If we use

1f + i0+

= P 1f− iπδ( f ), (III.17)

where P denotes the principal value and δ is the Dirac function, and then take the imaginary part of(III.16), we get

ℑχret.− (q, ω) = −4π

Ω ∑k

∫ +∞

−∞dx ( f (x)− f (x + ω)) Bk(x)Bk+q(x + ω). (III.18)

This equation is the starting point for the section (III.3) where we discuss deducing the anomalousspectral function from the experimental data. Notice also that by virtue of the Kramers-Kronig rela-tions, throwing away the real part of the data does not mean loss of information.

You can also check that if we would not assume the anomalous spectral function to be real, theequation (III.18) would modify into


Ω ∑k

∫ +∞

−∞dx ( f (x)− f (x + ω))ℜ

[B∗

k(x)Bk+q(x + ω)],

obviously a U(1)-invariant expression as it should be.

III.2 Comparison to the BCS model

In the previous section we derived formulas for the density-density and the spin-spin correlationfunctions. It is taking their sum (III.15) and the difference (III.15) that decouples the contribution of theordinary and of the anomalous spectral function, so we decided to study those. We start this sectionby deriving formulas for the BCS model [7]. The difference which is closely related to the anomalousspectral function will then be compared to the exact solution of the Eliashberg equations.


We start the calculation with evaluating the difference (III.16) of the correlation functions in theBCS model. Inserting the anomalous spectral functions (II.13) leads to

χBCS− (q, ω) =

4Ω ∑

k

∫ +∞

−∞dx∫ +∞

−∞dy

f (x)− f (y)x − y + ω + i0+

× ∆2

4EkEk+q[δ (x − Ek)− δ (x + Ek)]

[δ(y − Ek+q

)− δ

(y + Ek+q

)]=

4Ω ∑

k

∆2

4EkEk+q

f (+Ek)− f (+Ek+q)

Ek − Ek+q + ω + i0+−

f (−Ek)− f (+Ek+q)

−Ek − Ek+q + ω + i0+

−f (+Ek)− f (−Ek+q)

Ek + Ek+q + ω + i0++

f (−Ek)− f (−Ek+q)

−Ek + Ek+q + ω + i0+

.

We restrict ourselves to q = 0 only, hence the first and the last terms in the curly brackets are zero.The rest simplifies to

χBCS− (0, ω) =

4Ω ∑

k

∆2

(2Ek)2 [ f (−Ek)− f (+Ek)]

(1

ω + 2Ek + i0+− 1

ω − 2Ek + i0+

).

To simplify the formula even more, we will treat only the imaginary part. We will exploit formula(III.17) again. We shall also assume that2 ω > 0 so that δ(ω + 2Ek) is always zero. Then

ℑχBCS− (0, ω) =

4π

Ω ∑k

∆2

(2Ek)2 [ f (−Ek)− f (+Ek)] (−δ (ω + 2Ek) + δ (ω − 2Ek))

ω>0=

4π

Ω ∑k

∆2

(2Ek)2 [ f (−Ek)− f (+Ek)] δ (ω − 2Ek)

=4π

Ω ∑k

∆2

(2Ek)2 tanh(

Ek

2T

)δ (ω − 2Ek) .

For temperature kBT ≪ εF we can replace 1Ω ∑k 7→ N(0)

∫ +∞−∞ dε where N(0) is the density of states on

the Fermi surface. We also use that for the Dirac function

δ(g(x)) = ∑roots

δ(x − xi)

|g′(xi)|

with xi the roots of g(x). In our case

δ(

ω − 2√

ε2 + ∆2)=

ω

2√

ω2 − 4∆2

[δ(

ε − 12

√ω2 − 4∆2

)+ δ

(ε + 1

2

√ω2 − 4∆2

)](III.19)

so that

ℑχBCS− (0, ω) = 4πN(0)

∫ +∞

−∞dε

∆2

4√

ε2 + ∆2tanh

(√ε2 + ∆2

2T

)× ω

2√

ω2 − 4∆2

[δ(

ε − 12

√ω2 − 4∆2

)+ δ

(ε + 1

2

√ω2 − 4∆2

)].

Performing the integral we finally find

ℑχBCS− (0, ω) = 2πN(0)

∆2√

ω2 − 4∆2tanh

( ω

4T

). (III.20)

Now for the sum of the correlation functions which is given by the ordinary spectral function. Firstof all, refresh the equation (II.13) and notice that ABCS

−εk(−x) = ABCS

εk(x). This is a manifestation of the

2The case of ω < 0 can be deduced from the fact, that the real part of a retarded Green’s function is even and theimaginary part is odd.


perfect particle-hole symmetry of the BCS model. This means that in this case we may rewrite the sumof correlation functions (III.15) as

χBCS+ (q, ω) =

4Ω ∑

k

∫R×R

dxdyf (x)− f (y)

x − y + ω + i0+Ak(x)Ak+q(y).

Plugging (II.13) leads to

χBCS+ (q, ω) =

4Ω ∑

k

∫ +∞

−∞dx∫ +∞

−∞dy

f (x)− f (y)x − y + ω + i0+

×12

[(1 +

εk

Ek

)δ (x − Ek) +

(1 − εk

Ek

)δ (x + Ek)

]×1

2

[(1 +

εk+q

Ek+q

)δ(y − Ek+q

)+

(1 −

εk+q

Ek+q

)δ(y + Ek+q

)].

Integration over x and y can be perfomed immediately

χBCS+ (q, ω) =

1Ω ∑

k

f (+Ek)− f (+Ek+q)

Ek − Ek+q + ω + i0+

(1 +

εk

Ek

)(1 +

εk+q

Ek+q

)+

f (−Ek)− f (+Ek+q)

−Ek − Ek+q + ω + i0+

(1 − εk

Ek

)(1 +

εk+q

Ek+q

)+

f (+Ek)− f (−Ek+q)

Ek + Ek+q + ω + i0+

(1 +

εk

Ek

)(1 −

εk+q

Ek+q

)

+f (−Ek)− f (−Ek+q)

−Ek + Ek+q + ω + i0+

(1 − εk

Ek

)(1 −

εk+q

Ek+q

).

We will further restrict to the q = 0 case again. In such case the first and the last term vanish trivially.We have

χBCS+ (0, ω) =

1Ω ∑

k

f (−Ek)− f (+Ek)

−Ek − Ek + ω + i0+

(1 − εk

Ek

)(1 +

εk

Ek

)

+f (+Ek)− f (−Ek)

Ek + Ek + ω + i0+

(1 +

εk

Ek

)(1 − εk

Ek

)

=1Ω ∑

k[ f (−Ek)− f (+Ek)]

(1 −

ε2k

E2k

)[1

ω − 2Ek + i0+− 1

ω + 2Ek + i0+

]=

1Ω ∑

ktanh

(Ek

2T

)∆2

E2k

[1

ω − 2Ek + i0+− 1

ω + 2Ek + i0+

].

Taking the imaginary part leads to

ℑχBCS+ (0, ω) =

π

Ω ∑k

tanh(

Ek

2T

)∆2

E2k[−δ(ω − 2Ek) + δ(ω + 2Ek)] .

We will further assume ω > 0 so that the second Dirac function drops out. We replace 1Ω ∑k 7→

N(0)∫ +∞−∞ dε again and apply formula (III.19) so that

ℑχBCS+ (0, ω) = −πN(0)

∫ +∞

−∞dε tanh

(Ek

2T

)∆2

E2k

ω

2√

ω2 − 4∆2

×[δ(

ε − 12

√ω2 − 4∆2

)+ δ

(ε + 1

2

√ω2 − 4∆2

)]= −2πN(0)

∆2√

ω2 − 4∆2tanh

( ω

4T

). (III.21)


Figure III.3: This diagram illustrates two approaches that we use to obtain the imaginary part of the dierenceof the spin-spin and the density-density correlation function of the electron-phonon Hamiltonian. To keep thecalculations simple, we only study the function values at q = 0. The method in the right column which does theanalytic continuation in the end proceeded much faster.

