Anisotropic Migdal-Eliashberg theory using Wannier functions

Abstract

We combine the fully anisotropic Migdal-Eliashberg theory with
electron-phonon interpolation based on maximally-localized Wannier functions,
in order to perform reliable and highly accurate calculations of the
anisotropic temperature-dependent superconducting gap and critical temperature
of conventional superconductors. Compared with the widely used McMillan
approximation, our methodology yields a more comprehensive and detailed
description of superconducting properties, and is especially relevant for the
study of layered or low-dimensional systems as well as systems with complex
Fermi surfaces. In order to validate our method we perform calculations on two
prototypical superconductors, Pb and MgB$_2$, and obtain good agreement with
previous studies.

Full text

arXiv:1211.3345v1 [cond-mat.supr-con] 14 Nov 2012
Anisotropic Migdal-Eliashberg theory using Wannier functions
E. R. Margine and F. Giustino
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
We combine the fully anisotropic Migdal-Eliashberg theory with electron-phonon interpolation
based on maximally-localized Wannier functions, in order to perform reliable and highly accurate
calculations of the anisotropic temperature-dependent superconducting gap and critical temperature
of conventional superconductors. Compared with the widely used McMillan approximation, our
methodology yields a more comprehensive and detailed description of superconducting properties,
and is especially relevant for the study of layered or low-dimensional systems as well as systems
with complex Fermi surfaces. In order to validate our method we perform calculations on two
prototypical superconductors, Pb and MgB2, and obtain good agreement with previous studies.
I. INTRODUCTION
The prediction of superconducting properties such as
the critical temperature or the superconducting energy
gap remains one of the outstanding challenges in mod-
ern electronic structure theory.1–7 Owing to the com-
plex nature of the superconducting state, a quantita-
tive understanding of the pairing mechanism in conven-
tional superconductors requires a very detailed knowl-
edge of the electronic structure, the phonon dispersions,
and the interaction between electrons and phonons. In
conventional superconductors below the critical temper-
ature electron pairing results from a subtle interplay be-
tween the repulsive Coulomb interaction and the attrac-
tive electron-phonon interaction. Starting from the sem-
inal work of Bardeen, Cooper, and Schrieffer (BCS)8
several approaches to the calculation of the supercon-
ducting properties have been proposed, ranging from
semi-empirical methods such as the McMillan formula,9
to first-principles Green’s function methods such as the
Migdal-Eliashberg (ME) formalism,10,11 and more re-
cently also methods based on the density-functional the-
ory concept, such as the density functional theory for
superconductors (SCDFT).6,7
The vast majority of current investigations rely on the
semi-empirical McMillan’s approach.9In this approach
the entire physics of the electron-phonon interaction is
condensed into a single parameter called the electron-
phonon coupling strength λ. The McMillan’s method
works reasonably well for conventional bulk metals and
for anisotropic superconductors where the Fermi surface
anisotropy is smeared out by impurities.2,3 However, for
layered systems, systems of reduced dimensionality, and
those with complex multi-sheet Fermi surfaces, a care-
ful description of the pairing interactions is crucial and a
proper treatment of the anisotropic electron-phonon in-
teraction is required. The necessity of an anisotropic the-
ory has clearly been demonstrated in two important cases
such as magnesium diboride, MgB2, and the graphite in-
tercalation compound CaC6, using either the ME formal-
ism or the SCDFT.12–18
Unfortunately, the lack of adequate computational
tools prevents the research community from systemat-
ically exploring the importance of anisotropy in exist-
ing and well characterized superconductors, and for val-
idating computational predictions based on the semi-
empirical McMillan equation.19–22 This latter aspect is
particularly relevant in view of the increasingly im-
portant role that high-throughput materials design ap-
proaches are acquiring in the community.23,24
The momentum-resolved superconducting gap and the
quasiparticle density of states near the Fermi surface
can now be measured with unprecedented accuracy us-
ing high-resolution angle-resolved photoemission spec-
troscopy25 and scanning tunneling spectroscopy26 experi-
ments. In this context, the anisotropic Migdal-Eliashberg
formalism promises to be particularly useful for perform-
ing direct comparisons between theory and experiment,
and for helping establish unambiguously the symmetry
of the order parameter.
One critical point which arises when attempting to
solve the Eliashberg equations of the ME theory or the
Bogoliubov-de Gennes equations of the SCDFT is that
both sets of equations suffer from a strong sensitivity to
the sampling of the electron-phonon scattering processes
in the vicinity of the Fermi surface.27 The practical con-
sequence of this sensitivity is that, in order to achieve
numerical convergence, the electron-phonon matrix ele-
ments must be evaluated for extremely dense electron
and phonon meshes in the Brillouin zone. This long-
standing difficulty was overcome in Ref. 27 by developing
an efficient first-principles interpolation technique based
on maximally-localized Wannier functions (MLWF).28 In
this method one takes advantage of the spatial local-
ization of both electron and phonon Wannier functions
in order to evaluate only a small number of electron-
phonon matrix elements in the Wannier representation.
These matrix elements are subsequently interpolated to
arbitrary electron and phonon wavevectors in the Bloch
representation using a generalized Fourier transform.27
This method carries general validity and has been demon-
strated in several other areas, from Fermi surface calcu-
lations,29 to the anomalous Hall effect,30 and more re-
cently for GW calculations.31 A detailed introduction on
Wannier-based interpolation methods can be found in
Ref. 28. The scheme of Ref. 27 is adopted in the present
work since it provides a robust and efficient framework for
developing an algorithm to solve the anisotropic Eliash-

2
berg equations. Our current implementation enables the
calculation of the momentum- and band-resolved super-
conducting gap using a very fine Brillouin zone sampling.
Without our Wannier-based electron-phonon interpola-
tion this operation would not be possible owing to the
prohibitive computational cost.
This manuscript is organized as follows. In Sec. II we
review the Migdal-Eliashberg theory of superconductiv-
ity. A description of the computational techniques under-
pinning the electron-phonon interpolation implemented
in the EPW code32 is given in Sec. III. In Sec. IV we re-
port the numerical solutions of the Eliashberg equations
for two prototypical superconductors, Pb and MgB2. Fi-
nally we present our conclusions and outlook in Sec. V.
