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
Besöksadress:
Ångströmlaboratoriet
Lägerhyddsvägen 1
Hus 4, Plan 0
Postadress:
Box 536
751 21 Uppsala
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
The superconducting order parameter and the mass
renormalization function can be solved either at
discrete frequencies along the imaginary axis, or as a
function of continuous real frequencies. The latter is
done with a method called analytic continuation. The
analytic continuation can conveniently be done by
approximating a power series to the functions, the
Padè approximation. Studied in this project is the
difference between the Padè approximation, and a
formally exact analytic continuation of the functions.
As it turns out, the Padè approximant is applicable to
calculate the superconducting order parameter at
temperatures sufficiently below the critical
temperature. However close to the critical temperature
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
Populärvetenskaplig
Sammanfattning
Fenomenet supraledning är när ett material leder ström utan någon resistans. Det sker när vissa
material kyls ned till temperaturer nära den absoluta nollpunken, 0 Kelvin, då de övergår från den
fasta fasen till den supraledande fasen. Man kan tänka sig denna fasövergång på samma sätt som
att till exempel vatten övergår från flytande till fast form under 0◦ C eller att järn övergår till att
vara magnetiskt under 770◦ C.
För att undersöka om en sådan fasövergång har ägt rum studeras vanligen någon parameter som
är noll innan övergången, men antar något ändligt värde efter övergången. I fallet supraledning
kallas denna parameter ∆.
Supraledare upptäcktes år 1911 då H. Kamerlingh Onnes1 skulle studera resistansen hos kvicksilver i fast form under förhållanden där temperaturen närmade sig den absoluta nollpunkten.
Detta hade nyligen blivit möjligt att studera eftersom en teknik för att få fram flytande helium
precis uppfunnits och då kunde användas som kylmedel. Det han upptäckte var att när temperaturen kommit ned till cirka 4 K (-269◦ C) försvann resistansen helt.
Det dröjde sedan mer än 40 år av experimenterande innan en teori som kunde beskriva
fenomenet presenterades. J. Bardeen, L.N. Cooper och J.R. Schrieffer är namnen bakom BCSteorin2 som förklarar varför och hur ett material kan leda ström utan något elektriskt motstånd.
Grunden i denna teori är att valenselektronerna i materialt paras ihop två-och-två. Dessa
elektronpar konstituerar en ny slags ”superkanal” för elektrisk ström, och tillåter oändlig ledningsförmåga. Det vill säga materialet tillåts leda ström helt utan elektriskt motstånd. ∆ är ett
mått på denisiteten av sådana elektron-par, och kan alltså studeras för att påvisa om ett material
är i den supraledande fasen eller i den normal fasta fasen.
BCS-teorin har dock sina brister, och en mer fullständig teori har presenterats av Eliashberg.
Här införs ett tidsberoende som tidigare förbisågs av BCS-teorin. Detta gör att ∆ blir en tidsberoende variabel, eller genom Fouriertransform frekvensberoende.
Ekvationerna som beskriver tillståndet hos det supraledande materialet löses relativt enkelt
numeriskt med frekvenser från den imaginära axeln.
Tillståndsekvationerna för många av de supraledande materialen löses relativit enkelt med
imaginära frekvenser. För att resultatet ska vara till nytta och ge användbar fysikalisk information
måste lösningen översättas med värden på frekvenser från den reella axeln. Tekniken för att utöka
definitiosmängden för en analytisk funktion kallas analytisk fortsättning. Det kan användas för
att gå från det komplexa talplanet till den reella axeln.
Den teknik som vanligen används för att göra analytiska fortsättning kallas för Padèapproximationen,
och är som namnet antyder en approximation. Det som studeras i den här rapporten är emellertid
en metod som genomför denna analytiska fortsättning analytiskt, och kan därmed ses som en exakt
lösning av tillståndsekvationerna på den reella axeln. I rapporten jämförs även denna analytiska
metod med Padèapproximationen.
