TVE 14 023 juni Examensarbete 15 hp Juni 2014

Efficient Computational Procedure for the Analytic Continuation of Eliashberg Equations

Joakim Johansson Fredrik Lauren Abstract Efficient Computational Procedure for the Analytic Continuation of Eliashberg Equations Joakim Johansson & Fredrik Lauren

Teknisk- naturvetenskaplig fakultet UTH-enheten The superconducting order parameter and the mass renormalization function can be solved either at Besöksadress: discrete frequencies along the imaginary axis, or as a Ångströmlaboratoriet Lägerhyddsvägen 1 function of continuous real frequencies. The latter is Hus 4, Plan 0 done with a method called analytic continuation. The analytic continuation can conveniently be done by Postadress: approximating a power series to the functions, the Box 536 751 21 Uppsala Padè approximation. Studied in this project is the difference between the Padè approximation, and a Telefon: formally exact analytic continuation of the functions. 018 – 471 30 03

Telefax: As it turns out, the Padè approximant is applicable to 018 – 471 30 00 calculate the superconducting order parameter at temperatures sufficiently below the critical Hemsida: temperature. However close to the critical temperature http://www.teknat.uu.se/student the approximation fails, while the solution presented in this report remains reliable.

Handledare: Alexandros Aperis Ämnesgranskare: Henrik Olssson Examinator: Martin Sjödin ISSN: 1401-5757, TVE 14 023 juni Popul¨arvetenskaplig Sammanfattning

Fenomenet supraledning ¨arn¨arett material leder str¨omutan n˚agon resistans. Det sker n¨arvissa material kyls ned till temperaturer n¨araden absoluta nollpunken, 0 Kelvin, d˚ade ¨overg˚arfr˚anden fasta fasen till den supraledande fasen. Man kan t¨anka sig denna fas¨overg˚angp˚asamma s¨attsom att till exempel vatten ¨overg˚arfr˚anflytande till fast form under 0◦C eller att j¨arn¨overg˚artill att vara magnetiskt under 770◦C. F¨oratt unders¨oka om en s˚adanfas¨overg˚anghar ¨agtrum studeras vanligen n˚agonparameter som ¨arnoll innan ¨overg˚angen,men antar n˚agot¨andligtv¨ardeefter ¨overg˚angen. I fallet supraledning kallas denna parameter ∆. Supraledare uppt¨acktes ˚ar1911 d˚aH. Kamerlingh Onnes1 skulle studera resistansen hos kvick- silver i fast form under f¨orh˚allandend¨artemperaturen n¨armadesig den absoluta nollpunkten. Detta hade nyligen blivit m¨ojligtatt studera eftersom en teknik f¨oratt f˚afram flytande helium precis uppfunnits och d˚akunde anv¨andassom kylmedel. Det han uppt¨ackte var att n¨artempera- turen kommit ned till cirka 4 K (-269◦C) f¨orsvann resistansen helt. Det dr¨ojdesedan mer ¨an40 ˚arav experimenterande innan en teori som kunde beskriva fenomenet presenterades. J. Bardeen, L.N. Cooper och J.R. Schrieffer ¨arnamnen bakom BCS- teorin2 som f¨orklararvarf¨oroch hur ett material kan leda str¨omutan n˚agotelektriskt motst˚and. Grunden i denna teori ¨aratt valenselektronerna i materialt paras ihop tv˚a-och-tv˚a. Dessa elektronpar konstituerar en ny slags ”superkanal” f¨orelektrisk str¨om,och till˚atero¨andlig led- ningsf¨orm˚aga.Det vill s¨agamaterialet till˚atsleda str¨omhelt utan elektriskt motst˚and.∆ ¨arett m˚attp˚adenisiteten av s˚adanaelektron-par, och kan allts˚astuderas f¨oratt p˚avisaom ett material ¨ari den supraledande fasen eller i den normal fasta fasen. BCS-teorin har dock sina brister, och en mer fullst¨andigteori har presenterats av Eliashberg. H¨arinf¨orsett tidsberoende som tidigare f¨orbis˚agsav BCS-teorin. Detta g¨oratt ∆ blir en tids- beroende variabel, eller genom Fouriertransform frekvensberoende. Ekvationerna som beskriver tillst˚andethos det supraledande materialet l¨osesrelativt enkelt numeriskt med frekvenser fr˚anden imagin¨araaxeln. Tillst˚andsekvationerna f¨orm˚angaav de supraledande materialen l¨osesrelativit enkelt med imagin¨arafrekvenser. F¨oratt resultatet ska vara till nytta och ge anv¨andbarfysikalisk information m˚astel¨osningen¨overs¨attasmed v¨ardenp˚afrekvenser fr˚anden reella axeln. Tekniken f¨oratt ut¨oka definitiosm¨angdenf¨oren analytisk funktion kallas analytisk forts¨attning. Det kan anv¨andasf¨or att g˚afr˚andet komplexa talplanet till den reella axeln. Den teknik som vanligen anv¨andsf¨oratt g¨oraanalytiska forts¨attningkallas f¨orPad`eapproximationen, och ¨arsom namnet antyder en approximation. Det som studeras i den h¨arrapporten ¨aremellertid en metod som genomf¨ordenna analytiska forts¨attning analytiskt, och kan d¨armedses som en exakt l¨osningav tillst˚andsekvationerna p˚aden reella axeln. I rapporten j¨amf¨ors¨aven denna analytiska metod med Pad`eapproximationen. Resultatet fr˚anprojektet visar att Pad`eapproximationen ¨aren bra l¨osnigf¨oratt l¨osaden ana- lytiska forts¨attningenmen att den exakta l¨osningen¨arstabil under f¨orh˚allandend˚aPad`eapproximationen blir instabil.