Notice that this differs from result (III.20) only by the minus sign. We also note that results (III.20) and(III.21) are valid also for ω < 0.

Results (III.20) and (III.21) can be used to deduce the imaginary part of both studied correlationfunctions at q = 0.

ℑχBCSρρ (0, 0) = −2πN(0)

∆2√

ω2 − 4∆2tanh

( ω

4T

)ℑχBCS

ss (0, 0) = 0.

We will now compare the exact result (III.21) for the difference of the correlation functions at q = 0in the BCS model, to the solution of the electron-phonon Hamiltonian obtained by numerically solvingthe Eliashberg equations. To be sure about our numerical solution, we obtained it in two differentways:

1. We solve the Eliashberg equations on the imaginary axis. We do analytic continuation of theGreen’s function Fk(x) to obtain the spectral function Bk(x) as the cut on the real axis. This isidentical to our approach from section (II.3). We then numerically integrate according to formula(III.21) to obtain the result.

2. We solve the Eliashberg equations on the imaginary axis. We then use equations (III.12) and(III.13) to calculate the correlation functions on the imaginary axis. We finally do the analyticcontinuation to the real axis using the Padé approximation.

Comparison of these two methods is illustrated in figure (III.3).Results of these two procedures are shown in figure (III.4) where we also compare them to the

BCS result (III.21) with the same energy gap 2∆. Notice that the Eliashberg solution changes sign inthe vicinity of ω = ω0 while the BCS solution does not. This feature is a direct consequence of thenon-coherent weight of the anomalous spectral function which is not present in the simple BCS model.


Figure III.4: These graphs compare results for the electron-phonon Hamiltonian obtained by solving the Eliashbergequations to the result (III.20) for the BCS model. The red curve (Numerics) corresponds to the left column ofdiagram in gure (III.3), blue curve (Pade) corresponds to the right column. They coincide quite well exceptfor the sudden jump at the energy gap. While the red curve falls sharply to zero, blue curve exhibits a small"bump" just below the energy gap. It is a well-known feature of the Padé approximation that it fails to t sharpsteps in the spectral function, see e.g. article by Liu [12]. The green curve (BCS) is exact solution for the BCSmodel with the same gap. Notice that the Eliashberg calculation gives overall a smaller result. The importantobservation has to be done on the detail in the right picture: The Eliashberg solution changes sign while the BCSresult does not. This is a direct consequence of the non-coherent part of the anomalous spectral function, whichis not present in the overly simplied BCS solution.

Reversely, the change of the sign of the Eliashberg solution implies a weight of opposite sign in theanomalous spectral function, so some information obviously can be deduced from these data. As weshow in the next section, data in figure (III.4) together with data for nonzero momenta may lead tocomplete information about the anomalous spectral function.

III.3 Solving the inversion problem

Let us finally ponder over the problem of inverting equation (III.16) to obtain the anomalous spectralfunction. We want to milk as much information as possible from the data. As already mentionedbelow the equation, taking the imaginary part of the data does not mean loss of information. This isensured by the Kramers-Kronig relations. We can thus safely start our analysis with equation (III.18),that is


Ω ∑k

∫ +∞

−∞dx ( f (x)− f (x + ω)) Bk(x)Bk+q(x + ω). (III.22)

In the following paragraphs we will heavily use both spatial and temporal Fourier transformations.To avoid confusion I list here the adopted conventions about the normalization factor. We use

F(r) =1Ω ∑

kF(k)eik·r F(k) =

∫d3rF(r)e−ik·r

F(t) =∫ +∞

−∞

dω

2πF(ω)e−iωt F(ω) =

∫ +∞

−∞dtF(t)eiωt.

We would like to get rid of the sum and of the integral in (III.22). To do so, we try to reveal aconvolution on the right-hand side. We perform a spatial Fourier transformation and obtain

ℑχret.− (r, ω) =

1Ω ∑

qℑ(

χret.ss − χret.

ρρ

)(q, ω)eiq·r

= −4π∫ +∞

−∞dx ( f (x)− f (x + ω))

1Ω ∑

k

1Ω ∑

qBk(x)e−ik·rBk+q(x + ω)ei(k+q)·r

= −4π∫ +∞

−∞dx ( f (x)− f (x + ω)) B(r, x)B(r, x + ω) (III.23)


where in the last step we used that according to our discussion in the beginning of section (II.1) in boths-wave and d-wave superconductor Bk(x) = B−k(x). For p-wave the sign is opposite.

To eliminate the last integral we perform a trick. We know that the anomalous spectral function isodd in frequency. Throwing away its values for the negative argument therefore does not mean lossof information. The same assertion can be stated about the imaginary part of the susceptibility on theleft-hand side of equation (III.23). We thus multiply both sides of the equation by the Heaviside stepfunction θ(x) to obtain

θ(ω)ℑχret.− (r, ω) = −4π

∫ +∞

−∞dx ( f (x)− f (x + ω)) θ(ω)B(r, x)B(r, x + ω).

The trick is that if kBT = 0,3 then

( f (x)− f (x + ω)) θ(ω) = θ(−x)θ(x + ω)

as can be checked easily. This is great because if we introduce

B(r, ω) := B(r, ω)θ(ω) =

0 for ω < 0B(r, ω) for ω > 0

and use that the anomalous spectral function is odd in the frequency argument, we obtain

θ(ω)ℑχret.− (r, ω) = +4π

∫ +∞

−∞dxB(r,−x)B(r, x + ω)

which is a convolution again.The next step is obvious. Temporal Fourier transformation leads to∫ +∞

−∞

dω

2πθ(ω)ℑχret.

− (r, ω)e−iωt = 8π2∫ +∞

−∞

dω

2π

∫ +∞

−∞

dx2π

B(r,−x)B(r, x + ω)e−i(x+ω)teixt

= 8π2B2(r, t) (III.24)

where B(r, t) is temporal Fourier transform of B(r, x). It is generally complex-valued. The rest is clear.We take the square root of (III.24) and do the inverse temporal Fourier transform so that

B(r, x) =∫ +∞

−∞dtB(r, t)eixt (III.25)

B(r, x) = sign(x)B(r, |x|) = sign(x)∫ +∞

−∞dtB(r, t)ei|x|t. (III.26)

We conclude with the inverse spatial Fourier transform.There is a subtle complication that needs to be discussed. The square root function has two

branches! We expect that a physically meaningful spectral function B(r, t) changes smoothly as afunction of time t and does not contain "jumps". This is equivalent to demanding the spectral functionB(r, x) to be Lebesgue integrable in frequency x. This sounds reasonable too. This hints that whentaking the square root one is supposed to do it in a way to obtain a smooth result.

The algorithm breaks down however if a function B(r, t) crosses zero. It is not clear which branch tochoose in such case as both options correspond to a smooth function. Fortunately as B(r, t) is generallycomplex-valued, crossing zero is a very special case.

We postpone our algorithm to basic numerical verifications. Since performing the spatial Fouriertransformation is numerically demanding, we tested a simplified analogue. We make up an arbitraryreal-valued odd function B(x) which has most of its weight in some finite interval K around the origin.We then we calculate ∫ +∞

−∞dx ( f (x)− f (x + ω)) B(x)B(x + ω) ≡ F(ω) (III.27)

at zero temperature for a discrete set of frequencies ω in an interval several times broader than theinterval K. We obtained a set of data that we used as an input for the treatment discussed above to

3This is a good approximation also if kBT ≪ ∆.