II. MIGDAL-ELIASHBERG FORMALISM
A. General theory
A quantitative theory of the superconducting energy
gap can conveniently be formulated within the framework
of the Nambu-Gor’kov formalism.33,34 In this formalism
one introduces a two-component field operator:
Ψk=ck↑
c†
−k↓.(1)
The component ck↑(c†
−k↓) of the operator destroys (cre-
ates) an electron in a Bloch state of combined band and
momentum index k(−k) and spin up (down). A general-
ized 2×2 matrix Green’s functions ˆ
Gis then introduced
in order to describe electron quasiparticles and Cooper’s
pairs on an equal footing:2,4
ˆ
G(k, τ ) = −hTτΨk(τ)Ψ†
k(0)i,(2)
where Tτis Wick’s time-ordering operator for the imag-
inary time τand Ψk(τ) is obtained from Eq. (1) using
the Heisenberg picture. The braces indicate a grand-
canonical thermodynamic average. By replacing Eq. (1)
inside Eq. (2) we find:
ˆ
G(k, τ ) = −"hTτck↑(τ)c†
k↑(0)i hTτck↑(τ)c−k↓(0)i
hTτc†
−k↓(τ)c†
k↑(0)i hTτc†
−k↓(τ)c−k↓(0)i#.
(3)
Here the diagonal elements correspond to the standard
Green’s functions for electron quasiparticles and describe
the dynamics of single-particle electronic excitations in
the material. On the other hand, the off-diagonal ele-
ments represent Gor’kov’s anomalous Green’s functions
F(k, τ ) and F∗(k, τ). These functions describe the dy-
namics of Cooper’s pairs and are related to the supercon-
ducting energy gap.2,4,5 The off-diagonal elements of the
generalized Green’s function in Eq. (3) become nonzero
only below the critical temperature Tc, marking the tran-
sition to the superconducting state.
The generalized Green’s function ˆ
G(k, τ ) is periodic
in imaginary time, therefore it can be expanded using a
Fourier series as follows:
ˆ
G(k, τ ) = TX
iωn
e−iωnτˆ
G(k, iωn),(4)
where iωn=i(2n+ 1)πT (ninteger) stands for the
fermion Matsubara frequencies, and Tis the absolute
temperature. We use atomic units throughout the
manuscript, therefore we set ¯h=kB= 1. Following
Eq. (4) the matrix elements of the generalized Green’s
function read:
ˆ
G(k, iωn) = G(k, iωn)F(k, iωn)
F∗(k, iωn)−G(−k,−iωn).(5)
The study of the superconducting state involves the de-
termination of the matrix Green’s function in Eq. (5).
This can be achieved using Dyson’s equation:
ˆ
G−1(k, iωn) = ˆ
G−1
0(k, iωn)−ˆ
Σ(k, iωn),(6)
where ˆ
G0(k, iωn) is the electron Green’s function for
the normal state and ˆ
Σ(k, iωn) is the self-energy asso-
ciated with the pairing interaction. The normal-state
Green’s function is calculated by using the Kohn-Sham
states from density-functional theory to represent single-
particle excitations. If we denote by ǫkthe Kohn-Sham
eigenvalues measured with respect to the chemical po-
tential, and we introduce the Pauli matrices:
ˆτ0=1 0
0 1 ,ˆτ1=0 1
1 0 ,
ˆτ2=0−i
i0,ˆτ3=1 0
0−1,
(7)
then the normal-state matrix Green’s function acquires
the familiar form:
ˆ
G−1
0(k, iωn) = iωnˆτ0−ǫkˆτ3.(8)
Within the Migdal-Eliashberg approximation the elec-
tron self-energy leading to the superconducting pairing
consists of two terms, an electron-phonon contribution
ˆ
Σep(k, iωn) and a Coulomb contribution ˆ
Σc(k, iωn):2
ˆ
Σ(k, iωn) = ˆ
Σep(k, iωn) + ˆ
Σc(k, iωn),(9)
with
ˆ
Σep(k, iωn) = −TX
k′n′
ˆτ3ˆ
G(k′, iωn′)ˆτ3
×X
λ
|gkk′ν|2Dν(k−k′, iωn−iωn′),(10)
and
ˆ
Σc(k, iωn) = −TX
k′n′
ˆτ3ˆ
God(k′, iωn′)ˆτ3V(k−k′).(11)

3
In Eq. (10) Dν(q, iωn) = 2ωqν/(iωn)2−ω2
qνis the
dressed propagator for phonons with momentum qand
branch index ν, and gkk′νis the screened electron-phonon
matrix element for the scattering between the electronic
states kand k′through a phonon with wavevector q=
k′−k, frequency ωqνand branch index ν. In Eq. (11)
the V(k−k′)’s represent the matrix elements of the
static screened Coulomb interaction between the elec-
tronic states kand k′.
In writing the electron self-energy of Eqs. (9)-(11) the
following approximations are made: (i) only the first
term in the Feynman’s expansion of the self-energy in
terms of electron-phonon diagrams is included. This ap-
proximation corresponds to Migdal’s theorem10 and is
based on the observation that the neglected terms are
of the order (me/M)1/2(mebeing the electron mass
and Ma characteristic nuclear mass). (ii) Only the off-
diagonal contributions of the Coulomb self-energy are re-
tained. This is done in order to avoid double counting the
Coulomb effects that are already included in ˆ
G0(k, iωn).2
(iii) The self-energy is assumed to be diagonal in the
electron band index.35 This should constitute a reason-
able approximation for non-degenerate bands since the
energy involved in the superconducting pairing is very
small, therefore band mixing is not expected.