1Kamerling Onnes tilldelades Nobelpriset i fysik 1913 f¨orsin forskning p˚amaterial vid l˚agatemperaturer 2J. Bardeen, L.N. Cooper och J.R. Schieffer tilldelades ˚ar1972 nobelpriset f¨ordenna teori

iii Contents

1 Introduction 1 1.1 Description ...... 1 1.1.1 Goals ...... 2

2 Theory 3 2.1 Fourier transform properties ...... 3 2.2 Complex analysis ...... 3 2.2.1 Analytic functions ...... 3 2.2.2 Analytic continuation ...... 4 2.3 Solid state physics ...... 4 2.3.1 Solids ...... 5 2.3.2 Superconductivity ...... 6 2.4 Numerical methods ...... 7 2.4.1 Numerical integration ...... 7 2.4.2 Iterative methods ...... 7

3 Methods 8 3.1 The equations ...... 8 3.1.1 Properties of the Fourier transform ...... 8 3.1.2 Interval for the calculations ...... 9 3.1.3 The α2F (ω)-distribution ...... 9 3.2 Dirac delta function ...... 9 3.3 The Lorentzian ...... 11 3.4 Smearing factor ...... 12 3.5 Solving the system of equations ...... 13

4 Result 14 4.1 Results with delta Dirac function ...... 14 4.2 Results with the Lorenztian ...... 18

5 Discussion 21 5.1 The Dirac delta approach ...... 21 5.2 The Lorentzian approach ...... 21 5.3 Other approaches for solving the equations ...... 21 5.3.1 Convolution ...... 22

6 Conclusions 24

A Complex analysis 25 A.1 Analytic functions and power series ...... 25

B Additional plots 26 B.1 Additional plots for the Dirac delta approach ...... 26 B.2 Additional plots for the Lorentzian approach ...... 29

iv 1. Introduction

1.1 Description

When cooled down below critical temperature, a material may possess the ability to exhibit zero electrical resistance; the material is then said to be in a state of superconductivity. For a specific group of these superconducting materials, the superconducting state can be described with a system of coupled equations, known as the Eliashberg equations. These equations calculates the mass renormalization function, Z(t), and the superconducting order parameter ∆(t). The analytic solution to this system of equation is unfortunately not known. However, when Z(t) and ∆(t) are Fourier transformed, a discrete numerical solution can be obtained for the complex frequencies iωn = i(2n + 1)πT for Z(iωn) and ∆(iωn). The Eliashbergs equations are

|ωn0 |<ωc 1 X ωn0 Z(iω ) = 1 + λ(ω − ω 0 ) (1.1) n n n p 2 2n + 1 0 2 n0 ωn0 + ∆(iωn )

|ωn0 |<ωc X  ∗  ∆(iωn0 ) Z(iω )∆(iω ) = πT λ(ω − ω 0 ) − µ (ω ) . (1.2) n n n n c p 2 0 2 n0 ωn0 + ∆(iωn )

Where ωn = (2n+1)πT are the Matsubara frequencies with n ∈ Z , T is the temperature, λ and µ∗ are the electron-phonon and Coulomb coupling strengths. The summation for n’ is truncated for some cut off frequency, ωc. To get a useful solution and to be able to calculate real physical quantities the solutions need to be functions of real frequencies ω. The technique to expand the domain of an analytic function is called analytic continuation and it can be used to go from the complex plane to the real frequency axis. A commonly used technique to do the analytic continuation is with an approximation called Pad`eapproximation method. The disadvantages of this method is that it is an approximation and becomes unreliable under certain circumstances. A relatively recent technique provides a formally exact solution for the analytic continuation and the associated equations becomes:

X ∗
∆(iωn0 ) ∆(ω)Z(ω) = πT λ(ω − iωn0 ) − µ (ωc) p 0 n0 R(iωn ) ∞ (1.3) Z Z(ω − ω0)∆(ω − ω0) + iπ dω0Γ(ω, ω0)α2F (ω0) pZ(ω − ω0)R(ω − ω0) −∞

πT X ωn0 Z(ω) = 1 + i p )λ(ω − iωn0 ) ω 0 n0 R(iωn ∞ 0 (1.4) πT Z (ω − ω 0 )Z(ω − ω ) + i dω0Γ(ω, ω0)α2F (ω0) n ω pZ2(ω − ω0)R(ω − ω0) −∞ where

2 2 R(iωn) = ωn + ∆ (iωn) (1.5)

1 1 ω − ω0 ω0 Γ(ω, ω0) = tanh + coth
(1.6) 2 2T 2T

Z ∞ 2 0 0 α F (ω ) λ(ω − iωn) = − dω 0 (1.7) −∞ ω − iωn − ω and α2F (ω) varies for different approaches.

1.1.1 Goals The goal is to write a section of code in Matlab that solves (1.3) and (1.4), taking fixed parameters and values of Z(iω) and ∆(iω) from equation (1.1) and (1.2) as input and produces values of ∆ and Z for real frequencies. These results will also be compared to the known Pad`eapproximation of the solution to the equations.

2 2. Theory

2.1 Fourier transform properties

The physical quantities in this project is mainly treated in their frequency domain. The following properties of the Fourier transform are important for the method later in this report. Theorem 2.1.1 (Fourier transform symmetry). Consider a real function f of some real variable x and its fourier transform fˆ(ω). Then the following holds: if f is an even function then fˆ(ω) is also an even function. If f is an odd function then fˆ(ω) is also an odd function. Proof. Only the proof of the even function relationship is presented here, the proof for the odd function relationship can easily be done in a similar way. Substituting f(−x) with f(x) (assuming f is an even function) in the Fourier transform yields:

∞ Z fˆ(ω) = f(−x)e−i2πωxdx (2.1)

−∞

Substitute −x with u, and dx with −du yields:

−∞ Z fˆ(ω) = −f(u)e−i2πω(−u)du (2.2)

∞ ∞ Z = f(u)e−i2π(−ω)udu (2.3)

−∞ = fˆ(−ω) (2.4)

2.2 Complex analysis

This section includes necessary results from complex analysis and the idea is to make the underlying mathematics to the methods in the report more understandable. Only results that are directly used in the report will be stated in this section. Theorems that are not included in this section but are necessary for the proofs can be found in Appendix A.

