-
Eliashberg Theory: a short review
F. MarsiglioDepartment of Physics, University of Alberta,
Edmonton, AB, Canada T6G 2E1
(Dated: November 13, 2019)
Eliashberg theory is a theory of superconductivity that
describes the role of phonons in providingthe attractive
interaction between two electrons. Phonon dynamics are taken into
account, thusgiving rise to retardation effects that impact the
electrons, in the form of a frequency-dependentelectron
self-energy. In the superconducting state, this means that the
order parameter, generallyconsidered to be a static quantity in the
Bardeen-Cooper-Schrieffer (BCS) theory, also becomesfrequency
dependent. Here we review the finite temperature formulation of
Eliashberg theory, bothon the imaginary and real frequency axis,
and briefly display some examples of the consequences ofa
dynamical, as opposed to static, interaction. Along the way we
point out where further work isrequired, concerning the validity of
some of the assumptions used.
I. INTRODUCTION
Superconductivity is a remarkable phenomenon, notleast because
it represents a manifestation of the quan-tum world on a
macroscopic scale. It is spectacularlydemonstrated with levitating
train sets,1 and indeed thisproperty and many others of
superconductors are slowlybeing utilized in everyday applications.2
However, the es-tablished practice of incorporating superconductors
intothe real world should not be taken as an indication that“the
last nail in the coffin [of superconductivity]”3 hasbeen achieved.
On the contrary, in the intervening half-century since this quote
was written, many new super-conductors have been discovered, and we
have reached apoint where it is clear that a deep lack of
understanding4
of superconductivity currently exists. Reference [5] com-piles a
series of articles reviewing the various “families”or classes of
superconductors, where one can readily seecommon and different
characteristics. At the momentmany of these classes require a
class-specific mechanismfor superconductivity, a clearly untenable
situation, inmy opinion. Further classes have been discovered or
ex-panded upon since, such as nickelates,7 and the hydridesunder
pressure,6 for example.
Our “deep lack of understanding” should not be takento indicate
that theoretical contributions have not beenforthcoming. In fact
there have been remarkable con-tributions to key theoretical ideas
in physics that stemfrom research in superconductivity, starting
with Londontheory8 and Ginzburg-Landau9 theory, through to
BCStheory.10 When Gor’kov11 recast the BCS theory of
su-perconductivity in the language of Green functions, thenthe
stage was set for Eliashberg12,13 to formulate the the-ory that
bears his name. It is fitting that we honour thelasting impact of
his work with this brief review, on theoccasion of his 90th
birthday, and the 60th anniversaryof the publication of two papers
that paved the path forconsiderable future quantitative work in
superconductiv-ity. Based on an index I am fond of using for
famouspeople, his name appears in titles of papers 248 times,and in
abstracts and keywords 1439 times.16
Before proceeding further, we wish to make some re-marks about
the nature of this review. It will necessarily
repeat material from previous reviews, which we cata-logue as
follows. Scalapino17 and McMillan and Rowell18
perhaps gave one of the first comprehensive reviews ofboth
calculations and experiments that provide remark-able evidence for
the validity of Eliashberg theory forvarious superconductors. These
reviews were providedin the comprehensive monogram by Parks;19 the
readershould refer to this monogram and the references therein,as
we cannot possibly properly reference all the primaryliterature
sources before “Parks”, as this would consumetoo many pages here.
The author list in Parks is a who’swho of experts in
superconductivity, with two notable ex-ceptions, John Bardeen and
Gerasim (Sima) Eliashberg.
A subsequent very influential review was that of Allenand
Mitrović,20 where mostly superconducting Tc wasdiscussed. These
authors highlighted the expediency ofdoing many calculations on the
imaginary frequency axis,a possibility first noted in Ref. [21] and
utilized to greatadvantage in subsequent years.22–25
A few years later Rainer wrote a “state-of-the-union”address26
on first principles calculations of superconduct-ing Tc in which a
challenge was issued to both bandstructure and many-body theorists.