In the literature on the theory of superconductivity it
is common practice to decompose the matrix self-energy
ˆ
Σ(k, iωn) in a linear combination of Pauli matrices with
three scalar functions as coefficients. The scalar functions
are the mass renormalization function Z(k, iωn), the en-
ergy shift χ(k, iωn), and the order parameter φ(k, iωn),
and the decomposition reads:
ˆ
Σ(k, iωn) = iωn[1 −Z(k, iωn)] ˆτ0+χ(k, iωn)ˆτ3
+φ(k, iωn)ˆτ1.(12)
We note that here we have chosen the gauge where the
order parameter φis set to zero.2By replacing Eqs. (8),
(12) inside Eq. (6) and solving for the matrix Green’s
function we obtain:
ˆ
G(k, iωn) = − {iωnZ(k, iωn)ˆτ0+ [ǫk+χ(k, iωn)] ˆτ3
+φ(k, iωn)ˆτ1}/Θ(k, iωn),(13)
where the denominator is defined as:
Θ(k, iωn) = [ωnZ(k, iωn)]2+ [ǫk+χ(k, iωn)]2
+ [φ(k, iωn)]2.(14)
At this point the strategy is to impose self-consistency by
replacing the explicit expression for the Green’s function
in Eq. (13) inside the self-energy expressions Eqs. (9)-
(11). After equating the scalar coefficients of the Pauli
matrices this replacement leads finally to the anisotropic
Eliashberg equations:
Z(k, iωn) = 1 + T
ωnNFX
k′n′
ωn′Z(k′, iωn′)
Θ(k′, iωn′)λ(k,k′, n −n′),
(15)
χ(k, iωn)= −T
NFX
k′n′
ǫk′+χ(k′, iωn′)
Θ(k′, iωn′)λ(k,k′
, n −n′),
(16)
φ(k, iωn) = T
NFX
k′n′
φ(k′, iωn′)
Θ(k′, iωn′)
×[λ(k,k′, n −n′)−NFV(k−k′)] .(17)
In Eqs. (15)-(17) NFrepresents the density of states per
spin at the Fermi level, and λ(k,k′, n −n′) is an auxil-
iary function describing the anisotropic electron-phonon
coupling and defined as follows:
λ(k,k′, n −n′) = Z∞
0
dω 2ω
(ωn−ωn′)2+ω2α2F(k,k′, ω),
(18)
with α2F(k,k′, ω) the Eliashberg electron-phonon spec-
tral function:
α2F(k,k′, ω) = NFX
ν
|gkk′ν|2δ(ω−ωk−k′,ν ).(19)
The superconducting gap is defined in terms of the renor-
malization function and the order parameter as:
∆(k, iωn) = φ(k, iωn)
Z(k, iωn).(20)
From Eqs. (17),(20) we see that the Eliashberg equations
admit the trivial solution ∆(k, iωn) = 0 at all tempera-
tures. The highest temperature for which the Eliashberg
equations admit nontrivial solutions ∆(k, iωn)6= 0 de-
fines the critical temperature Tc.
B. Standard approximations
After having presented a concise derivation of Eliash-
berg’s equations in the previous Section, we now discuss
technical aspects which need to be addressed in order to
actually solve these equations.
Since the superconducting pairing occurs mainly
within an energy window of width ωph around the Fermi
surface (ωph being a characteristic phonon energy), it
is standard practice to simplify the Eliashberg equa-
tions by restricting the description to electron bands
near the Fermi energy.2–4,36 This simplification can be
achieved in the formalism by introducing the identity
R∞
−∞ dǫ′δ(ǫk′−ǫ′)= 1 on the right hand side in Eqs. (15)-
(17). The rapid changes of Θ(k′, ωn′) and the numerator
of Eq. (16) with the energy ǫ′can be integrated analyt-
ically, while for the other quantities we can set ǫ′to the
Fermi energy since the associated variations take place
on a much larger energy scale.4,36 Under this approxima-
tion the energy shift becomes χ(k, iωn) = 0 and only two
equations are left to solve, one for the renormalization
function and one for the order parameter (or equivalently

4
the superconducting gap):
Z(k, iωn) = 1 + πT
ωnX
k′n′
Wk′
ωn′
R(k′, iωn′)λ(k,k′, n−n′),
(21)
Z(k, iωn)∆(k, iωn) = πT X
k′n′
Wk′
∆(k′, iωn′)
R(k′, iωn′)
×[λ(k,k′,n −n′)−NFV(k−k′)] ,(22)
where R(k, iωn) and Wkare given by:
R(k, iωn) = pω2
n+ ∆2(k, iωn) and Wk=δ(ǫk)/NF.
(23)
Equations (21) and (22) form a coupled nonlinear system
and need to be solved self-consistently at each tempera-
ture T. The approximations leading to Eqs. (21) and (22)
imply that Z(k, iωn) and ∆(k, iωn) are only meaningful
for the momentum/band index kat or near the Fermi
surface. Away from the Fermi surface the energy depen-
dence of these quantities is weak and is neglected.4,36 In
addition Eqs. (21)-(22) implicitly assume that the elec-
tron density of states is approximately constant near the
Fermi energy. This simplification may break down for
materials with narrow bands or critical points in proxim-
ity of the Fermi level.2,35
In order to solve Eqs. (21),(22) numerically it is nec-
essary to truncate the sum over Matsubara frequencies.
It is standard practice to restrict the sum to frequencies
smaller than a given cutoff ωc, with the cutoff of the or-
der of 1 eV and typically set to 4-10 times the largest
phonon energy. In addition, it is convenient to introduce
a dimensionless Coulomb interaction parameter µ∗
cde-
fined as the double Fermi surface (FS) average over k
and k′of the term V(k−k′) in Eq. (22):
µc=NFhhV(k−k′)iiFS.(24)
By performing the energy integral analytically up to the
cutoff frequency it can be shown that the NFV(k−k′)
term in Eq. (22) can be replaced by the Morel-Anderson
pseudopotential µ∗
cgiven by:37
µ∗
c=µc
1 + µcln(ǫF/ωc).(25)
Following this replacement, µ∗
cis used as a semi-empirical
parameter in the subsequent numerical solution of the
Eliashberg equations. For a large class of superconduc-
tors Eqs. (24)-(25) yields values of µ∗
cin the range 0.1-
0.2. However, it is clear by now that in several cases
values of the Coulomb parameter outside of this range
are necessary for explaining experimental data.27,39,40 In
addition, the anisotropic nature of the Coulomb interac-
tion cannot be neglected for an accurate description of
the superconducting properties.12,14,15,38 These observa-
tions should make it clear that the simplification provided
by Eqs. (24)-(25) is not optimal, and a fully ab-initio
approach to the solution of the Eliashberg equations is
highly desirable.2,35 A description of the electron-phonon
and the electron-electron interactions on the same footing
is achieved in the SCDFT approach.6,7 While it is clear
that the Eliashberg approach considered in this work
should be extended in order to incorporate Coulomb ef-
fects from first-principles, this is beyond the scope of the
present investigation.
C. Superconducting gap along the real energy axis
In Eqs. (13)-(23) the dynamical aspects of the super-
conducting pairing are described using the imaginary
Matsubara frequencies iωn. The reason for this choice
is that the resulting formulation is computationally ef-
ficient since it only involves sums over a finite number
of frequencies. While the imaginary axis formulation is
adequate for calculating the critical temperature as de-
scribed in Sec. II A, the calculation of spectral properties
such as the quasiparticle density of states requires the
knowledge of the superconducting gap along the real fre-
quency axis.