Resultatet från projektet visar att Padèapproximationen är en bra lösnig för att lösa den analytiska fortsättningen men att den exakta lösningen är stabil under förhållanden då Padèapproximationen
blir instabil.
1 Kamerling
2 J.
Onnes tilldelades Nobelpriset i fysik 1913 för sin forskning på material vid låga temperaturer
Bardeen, L.N. Cooper och J.R. Schieffer tilldelades år 1972 nobelpriset för denna teori
iii
Contents
1 Introduction
1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2 Theory
2.1 Fourier transform properties .
2.2 Complex analysis . . . . . . .
2.2.1 Analytic functions . .
2.2.2 Analytic continuation
2.3 Solid state physics . . . . . .
2.3.1 Solids . . . . . . . . .
2.3.2 Superconductivity . .
2.4 Numerical methods . . . . . .
2.4.1 Numerical integration
2.4.2 Iterative methods . . .
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3
3
3
3
4
4
5
6
7
7
7
3 Methods
3.1 The equations . . . . . . . . . . . . . . . .
3.1.1 Properties of the Fourier transform
3.1.2 Interval for the calculations . . . .
3.1.3 The α2 F (ω)-distribution . . . . . .
3.2 Dirac delta function . . . . . . . . . . . .
3.3 The Lorentzian . . . . . . . . . . . . . . .
3.4 Smearing factor . . . . . . . . . . . . . . .
3.5 Solving the system of equations . . . . . .
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8
8
8
9
9
9
11
12
13
4 Result
4.1 Results with delta Dirac function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Results with the Lorenztian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
14
18
5 Discussion
5.1 The Dirac delta approach . . . . . . . . . .
5.2 The Lorentzian approach . . . . . . . . . .
5.3 Other approaches for solving the equations
5.3.1 Convolution . . . . . . . . . . . . . .
21
21
21
21
22
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6 Conclusions
24
A Complex analysis
A.1 Analytic functions and power series . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
B Additional plots
B.1 Additional plots for the Dirac delta
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Additional plots for the Lorentzian approach . . . . . . . . . . . . . . . . . . . . .
26
iv
26
29
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
1
Z(iωn ) = 1 +
2n + 1
|ωn0 |<ωc
X
Z(iωn )∆(iωn ) = πT
n0
|ωn0 |<ωc
ωn0
λ(ωn − ωn0 ) p 2
ωn0 + ∆(iωn0 )2
(1.1)
∆(iωn0 )
.
λ(ωn − ωn0 ) − µ∗ (ωc ) p 2
ωn0 + ∆(iωn0 )2
(1.2)
X
n0
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
Padè approximation 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
∆(ω)Z(ω) = πT
n0
Z∞
+ iπ
−∞
∆(iωn0 )
λ(ω − iωn0 ) − µ∗ (ωc ) p
R(iωn0 )
Z(ω − ω 0 )∆(ω − ω 0 )
dω Γ(ω, ω )α F (ω ) p
Z(ω − ω 0 )R(ω − ω 0 )
0
0
2
(1.3)
0
πT X
ω 0
p n
)λ(ω − iωn0 )
ω 0
0
R(iω
n
n
Z∞
πT
(ω − ωn0 )Z(ω − ω 0 )
+i
dω 0 Γ(ω, ω 0 )α2 F (ω 0 ) p
ω
Z 2 (ω − ω 0 )R(ω − ω 0 )
Z(ω) = 1 + i
(1.4)
−∞
where
R(iωn ) = ωn2 + ∆2 (iωn )
1
(1.5)
1
ω − ω0
ω0 tanh
+ coth
2
2T
2T
Z ∞
α2 F (ω 0 )
λ(ω − iωn ) = −
dω 0
ω − iωn − ω 0
−∞
Γ(ω, ω 0 ) =
(1.6)
(1.7)
and α2 F (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 Padè approximation
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:
fˆ(ω) =
Z∞
f (−x)e−i2πωx dx
(2.1)
−∞
Substitute −x with u, and dx with −du yields:
fˆ(ω) =
−∞
Z
−f (u)e−i2πω(−u) du
∞
Z∞
=
f (u)e−i2π(−ω)u du
(2.2)
(2.3)
−∞
= fˆ(−ω)
2.2
(2.4)
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 introduced, basic knowledge about analytic functions are needed. The Padé approximation 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
f (z) =
∞
X
ai (z − z0 )i
(2.5)
i=0
is an analytic function on the disk DR (z0 ) with radius R centered at z0 .