1 2 3 4x

-4

-3

-2

-1

0

1

BHxL

input

1 2 3 4 5 6x0

20

40

60

80

100

120

FHxL

1 2 3 4 5 6x

-0.2

-0.1

0.0

0.1

0.2

noiseHxL

1 2 3 4x

-4

-3

-2

-1

0

1

BHxL

output

input

Figure III.5: Illustration of the test of the inversion procedure. In upper left picture we conceive an arbitraryreal-valued odd function B(x), here plotted only for positive arguments. We then use integral (III.27) to calculateexact vaues of F(ω) in the upper right graph. This function corresponds to the imaginary part of the dierenceof the susceptibilities. We add randomly-generated noise shown in the lower left picture to function F(ω). Wenally apply the described procedure to reproduce the function B(x) from the data with noise. The lower rightpicture compare the reproduced function to the original one. If no noise were added to F(ω), perfect reproductionwould be achieved. Notice however that the very small noise in our example already deteriorates the method.

reproduce function B(x). Perfect match was obtained. This is not a surprise because our algorithm isanalytic. In about half of the tests the global sign of the reproduced function was flipped what makesperfect sense.

We also modified the test to include noise in the input data. As the method is not linear, noisedeteriorates the reproduced data significantly. On the other hand if the correlation length of the noisewas sufficiently big or if its amplitude was small enough, the reproduction proceeded quite successfullyas demonstrated in figure (III.5). We did not test the method on the physical data so far.

Conclusion

We studied the anomalous spectral function of a superconductor. The main contribution of this workwas raising the right questions in the first place. The fact that such a central object in the theory ofsuperconductivity is not studied much in literature is surprising to us. In this work we attempted tofill in this gap by trying to answer some of the questions.

What does the function typically look like? The sum rules were useful to make certain observation.We showed that for (i) a d-wave superconductor in the Hubbard model and for (ii) a superconductorin the electron-phonon Hamiltonian without Coulomb repulsion, all 0th , 1st and 2nd moments of theanomalous spectral function are zero. This led us to a hypothesis that the function consists of a peakat the quasiparticle excitation and a non-coherent part of opposite sign. Our numerical calculationsproved this conjecture for case (ii). We also discussed parity of the anomalous spectral function in boththe momentum and the frequency arguments.

How does the function encode information about the pairing mechanism of electrons? The anomalous spec-tral function encodes all information about the anomalous propagator via the spectral representation,and the anomalous propagator contains all information about correlations of the superconducting con-densate. The statement that knowledge of the anomalous spectral function might be used to identifythe agent of electron pairing mechanism thus sounds very convincingly. If true, this function mightopen a new experimental window into the study of superconductivity. In spite of all these persuasiveclaims, the finite amount of time did not allow us to address this question in our work.

How does the anomalous spectral function influence the physical observables? Here we came to someconclusions, though keep in mind that our approach was not really a fair-play. Instead of physicallyobservable reducible susceptibilities we studied only their irreducible counterparts. Elaborate reason-ing is needed to deduced these from the measurable quantities. But if this is somehow achieved,information about the anomalous spectral function might be obtained rather easily. We showed thatthe non-coherent part of the anomalous spectral function leaves fingerprints in the difference of thespin-spin and the density-density correlation functions, and we compared the signatures of the simpleBCS model and of the fuller treatment Eliashberg solution.

Is there an experiment that might serve us to deduce this function? In the work we showed that ifwe know values of the spin-spin and the density-density correlation functions for all momenta andfrequencies, then it really might be possible to deduce the anomalous spectral function. There werecertain subtleties to be discussed though. The fact that the square root function has two branches wasthe major complication, causing that even small noise in the data corrupted our method severely. Abetter, more robust method of experimental determination of the anomalous spectral function shouldthus be sought.

35

Appendices

A Uniqueness of analytic continuation

It is well-known that any Green’s function of complex frequency argument is separately analytic in theupper and the lower half-plane, it has a cut on the real axis, and goes to zero as z approaches infinityalong any straight line. If we express the Green’s function using the corresponding spectral function

G(ω) =∫ +∞

−∞

A(x)dxx − ω

,

the cut (as a function of real axis position x) is according to Plemelj [3] equal to −2πiA(x). Identity

1x ± i0+

= P 1x∓ iπδ(x)

is a handy tool to derive the Sokhotskij-Plemelj relations

G(ω ± i0+) = P∫ +∞

−∞

A(x)dxx − ω

∓ iπA (ω) ,

with ω ∈ R.In condensed matter physics we compute Green’s functions at the Matsubara frequencies ωℓ =

i(2ℓ+ 1)π/β, ℓ ∈ Z on imaginary axis. The question is: If we are given exact values of a Green’s function atall Matsubara frequencies in the upper half-plane, does the Green’s function have a unique analytic continuationinto entire upper half-plane? The answer is yes, i.e. there is only one solution to the problem, though theargument is a bit lengthy.

There is the following theorem in complex calculus: Two functions which are analytic in a domainD and which coincide on some set with a limit point in D, coincide throughout D (are identical) [13].This theorem might be straightforwardly reversed: If we study a function defined on a set of points indomain D with a limit point in D, then there is a unique (or none) analytic continuation of this functioninto the domain D.

Does the reversed theorem apply to the problem of analytic continuation of Matsubara frequenciesdata? Sadly, it does not. Matsubara points have a limit point ω = ∞ which is unfortunately generallyoutside the domain of analyticity of The Green’s function. The previous paragraph is therefore notapplicable for our problem. To prove the uniqueness we have to proceed differently. We will show theoriginal proof by Baym and Mermin [14] which uses the Carleman’s theorem.

The proof

The Carleman’s theorem is a handy tool to prove the uniqueness of the analytic continuation problem.It states that if a function g(z) is analytic and bounded in the upper half-plane including the real axis,and has zeroes in the upper half-plane at isolated points rneiθn , then the series

S = ∑n

sin θn

rn

is convergent. Proof of this theorem is rather awkward and we leave it to the end of this appendix. Atthe moment we take its validity as granted.

36


Figure A.1: Integration contour used to prove the Carleman's theorem. The radii ρ and R are taken so that noroot lie on the contour.

The Green’s function G(ω) is in general bounded only away from real axis, which contradictsthe assumptions of the Carleman’s theorem. Vertical shift solves the problem: The function ϕ(z) =G(ω + 2π

β ) is bounded in the upper half plane and on the real axis as required.We will prove the uniqueness by a contradiction: Suppose we found two different analytic contin-

uations ϕ1(z) and ϕ2(z) of function ϕ(z). Let us study their difference ϕ1(z)− ϕ2(z) ≡ g(z). This newfunction satisfies assumptions of Carleman’s theorem, and has isolated zeroes at rneiθn = iπ(2n − 1)/βwith n ∈ 1, 2, 3, . . . and maybe also some other. In the upper half-plane θn ∈ (0, π) meaning that allsummands are positive. We have

S = ∑roots

sin θn

rn> ∑

Mats. points

sin θn

rn=

∞

∑n=1

β

π(2n − 1)= +∞

so S diverges. This is the contradiction. We could not have started with two different analytic contin-uations; they both must have been the same function. Under such circumstances the difference g(z)is a zero function that does not have isolated roots, hence the theorem does not apply. The analyticcontinuation is unique indeed.

Carleman’s theorem

The Carleman’s theorem worked as a miracle. But where does this sorcery come from? Here weinclude a proof as given in Titchmarsh’s book [15]. Let g(z) be an analytic and bounded function inthe upper half-plane with zeroes at rneiθn . Take a contour ∂D consisting of semicircles |z| = ρ, |z| = R,0 ≤ arg z ≤ π, and the parts of the real axis joining them, see the figure (A.1). Take the radii so thatg(z) does not have roots on the contour, i.e R = rn = ρ.

We shall study the imaginary part of integral

I =1

2πi

∮∂D

dz(

1R2 − 1

z2

)log g(z). (A.1)

Don’t forget that complex logarithm is multivalued! The function log g(z) is analytic everywhereexcept at the roots of g(z). If we move along a closed loop around any of these points we will find 2πidifference between the initial and final value of the function. We thus have to define where is the cutand which branch do we consider. It turns out that a convenient choice is a cut passing ∂D at a singlepoint A.4 The cut at point A will be 2πi ∑n∈D αn where αn is multiplicity of root rneiθn .