It is in principle possible to calculate the supercon-
ducting gap along the real axis, however this procedure
involves the evaluation of many principal value integrals
and hence is numerically demanding.1,41 In this work we
prefer instead to determine the solutions of the Eliash-
berg equations on the real axis by analytic continuation
of our calculated solutions along the imaginary axis. The
analytic continuation can be performed either by using
Pad´e approximants as in Refs. 42,43 or by means of an
iterative procedure as in Ref. 44.
The continuation based on Pad´e approximants involves
a very light computational workload, however it is very
sensitive to the numerical precision of the solutions on the
imaginary axis.42,43,45 As a rule of thumb the analytic
continuation based on Pad´e approximants exhibits the
correct gross structure of the superconducting gap on the
real frequency axis, however fine spectral features are not
always captured completely.
The iterative analytic continuation, on the other hand,
is generally rather accurate but involves a high compu-
tational workload. In fact, as shown in Ref. 44, the iter-
ative analytic continuation requires solving the following
equations self-consistently:
Z(k, ω) = 1+iπT
ωX
k′n′
Wk′
ωn′
R(k′, iωn′)λ(k,k′, ω−iωn′)
+iπ
ωZ∞
−∞
dω′Γ(ω, ω′)X
k′
Wk′α2F(k,k′, ω′)
×(ω−ω′)Z(k′, ω −ω′)
P(k′, ω −ω′),(26)

5
Z(k, ω)∆(k, ω) = πT X
k′n′
Wk′[λ(k,k′, ω −iωn′)−µ∗
c]
×∆(k′, iωn′)
R(k′, iωn′)+iπ Z∞
−∞
dω′Γ(ω, ω′)X
k′
Wk′α2F(k,k′, ω′)
×Z(k′, ω −ω′)∆(k′, ω −ω′)
P(k′, ω −ω′),(27)
where the following quantities have been introduced:
P(k, ω) = pZ2(k, ω) [ω2+ ∆2(k, ω)],(28)
Γ(ω, ω′) = 1
2tanh ω−ω′
2T+ coth ω′
2T,(29)
λ(k,k′, ω −iωn) = −Z∞
−∞
dω′α2F(k,k′, ω′)
ω−iωn−ω′,(30)
α2F(k,k′,−ω) = −α2F(k,k′, ω).(31)
In the case where the square-root on the right hand side
of Eq. (28) is complex, the root with positive imaginary
part is chosen.
Once determined the mass renormalization function
Z(k, ω) and the superconducting gap ∆(k, ω) on the real
frequency axis, one can examine the poles of the diagonal
component of the single-particle Green’s function:3
G11(k, ω) = ωZ(k, ω) + ǫk
[ωZ(k, ω)]2−ǫ2
k−[Z(k, ω)∆(k, ω)]2,(32)
in order to obtain the quasiparticle energy Ek:
E2
k=ǫk
Z(k, Ek)2
+ ∆2(k, Ek).(33)
At the Fermi level ǫk= 0 and the quasiparticle shift is
Ek= Re∆(k, Ek). As a result the leading edge ∆kof the
superconducting gap for the combined band/momentum
index kat the Fermi surface is given by:
Re[∆(k,∆k)] = ∆k.(34)
D. Isotropic approximation
For conventional bulk metals or superconductors where
the Fermi surface anisotropy is either weak or smeared
out by impurities, it is possible to resort to a simplified
isotropic formulation of the Eliashberg equations. Such
formulation is obtained from Eqs. (21),(22) by averaging
kover the Fermi surface. We obtain:
Z(iωn) = 1 + πT
ωnX
n′
ωn′
R(iωn′)λ(n−n′),(35)
Z(iωn)∆(iωn) = πT X
n′
∆(iωn′)
R(iωn′)[λ(n−n′)−µ∗
c],(36)
where R(iωn) and λ(n−n′) are given by:
R(iωn) = pω2
n+ ∆2(iωn),(37)
λ(n−n′) = Z∞
0
dω 2ωα2F(ω)
(ωn−ωn′)2+ω2,(38)
and α2F(ω) is the isotropic Eliashberg spectral function:
α2F(ω) = X
k,k′
WkWk′α2F(k,k′, ω).(39)
The isotropic Eliashberg equations on the real axis can be
obtained similarly by starting from Eqs. (26),(27). From
the isotropic superconducting gap on the real axis we can
obtain the normalized quasiparticle density of states in
the superconducting state NS(ω):
NS(ω)
NF
= Re "ω
pω2−∆2(ω)#.(40)
III. COMPUTATIONAL METHODOLOGY
A. Electron-phonon Wannier interpolation
We now describe the combination of the anisotropic
Eliashberg formalism of Sec. II A with the electron-
phonon Wannier interpolation of Ref. 27. The numer-
ical solution of Eqs. (21),(22) and Eqs. (26),(27) requires
an extremely careful description of the electron-phonon
scattering processes, especially in proximity of the Fermi
surface. This requirement translates into the necessity
of evaluating electronic eigenvalues ǫk, phonon frequen-
cies ωqν, and electron-phonon matrix elements gkk′νfor
a very large set of electron and phonon wavevectors in
the Brillouin zone, of the order of tens of thousands.
While it is practically impossible to evaluate so many
electron-phonon matrix elements directly using standard
density-functional calculations, it is possible to perform
an optimal ab-initio interpolation of the matrix elements
by exploiting localization in real space. The key idea is to
first evaluate a small number of electron-phonon matrix
elements in the maximally-localized Wannier representa-
tion, and then perform a generalized Fourier interpola-
tion into the momentum space, i.e. into the Bloch repre-
sentation. The relation between the matrix elements in
the Wannier representation gRR and those in the Bloch
representation gkk′is:
gkk′=1
NX
R,R′
ei(k·R+q·R′)Uk′gRR′U†
kuq,(41)
where Nis the size of the discrete Brillouin-zone mesh,
Ukis a band-mixing matrix which maps electron Bloch
bands into Wannier functions, uqis a branch-mixing ma-
trix which maps phonon branches into individual atomic

6
displacements, q=k′−k, and R,R′are vectors of the di-
rect lattice. In Eq. (41) the band and branch indices are
absorbed in k,k′and in R,R′. More detailed expressions
for implementation purposes can be found in Refs. 27,32.