Proof. Each term of the series ai (z − z0 )i 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 Padè approximation). Given a function f (z) with the power series representation:
∞
X
f (z) =
ci z i
(2.6)
i=0
A Padè approximation, [m/n], given the two integers m and n is the fractional function:
Pm
j
j=0 aj z
P
.
[m/n] =
n
1 + k=1 bk z k
(2.7)
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(z m+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 = n1 a1 + n2 a2 + · · · + ni ai .
(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 vibrations. [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 BardeenCooper-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 superconducting, 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:
Zb
f (x)dx ≈
a
N
b−a X
(f (xn ) + f (xn+1 ))
2N n=1
(2.10)
The error in the trapezium method is proportional to h3 , where h is the stepsize in the integration. [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:
Z(iωn ) = 1 +
ωn0
1 X
λ(ωn − ωn0 ) p 2
2n + 1 n=1
ωn0 + ∆(iωn0 )2
|ωn0 |<ωc
X
Z(iωn )∆(iωn ) = πT
n0
∆(iωn0 )
λ(ωn − ωn0 ) − µ∗ (ωc ) p 2
.
ωn0 + ∆(iωn0 )2
(3.1)
(3.2)
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].
∆(ω)Z(ω) = πT
X
n0
Z∞
+ iπ
−∞
∆(iωn0 )
λ(ω − iωn0 ) − µ∗ (ωc ) p
R(iωn0 )Z 2 (ω − ω 0 )
Z(ω − ω 0 )∆(ω − ω 0 )
dω Γ(ω, ω )α F (ω ) p
Z(ω − ω 0 )R(ω − ω 0 )
0
0
2
(3.3)
0
ω 0
πT X
p n
λ(ω − iωn0 )
ω 0
R(iω
n0
n
Z∞
πT
(ω − ωn0 )Z(ω − ω 0 )
+i
dω 0 Γ(ω, ω 0 )α2 F (ω 0 ) p
ω
Z 2 (ω − ω 0 )R(ω − ω 0 )
Z(ω) = 1 + i
(3.4)
−∞
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)
and
8
(3.5)
Z [eV]
2.5
3
∆ [eV]
·10−3
2
2
1
1.5
0
1
−20
−10
0
10
−1
−20
20
−10
iω/ω D
0
10
20
iω/ω D
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
Z 0 (−ω) = Z 0 (ω) and Z 00 (−ω) = −Z 00 (ω)
(3.6)
∆(ω) = ∆0 (ω) + i∆00 (ω)
Z(ω) = Z 0 (ω) + iZ 00 (ω).
(3.7)
where
3.1.2
Interval for the calculations
The Padè approximant 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 Padè approximation. The Padé approximation 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 α2 F (ω)-distribution
The function α2 F (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 α2 F (ω) in equation (3.3) and (3.4) is approximated to be a
composition of Dirac delta functions of the form:
9
Z
real
imaginary
8
∆
·10−2
real
imaginary
1
6
0.5
4
0
2
−0.5
0
0
2
4
6
8
0
ω/ω D
2
4
6
8
ω/ω D
Figure 3.2: Pade Approximant of both Z and ∆ for real frequencies ω = [08ωD ]. The Padé
approximant is evaluated for the temperature T = 1K.