We shall evaluate the integrals in two different ways. At first, let us divide the contour into four

4Other convenient choices are that the cut passes ∂D at B, C or D. Any other choice would corrupt substitutions used inevaluating the integrals in the following calculations.


parts. On AB we can substitute z = x ∈ R and we will easily find

ℑIA→B = ℑ

12πi

∫ −ρ

−Rdx(

1R2 − 1

x2

)log g(x)

= ℑ

1

2πi

∫ R

ρdx(

1R2 − 1

x2

)log g(−x)

= ℑ

1

2πi

∫ R

ρdx(

1R2 − 1

x2

)(log |g(−x)|+ i arg g(−x))

= − 1

2π

∫ R

ρdx(

1R2 − 1

x2

)log |g(−x)|.

In similar manner we shall find the integral along CD

ℑIC→D = − 12π

∫ R

ρdx(

1R2 − 1

x2

)log |g(+x)|,

so that

ℑ[

IA→B + IC→D]= − 1

2π

∫ R

ρdx(

1R2 − 1

x2

)log |g(+x)g(−x)|. (A.2)

Since g(z) is bounded by some M, expression (A.2) is bounded too5

∣∣∣ℑ [IA→B + IC→D]∣∣∣ ≤ 1

2π

∫ R

ρdx∣∣∣∣ 1R2 − 1

x2

∣∣∣∣ 2 log M =log M

π

(ρ

R2 +1ρ

)which means that ℑ

[IA→B + IC→D] is bounded for any R and also in the limit R → ∞.

As for IB→C, it is an integral of bounded function along a finite curve and is thereby triviallybounded. For the integral along the large semicircle we use substitution z = Reiθ with θ going from 0to π. That leads to

ℑID→A = ℑ

12πi

∫ π

0Reiθidθ

(1

R2 − e−2iθ

R2

)log g(eiθ)

= ℑ− 1

πiR

∫ π

0dθ sin θ log g(eiθ)

= ℑ

− 1

πiR

∫ π

0dθ sin θ

(log∣∣∣g(eiθ)

∣∣∣+ i arg g(eiθ)

=1

πR

∫ π

0dθ sin θ log

∣∣∣g(eiθ)∣∣∣ .

This can be bounded as ∣∣∣ℑID→A∣∣∣ ≤ 1

πR

∫ π

0dθ sin θ log M =

2 log MπR

what goes to zero as R → ∞. Altogether, we have shown that ℑI is bounded for R both finite andinfinite.

Let us now tackle the integral (A.1) differently. Integrating by parts gives

I =1

2πi

[(z

R2 +1z

)log g(z)

]finish

start− 1

2πi

∫∂D

dzg′(z)g(z)

(1z+

zR2

).

In the first term, the start and the finish mean the same point A which is approached from both sidesof the cut. As we go along the contour, the logarithm changes by 2πi ∑n∈D αn, so that the value of thefirst term is

12πi

[(z

R2 +1z

)log f (z)

]finish

start=

12πi

(2πi ∑

nαn

)(−RR

+1−R

)5According to the assumptions, the contour is not crossing zero so ℜ [log z] ≥ −m > −∞ with m some finite positive

number. In the proof we assume M to be maximum of M, em .


which obviously has zero imaginary part. Applying Cauchy’s residue theorem to the second termgives

+∞ > ℑI = ℑ− ∑

n∈Dαn

(1

rneiθn+

rneiθn

R2

)= ∑

n∈Dαn

(1rn

− rn

R2

)sin θn ≥ ∑

n∈D

(1rn

− rn

R2

)sin θn > 0,

where the inequalities follow from the fact that all summands are positive (remember that rn < R andθ ∈ [0, π]) and αn ≥ 1.

Now restrict ourselves to roots with rn < R/2 only. Since all summands are positive, we have

+∞ > ∑ρ<rn<R/2

(1rn

− rn

R2

)sin θn >

34 ∑

ρ<rn<R/2

sin θn

rn> 0.

We finally take the limit R → ∞, so that the sum goes over all roots. We find that

S = ∑n

sin θn

rn

is convergent.

B Maximum entropy method

Imagine that we experimentally measure the dependence of an observable f on a parameter x. Aftersome time we finish with a set of pairs (xi, fi) that can be plotted in a graph. If we have prior informa-tion that the dependence is linear f (x) = ax + b, we may employ the least-squares method to find theoptimal values of coefficients a and b. If we have the knowledge that the dependence is different (e.g.exponential or higher order polynomial) but specified, the least-squares method remains a handy toolto find values of all free parameters. But how are we supposed to proceed if we do not have any priorinformation about the character of the function? The maximum entropy method (MEM) is a Bayesianmethod for estimating the function f if it can be interpreted as a probability distribution of something.6 Westart with a very simple problem of description of a random number generator. Then we show an ob-servation by Jaynes [16] that the statistical ensembles (microcanonical, canonical, and grandcanonical)can be thought of as special cases of the random generators and that MEM reproduces the formulaswe know from statistical physics. In the end we foreshadow that the method may in principle be usedto the problem of analytic continuation of the particle propagator.

Inference of a probability distribution

Let us begin the discussion with an easy case of a random number generator. By experimental exami-nation we may find some of its properties like average value or standard deviation, while not findingout the exact distribution. There are obviously infinitely many probability distributions that are compat-ible with these experimental observations as well as with our prior knowledge (if we have some). Theidea of MEM is to select that distribution which can be encoded in the least possible number of bits. By doingso we minimize the amount of extra information that might be unintentionally put into the probabilitydistribution and which is not statistically relevant. Equivalent to minimizing the information in bitswe may speak of maximizing our ignorance – the entropy. Hence the name of the method.

Depending on whether we work with a discrete or a continuous variable, the entropy of a proba-bility distribution p is defined as either

S[p(x)] = −∑i

p(xi) log[p(xi)] or S[p(x)] = −∫ +∞

−∞dxp(x) log[p(x)].

6So it is not applicable to the linear function from our example, though there are many applications when this can beused.


Both the prior knowledge and experimental results can be incorporated into the formalism as con-straints via the Lagrange multipliers. To see how the method works, let us discuss few easy examples.

Example 1: Think of a random generator that gives numbers in interval [0, 1] – that is the priorknowledge. The question is: What is the best assumption for the probability distribution before we do the firstexperiment?

In this case, the only constraint we have is that probabilities sum to one. According to MEM wehave to minimize the functional

F[ f ] = −∫ 1

0dx f (x) log[ f (x)]− µ

[1 −

∫ 1

0dx f (x)

].

You can check that extremizing δF[ f ]/δ f (x) != 0 leads to f (x) = eµ−1 for x ∈ [0, 1] and f = 0 otherwise.

Multiplier µ is to be determined by normalization of f (x). We see that MEM gives a homogeneousprobability distribution. This makes sense – before performing any experiment there is no reason tohave a bias to some point in the interval.

Example 2:7 Think of a random generator of positive integers. We know8 that it gives x = 2.87 onaverage and that p(2) + p(3) = 0.6. What is the best guess about the probability distribution of thismachine?

In this case we have three constraints. The functional we have to minimize is

F[p] = −∞

∑i=1

p(i) log[p(i)]− λ

(2.87 −

∞

∑i=1

ip(i)

)− ν (0.6 − p(2)− p(3))− µ

(1 −

∞

∑i=1

p(i)

). (B.1)

The calculation is more difficult than in the first example but argumentation is the same. By imposing∂F

∂p(i)!= 0 we find

p(2) = e2λ−1+µ+ν p(3) = e3λ−1+µ+ν p(i = 2, 3) = eiλ−1+µ.

The multipliers λ, µ, ν are to be determined from equations

p(2) + p(3) =(

eλ + 1)

e2λ−1+µ+ν = 0.6∞

∑i=1

p(i) =eλ+µ−1

1 − eλ+(1 − e−ν

) (eλ + 1

)e2λ−1+µ+ν = 1

∞

∑i=1

ip(i) =eλ+µ−1

(1 − eλ)2 +

(1 − e−ν

) (3eλ + 2

)e2λ−1+µ+ν = 2.87.