2.2.1 Analytic functions Analytic continuation is one of the key-concepts in this project, but before this technique is intro- duced, basic knowledge about analytic functions are needed. The Pad´eapproximation is defined here as well. Definition 2.2.1. A function f of a complex variable is said to be analytic in an open set A if it is complex differentiable at every point z0 ∈ A. If B is a non-open set, then f is said to be analytic in B if f is analytic on some open set containing B. We say that f is analytic at a point z0 if f is analytic on some neighborhood of z0. [1]

3 Theorem 2.2.1. Given a power series representation of a function f, (see A.1.2 in Appendix A) of a complex variable z and a positive radius of convergence, R, then

∞ X i f(z) = ai(z − z0) (2.5) i=0 is an analytic function on the disk DR(z0) with radius R centered at z0.

i Proof. Each term of the series ai(z − z0) is entire (analytic function whose domain is the whole complex plane). It follows that f(z) is analytic on DR(z0)if the sum (A.3) converges to f(z) there, which it does by definition (see A.1.2). Theorem 2.2.2 (Identity theorem). Let f be a function of a complex variable and analytic on a domain, Ω, then f is uniquely determined over Ω by it’s values on some non-empty subset of Ω. Proof. This proof can be done making use of the fact that an analytic function can be represented by it’s Taylor series. The full proof is excluded in this report but can be found here [2]. Definition 2.2.2 (The Pad`eapproximation). Given a function f(z) with the power series repre- sentation: ∞ X i f(z) = ciz (2.6) i=0 A Pad`eapproximation, [m/n], given the two integers m and n is the fractional function:

Pm j j=0 ajz [m/n] = Pn k . (2.7) 1 + k=1 bkz that agrees with f(z) to the highest possible order, or equivalently the first m + n terms of the Maclaurin expansion of [m/n] agrees with the first m + n terms of (2.6).[3] Given (2.7), f(z) can be expressed as: f(z) = [m/n] + O(zm+n+1). (2.8)

2.2.2 Analytic continuation Analytic continuation is the technique provided to extend the domain over which an analytic function is defined. This technique is built upon the results of the identity theorem 2.2.2 in the previous section.

Definition 2.2.3. Let Ω1 and Ω2 be two overlapping domains, i.e. Ω1 ∩ Ω2 6= ∅, and let the functions f1 and f2 be defined on Ω1 and Ω2 respectively, such that f1 = f2 on Ω1 ∩ Ω2. Then we call f2 the analytic continuation of f1 into Ω2 and vice versa.[1] As illustrated in example 2.2.1 below, one could use the technique of analytic continuation repeatedly to obtain an overlapping series of disks to greatly extend the domain of an analytic function. Example 2.2.1. Consider an analytic function f of a complex variable z defined on an arbitrary domain Ω1 with the boundary C1. According to theorem 2.2.1 for some arbitrary point z0 in Ω1, f can be expressed as a power series, with some radius of convergence, R. In the case that R exceeds the distance between z0 and the boundary C1, the domain Ω2 : |z − z0| < R is not a sub domain of Ω1 and the series expansion defines an analytic function f2 on Ω2 such that f2 = f1 for all z in Ω1 ∩ Ω2 which is clearly an analytic expansion of f1 into Ω2. Now one could start from Ω2, pick a new point z1 ∈ Ω2 and repeat this procedure to create an overlapping chain of these domains, see figure 2.1.

2.3 Solid state physics

The intention is to present some basic theory from the field solid state physics to make the results in this report more understandable. The sections will be kept basic and stripped from complicated equations since there is no point going into details. Results corresponding to a first course in quantum physics will not be presented.

4 Ω2

Ω1 z1 z0

R

C1

Figure 2.1: A figure of the repeated process of analytic continuation using power series expansions of an arbitrary function. The function is originally defined on the domain Ω1, then by expanding the function in a point z0 the radius of convergence is overlapping the original domain. The series expansion is an analytic continuation of the function into the new domain Ω2.

2.3.1 Solids To better understand the following section about superconductivity, 2.3.2, one need to have some knowledge about how atoms behaves in a solid. This section will give an introduction to solids and establish some useful terminology.

Crystal lattice Any physical crystal can be described by defining a lattice with a basis, Bravais lattice (2.3.1), and information about how the atoms, ions etc. are arranged in this lattice. [4] Definition 2.3.1. The Bravais lattice (in i-dimensions) consists of all the points that can be reached by the postion vector:

R = n1a1 + n2a2 + ··· + niai. (2.9)

Where (a1, a2,..., ai) are any i vectors not in the same plane spanning the lattice, and (n1, n2, . . . , ni) are integers. For metals, each point in the Bravais lattice is occupied by an atom and the valence electrons of each atom are detached from it’s ”parent” nucleus. It can be thought of as the valence electrons form an electron gas over the nucleus. These electrons are free to wander throughout the solid subjected to the combined potential of the entire crystal structure, and each electron can be treated, in a primitive picture, as a free particle in a box.[4] Definition 2.3.2. In k-space the surface that are separating the occupied and unoccupied states is called the Fermi surface, and is indicated with a subscript F. [4]

Definition 2.3.3. The energy of the electrons associated with the highest occupied quantum state is called the Fermi energy, and is denoted with EF . [4] Definition 2.3.4. When the atoms in the crystal lattice get perturbed, they start to oscillate around their equilibrium position. The Debye frequency is the highest frequency for these vibra- tions. [4]

Band theory The energy levels corresponding to a free atom are discrete values. For a material on the other hand, the energy levels are not discrete values but continuous within certain energy intervals containing allowed energy levels for the atoms. For that reason the energy levels for a material are called energy bands. Between the intervals are gaps separating each energy interval. The

5 conduction band is the energy band for the material where the energy is high enough to free an electron from its atom. What determines whether a material is a conductor or an insulator is if there are electrons in the conduction band. For insulators, the gap between the Fermi energy and the conduction band is big, unlike a conductor where the Fermi level is the same as the lowest energy of the conduction band. For semi conductors the gap is small and the electrons may reach the conduction band under certain circumstances (for example increasing the temperature). [4]

2.3.2 Superconductivity The derivation of the Eliashberg equations, which are the subject of the calculations in the report, requires knowledge of advanced theory which is out of the scope of this project. However the following part will hopefully contribute to the understanding of the basic concepts of the Bardeen- Cooper-Schrieffer (BSC) theory, which is the standard model for the microscopic description of superconductivity. A more rigorous explanation of superconductivity can be found in [4] Superconductivity is the phenomenon where a material can carry a current with zero resistivity. A metal become a superconductor below a critical temperature, TC , and when this happens, a phase transition takes place at TC . Below TC , the material is now in a new state (phase) of matter which has different physical properties than its previous metallic phase. The latter state can be recovered if the metal is heated above TC . An every day example of a phase transition is witnessed in water. Below 0 ◦C water turns into ice; thus a new state with different physical properties. [4]