Once obtained the matrix elements in the Wannier rep-
resentation, the evaluation of Eq. (41) for any pairs of
initial and final electron wavevectors is inexpensive since
it involves only very small matrix multiplications.
The matrix elements in the Wannier representation are
computed by first calculating the corresponding elements
in the Bloch representation on a coarse Brillouin zone
mesh using density-functional perturbation theory46 and
then transforming into the maximally localized Wannier
representation47,48 using the inverse relation of Eq. (41).
All our density-functional and density-functional per-
turbation theory calculations are performed using the
Quantum ESPRESSO package,49 and maximally-localized
Wannier functions are determined using the Wannier90
program.50 The subsequent electron-phonon interpola-
tion is performed using the EPW program,32 which ex-
tracts and processes information from both Quantum
ESPRESSO and Wannier90. Further details on the no-
tion of Wannier interpolation and its use in the study of
electron-phonon interactions can be found in Ref. 28 and
Refs. 27,32, respectively.
Even when using electron-phonon Wannier interpola-
tion the computational workload can become quite sub-
stantial when one evaluates hundreds of thousands of ma-
trix elements. In order to reduce the computational load
we exploit the crystal symmetries and only evaluate the
gap function ∆(k, iωn) and the renormalization function
Z(k, iωn) in the irreducible wedge of the Brillouin zone.
On the other hand, the sums over k′in Eqs. (21),(22)
and Eqs. (26),(27) are performed over the entire Brillouin
zone. The meshes of wavevectors kand q=k′−kare cho-
sen to be uniform and commensurate, in such a way that
the grid of electron wavevectors in the final state k′maps
into the grid of the initial wavevectors k. Since the con-
tributions to the superconducting gap arising from elec-
tronic states away from the Fermi energy are essentially
negligible, the matrix elements of Eq. (41) are evaluated
only for electronic states such that ǫkand ǫk′are near the
Fermi energy. Numerical convergence can be achieved
typically by restricting the sums in Eqs. (21),(22) and
Eqs. (26),(27) to an energy window around the Fermi
level of width corresponding to 4-10 times the character-
istic phonon frequency.
B. Self-consistent solution of the nonlinear system
and analytic continuation
In order to solve iteratively the Eliashberg equations on
the imaginary axis Eqs. (21),(22), we start from an initial
guess ∆0(iωn) for the superconducting gap. The starting
guess ∆0(iωn) is chosen to be a step function vanishing
for iωn>2ωmax
ph ,ωmax
ph being the largest phonon energy in
the system. The magnitude of ∆0(iωn) is estimated from
the BCS formula8at zero temperature 2∆0(iωn)/Tc=
3.52, with Tcgiven by Allen-Dynes equation.51
Our experience shows that the convergence of the iter-
ative self-consistent solution is significantly improved by
using the Broyden mixing scheme commonly employed in
standard density-functional calculations.52,53 For the test
cases considered in Sec. IV below we find that around 15-
20 iterations are sufficient to achieve convergence when-
ever T<
∼0.8Tc. The number of iterations increases to
40-60 for temperatures between 0.8-0.95Tc, and may ex-
ceed 100 for T>
∼0.95Tc. In order to accelerate the con-
vergence we use the gap functions calculated at a given
temperature as seeds for the iterations at the next tem-
perature. An alternative strategy for solving the equa-
tions when T≃Tcwould be to use the linearized form of
the Eliashberg equations and determine the critical tem-
perature by solving an eigenvalue problem,2,17 however
we did not explore this possibility.
In order to determine the superconducting gap along
the real energy axis we consider two possibilities. The
first possibility is to perform an approximate analytic
continuation using Pad´e functions.42,43,45 This procedure
works well if the Pad´e functions are constructed using the
Matsubara frequencies on the imaginary axis. The sec-
ond possibility consists of performing the exact analytic
continuation of the imaginary solution to the real energy
axis as described in Sec. III B. Since this latter approach
is computationally very demanding, we speed up the con-
vergence of the iterative analytic continuation by using
the approximate Pad´e continuation as an initial guess.
IV. APPLICATIONS
In order to validate the computational methodology
developed within EPW, we investigate two prototypical
systems: the nearly-isotropic lead (Pb) superconductor,
and the anisotropic magnesium diboride (MgB2) super-
conductor.
A. Computational details
The calculations are performed within the local density
approximation (LDA) to density-functional theory54,55
using Quantum ESPRESSO.49 The valence electronic wave-
functions are expanded in planewaves basis sets with ki-
netic energy cutoff of 80 Ry and 60 Ry for Pb and MgB2,
respectively. The core-valence interaction is taken into
account by using norm-conserving pseudopotentials.56,57
For Pb we consider four valence electrons and a scalar-
relativistic pseudopotential. In order to facilitate the
comparison with previous theoretical studies we use the
LDA theoretical lattice parameters for Pb (a= 4.778 ˚
A)
and the experimental lattice parameters for MgB2(a=
3.083 ˚
A and c/a = 1.142). The charge density is com-
puted using Γ-centered Brillouin-zone mesh with 163
and 243k-points for Pb and MgB2, respectively, and a

7
0 1 2 3 4 5 6 7 8 9 10
0.0
0.5
1.0
1.5
2
F( )
(
meV
)
FIG. 1: (Color online) Calculated isotropic Eliashberg spec-
tral function α2F(black solid line) of Pb, and cumulative
contribution to the electron-phonon coupling strength λ(red
dashed line). The top of the red dashed curve corresponds to
λ=1.24.
Methfessel-Paxton smearing58 of 0.02 Ry. The dynami-
cal matrices and the linear variation of the self-consistent
potential are calculated within density-functional pertur-
bation theory46 on the irreducible set of a regular 83(Pb)
and 63(MgB2)q-point meshes. The electronic wavefunc-
tions required for the Wannier interpolation within EPW
are calculated on uniform and Γ-centered k-point meshes
of sizes 83and 63for Pb and MgB2.
In the case of Pb four Wannier functions are used to
describe the electronic structure near the Fermi level.
These states are sp3-like functions localized along each
one of the Pb-Pb bonds, with a spatial spread of 2.40 ˚
A.
In the case of MgB2we consider five Wannier functions
in order to describe the band structure around the Fermi
level. Two functions are pz-like states and are associated
with the B atoms, and three functions are σ-like states
localized in the middle of B-B bonds. The spatial spread
of the MLWFs in MgB2are 2.02 ˚
A (pz) and 1.16 ˚
A (σ).