Solution has to be found numerically, it is λ ≈ −0.383, µ ≈ −0.225, ν ≈ 0.960, the maximized entropyis 1.748 nat.9 The corresponding probability distribution is shown in figure (B.1).

Example 3: Consider the same settings as in example 2 without the condition p(2) + p(3) = 0.6.We now have two constraints, namely the average value and the normalization condition. We have toextremize the same functional as (B.1) with the penultimate term dropped out. Zero variation leads to

p(i) = eiλ−1+µ.

This means, that the probabilities are given by an exponential function with characteristic length −λ.Lagrange multipliers are to be determined from equations

∞

∑i=1

p(i) =eλ+µ−1

1 − eλ

!= 1 and

∞

∑i=1

ip(i) =eλ+µ−1

(1 − eλ)2

!= 2.87. (B.2)

7Taken from Wikipedia [17].8Either we found out experimentally or the manufacturer guarantees so.9"Nat" is a unit of entropy measure that equals ≈ 1.44 bits.


Figure B.1: (left) Result of the problem in example 2. The graph shows the resulting probability distribution ofpositive integers. It is easy to see that p(2) + p(3) = 0.6, the other probabilities are distributed exponentially.

Figure B.2: (right) Result of the problem in example 3. The probabilities are exponentially distributed.

They may be solved analytically, the solution is

λ = log(

1 − 12.87

)≈ −0.428 µ = log

(1

2.87 − 1

)+ 1 ≈ 0.374.

The entropy can be calculated to be 1.855 nat. That is more than in example 2 as it is supposed to be.

How does this relate to the statistical ensembles? With a little abstraction we can think of amacrosystem as of a random generator that gives a specific microstate if asked. Determining the prob-ability of a specific microstate to be realized is pretty much the same as determining the probability ofcertain integer to be generated in the above examples:

I If we are given an isolated system, we are choosing one microstate from many with the sameenergy. The only constraint is the normalization condition and the problem simplifies to example1. The result is a homogeneous distribution, i.e. the microcanonical ensemble.

I If we are given a closed system that is in contact with a reservoir, we allow for microstateswith various energies. We impose a constraint on the average energy ⟨E⟩. We can obtain thisif we replace (2.87 − ∑∞

i=1 ip(i)) 7→ (⟨E⟩ − ∑∞i=1 Ei p(i)) in equation (B.2). This will in a complete

analogy lead to an exponential distribution p(i) ∝ exp [λEi], i.e. the canonical ensemble withtemperature −1/λ.

I If we are given an open system that can exchange both energy and particles with the environment,observed averages ⟨E⟩ and ⟨N⟩ are imposed. The MEM reasoning leads to a generalization ofexample 3. The result is p(i) ∝ exp [−λ1Ei − λ2Ni]. That is the grandcanonical ensemble withtemperature −1/λ and chemical potential λ2/λ1.

The surprising result that observed values of energy and number of particles correspond so accuratelywith the averages ⟨E⟩ and ⟨N⟩ follow automatically from the extremely high dimensionality of theproblem. More thorough discussion can be found in the article by Jaynes [16].

Application to analytic continuation

We now return to the problem of analytic continuation. As mentioned in equation (I.14), the ordinaryparticle propagator can be expressed using the spectral function as

Gk(ω) =∫ +∞

−∞dx

Ak(x)ω − x

. (B.3)


This equation ensures that the Green’s function has the correct analytic properties. Notice that ifone knows the spectral function, he or she also knows the Green’s function at any ω ∈ C. Thismeans that performing the analytic continuation is equivalent to deducing the spectral function. Herethe maximum-entropy method gets on the scene. As mentioned below equation (I.14), the ordinaryspectral function has a probabilistic interpretation, especially we know that it is always non-negativeand that ∫ +∞

−∞dxAk(x) = 1. (B.4)

Equation (B.4) can be checked by inserting (I.14) and directly integrating. One only has to keep in mindthat for electrons c†

kσckσ + ckσc†kσ = 1, the partition sum is Z = ∑m e−Em/T, and that ∑n |n⟩⟨n| = 1 is

the identity operator.Correct usage of the method is intricate. To realize the subtleties we now intentionally reason in

an incorrect way. After all, it seems to be completely clear what to do. For simplicity we suppress themomentum and spin index of the spectral function and of the Green’s function. From all those spectralfunctions that satisfy ∫ +∞

−∞dx

A(x)iωℓ − x

!= G(iωℓ) (B.5)

for the given set of Matsubara frequencies, we will take the one that maximizes the entropy. Takinginto account the constraints, we are left with maximizing with respect to A(x) functional

F[A] = −∫ +∞

−∞dxA(x) log[A(x)]− ∑

ℓ

λℓ

(G(iωℓ)−

∫ +∞

−∞dx

A(x)iωℓ − x

)− µ

(1 −

∫ +∞

−∞dxA(x)

).

The spectral function can be determined by imposing δF/δA(x) != 0. After a little calculation we find

A(x) = exp

[µ − 1 + ∑

ℓ

λℓ

iωℓ − x

]. (B.6)

The multipliers µ and λℓ are to be found from the set of equations (B.5) and from the normalizationcondition.

This all sounds great but it is wrong. The integral in (B.6) diverges and there are some nastycomplex numbers all over the place. We were too hasty. Without going into the details we willmention three enhancements that are usually implemented.

1. To get rid of the complex numbers, one does the Fourier transformation from the imaginaryfrequencies to imaginary time. One can show that for τ ∈ [0, β]

G(τ) =∫ +∞

−∞dx

A(x)e−ωτ

1 + e−ωβ.

Values of this Green’s function at discrete set of points are then taken and used as the constraintsfor A(x).

2. To get rid of the divergences, the expression for entropy is reexpressed as

S[A] = −∫ +∞

−∞p(x) log

p(x)m(x)

where m(x) is the so-called default model. It is a smooth function that is chosen so that it satisfiesthe sum rules and may be used to introduce a cut-off to the problem.

3. We want to generalize the method to include cases with numerical error in the data. In suchcases we do not impose constraints (B.5). One instead does a compromise between maximizingthe entropy and minimizing the sum of squares

χ2 = ∑k

[G(τk)− GA(x)(τk)

]2

σ2k


where τk is the discretized set of times from the interval [0, β], GA(x)(τk) is the Green’s functionat τk given by spectral function A(x), and σk is standard deviation of the data that we know if thedata were obtained using the Quantum Monte Carlo method. One is thus left with maximizingaS[A]− χ2[A]. The coefficient a has to be determined from further Bayesian reasoning.

It is clear that the probabilistic interpretation of the spectral function is essential for the legitimacyof the procedure. This means that although popular, the method is inapplicable for the problem ofanalytic continuation of the off-diagonal Green’s function. This is because the spectral function doesnot have a straightforward probabilistic interpretation.10 Despite this, the method still enjoys a widerange of applications that cover various kinds of image reproduction including police radar imagesdeconvolution, radio interferometry, and tonography [18]. A curious reader is invited to study furtherdetails about the method in the articles by Silver et. al. [19] and Gubernatis et. al. [20] by himself.

C Padé Approximation

Since the maximum entropy method is not applicable to our problem of analytic continuation, wesought an alternative. The method we chose uses the so-called Padé approximation. For the first timeit was proposed to the problem of analytic continuation in condensed matter physics by Vidberg andSerene [21] in 1977. In our work we use the implementation of Beach et. al. [22].

What is the main idea of this method? If you have ever laughed at freshmen who used computersoftware to fit n experimentally obtained data points with a polynomial of order n− 1 and were amazedby the perfect match – that’s just it! Padé approximation is based on fitting a function in such a waythat there are actually no free parameters. The only substantial difference from the "freshmen example"is that Padé approximation uses a rational rather than polynomial fit.