Definition 2.3.5 (Critical temperature). When cooled down below the critical temperature, TC , the material possesses superconducting properties. Above TC the material behaves like a normal metal with associated properties. [4] The standard theory available today to describe superconductivity on a microscopic level is the BSC theory. The BSC theory works well for a large number of known superconductors, the so-called conventional superconductors. The fundamental idea of the BCS theory is that electrons traveling through the solid become correlated pair-wise and end up forming the Cooper-pairs. [4] The pairing arises as an electron travels through a solid. The electron will, due to its negative charge, attract the positive ions in the lattice and leave behind a increased density of positive charge. The accumulation of positive charge will attract another electron behind the first one creating a correlation between the electrons. Electrons that participate in such processes have energies lying in the Fermi surface. The vibrations of the perturbed ions by the electrons is characterized by their Debye frequency and this naturally affects how electrons correlate. [4]

Cooper pairs When two electrons pair up, they form what is called a Cooper pair. As the temperature gradually lowered below TC , more electrons form Cooper pairs and at absolute zero, all the electrons have been paired-up. Picture it like a two fluid model, where electrons form the normal fluid and the Cooper pairs the ”superfluid”. [4] The electron, which is a fermion, by itself has half-integer valued spin, either 1/2 or −1/2 corresponding to spin up (↑) or spin down (↓) respectively. However the Cooper pairs has a whole integer spin (0 or 1) and behaves, statistically, like a boson. This means that the Cooper pairs are allowed to occupy the same quantum state without violating the Pauli exclusion principle. [4]

Infinite current

When the superfluid density is finite, that is below TC i.e. when the material becomes super- conducting, the material now has two ”channels” through which it can conduct: either through the normal electron fluid or through the superfluid. The superfluid allows infinite current, thus a supercurrent (a current with zero resistance) passes through the material. [4]

Phase transition parameters, ∆ and Z In order to model theoretically a phase transition, the quantity that can act like an indicator of the transition needs to be chosen. A variable that is zero before the transition occurs, and finite afterwards (below the critical temperature) is a good choice. In the case of superconductivity, this parameter is called ∆ and it measures (in the simplest cases) whether there are Cooper pairs formed

6 in the metal. Another variable that will be encountered below is the renormalization function Z. Z measures how electrons get affected by their interaction with the ion lattice. [4]

Eliashberg The BSC model assume that the pairing interaction between two electrons is instantaneous. This approximation works well for many conventional superconductors but the full problem including time retardation is described by the theory first formulated by Eliashberg. [4]

2.4 Numerical methods

2.4.1 Numerical integration Definition 2.4.1 (The trapezium method). The trapezium method approximates an integral over an interval by breaking down the area under the graph of the function f(x) into trapezoids and calculate their total area. The approximation becomes:

b N Z b − a X f(x)dx ≈ (f(x ) + f(x )) (2.10) 2N n n+1 a n=1 The error in the trapezium method is proportional to h3, where h is the stepsize in the inte- gration. [5]

2.4.2 Iterative methods The system of equations in this project is a nonlinear system of equations including integrals. To solve the system, an iterative method is used that takes the solution from the previous iteration when calculating the solution to the present iteration

xi+1 = f(xi)x0 = g(x) (2.11) where x is the solution to the system and f(x) represents to the system of equations. g(x) is the initial guess. To determine if the solution has converged, two subsequent iterations are compared and if the difference is smaller than a specific tolerance the solution has converged.

7 3. Methods

This section treats assumptions and techniques that are used to get the results. The section also explains consequences of some of the theorems presented in the theory section.

3.1 The equations

Remember the equations (1.1) and (1.2) from the introduction, which are restated down below to increase the readability of this section:

1 X ωn0 Z(iω ) = 1 + λ(ω − ω 0 ) (3.1) n n n p 2 2n + 1 0 2 n=1 ωn0 + ∆(iωn )

|ωn0 |<ωc X  ∗  ∆(iωn0 ) Z(iω )∆(iω ) = πT λ(ω − ω 0 ) − µ (ω ) . (3.2) n n n n c p 2 0 2 n0 ωn0 + ∆(iωn ) These are the Eliashberg equations, see section 2.3.2 and their solutions are plotted in figure 3.1. Since the analytical solutions are not known for Z(iω) and ∆(iω), only the values of the equations in iωn = i(2n + 1)πT (the Matsubara frequencies) are known. Using the theory of analytic continuation and complex analysis (see section 2.2) the equations (3.1) and (3.1) can analytically be rewritten as the equations (1.3) and (1.4) in the introduction, which are also restated below. However the derivation is outside the scope of this project but it is done in an article by F. Marsiglio, M. Schossmann, and J. P. Carbotte[6].

X ∗
∆(iωn0 ) ∆(ω)Z(ω) = πT λ(ω − iωn0 ) − µ (ωc) p 0 2 0 n0 R(iωn )Z (ω − ω ) ∞ (3.3) Z Z(ω − ω0)∆(ω − ω0) + iπ dω0Γ(ω, ω0)α2F (ω0) pZ(ω − ω0)R(ω − ω0) −∞

πT X ωn0 Z(ω) = 1 + i p λ(ω − iωn0 ) ω 0 n0 R(iωn ∞ 0 (3.4) πT Z (ω − ω 0 )Z(ω − ω ) + i dω0Γ(ω, ω0)α2F (ω0) n ω pZ2(ω − ω0)R(ω − ω0) −∞

3.1.1 Properties of the Fourier transform By looking at figure 3.1 it is clear that both ∆(iω) and Z(iω) are even functions of ω, which implies that ∆(t) and Z(t) must be even functions of t, 2.1.1. Furthermore, because ∆(t) and Z(t) are physical quantities they are real. Since ∆(ω) and Z(ω) are their Foureier transform, the following results hold from properties of the Fourier transform:

∆0(−ω) = ∆0(ω) and ∆00(−ω) = −∆00(ω) (3.5) and

8 Z [eV] ∆ [eV] ·10−3 2.5 3

2 2

1

1.5 0

1 −1 −20 −10 0 10 20 −20 −10 0 10 20

iω/ωD iω/ωD

Figure 3.1: The solutions of the Eliashberg equations for the imaginary axis. To the left Z(iωn). To the right ∆(iωn). Both functions are plotted for the Matsubara frequencies, iωn = i(2n + 1)πT

Z0(−ω) = Z0(ω) and Z00(−ω) = −Z00(ω) (3.6) where

∆(ω) = ∆0(ω) + i∆00(ω) (3.7) Z(ω) = Z0(ω) + iZ00(ω).