In order to solve the Eliashberg equations we evalu-
ate electron energies, phonon frequencies, and electron-
phonon matrix elements on fine grids using the method
of Ref. 27. The fine grids contain (403,403)k- and q-
points for Pb (random grids), and (603,303) points for
MgB2(uniform Γ-centered grids). Such an extremely
fine sampling of the Brillouin zone is found to be cru-
cial for the convergence of the superconducting energy
gap in the fully anisotropic case. The frequency cut-
off ωcin Eqs. (21),(22) and Eqs. (35),(36) is set to ten
times the maximum phonon frequency of the system:
ωc= 10ωmax
ph . The calculations are performed using
smearing parameters in the Dirac delta functions cor-
responding to 100 meV and 50 meV for electrons and
phonons, respectively.
B. Lead
Bulk lead is the best known example of a strong-
coupling superconductor, exhibiting a superconducting
0 10 20 30 40 50 60
-0.4
0.0
0.4
0.8
1.2
0 10 20 30 40 50 60
-1.5
0.0
1.5
3.0
(a)
(meV)
Re - Pade approx.
Im - Pade approx.
Re - analytic cont.
Im - analytic cont.
(meV)
(meV)
(b)
FIG. 2: (Color online) Calculated energy-dependent super-
conducting gap of Pb at T=0.3 K. The gap is obtained by
solving the isotropic Eliashberg equations with µ∗
c= 0.1. (a)
Superconducting gap along the imaginary energy axis (black
solid line). (b) Superconducting gap along the real energy
axis. We show both the solutions obtained from the approxi-
mate analytic continuation using Pad´e functions (black solid
line and red dashed line), and the solutions obtained using
the iterative analytic continuation (green dash-dotted line and
blue dotted line).
transition temperature Tc= 7.2 K.59 Although Pb is
known to be a two-band superconductor, the supercon-
ducting gap function is only very weakly anisotropic,60–63
therefore for the sake of testing our method we use the
isotropic approximation to the Migdal-Eliashberg formal-
ism described in Sec. II D.
Figure 1 shows the calculated Eliashberg spectral func-
tion α2F(ω) and the corresponding electron-phonon cou-
pling parameter λ. We find an overall good agreement
with experimental results,59 although we observe a small
(≃0.5 meV) but non-negligible blueshift of the two peaks
in the Eliashberg function. This blueshift is well un-
derstood now and arises from the overestimation of the
phonon frequencies in the absence of spin-orbit coupling
in our calculation.64,65 Our calculated electron-phonon
coupling λ=1.24 lies in between the values reported in
previous theoretical studies,39,62,63 although it is some-
what smaller than the value 1.55 obtained from tunneling
measurements59 owing to the neglect of spin-orbit cou-
pling.
Figure 2 shows the solutions of the isotropic Eliashberg
equations Eqs. (35),(36) for µ∗
c= 0.1 and T=0.3 K. Along
the imaginary axis the superconducting gap function is
purely real and displays a frequency dependence similar

8
0 2 4 6 8 10 12 14 16
0
2
4
6
0 2 4 6 8 10 12 14 16
1.0
1.1
1.2
(a)
N
S
( )/N
F
N
S
()/N
F
(meV)
(b)
FIG. 3: (Color online) (a) Calculated quasiparticle density
of states of Pb at T=0.3 K (black solid lines). The supercon-
ducting gap is obtained from Fig. 2. (b) Same quantity as in
(a), magnified in order to show the structure which is used in
tunneling experiments for extracting the Eliashberg spectral
function.
to standard plasmon-pole models [Fig. 2(a)]. The contin-
uation of the calculated superconducting gap to the real
energy axis is shown in Fig. 2(b). We see that the approx-
imate analytic continuation using Pad´e functions and the
exact iterative analytic continuation yield very similar
results. As expected the approximate Pad´e continuation
misses some of the fine features which are instead ob-
served in the exact iterative continuation. Our calculated
superconducting gap is in very good agreement with solu-
tions of the Eliashberg equations obtained directly on the
real energy axis.62 In Fig. 2(b) we see that a two-peak
structure emerges both in the real part and the imagi-
nary part of ∆(ω). These two peaks occur on the scale
of the phonon energies and correlate with those observed
in the Eliashberg spectral function of Fig. 1.5A detailed
analysis shows that the peaks in the real part of the gap
function are blueshifted by approximately ∆0= ∆(ω=0)
with respect to the corresponding peaks in α2F(ω).
Figure 3(a) shows the normalized quasiparticle density
of unoccupied states obtained from Eq. (40) using the
gap function of Fig. 2. As expected the strong van Hove
singularity marks the leading edge ∆0of the supercon-
ducting gap. The fine structure of the density of states
around the van Hove singularity [Fig. 3(b)] is the direct
signature of the electron-phonon physics and is precisely
the basis for direct measurements of the Eliashberg func-
tion using tunneling spectroscopy.
Figure 4 shows the superconducting gap function at the
0 1 2 3 4 5 6 7
0.0
0.4
0.8
1.2
0.00 0.06 0.12
1.2
1.6
2.0
0.00 0.06 0.12
6
7
8
9
(a)
0
(meV)
T (K)
(b)
*
c
0
(T=0) (meV)
(c)
T
c
(K)
*
c
FIG. 4: (Color online) (a) Calculated superconducting gap
of Pb at the Fermi level as a function of temperature (disks).
The Coulomb parameter is set to µ∗
c= 0.1. (b) Calculated
superconducting gap of Pb at the Fermi level for T=0 K as a
function of the Coulomb parameter µ∗
c(disks). (c) Calculated
critical temperature as a function of the Coulomb parameter
µ∗
c(disks). In all panels the solid lines are guides to the eye.
Fermi level as a function of temperature, calculated for
µ∗
c= 0.1. The leading edge of the gap at T=0 K is found
to be ∆0=1.24 meV, in good agreement with tunneling
measurements yielding 1.33 meV.59 The superconduct-
ing Tcis identified as the temperature at which the gap
vanishes. From Fig. 4(a) we find Tc=6.8 K, in very good
agreement with previous theoretical studies,7,62 and only
slightly lower than the experimental datum of 7.2 K. For
completeness in Fig. 4(b) and (c) we also explore the sen-
sitivity of the calculated gap and critical temperature to
the choice of the Coulomb parameter µ∗
c. As expected, a
reduction of the effective Coulomb interaction results in
an increase of both ∆0and Tc.