...
i
. . . iωD . . .
endi
A ωC . . .
0
. . . ωD . . .
ωC
1
. . . kωD . . .
endk
0
. . . ωD . . .
5ωD
1
B
Figure 3.3: Illustration of how the interesting interval is handled using vectors. The above vector, A, is the data from the Padè approximation 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 argument 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
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:
α2 F (ω) = λ
i
2
X ∆(ωn0 ) h
2λωD
∗
p
−
µ
(ω
)
C
2 − (ω − iω 0 )2
R(iωn0 ) ωD
n
n0
h
1
ωD ∆(ω + ωD )
ω + ωD
p
+ i λωD π − tanh
+ coth
4
2T
2T
R(ω + ωD )
ω − ωD
ωD ∆(ω − ωD ) i
+ tanh
+ coth
) p
2T
2T
R(ω − ωD )
∆(ω)Z(ω) = πT
2
πT X
ω 0
2λωD
p n
2 − (ω − iω 0 )2
ω 0
R(iωn0 ) ωD
n
n
h
ω + ωD
ωD (ω + ωD )
π
1
p
− tanh
+ coth
+ i λωD
4
ω
2T
2T
R(ω + ωD )
ωD (ω − ωD ) i
ω − ωD
+ coth
) p
+ tanh
.
2T
2T
R(ω − ωD )
(3.9)
Z(ω) = 1 + i
(3.10)
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 α2 F (ω).
α2 F (ω) ∝ λ
γ2
γ2
−
γ 2 + (ω − Ω)2
γ 2 + (ω + Ω)2
(3.11)
where γ 2 ∈ (0, 1]. Frequencies close to Ω will get increased amplitude while frequencies far away
vanishes. This effect increases as γ 2 → 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.
∆(ω)Z(ω) = πT
X
n0
∞
Z
+ iπ
−∞
∆(iωn0 )
λ(ω − iωn0 ) − µ∗ (ωc ) p
R(iωn0 )
∆(ω − ω 0 )
Γ(ω, ω )α F (ω ) p
dω 0
R(ω − ω 0 )
0
2
(3.12)
0
ω 0
πT X
p n
λ(ω − iωn0 )
ω 0
R(iωn0
n
Z∞
πT
(ω − ωn0 )
+i
Γ(ω, ω 0 )αF (ω 0 ) p
dω 0
ω
R(ω − ω 0 )
Z(ω) = 1 + i
−∞
11
(3.13)
Finally, after using the property of the Fourier transform, equation (3.12) and (3.13) becomes
∆(iωn0 )
λ(ω − iωn0 ) − µ∗ (ωc ) p
R(iωn0 )
X
∆(ω)Z(ω) =πT
n0
Z∞
+ iπ
dω 0 α2 F (ω 0 )
0
+
− tanh
ω − ω0
1 n
ω 0 ∆(ω − ω 0 )
p
tanh
+ coth
2
2T
2T
R(ω − ω 0 )
(3.14)
ω + ω0
ω 0 ∆(ω + ω 0 ) o
p
+ coth
2T
2T
R(ω + ω 0 )
πT X
ω 0
p n
λ(ω − iωn0 )
ω 0
R(iωn0
n
Z∞
ω − ω0
1 n
ω0 ω − ω0
πT
p
tanh
dω 0 α2 F (ω 0 )
+ coth
+i
ω
2
2T
2T
R(ω − ω 0 )
Z(ω) = 1 + i
(3.15)
0
+
− tanh
ω + ω0
ω0 ω + ω0 o
p
+ coth
2T
2T
R(ω − ω 0 )
and since only positive frequencies are used
0
Z∞
λ(ω − ω ) =
dω 0 α” F (ω 0 )
ω 02
2ω 0
− (ω − iωn )2
(3.16)
0
and
α2 F (ω) ∝ λ
γ2
γ 2 + (ω − Ω)2
(3.17)
When implementing the Lorentztian as it is written in (3.17) numerically, it is rewritten in the
following form:
α2 F (ω) = λ
i
Ωh
1
1
−
h(γc − |ω − Ω|).