3.1.2 Interval for the calculations

The Pad`eapproximant for Z(ω) and ∆(ω) is plotted in figure 3.2 on the interval [0, 8ωD] for a specific temperature. As seen in the figures the functions converge to a constant value when ω > 5ωD. For other temperatures the function could converge for bigger ω and for that reason the interval for the calcualtions are restricted to ω ∈ [−8ωD, 8ωD]. Still, due to the shift in the argument in (3.3) and (3.4), values outside this interval is needed for the calculations. One approach is to use values from the solution of the Pad`eapproximation. The Pad´eap- proximation can be calculated on any interval, say ω ∈ [−10ωD, 10ωD], and since it is a good approximation for the functions ∆(ω) and Z(ω), they are considered to be a good guess for the functions outside the interval [−8ωD, 8ωD]. Furthermore since the behaviour of the functions are known in the negative frequency domain, see 3.1.1, the interval of interest can be constrained to the positive axis only, see figure 3.3. Because of (3.5) and (3.6) the calculations can be restricted to the interval ω ∈ [0, 8ωD]. Outside of this interval, (3.5) and (3.6) are used in the calculations.

3.1.3 The α2F (ω)-distribution The function α2F (ω) corresponds to the spectrum of how the electrons are affected for different frequencies in the lattice. In this report the function is approximated with a one-peak Dirac-delta function and then a Lorentzian function. The two different approaches should give equal results in the limit when the Lorentzian width becomes infinitely small.

3.2 Dirac delta function

As a first approach the function α2F (ω) in equation (3.3) and (3.4) is approximated to be a composition of Dirac delta functions of the form:

9 Z ∆ ·10−2 real real 1 8 imaginary imaginary

6 0.5

4 0

2 −0.5

0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 3.2: Pade Approximant of both Z and ∆ for real frequencies ω = [08ωD]. The Pad´e approximant is evaluated for the temperature T = 1K.

1 ... i ... iωD ... endi

A ωC ... 0 ... ωD ... ωC

1 ... kωD ... endk

B 0 ... ωD ... 5ωD

Figure 3.3: Illustration of how the interesting interval is handled using vectors. The above vec- tor, A, is the data from the Pad`eapproximation and is defined between two cut off frequencies [−ωC , ωC ]. The lower vector, B, is the restricted interval for the calculations, [0, 8ωD]. Since the behaviour on the negative frequency axis is known, only the real part is included. When the ar- gument is shifted outside of B the values from A are used instead. The indices iωD and kωD mark the Debye frequency in A and B respectively.

10 Ω α2F (ω) = λ δ(ω − Ω) − δ(ω + Ω)
, (3.8) 2 where λ and Ω are the electron-phonon coupling and Debye frequency respectively, which are specific constants for different materials. This approach reduces the integral equations to regular equations due to the properties of the Dirac delta function. In addition; imposing the consequence of the Fourier transform 3.1.1, and using equation (1.6) one finally arrive at the simplified forms of (3.3) and (3.4) below:

2 X ∆(ωn0 ) h 2λω i ∆(ω)Z(ω) = πT D − µ∗(ω ) p 2 2 C 0 ω − (ω − iωn0 ) n0 R(iωn ) D 1 h ω + ωD ωD
∆(ω + ωD) + i λωDπ − tanh + coth p (3.9) 4 2T 2T R(ω + ωD) ω − ωD ωD
∆(ω − ωD) i + tanh + coth ) p 2T 2T R(ω − ωD)

2 πT X ωn0 2λω Z(ω) = 1 + i D p 2 2 ω 0 ω − (ω − iωn0 ) n0 R(iωn ) D 1 π h ω + ωD ωD
(ω + ωD) + i λωD − tanh + coth p (3.10) 4 ω 2T 2T R(ω + ωD) ω − ωD ωD
(ω − ωD) i + tanh + coth ) p . 2T 2T R(ω − ωD)

For the positive frequency axis. From 3.1.1, the calculations can be restricted to ω ∈ [0, 8ωD]. Hereby this approach will be refered to as the ”Dirac delta approach”.

3.3 The Lorentzian

Another approach to solve the integrals in (3.4) and (3.3) is to use the Lorentzian in the evaluation of the function α2F (ω).

 γ2 γ2  α2F (ω) ∝ λ − (3.11) γ2 + (ω − Ω)2 γ2 + (ω + Ω)2 where γ2 ∈ (0, 1]. Frequencies close to Ω will get increased amplitude while frequencies far away vanishes. This effect increases as γ2 → 0. The Lorentzian is used instead of (3.8) because it will make the solution more ”smooth”, and it is a better reflection of the reality. However using the Lorentzian comes with a price, and the equations (3.4) and (3.3) can no longer be simplified due to the fortunate properties of the Dirac delta function.

X ∗
∆(iωn0 ) ∆(ω)Z(ω) = πT λ(ω − iωn0 ) − µ (ωc) p 0 n0 R(iωn ) ∞ (3.12) Z ∆(ω − ω0) + iπ Γ(ω, ω0)α2F (ω0) dω0 pR(ω − ω0) −∞

πT X ωn0 Z(ω) = 1 + i p λ(ω − iωn0 ) ω 0 n0 R(iωn ∞ (3.13) πT Z (ω − ω 0 ) + i Γ(ω, ω0)αF (ω0) n dω0 ω pR(ω − ω0) −∞

11 Finally, after using the property of the Fourier transform, equation (3.12) and (3.13) becomes

X ∗
∆(iωn0 ) ∆(ω)Z(ω) =πT λ(ω − iωn0 ) − µ (ωc) p 0 n0 R(iωn ) ∞ Z 1n ω − ω0 ω0  ∆(ω − ω0) + iπ dω0α2F (ω0) tanh + coth (3.14) 2 2T 2T pR(ω − ω0) 0  ω + ω0 ω0  ∆(ω + ω0) o + − tanh + coth 2T 2T pR(ω + ω0)