C. Magnesium diboride
Within the class of phonon-mediated superconductors
MgB2holds the record of the highest critical tempera-
ture, with Tc=39 K.66 After a decade of intense exper-
imental and theoretical investigations since its discov-
ery, it is now understood that MgB2is an anisotropic
two-gap electron-phonon superconductor.15,17,67–69 The
anisotropy of the superconducting gap is a consequence
of the multi-sheet Fermi surface of MgB2, consisting of
two hole-like coaxial cylinders arising from the σbonding
bands, and two hole-like tubular networks arising from
the πbonding and antibonding bands (see for example
Fig. 3 of Ref. 67).
Figure 5(a) shows our calculated isotropic Eliash-

9
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
( )
(b)
k
0 10 20 30 40 50 60 70 80 90 100
0.0
0.5
1.0
1.5
2
F( )
(
meV
)
(a)
FIG. 5: (Color online) (a) Calculated isotropic Eliashberg
spectral function α2Fof MgB2(black solid line) and cumu-
lative contribution to the electron-phonon coupling strength
λ(red dashed line). (b) Distribution of the electron-phonon
coupling strenght λkfor MgB2(black solid line).
berg spectral function α2F(ω) and electron-phonon cou-
pling strength λfor MgB2.α2F(ω) displays a large
dominant peak centered around 60 meV and a sec-
ond weaker peak centered around 86 meV. The corre-
sponding isotropic electron-phonon coupling strength is
λ=0.748. In order to quantify the different contributions
to the coupling strength associated with the σsheets and
from the πsheets of the Fermi surface we evaluate a
band- and wavevector- dependent electron-phonon cou-
pling strength defined by λk=Pk′Wk′λ(k,k′, n = 0).
Figure 5(b) shows that the calculated λkcluster into two
separate ranges. The lower range λk=0.35-0.50 corre-
sponds to the coupling of the πFermi surface sheets, and
the higher range 0.95-1.35 corresponds to the coupling of
the σsheets. The wider range of λkin the σsheets re-
flects a more pronounced anisotropy with respect to the π
sheets. The structure of α2F(ω) and the calculated cou-
pling strength are in good agreement with previous cal-
culations.15,17,70–72 In particular our results are in very
good agreement with those reported in Ref. 71 where a
related interpolation scheme was used. For completeness
we show in Table I the sensitivity of the calculated aver-
age coupling strength λon the underlying Brillouin-zone
grids, and we compare with previous first-principles cal-
culations.
Figure 6(a) shows the anisotropic superconducting gap
function ∆(k, ω) of MgB2at T=10 K calculated along
the imaginary axis using the anisotropic Eliashberg equa-
tions Eqs. (21),(22) and µ∗
c=0.16. Figure 6(b) shows the
real part of the superconducting gap function along the
real energy axis, as obtained from the imaginary axis
solutions of Fig 6(a) via the approximate analytic con-
tinuation using Pad´e functions. The two-gap nature of
MgB2emerges in a completely natural way from our im-
plementation. Indeed for each energy two distinct sets
of superconducting gaps can be identified and associated
with the σand the πsheets of the Fermi surface. The two
gaps are both anisotropic, and the corresponding Fermi-
surface averages are ∆π= 1.8 meV and ∆σ= 8.5 meV,
respectively. For comparison the experimental values for
the gaps lie in the range 2.3-2.8 meV for π-band, and 7.0-
7.1 meV for σ-band.73–75 As in the case of Pb, the struc-
ture that can be observed both in the σand πsupercon-
ducting gaps reflect the peaks occuring in the Eliashberg
spectral function of Fig. 5.
Figure 7(a) shows the calculated leading edges ∆kof
the superconducting gaps as a function of temperature.
Both the πand σgaps vanish at the critical tempera-
ture Tc=50 K. The corresponding quasiparticle density
of states, presented in Fig. 7(b), clearly shows the two-
gaps structure of MgB2, in agreement with experiment.69
Our calculated critical temperature is larger than the
experimentally measured Tcof 39 K,66 however it is in
very good agreement with previous first-principles cal-
culations based on the ME formalism17 or the SCDFT
formalism.15 At this time it is still unclear whether the
overestimation of the experimental critical temperature
is due to possible anharmonic effects17 or to the use
of an isotropic Coulomb parameter.15 In fact, using the
anisotropic Eliashberg formalism and a Coulomb param-
eter µ∗
c= 0.12, the authors of Ref. 17 find that the cal-
culated Tcdecreases from 55 K to 39 K if phonon anhar-
Reference k-mesh q-mesh λ
Bohnen et al. (Ref. 70) 363630.73
Choi et al. (Ref. 17) 12×18212×1820.73
Floris et al. (Ref. 15) 2432030.71
Eiguren et al. (Ref. 71) 4034030.776
Calandra et al. (Ref. 72) 8032030.741
This work 403203(403) 0.735
803203(403) 0.739
603303(603) 0.748
3033030.782
5035030.744
64000 8000 0.757
216000 27000 0.726
TABLE I: Electron-phonon coupling strength λof MgB2cal-
culated using various meshes of k- and q-points in the Bril-
louin zone. The numbers in the brackets correspond to a sec-
ond choice of q-mesh while keeping the k-mesh unchanged.
The two bottom rows correspond to uniform random distri-
butions of k- and q-points.

10
0 20 40 60 80 100 120 140 160 180
-3
0
3
6
9
0 20 40 60 80 100 120 140 160 180
-20
-10
0
10
20
30
(a)
(meV)
Re - Pade approx.
(meV)
(meV)
(b)
FIG. 6: (Color online) Calculated energy-dependent super-
conducting gap of MgB2at T=10 K. The gap is obtained
by solving the fully anisotropic Eliashberg equations with
µ∗
c= 0.16. (a) Superconducting gap along the imaginary
energy axis (black dots). (b) Superconducting gap along the
real energy axis, obtained from the approximate analytic con-
tinuation using Pad´e functions (black dots). Only a represen-
tative sample of 105data points out of entire set of calculated
gaps (107points) is shown for clarity.
monicity is taken into account. On the other hand, using
the SCDFT formalism the authors of Ref. 15 calculate a
critical temperature Tc=22 K when using the complete
wavevector-dependent superconducting gap. When em-
ploying a band-averaged superconducting gap and var-
ious levels of approximations for the Coulomb interac-
tion, the same authors find critical temperatures in the
range 30-50 K.15 A similar sensitivity of the calculated
Tcto fine details of the calculations are reported in other
studies based on a two-bands approximation to the ME
formalism.14,38 The origin of the discrepancy between
first-principles calculations of the critical temperature of
MgB2and experiment clearly deserves further investi-
gation, however this is beyond the scope of the present
manuscript.