2 γ 2 + (ω − Ω)2
γ 2 + γc2
(3.18)
The reason for this is to emulate the behaviour of the Dirac pulse. Where γ and γc are
parameters chosen to satisfy:
Z∞
Ω
0
i
dω h
1
1
−
h(γc − |ω − Ω|) = 1,
ω γ 2 + (ω − Ω)2
γ 2 + γc2
(3.19)
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 Padé
approximant, figure 3.2, the solution with the smearing factor is a better fit.
12
Re(∆) [eV]
·10−2
Im(∆) [eV]
·10−3
Real
Imaginary
1
Real
Imaginary
5
0.5
0
0
−5
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
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 ))|
; |real(Z(ωi ) − Z(ωi+1 ))|; |imag(Z(ω) − Z(ωi+1 ))|
(3.20)
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 α2 F -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 Padè approximation in electron-volts [eV ]. A small smearing factor has been added to each
solution in the manner discussed in 3.4. The Padè approximation 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 Padè approximation and the difference between them are small. For the figures 4.6 and 4.8 the Padè approximation diverges in
comparison to the analytic solution, which is stable and gives a reasoable result.
Re(Z) [eV]
Im(Z) [eV]
8
Analytic
Approximation
8
Analytic
Approximation
6
6
4
4
2
2
0
0
2
4
6
8
ω/ω D
0
0
2
4
6
8
ω/ω D
Figure 4.1: The analytic solution (solid line) and the Padè approximation (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.
14
Re(∆) [eV]
·10−2
1
Im(∆) [eV]
·10−2
Analytic
Approximation
Analytic
Approximation
1
0.5
0.5
0
0
−0.5
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
Figure 4.2: The analytic solution (solid line) and the Padè approximation (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]
4
Analytic
Approximation
Analytic
Approximation
2
3
1
2
0
1
0
2
4
6
8
ω/ω D
0
2
4
6
8
ω/ω D
Figure 4.3: The analytic solution (solid line) and the Padè approximation (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.
15
Re(∆) [eV]
·10−3
6
6
Im(∆) [eV]
·10−3
Analytic
Approximation
4
Analytic
Approximation
4
2
2
0
−2
0
−4
−6
−2
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
Figure 4.4: The analytic solution (solid line) and the Padè approximation (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.
Re(Z) [eV]
Im(Z) [eV]
3.5
2
Analytic
Approximation
Analytic
Approximation
3
1.5
2.5
1
2
0.5
1.5
0
1
0
2
4
6
8
−0.5
ω/ω D
0
2
4
6
8
ω/ω D
Figure 4.5: The analytic solution (solid line) and the Padè approximation (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.
16
Re(∆) [eV]
·10−2
Im(∆) [eV]
·10−3
Analytic
Approximation
Analytic
Approximation
5
0
−0.5
0
−1
−5
0
2
4
6
0
8
2
ω/ω D
4
6
8
ω/ω D
Figure 4.6: The analytic solution (solid line) and the Padè approximation (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.
Im(Z) [eV]
Re(Z) [eV]
3.5
1.5
Analytic
Approximation
Analytic
Approximation
3
1
2.5
2
0.5
1.5
0
1
0
2
4
6
8
ω/ω D
0
2
4
6
8
ω/ω D
Figure 4.7: The analytic solution (solid line) and the Padè approximation (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.
17
Re(∆) [eV]
·10−13
0.5
Im(∆) [eV]
·10−14
Analytic
Approximation
Analytic
Approximation
0
−2
0
−4
−0.5
−6
−1
0
2
4
6
8
ω/ω D
0
2
4
6
8
ω/ω D
Figure 4.8: The analytic solution (solid line) and the Padè approximation (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.