πT X ωn0 Z(ω) = 1 + i p λ(ω − iωn0 ) ω 0 n0 R(iωn ∞ πT Z 1n ω − ω0 ω0  ω − ω0 + i dω0α2F (ω0) tanh + coth (3.15) ω 2 2T 2T pR(ω − ω0) 0  ω + ω0 ω0  ω + ω0 o + − tanh + coth 2T 2T pR(ω − ω0)

and since only positive frequencies are used

∞ Z 0 0 0 ” 0 2ω λ(ω − ω ) = dω α F (ω ) 02 2 (3.16) ω − (ω − iωn) 0 and

γ2 α2F (ω) ∝ λ (3.17) γ2 + (ω − Ω)2 When implementing the Lorentztian as it is written in (3.17) numerically, it is rewritten in the following form:

2 Ωh 1 1 i α F (ω) = λ 2 2 − 2 2 h(γc − |ω − Ω|). (3.18) 2 γ + (ω − Ω) γ + γc

The reason for this is to emulate the behaviour of the Dirac pulse. Where γ and γc are parameters chosen to satisfy:

∞ Z dω h 1 1 i Ω 2 2 − 2 2 h(γc − |ω − Ω|) = 1, (3.19) ω γ + (ω − Ω) γ + γc 0 i.e. γ and γc are dependent of Ω. Again, the calculations are restricted to the interval ω ∈ [0, 8ωD]. Hereby this approach will be referred to as the ”Lorentzian approach”.

3.4 Smearing factor

A problem with singularities arises when the equations are implemented as they are stated above (equation (3.9) - (3.10) and (3.14) - (3.15)). To avoid the effect of these singularities a small imaginary part i is added to the frequency vector, acting as an smearing factor. The constant  is in the regime [0.0001, 0.0015]. The effect is shown in figure 3.4. Comparing with the Pad´e approximant, figure 3.2, the solution with the smearing factor is a better fit.

12 Re(∆) [eV] Im(∆) [eV] ·10−2 ·10−3 Real Real 1 Imaginary Imaginary 5

0.5

0

0

−5 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 3.4: The figure shows the effect of the smearing factor i. Plotted to the left is the solution of ∆(ω) with no smearing factor calculated with the Dirac delta approach. In the figure to the right a smearing factor has been added to smooth out the singularities. The solid line is the real part of each solution, and the dashed line is the imaginary part.

3.5 Solving the system of equations

When finally arrived at the system of equations an iterative method is chosen to find a solution. The iteration algorithm follows • while error > tolerance

– solve the equations with ∆i and Zi to get ∆i+1 and Zi+1

• error = max(|Zi − Zi+1|, |∆i − ∆i+1|) The error is compared termwise in the solution vectors. This comparison lets no single value in the solution differ much from the previous. Both the convergence in the real and imaginary part for ∆(ω) and Z(ω) compared, this is to ensure that that all the values in the solution vectors are close to the previous iteration. The difference between two values is taken as the absolute value.

 error = max |real(∆(ωi) − ∆(ωi+1))|; |imag(∆(ωi) − ∆(ωi+1))| (3.20) ; |real(Z(ωi) − Z(ωi+1))|; |imag(Z(ω) − Z(ωi+1))|

The integrals in the equation using the Lorentzian approach is evaluated with the trapezodial method, see section 2.4.1.

13 4. Result

4.1 Results with delta Dirac function

Solving the analytic continuation using the α2F -function as it is stated in (3.8) results in the ∗ following plots in this section. The parameters: µ = 0.1, λ = 1.5, ωD = 100 is kept fixed while the temperature T varies for different plots. Here the plots for T = 1K, 12K and 20K for both the mass renormalization function and the superconducting parameter are presented in figure 4.1 - 4.2, 4.5 - 4.6 and 4.7 - 4.8 respectively. The x-axis of the figures shows the frequency for the ion lattice and is scaled by the Debye frequency, ωD. On the y-axis are the values of each part of the functions ∆(ω) and Z(ω) and the Pad`eapproximation in electron-volts [eV ]. A small smearing factor has been added to each solution in the manner discussed in 3.4. The Pad`eapproximation for each temperature is added for comparison. Results for T =2K, 5K, and 10K can be found in appendix B. These plots does not show any unexpected features, but can still be interesting. Looking at figure 4.1 - 4.6 the analytic solution follows the Pad`eapproximation and the dif- ference between them are small. For the figures 4.6 and 4.8 the Pad`eapproximation diverges in comparison to the analytic solution, which is stable and gives a reasoable result.

Re(Z) [eV] Im(Z) [eV] 8 Analytic Analytic 8 Approximation Approximation 6 6

4 4

2 2

0 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.1: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 1K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

14 Re(∆) [eV] Im(∆) [eV] ·10−2 ·10−2 1 Analytic Analytic 1 Approximation Approximation

0.5 0.5

0 0

−0.5

0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.2: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 1K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 4 Analytic Analytic Approximation 2 Approximation

3

1

2

0

1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.3: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 11K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

15 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 6 6 Analytic Analytic 4 Approximation Approximation 4 2 2 0

−2 0

−4 −2 −6 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.4: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 11K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 2 Analytic Analytic Approximation Approximation 3 1.5

2.5 1

2 0.5

1.5 0

1 −0.5 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.5: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 12K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

16 Re(∆) [eV] Im(∆) [eV] ·10−2 ·10−3 Analytic Analytic Approximation Approximation 5 0

−0.5 0

−1 −5

0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.6: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 12K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 Analytic 1.5 Analytic Approximation Approximation 3

1 2.5

2 0.5

1.5 0 1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.7: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 20K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

17 Re(∆) [eV] Im(∆) [eV] ·10−13 ·10−14 0.5 Analytic Analytic Approximation 0 Approximation

0 −2

−4 −0.5

−6

−1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.8: The analytic solution (solid line) and the Pad`eapproximation (dashed line) at tem- perature T = 20K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

4.2 Results with the Lorenztian

This section presents the result when α2F (ω) is a Lorentzian function, as discussed in 3.3. The ∗ parameters: µ = 0.1, λ = 1.5, ωD = 100 is kept fixed while the temperature T varies for different plots. The results from the Lorentzian appraoch are for comparing the results from the Delta dirac apoproach. For this reason, only results from T = 12 and T = 20 is included in this section and the rest can be found in Appendix B. The x-axis of the figures shows the frequency for the ion lattice and is scaled by the Debye frequency, ωD. On the y-axis are the values of each part of the functions ∆(ω) and Z(ω) both from the Delta dirac approach and from the Lorentzian approach in electron volts, [eV ]. A small smearing factor has been added to each solution in the manner discussed in 3.4. The results from the Dirac delta approach are for comparison. Looking at figure 4.9 and 4.10, when the temperature is lower than TC the results form the Lorentzian approach follows the results from the Dirac delta approach well. In figure 4.11 and 4.12 the results from the Lorentzian approach differs when compared to the results from the Dirac delta approach.