Whatever cheap might this method seem to be, it proves to be very powerful. There is just oneminor but essential detail we did not mention so far – the method is extremely sensitive to the precisionof the input data as is thoroughly discussed in the work by Beach et. al. [22]. This is not a surprise ifone thinks of a specific example – imagine that you are trying to recognize a cosine function, but youonly know its values close to x = 0. It is obviously easy to find the general parabolic behavior of thefunction in the vicinity of the origin. But if one wants to reproduce the oscillatory behavior furtherfrom the origin, very precise values of the function close to the origin are necessary, indeed.

What follows is a very brief excerpt from [22]. We know from the spectral representation11 that atlarge frequencies the Green’s function has an asymptotic behavior

G(ω) ∝1ω

.

We will therefore approximate the spectral function by

G(r)(ω) =p1 + p2ω + · · ·+ prωr−1

q1 + q2ω + · · ·+ qrωr−1 + ωr

as this has the correct assymptotic behavior. The coefficients pn, qn may be determined by specifyingthe value of G(r)(ω) at some 2r points. The points that we shall use are the Matsubara frequencies iωℓ,values of G(r)(ω) are provided by numerical calculation.

How to obtain those 2r coefficients? Let us define 2r-component column vectors

[pq

]=

p1...

prq1...

qr

and g =

g1(iω1)

r

g2(iω2)r

...g2r(iω2r)r

10It is not a usual property of probability distributions to have negative or complex values.11More exactly, we know this from the mere existence of the spectral representation until the spectral function is non-zero

in some finite interval on the real axis and drops down sufficiently quickly outside the interval.


where gℓ = G(iωℓ) with ωℓ the Matsubara frequencies. We further define a matrix

X =

1 iω1 · · · (iω1)

r−1 −g1 · · · −g1(iω1)r−1

1 iω2 · · · (iω2)r−1 −g2 · · · −g2(iω2)r−1

......

......

...1 iω2r · · · (iω2r)r−1 −g2r · · · −g2r(iω2r)r−1

. (C.1)

With these the equations for coefficients pn, qn can be written concisely as

X ·[

pq

]= g ⇒

[pq

]= X−1 · g.

So the problem simplifies to finding the inverse of X. After we obtain the coefficients [ p q ] theanalytic continuation of the Green’s function is

G(r)(z) =

[1 z z2 · · · zr−1 ] · p[

1 z z2 · · · zr−1 ] · q + zr.

Finding the inverse of X is complicated by the fact that its determinant is roughly of order (iω)r(r−1)

where ω is some typical frequency. Remember that a large number of input data is necessary to obtaina reliable analytic continuation (to be specific, we used 20 ≤ r ≤ 40 in our numerical treatment, and10 ≤ r ≤ 80 was used in the article [22]. The complication is that powers to such high exponents areeither very small if the typical frequency |ω| < 1 or very high if |ω| > 1. This implies that one has to gobeyond the usual double precision to be able to compute the inverse matrix precisely. We used softwareMathematica 8.0.4 to implement the method because it offers an easy control over the precision ofthe data. It also offers a wide range of handy numerical techniques, e.g. numerical integration.

The aim of the following pictures is only to persuasively demonstrate the efficacy of Padé ap-proximation. To observe the precision of this method we devised the following test: We generate anarbitrary trial spectral function Borig.(x). We then use integral formula

G(iωℓ) =∫ +∞

−∞dx

Borig.(x)iωℓ − x

to obtain the temperature Green’s function at a set of Matsubara frequencies. We follow with usingthese numerical values as an input for the Padé procedure to reproduce some output spectral functionBrepr.(x). We then compare Borig.(x) to Brepr.(x) qualitatively by plotting them next to each other.

We now present results of our test. The trial function we used and its Padé approximations areshown in figure (C.1). In all shown reproductions we kept 2r = 60 and temperature t = 0.008. Weintentionally used an odd spectral function because, as discussed in section II.1, the anomalous spectralfunction of a superconductor has this property. What cannot be seen so easily is that this spectralfunction also has a vanishing first moment, to resemble the object of study even more. We have tomention though that generalizing the test to arbitrary spectral functions was equally successful.

D Calculation of sum rules

Calculation of the (anti)commutators on the right-hand side of the sum rules (II.11) is a tedious torture.We only list here several notes leading to the results. We begin the listing with the Hubbard model(I.48). If r denotes momentum, then[

cr↑,HT

]= εrcr↑[

cr↑,HU

]= −U

N ∑p,q

c†p+q↓cq+r↑cp↓[

cr↑,H]

=[cr↑,HT

]+[cr↑,HU

]= εrcr↑ −

UN ∑

p,qc†

p+q↓cq+r↑cp↓


-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

10 digits precision

-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

15 digits precisions

-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

20 digits precision

-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

30 digits precision

-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

40 digits precision

-3 -2 -1 1 2 3x

-1.0

-0.5

0.5

1.0AHxL

reproduced

original

50 digits precision

Figure C.1: The original spectral function (shown in blue) and the reproduced (shown in red). Double precisioncorrespond to the second picture. Precision used in the numerical solution of the Eliashberg equations correspondsto the fourth picture. In all reproductions we kept 2r = 60 Matsubara points and we worked at temperaturet = 0.008. Reproduction of other trial functions was equally successful.


[cr↑,H

], c−r↓

= −U

N ∑q

cq↑c−q↓[[cr↑,HT

],H]

= ε2rcr↑ − εr

UN ∑

p,qc†

p+q↓cq+r↑cp↓[[cr↑,HU

],HT

]= −U

N ∑p,q

(εp + εq+r − εp+q

)c†

p+q↓cq+r↑cp↓

[[cr↑,HU

],HU

]= −

(UN

)2

∑p,q,p′,q′

[c†

p+q↓cq+r↑c†p′−q′↑cp′↑cp−q′↓

−c†p+q↓c†

p′+q′↓cq+q′+r↑cp′↓cp↓ − c†p+q+q′↓c†

p′−q′↑cp′↑cq+r↑cp↓

][[

cr↑,HT

],H]

, c−r↓

= −εr

UN ∑

pcp↑c−p↓[[

cr↑,HU

],HT

], c−r↓

= −U

N ∑p

(2εp − εr

)cp↑c−p↓

[[cr↑,HU

],HU

], c−r↓

= −U

NU ∑

pcp↑c−p↓ − 2

(UN

)2

∑p,q,p′

c†p+q↓cp↓cq−p′↑cp′↓[[

cr↓,H]

,H]

, c−r↓

=

[[cr↑,HT

],H]

, c−r↓

+[[

cr↑,HU

],H]

, c−r↓

=

= −UN ∑

p

(2εp + U

)cp↑c−p↓ − 2

(UN

)2

∑p,q,p′

c†p+q↓cp↓cq−p′↑cp′↓ (D.1)

where we used εk = ε−k, i.e. the dispersion relation is symmetric.The last term in the last expression simplifies if we transform to the real space operators. The

corresponding transformation relations are

c†kσ =

1√N

∑i

eip·Ri c†Riσ

and ckσ =1√N

∑i

e−ip·Ri cRiσ.

Appying these relations we find

∑p,q,p′

c†p+q↓cp↓cq−p′↑cp′↓ =

1N2 ∑

p,q,p′∑ijkℓ

eip·(Ri−Rj)eiq·(Ri−Rk)eip′·(Rk−Rℓ)c†Ri↓cRj↓cRk↑cRℓ↓

= N ∑i

c†Ri↓cRi↓cRi↑cRi↓ = 0

where the last equality follows from consecutively repeating the operator cRi↓. As a consequence wecan rewrite equation (D.1) as[[

c−r↓, H]

, H]

, cr↑

= −U

N ∑p

(2εp + U

)cp↑c−p↓.