4.2
Results with the Lorenztian
This section presents the result when α2 F (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
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
0
1
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
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.
4
Re(∆) [eV]
·10−3
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
Lorentzian
Dirac Delta
4
2
2
0
0
−2
−4
0
2
4
6
8
−2
ω/ω D
0
2
4
6
8
ω/ω D
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
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
3
1
2.5
0.5
2
1.5
0
1
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
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.
8
Re(∆) [eV]
·10−15
6
Lorentzian
Dirac Delta
6
2
2
0
0
−2
−2
−4
0
2
4
6
Lorentzian
Dirac Delta
4
4
−4
Im(∆) [eV]
·10−15
8
−6
ω/ω D
0
2
4
6
8
ω/ω D
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 Padé approximant.
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 Padè
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 Padè approximation 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 Padè approximation.
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
4
∆ [eV]
·10−3
∆ [eV]
·10−15
Real
Imaginary
Real
Imaginary
4
2
2
0
0
−2
−2
0
2
4
6
8
0
2
ω/ω D
4
6
8
ω/ω D
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 − ω 0 )∆(ω − ω 0 )
dω 0 Γ(ω, ω 0 )α2 F (ω 0 ) p
Z 2 (ω − ω 0 )R(ω − ω 0 )
−∞
(5.1)
when inserting Γ(ω, ω 0 ) and sort the terms with respect to their argument, it becomes more
obvious how the convolution will proceed
1
2
∞
ω 0 Z(ω − ω 0 )∆(ω − ω 0 )
p
dω 0 α2 F (ω 0 ) coth
2T Z 2 (ω − ω 0 )R(ω − ω 0 )
−∞
ω − ω 0 Z(ω − ω 0 )∆(ω − ω 0 ) p
+α2 (F (ω 0 ) tanh
2T
Z 2 (ω − ω 0 )R(ω − ω 0 )
Z
(5.2)
Applying convolution leads to
1 2
ω − ω 0 Z(ω − ω 0 )∆(ω 0 )
p
α F (ω 0 ) ? tanh
2
2T
Z 2 (ω 0 )R(ω 0 )
1
ω 0 Z(ω − ω 0 )∆(ω 0 )
+ α2 F (ω 0 ) coth
? p
2
2T
Z 2 (ω 0 )R(ω 0 )
(5.3)
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
R(iωn0 )
R(iωn0 )
n0
(5.4)
were the first term can be evaluated with convolution
∆(iωn0 )
λ(iωn0 ) ? p
R(iωn0 )
22
(5.5)
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 Padè approximation shows that they are
similar for low frequencies and low temperatures. When the temperature is closer to TC the Padè
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:
f 0 (z0 ) = lim
∆z→0
f (z0 + ∆z) − f (z)
∆z
(A.1)
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 function, then u and v satisfy the Cauchy-Riemann equations:
∂u
∂x
=
∂v
∂y ,
∂v
∂x
= − ∂u
∂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
f (z) =
∞
X
ai (z − z0 )i = a0 + a1 (z − z0 ) + a2 (z − z0 )2 + . . .
(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
h(z) =
f (z)
g(z)
(A.4)
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 γ 0 without leaving A, then γ is said to be homotopic to
γ 0 in A.
25
B. Additional plots
B.1
Additional plots for the Dirac delta
approach
Re(Z) [eV]
Im(Z) [eV]
6
Analytic
Approximation
Analytic
Approximation
4
5
3
4
2
3
1
2
0
1
0
2
4
6
8
ω/ω D
0
2
4
6
8
ω/ω D
Figure B.1: The analytic solution (solid line) and the Padé approximant (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.
26
Re(Z) [eV]
Im(Z) [eV]
6
4
Analytic
Approximation
Analytic
Approximation
5
3
4
2
3
1
2
0
1
0
2
4
6
8
0
2
4
ω/ω D
6
8
ω/ω D
Figure B.3: The analytic solution (solid line) and the Padé approximant (dashed line) at temperature T = 5K for the mass renormalization function, Z(ω). The real part to the left and the
imaginary part to the right.