18 Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2 0.5

1.5 0 1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.9: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 12K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 4 Lorentzian Lorentzian Dirac Delta 4 Dirac Delta 2

2 0

0 −2

−4 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.10: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 12K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

19 Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta Dirac delta 3 1

2.5

0.5 2

1.5 0

1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.11: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 20K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

Re(∆) [eV] Im(∆) [eV] ·10−15 ·10−15 8 6 Lorentzian Lorentzian 6 Dirac Delta 4 Dirac Delta

4 2

2 0

0 −2

−2 −4

−4 −6 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 4.12: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 20K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

20 5. Discussion

5.1 The Dirac delta approach

In the analytic solution in figure 4.1 to 4.8, distinctive steps are noticed. Also, these steps seems to occur repeatedly with a period of ωD. This behaviour cannot be found in the Pad´eapproximant. The origin of this is not clear, except that it is a consequence of the singularities in the equations (3.10) and (3.9). This can be confirmed by looking at the effect of the smearing factor, 3.4, where the addition of this factor heavily reduces these steps. The real power of the analytic method tested in this project is shown in figure 4.6 and 4.8. In the latter the temperature, T = 20K, is clearly above the critical temperature since the superconducting order parameter is in the order of 10−14 which is numerically zero. For this temperature the Pad`e approximation collapses and diverges at roughly 2ωD while the analytic solution does not. In fact the analytic solution keeps its characteristic shape even though the temperature is above (or close to) the critical temperature which is shown in figure 5.1. In figure 4.6 the temperature, T = 12K, is not above the critical temperature (the superconducting parameter is finite). However, for T = 12K the approximation still collapses and shows only nonsense in comparison with the analytic solution. In addition, as can be seen in figure 4.4, this behaviour of the approximation is not observed. The stability of the analytc solution suggests that this is not something that comes gradually with increasing temperature, but rather something that happens above some specific fraction of the critical temperature. Even though the Pad`eapproximation for the superconducting order parameter behaves strange for the higher temperatures tested, this is not true for the mass renormalization function. Except for some difference in amplitude, which is due to the smearing factor which flattens out the peaks of the function, the approximation and the analytic solution fits perfectly for all temperatures.

5.2 The Lorentzian approach

The solution of the Lorentzian approach fits the solution of the Dirac delta approach. This can be interpreted as evidence of the correctness of the solutions. Just as in the case with the Dirac delta approach this solution is convergent for all temperatures. The Lorentzian approach yields a smoother function than the Dirac delta approach, which is to be expected since the Lorentzian lacks the Dirac pulse’s spikiness. Both the Lorentzian- and the Dirac delta approach are just ways to model the reality. No physical quantity can behave lika a dirac pulse, therefore arguably the Lorentzian approach better approximates the real system. However the downside of this approach is that the system of equation to solve is more computationally heavy, compared to the Dirac delta counterpart. With this in consideration the Dirac delta approach does an impressively good job modelling both the mass renormalization function and the superconducting order parameter. In addition, both methods supply a much more detailed spectrum than the commonly used Pad`eapproximation.

5.3 Other approaches for solving the equations

This section is about how to continue working to solve the equations (1.3) and (1.4).

21 ∆ [eV] ∆ [eV] ·10−3 ·10−15 4 Real Real Imaginary 4 Imaginary

2 2

0 0

−2 −2

0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure 5.1: The anlytic solution of the superconducting order parameter, ∆(ω) for T = 12K to the left, and T = 20K to the right. Both the real part (solid line) and the imaginary part (dashed line) is shown.

5.3.1 Convolution Remember the integral in the equation (1.3)

Z ∞ Z(ω − ω0)∆(ω − ω0) dω0Γ(ω, ω0)α2F (ω0) (5.1) p 2 0 0 −∞ Z (ω − ω )R(ω − ω ) when inserting Γ(ω, ω0) and sort the terms with respect to their argument, it becomes more obvious how the convolution will proceed

1 Z ∞  ω0 Z(ω − ω0)∆(ω − ω0) dω0 α2F (ω0) coth p 2 0 0 2 −∞ 2T Z (ω − ω )R(ω − ω ) (5.2) ω − ω0 Z(ω − ω0)∆(ω − ω0)  +α2(F (ω0) tanh 2T pZ2(ω − ω0)R(ω − ω0)

Applying convolution leads to

1 ω − ω0 Z(ω − ω0)∆(ω0) α2F (ω0) ? tanh 2 2T pZ2(ω0)R(ω0) (5.3) 1 ω0 Z(ω − ω0)∆(ω0) + α2F (ω0) coth ? 2 2T pZ2(ω0)R(ω0)

Then apply the same technique for the integral in equation (1.4). The convolution will probably decrease the running time since it will lead to less iterations during the computation and in Matlab this will reduce the running time. It is also possible to use discrete convolution to calculate the sum in equation (1.3)

X  ∆(iωn0 ) ∗ ∆(iωn0 )  λ(ω − iωn0 )p + µ (ωc)p (5.4) 0 0 n0 R(iωn ) R(iωn ) were the first term can be evaluated with convolution

∆(iωn0 ) λ(iωn0 ) ? p (5.5) R(iωn0 )

22 The second term in (5.4) needs to bee calculated without convolution. The sum in equation (1.4) can also be evaluated with convolution.