The use of electron-phonon Wannier interpolation al-
lows us to investigate the sensitivity of the superconduct-
ing gap to the electron and phonon meshes used for the
calculations. Figure 8 shows the energy distribution of
the σ-gap at T=30 K for eight sets of electron and phonon
meshes. If we compare the gaps shown in Fig. 8 and the
average coupling strengths reported in Table I we see that
obtaining converged results for ∆kis considerably more
challenging than for λ. For example, when using the same
k-points mesh (803) and different q-points meshes (203
-9 -6 -3 0 3 6 9
0
1
2
0 10 20 30 40 50
0
3
6
9
(a)
k
(meV)
T (K)
(b)
S F
(meV)
45 K
30 K
15 K
FIG. 7: (Color online) (a) Calculated anisotropic supercon-
ducting gaps of MgB2on the Fermi surface as a function of
temperature. The Coulomb potential is set to µ∗
c= 0.16. (b)
Corresponding quasiparticle density of states for a few repre-
sentative temperatures (15 K black solid line, 30 K red dashed
line, 45 K blue dash-doted line).
and 403), the spread of the superconducting gap distri-
bution changes from ≃2.5 meV to ≃1.5 meV, while the
average coupling strength λis essentially unaffected. The
same observation applies when we compare results for the
same q-points mesh (403) but different k-points meshes
(403and 803). These differences highlight the difficulty of
describing anisotropic quantities, and point out the need
of having a very dense sampling of the Brillouin zone not
only for the electrons but also for the phonons. For this
reason the combination of the Eliashberg formalism with
electron-phonon Wannier interpolation demonstrated in
this work provides an ideal computational tool for inves-
tigating anisotropic superconductors.
V. CONCLUSIONS
In summary we developed a computational method
which combines the anisotropic Migdal-Eliashberg for-
malism with electron-phonon interpolation based on
maximally-localized Wannier functions (EPW). Our new
method allows us to calculate the momentum- and band-
resolved superconducting gap both effectively and accu-
rately using a very fine description of electron-phonon
scattering processes on the Fermi surface. In order to
demonstrate our methodology we reported a compre-
hensive set of tests on two representative superconduc-
tors, namely Pb and MgB2, and validated our approach

11
6
7
8
9
k
(meV)
60
3
60
3
50
3
50
3
40
3
80
3
40
3
40
3
30
3
60
3
20
3
80
3
q = 20
3
k = 40
3
30
3
30
3
FIG. 8: (Color online) Calculated energy distribution of the
superconducting gaps on the σsheets of the Fermi surface of
MgB2at T=30 K (black lines). The gaps are obatined for
various k- and q-points meshes, as indicated by the labels
on the horizontal axis. The Coulomb parameter is set to
µ∗
c= 0.16.
against previous first-principles calculations as well as ex-
periment. We discussed the performance of two analytic
continuation methods for obtaining the superconducting
gap on the real energy axis, and we investigated the sen-
sitivity of the calculated gaps on the underlying choice
of the Brillouin-zone grids for electrons and phonons.
In order to set a road map for first-principles studies of
superconductors we now discuss the key approximations
involved in the present approach and suggest possible
avenues for future developments. The main approxima-
tions leading to Eqs. (21),(22) are as follows: (i) vertex
corrections in the diagrammatic expansion of the electron
self-energy are neglected following Migdal’s theorem, (ii)
the self-energy is assumed to be diagonal in band indices,
(iii) the Eliashberg equations are restricted to bands near
the Fermi level, (iv) the density of states near the Fermi
level is assumed to be constant, and (v) the Coulomb
interaction is described by an empirical parameter.2,35,36
Regarding approximation (i), it is well known that
Migdal’s theorem can break down for large electron-
phonon coupling or in the presence of Fermi-surface nest-
ing.2,35 In this area the availability of electron-phonon
matrix elements at a very small computational cost, as
provided by our method, could be used as a starting
point to explore the effects of vertex corrections beyond
Migdal’s approximation. For example the evaluation of
the first non-crossing diagram should not constitute a
major challenge, at least in the normal state. This may
help assessing the numerical error introduced by Migdal’s
approximation.
Going beyond approximation (ii) is computationally
challenging since the Green’s function and the self-energy
should be treated as matrices in the band indices.76,77
This step would increase substantially the complexity of
the formalism since the inversion of the Dyson’s equa-
tion would require matrix operations. Given the small
energies associated with the superconducting gap it is
reasonable to expect that off-diagonal terms should not
play an important role in simple cases. However in the
presence of degeneracies, and in particular in Jahn-Teller
systems, the correct description of these terms may prove
critical.78
It should be possible, at least in principle, to remove
approximation (iii) by including bands away from the
Fermi level in the calculations. Here the difficulty is
purely on the computational side, and for simple systems
this should be doable. However relaxing this constraint
would also introduce one additional equation [Eq. (16)]
in order to calculate the correct quasiparticle shifts and
impose the conservation of charge in the system.
The assumption (iv) of a constant density of states
near the Fermi level obviously breaks down for materi-
als exhibiting structure in the density of states on the
scale of the phonon energy. In theses cases it has been
shown that the energy dependence of density of states
can be retained within the isotropic approximation to
the Eliashberg equations.2,35,79 We are planning to in-
clude this possibility in our methodology in the future.
Lastly, approximation (v) means that in the present
implementation the Coulomb repulsion between electrons
remains described at the empirical level as an adjustable
parameter. In order to remove this limitation our first
step will be to evaluate the Coulomb parameter using the
dielectric matrix in the random-phase approximation.80
In the longer term it would be desirable to introduce in-
side Eq. (22) matrix elements of the screened Coulomb in-
teraction calculated using the Sternheimer-GW method
of Ref. 81.
We hope that the method reported here will prove use-
ful to the superconductivity community as a robust and
rigorous procedure for shedding light on existing super-
conductors and possibly for predicting new superconduc-
tors yet to be discovered.
Acknowledgments
E.R.M. was funded by Marie Curie IEF project FP7-
PEOPLE-2009-IEF-252586. F.G. acknowledges support
from the European Research Council under the EU
FP7/ERC grant no. 239578.
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