1
Re(∆) [eV]
·10−2
1
Analytic
Approximation
Im(∆) [eV]
·10−2
Analytic
Approximation
0.8
0.5
0.6
0.4
0
0.2
0
−0.5
0
2
4
6
8
ω/ω D
−0.2
0
2
4
6
8
ω/ω D
Figure B.2: The analytic solution (solid line) and the Padé approximant (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.
27
Re(∆) [eV]
·10−3
1
Analytic
Approximation
Im(∆) [eV]
·10−2
Analytic
Approximation
0.8
5
0.6
0.4
0
0.2
0
−5
0
2
4
6
−0.2
8
0
2
ω/ω D
4
6
8
ω/ω D
Figure B.4: The analytic solution (solid line) and the Padé approximant (dashed line) at temperature T = 5K for the super conducting order parameter, ∆(ω). The real part to the left and the
imaginary part to the right.
Im(Z) [eV]
Re(Z) [eV]
4
Analytic
Approximation
Analytic
Approximation
2
3
1
2
0
1
0
2
4
6
8
ω/ω D
0
2
4
6
8
ω/ω D
Figure B.5: The analytic solution (solid line) and the Padé approximant (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.
28
Re(∆) [eV]
·10−3
Im(∆) [eV]
·10−3
Analytic
Approximation
Analytic
Approximation
6
5
4
2
0
0
−5
0
2
4
6
−2
8
0
2
ω/ω D
4
6
8
ω/ω D
Figure B.6: The analytic solution (solid line) and the Padé approximant (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.
B.2
Additional plots for the Lorentzian approach
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
1
0
2
4
6
0
8
ω/ω D
0
2
4
6
8
ω/ω D
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
6
Re(∆) [eV]
·10−3
8
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
4
Lorentzian
Dirac Delta
6
2
4
0
2
−2
0
−4
−6
0
2
4
6
8
−2
0
2
ω/ω D
4
6
8
ω/ω D
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.
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
1
0
2
4
6
8
ω/ω D
0
0
2
4
6
8
ω/ω D
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
6
Re(∆) [eV]
·10−3
8
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
4
Lorentzian
Dirac Delta
6
2
4
0
2
−2
0
−4
−6
0
2
4
6
8
−2
0
2
ω/ω D
4
6
8
ω/ω D
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.
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
1
0
2
4
6
8
ω/ω D
0
0
2
4
6
8
ω/ω D
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
6
Re(∆) [eV]
·10−3
8
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
4
Lorentzian
Dirac Delta
6
2
4
0
2
−2
0
−4
−6
0
2
4
6
8
−2
0
2
ω/ω D
4
6
8
ω/ω D
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.
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
1
0
2
4
6
8
ω/ω D
0
0
2
4
6
8
ω/ω D
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]
·10−3
Lorentzian
Dirac Delta
4
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
6
2
4
0
2
−2
0
−4
0
2
4
6
8
−2
0
2
ω/ω D
4
6
8
ω/ω D
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.
Im(Z) [eV]
Re(Z) [eV]
3.5
Lorentzian
Dirac delta
Lorentzian
Dirac delta
1.5
3
2.5
1
2
0.5
1.5
1
0
2
4
6
8
ω/ω D
0
0
2
4
6
8
ω/ω D
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
4
Re(∆) [eV]
·10−3
6
Im(∆) [eV]
·10−3
Lorentzian
Dirac Delta
Lorentzian
Dirac Delta
2
4
0
2
−2
0
−4
0
2
4
6
8
−2
ω/ω D
0
2
4
6
8
ω/ω D
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
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Weisstein,
Eric
W.
Padé
http://mathworld.wolfram.com/PadeApproximant.html.
approximant,
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35