23 6. Conclusions

Comparing the analytic results with the results from the Pad`eapproximation shows that they are similar for low frequencies and low temperatures. When the temperature is closer to TC the Pad`e approximation fails and is not reliable. The analytic solution is, on the other hand, convergent for all temperature and gives good results for temperatures close to TC . The approach with the Lorentzian is in good agreement with the results from the Dirac delta approach. This verifies that the present code is reliable and works well with functional forms of the Eliashberg function. Moreover, realistic calculations involve the use of generalized Lorentzian approaches where more than two Lorentzians are used to describe how the interaction between electrons and ions depends on the frequency. Therefore, the Lorentzian method presented in this report and the written code represent the most accurate technique for this type of calculations. On the other hand, since the two approaches for the analytic solution gives similar results, it is in some cases better to use the Dirac delta approach to save computational resources.

24 A. Complex analysis

A.1 Analytic functions and power series

Definition A.1.1. Let f be a function of a complex variable z, then f is complex differentiable at z0 and has the derivative:

0 f(z0 + ∆z) − f(z) f (z0) = lim (A.1) ∆z→0 ∆z there, provided that this limit exists. Theorem A.1.1 (Cauchy-Riemann condition). Let f(z) = u(x, y) + iv(x, y) be an analytic func- tion, then u and v satisfy the Cauchy-Riemann equations:

∂u ∂v ∂v ∂u ∂x = ∂y , ∂x = − ∂y (A.2) Proof. This follows from the definition of complex differentionation A.1.1, by putting ∆z = η + iζ and letting ∆z approach zero along the real and imaginare axis respectivly. Definition A.1.2. A power series is an infinit series of the form

∞ X i 2 f(z) = ai(z − z0) = a0 + a1(z − z0) + a2(z − z0) + ... (A.3) i=0 being centered at z0, where ai is called the ith coefficient of the power series. All power series may converge for only z = z0, or for all z or for some z within a radius of convergence, R: |z − z0| < R. Theorem A.1.2 (removable isolated singularities). Let f(z) and g(z) be two analytic functions and both have zeros at z0 of the same order, then f(z) h(z) = (A.4) g(z) has a removable isolated singularity at z0.

Definition A.1.3. Let A ∈ C, and let γ be a simple close curve in A. If it is possible to continuously deform γ into another simple closed curve γ0 without leaving A, then γ is said to be homotopic to γ0 in A.

25 B. Additional plots

B.1 Additional plots for the Dirac delta approach

Re(Z) [eV] Im(Z) [eV] 6 Analytic Analytic 4 Approximation Approximation 5 3 4 2 3 1 2 0 1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.1: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at tem- perature T = 2K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

26 Re(Z) [eV] Im(Z) [eV] 6 4 Analytic Analytic Approximation Approximation 5 3

4 2

3 1

2 0 1 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.3: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at tem- perature T = 5K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

Re(∆) [eV] Im(∆) [eV] ·10−2 ·10−2 1 1 Analytic Analytic Approximation 0.8 Approximation

0.5 0.6

0.4

0 0.2

0 −0.5 −0.2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.2: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at temper- ature T = 2K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

27 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−2 1 Analytic Analytic Approximation 0.8 Approximation 5 0.6

0.4 0 0.2

0 −5 −0.2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.4: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at temper- ature T = 5K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 4 Analytic Analytic Approximation 2 Approximation 3

1 2

1 0

0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.5: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at temper- ature T = 10K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

28 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 Analytic Analytic 6 Approximation Approximation 5 4

0 2

0

−5 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.6: The analytic solution (solid line) and the Pad´eapproximant (dashed line) at temper- ature T = 10K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

B.2 Additional plots for the Lorentzian approach

Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2

0.5 1.5

1 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.7: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 1K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

29 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 6 8 Lorentzian Lorentzian 4 Dirac Delta Dirac Delta 6

2 4 0 2 −2

0 −4

−6 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.8: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 1K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2

0.5 1.5

1 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.9: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 2K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

30 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 6 8 Lorentzian Lorentzian 4 Dirac Delta Dirac Delta 6

2 4 0 2 −2

0 −4

−6 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.10: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 2K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2

0.5 1.5

1 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.11: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 20K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

31 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 6 8 Lorentzian Lorentzian 4 Dirac Delta Dirac Delta 6

2 4 0 2 −2

0 −4

−6 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.12: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 20K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2

0.5 1.5

1 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.13: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 10K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

32 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 Lorentzian Lorentzian 4 6 Dirac Delta Dirac Delta

2 4

0 2

−2 0 −4 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.14: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 10K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

Re(Z) [eV] Im(Z) [eV] 3.5 Lorentzian Lorentzian Dirac delta 1.5 Dirac delta 3

2.5 1

2

0.5 1.5

1 0 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.15: The solution, using the Lorentzian approach (solid line) and the solution using the Dirac Delta approach (dashed line) at temperature T = 11K for the mass renormalization function, Z(ω). The real part to the left and the imaginary part to the right.

33 Re(∆) [eV] Im(∆) [eV] ·10−3 ·10−3 4 6 Lorentzian Lorentzian Dirac Delta Dirac Delta 2 4

0 2

−2 0

−4 −2 0 2 4 6 8 0 2 4 6 8

ω/ωD ω/ωD

Figure B.16: The solution, using the Lorentzian approach (solid line) and the analytic solution using the Dirac delta approach (dashed line) at temperature T = 11K for the super conducting order parameter, ∆(ω). The real part to the left and the imaginary part to the right.

34 Bibliography

[1] George B. Arfken and Hans-Jurgen Weber. Mathematical methods for physicists. Elsevier, San Diego, Calif., 6. ed. edition, 2005. [2] Joseph Bak and Donald J. Newman. Complex analysis. Springer, New York, 3rd ed. edition, 2010. [3] MathWorld Weisstein, Eric W. Pad´e approximant, May 2014. http://mathworld.wolfram.com/PadeApproximant.html.

[4] Neil W. Ashcroft and N. David Mermin. Solid state physics. Saunders College, Philadelphia, 1976. [5] S.R. Otto and J.P. Denier. An Introduction to Programming and Numerical Methods in MAT- LAB [electronic resource]. Springer-Verlag London Limited, London, 2005.

[6] M. Schossmann F. Marsiglio and J. P. Carbotte. Iterative analytic continuation of the electron self-energy to the real axis. Physical Review B, 37(10), 1988.

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