To sum it up, the anomalous spectral function for the Hubbard model have to exactly satisfy∫R

dxB(r, x) = 0∫R

dxxB(r, x) = −UN ∑

q

⟨cq↑c−q↓

⟩∫

Rdxx2B(r, x) = −U

N ∑q

(2εq + U

) ⟨cq↑c−q↓

⟩.

A notable observation is that right-hand side of these equations does not contain the state of momen-tum r which is manifestly present on the left-hand side.


Let us now proceeed with the electron-phonon Hamiltonian given in equation (I.23). We list hereagain some notes leading to the sum rules.

[cr↑,H1

]= εrcr↑[

cr↑,H1

], c−r↓

= εr

cr↑, c−r↓

= 0[

cr↑,H2

]= 0 =

[cr↑,H2

], c−r↓

[cr↑,H3

]=

1√Ω

∑q,s

gsr−q,r

(aqs + a†

−qs

)cr−q↑[

cr↑,H3

], c−r↓

=

1√Ω

∑q,s

gsr−q,r

(aqs + a†

−qs

) cr−q↑, c−r↓

= 0

[cr↑,H4

]=

1Ω ∑

q =0∑pσ

Vq + V−q

2︸︷︷︸ℜ[Vq]

c†p+qσcpσcq+r↑

[cr↑,H4

], c−r↓

=

1Ω ∑

q =0ℜVq c−(q+r)↓c(q+r)↑ =

[cr↑,H

], c−r↓

[cr↑,H

]= εrcr↑ +

1√Ω

∑q,s

gsr−q,r

(aqs + a†

−qs

)cr−q↑ +

1Ω ∑

q =0∑pσ

ℜ[Vq]

c†p+qσcpσcq+r↑[[

cr↑,H1

],H]

= εr

[cr↑,H

]= ε2

rcr↑ +1√Ω

∑q,s

εr gsr−q,r

(aqs + a†

−qs

)cr−q↑ +

1Ω ∑

q =0∑pσ

εrℜ[Vq]

c†p+qσcpσcq+r↑[[

cr↑,H1

],H]

, c−r↓

=

1Ω ∑

q =0εrℜ

[Vq

]c−(q+r)↓c(q+r)↑[[

cr↑,H2

],H]

= 0 =[[

cr↑,H2

],H]

, c−r↓

[[

cr↑,H3

],H1

]=

1√Ω

∑q,s

εr−qgsr−q,r

(aqs + a†

−qs

)cr−q↑[[

cr↑,H3

],H2

]=

1√Ω

∑q,s

ωqsgsr−q,r

(aqs − a†

−qs

)cr−q↑[[

cr↑,H3

],H1

], c−r↑

=

[[cr↑,H3

],H2

], c−r↑

= 0[[

cr↑,H3

],H3

]=

1Ω ∑

q,s∑q′,s′

gsr−q,rgs′

r−q−q′,r−q

(aqs + a†

−qs

) (aq′s′ + a†

−q′s′

)cr−q−q′↑[[

cr↑,H3

],H3

], c−r↓

= 0[[

cr↑,H3

],H4

]=

1√Ω

∑q,s

gsr−q,r

1Ω ∑

q′ =0ℜ[Vq′

]∑pσ

(aqs + a†

−qs

)c†

p−q′σcpσcr−q−q′↑[[cr↑,H3

],H4

], c−r↓

=

[[cr↑,H3

],H]

, c−r↓

=

1√Ω

∑q,s

gsr−q,r

1Ω ∑

q′ =0ℜ[Vq′

] (aqs + a†

−qs

)cq′−r↓cr−q−q′↑[[

cr↑,H4

],H1

]=

1Ω ∑

q =0∑pσ

ℜ[Vq

] (εq+r + εp − εp+q

)c†

p+qσcpσcq+r↑[[cr↑,H4

],H1

], c−r↓

=

1Ω ∑

q =0ℜ[Vq

] (εq+r + εp − εp+q

)c−q−r↓cq+r↑

CHAPTER III. EXPERIMENTAL DETERMINATION OF BK(X) 48[[cr↑,H4

],H2

]=

[[cr↑,H4

],H2

], c−r↓

= 0[[

cr↑,H4

],H3

]=

1Ω ∑

q =0∑pσ

ℜ[Vq

] 1√Ω

∑q′,s′

(aq′s′ + a†

−q′s′

) (gs′

q+r−q′,q+rc†p+qσcpσcq+r−q′↑

+ gs′p−q′,pc†

p+qσcp−q′σcq+r↑ − gs′p+q,p+q+q′c†

p+q+q′σcpσcq+r↑

)[[

cr↑,H4

],H3

], c−r↓

=

1Ω ∑

q =0ℜ[Vq

] 1√Ω

∑q′,s

(aq′s + a†

−q′s

) (gs

q+r−q′,q+rc−q−r↓cq+r−q′↑

+ gs−q−r−q′,−q−rc−q−r−q′↓cq+r↑ − gs

−r−q′,−rc−r−q−q′↓cq+r↑

)where we used that ωqs = ω−qs, i.e. the phonon dispersion is symmetric too.

E Solving the Eliashberg equations

In chapter II we need to solve the Eliashberg equations. For reasons of conciseness we did not includediscussion of the appropriate techniques there. This appendix is supposed to fill in this gap. Westart by showing our implementation of the Eliashberg equations (I.44) and (I.45). We further studyhow changing the number of Matsubara frequencies and changing the overall precision influences ourresults for variables Zωn , ∆ωn . We also investigate how the analytic continuation and thus also thededuced spectral function Bk(x) are affected by changing the same parameters.

We start with rewriting the Eliashberg equations into dimensionless variables. For variables thathave a dimension of energy this is done by dividing by the Debye energy12 ω0. We define

t := 1/βω0

xn := ωn/ω0 = (2n + 1)πtdn := ∆ωn /ω0.

The electron mass renormalization function Zk(ω) is dimensionless, though we simplify the notationas Zωn ≡ Zn.

We will further consider the Debye pairing function gDebye(ωn − ωm) from equation (I.42).13 If weinsert it into (I.44) and (I.45), application of the introduced dimensionless parameters leads to

Zn = 1 +λπtxn

∑m

[1 − (xn − xm)

2 log

(1 +

1

(xn − xm)2

)]xm√

x2m + d2

m(E.1)

dn =λπtZn

∑m

[1 − (xn − xm)

2 log

(1 +

1

(xn − xm)2

)]dm√

(xm)2 + d2m

(E.2)

where we assumed dn to be real as is motivated in the paragraph before (II.1). We also fix value of λto be 0.5 throughout the work.

When solving equations (E.1) and (E.2) we are forced to introduce an artificial cutoff for the num-ber of considered Matsubara frequencies. We take this number always even and denote it N. Thecorresponding extremal Matsubara frequencies are ±(N − 1)πt. The equations themselves are solvediteratively. We define the initial values

Z(0)n = 1 and d(0)n = dBCS = 2e−1/λ

for all n. Here dBCS is the BCS result for the energy gap taken from chapter 6 of lecture notes [6]. Inthe entire work we fix λ = 0.5 which slightly exaggerates the typical physical values.

The iteration then proceeds followingly: Having k-th step values d(k)n use equation (E.1) to calculateZ(k+1)

n . Then use values of Z(k+1)n and d(k)n as an input for equation (E.2). Reiterate this equation several

times14 to obtain d(k+1)n . Repeat this procedure until the values are converged.

12Since we put h = 1 we do not distinguish between energy and frequency.13Validity of our calculations for Einstein model is not tested.14We reiterated the second equation four times per each usage of the first one. This choice seems to optimize the time of

the calculation for our input parameters which are mentioned later in the text.


Figure E.1: Dependence of energy gap ∆ on temperature for varying number of considered Matsubara frequenciesN. You can observe that the numerics fails at temperatures close to absolute zero. This happens when thelargest Matsubara frequency decreases below the Debye frequency, i.e. when Nπt . 1. This relation claries whyincreasing N enhances the results for small temperatures. The values close to the phase transition seems to berather well determined already for a small N. This region however has a serious drawback that cannot be seenin the picture the equations converge very slowly there. In the work we decided to do the calculations at xedtemperature t = 0.015 (dashed line) which is away enough to avoid both of these complications. This graph wasproduced for F = 14 (double) and p = 9.

What do we mean by "converged values" depends on the accuracy of our calculation. There aretwo aspects we have to distinguish:

1. What is the precision (number of digits) of variables used in the computer software?

2. What accuracy (difference from the exact value) do we want to achieve?

We address both of them. We denote the number of digits in the calculations as F, i.e. the precision is10−F. This is kept 40 in the entire work unless otherwise stated. As for the accuracy of the results, ateach step of the iteration we calculate

D(k) = ∑n

(d(k+1)

n − d(k)n

)2. (E.3)

We consider the values to be converged when D(k) decreases below some given number 10−p, p isusually taken to be 30. We used software Wolfram Mathematica 8.0.4 because it offers a comfortableway of setting the data precision.

Now that we explained proceedings of the numerical computation, let us study the influence of

I the number of considered Matsubara frequencies,

I precision F and accuracy p

on the solution of the Eliashberg equations and on the analytic continuation. We start with determiningthe interval of temperatures at which the system exhibits superconductivity. This is done in figure(E.1) which shows the dependence of d1 at frequency ω1 = πt on temperature.15 You can see that thenumerics fails if the temperature is very low. As will be clarified in a moment, this failure happenswhen ωmax . ω0. There is also a problem at temperature close to the phase transition where thenumerics converges very slowly. We decided to do all the calculations at fixed temperature t = 0.015which is far enough to avoid both of the mentioned complication.


Figure E.2: (left) Plot of Zn for varying number of Matsubara frequencies N. Notice that for all N there isalways a problem with the tails. On the other hand the solutions for all values of N seem to be well converged inthe center. As we use those values for as input for the Padé procedure, it should be no surprise that the analyticcontinuation does not depend on N much. These calculations were done at t = 0.015 with precision F = 40 andp = 30.

Figure E.3: (right) Plot of ∆ω for varying number of Matsubara frequencies N. The results seems to be veryaccurate already for small N. The only notable dierence occurs around zero where we can see that taking smallN leads to a slightly exaggerated results.

Figure E.4: (left) Inuence of variable number N of Matsubara frequencies on the deduced normal spectralfunction. We xed precision to F = 40, p = 30 and Padé approximation parameters to r = 20, s = 1. Noticethat the curves are almost identical. This is rather surprising because we mentioned in appendix C that enormousprecision is needed to do the analytic continuation reliably. Explanation is that even though accuracy of our resultmay not be perfect, the error is in a way consistent rather than in a form of a randomly spread noise.

Figure E.5: (right) Results of the same test on the anomalous spectral function.

Let us now observe how varying the number of Matsubara frequencies N influences the set ofvalues Zn, dn. The solutions for four different values of N are plotted in figures (E.2) and (E.3). Onecan observe that there are obvious errors in the tails of Zn. Fortunately, as we shall reveal shortly, thesetails do not corrupt the analytic continuation because this is performed from the values close to theorigin. In that region the data seem to be quite stable. The data for dn look even better. The only effectof finite N is a slightly exaggerated value of dn around zero frequency. Again though – as the overallshape is the same, this rescaling does not spoil the analytic continuation. Influence of the varying Non the analytic continuation at fixed precision is shown in figures (E.4) and (E.5). The data in the workwere calculated for N = 1500 Matsubara frequencies.

Let us finally keep the number of Matsubara frequencies fixed at N = 400 and investigate effect of

15Points ω1 is the positive Matsubara frequency that lies closest to zero. Figure (E.3) shows that the value of d1 is alwaysthe largest of all dn.


Figure E.6: This graph shows inuence of the precision of the calculation on the precision parameters F andp. The number of Matsubara frequencies was kept at N = 400 and Padé approximation parameters at r = 20and s = 1. All curves lie on top of each other, meaning that the lowest of these precisions is already sucient.It is not possible though to lower the precision even more because that leads to crash of the calculation. Forexplanation see the text.

the precision. As we discussed in appendix (C) about the Padé approximants, there is certain boundaryprecision below which the computation crashes – it fails to find inverse of the matrix (C.1). As shown infigure (E.6), results obtained at this "threshold" precision are so accurate, that increasing the precisiondoes not change the spectral function at all.

Based on this analysis we decided to do most of the calculations for parameters listed in the workin table (II.1). If other parameters were used we state so explicitely.

Bibliography

[1] S. R. Park and Y. Cao et al. Broken relationship between superconducting pairing interaction andelectronic dispersion kinks in La2−xSrxCuO4, 2013. arXiv:1304.0505.

[2] Adrian Cho. High Tc: The Mystery That Defies Solution. Science, 314(5802):1072–1075, 2006.

[3] Wikipedia. Sokhotski–Plemelj theorem — Wikipedia, The Free Encyclopedia, 2004. [Online; ac-cessed 25-April-2013].

[4] Gerald Rickayzen. Green’s Functions and Condensed Matter. Academic Pr., Waltham, United States,1984.

[5] Yoichiro Nambu. Quasi-Particles and Gauge Invariance in the Theory of Superconductivity. Phys-ical Review, 117(3):648–663, 1960.

[6] Richard Hlubina. Kvantová teória mnohocasticových systémov. lecture notes, 2012.

[7] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of Superconductivity. Physical Review,108(5):1175–1204, 1957.

[8] Steven R. White. Spectral weight function for the two-dimensional Hubbard model. PhysicalReview B, 44(9):4670–4673, 1991.

[9] Jerry Orloff. 18.03 Uniqueness of Laplace Transform, 2013. lecture notes on Differential equations– accessed 27-April-2013.

[10] Wikipedia. Stone–Weierstrass theorem — Wikipedia, The Free Encyclopedia, 2004. [Online; ac-cessed 27-April-2013].

[11] Wikipedia. Schwarz reflection principle — Wikipedia, The Free Encyclopedia, 2004. [Online;accessed 27-April-2013].

[12] Jun Liu. A general method for testing validity of one-particle spectral functions. Computer PhysicsCommunications, 182(8):1585–1588, 2011.

[13] Encyclopedia of Mathematics. Analytic function. [Online; accessed 27-April-2013].

[14] Gordon Baym and N. David Mermin. Determination of Thermodynamic Green’s Funcions. Journalof Mathematical Physics, 2(2):232–234, 1961.

[15] Edward Charles Titchmarsh. The Theory of Functions. Oxford University Press, London, England,1939.

[16] E. T. Jaynes. Information Theory and Statistical Mechanics. Physical Review, 106(4):620–630, 1957.

[17] Wikipedia. Principle of maximum entropy — Wikipedia, The Free Encyclopedia, 2004. [Online;accessed 30-April-2013].

[18] S. F. Gull and J. S. Killing. Maximum entropy method in image processing. Communications, Radarand Signal Processing, IEE Proceedings F, 131(6):646–659, 1984.

52

BIBLIOGRAPHY 53

[19] R. N. Silver, D. S. Sivia, and J. E. Gubernatis. Maximum-entropy method fot analytic continuationof quantum monte carlo data. Physical Review B, 41(4):2380–2389, 1990.

[20] J. E. Gubernatis, Mark Jarrell, R. N. Silver, and D. S. Sivia. Quantum Monte Carlo simulationsand maximum entropy: Dynamics from imaginary-time data. Physical Review B, 44(12):6011–6029,1991.

[21] H. J. Vidberg and J. W. Serene. Solving the Eliashberg Equations by Means of N-point PadéApproximants. Journal of Low Temperature Physics, 29(3/4):179–192, 1977.

[22] K. S. D. Beach, R. J. Gooding, and F. Marsiglio. Reliable padé analytical continuation methodbased on a high-accuracy symbolic computation algorithm. Physical Review B, 61(8):5147–5157,2000.

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