For the former, themissing ingredient was a complete (italics are
mine) cal-culation of the electron-phonon coupling. These werefirst
calculated in the 1960’s (e.g. Ref. [27]) but haveexperienced vast
improvement over the past 50 years,through the adoption and
improvement of Density Func-tional Theory methods, plus the
increased computationalability achieved in the intervening decades.
Excellentsummaries of this progress is provided in Refs.
[28,29,30],where two alternative procedures are described. The
firstfollows the original route of determining the electron-phonon
interaction and including this as input to theEliashberg equations,
while the second aims to treatboth the electron-phonon-induced
electron-electron in-teraction and the direct Coulomb interaction
on an equalfooting. The result is advertised to meet Rainer’s
chal-lenge and calculate Tc and other superconducting prop-erties
without any experimental input. It is noteworthythat in the second
formulation (see also Refs. [31] and[32]) the equations are
BCS-like and do not depend onfrequency, but only momentum. In the
usual formula-
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tion, the calculation of µ∗, the effective direct
electron-electron Coulomb interaction, is often simply assigneda
(small) numerical value, and therefore is treated
phe-nomenologically as a fitting parameter. The challengeto
many-body theorists remains, as more superconduc-tors have been
discovered that seem to extend beyondthe weak coupling regime, and
likely require descriptionsbeyond BCS and Eliashberg theory.
In 1990 the review by Carbotte33 provided a compre-hensive
update for a number of thermodynamic proper-ties of various
superconductors known at the time, in-cluding the high temperature
cuprate materials. Partlyfor this reason his review is titled
“Properties of boson-exchange superconductors,” since there was a
feeling atthe time (and still is in parts of the community) that
theEliashberg framework might apply to these superconduc-tors, but
with exchange of a boson other than the phonon.Quite a few years
later we wrote a review jointly,34 fo-cusing on “electron-phonon
superconductivity.” This re-view summarized known properties and
extended resultsto dynamical properties such as the optical
conductivity,building on earlier work in Ref. [35] and mini-reviews
inRef. [36]. More recently Ummarino has published a mini-review
with some generalizations to multi-band and theiron pnictide
superconductors.37
The other remark we should make is that while Eliash-berg theory
has been extremely successful, we will alsopoint out the
limitations that exist. Indeed, thesewere recognized right from the
beginning, with bothEliashberg12 and Migdal38 emphasizing that
limitationsexist on the value of the dimensionless coupling
parame-ter, λ, due to the expected phonon softening that wouldoccur
as λ increases. They claimed an upper limit ofλ ≈ 1, which then
significantly restricts the domain ofvalidity of the theory.
Constraints on the parameterswould be a constant theme over the
ensuing years. In1968 McMillan39 gave more quantitative arguments
fora maximum Tc, based on the expected relationship be-tween the
coupling strength and the phonon frequency.This was reinforced by
Cohen and Anderson40 and hasbeen discussed critically a number of
times since.41,42
Alexandrov43 has also raised objections, based on po-laron
collapse, a topic we will revisit later.
Some of the early history regarding the origins of
theelectron-phonon interaction was provided in Ref. [34]and will be
omitted here. By the early to mid 1950’sFröhlich44 and Bardeen and
Pines45 had established thatthe effective Hamiltonian for the
electron-phonon inter-action had a potential interaction of the
form46
V effk,k′ =4πe2
(k− k′)2 + k2TF
[1+
h̄2ω2(k− k′)(�k − �k′)2 − h̄2ω2(k− k′)
],
(1)where kTF is the Thomas–Fermi wave vector, and ω(q)
is the dressed phonon frequency. This part of theHamiltonian
represents the pairing interaction betweentwo electrons with wave
vectors k and k′ in the FirstBrillouin Zone (FBZ) and energies �k
and �k′ . Theinteraction Hamiltonian written in this form is
oftensaid to have “the phonons integrated out.” It was onthe basis
of this Hamiltonian that Bardeen, Cooper andSchrieffer (BCS)10
formulated a model Hamiltonian withan attractive (negative in sign)
interaction for electronenergies near the Fermi energy, �F .
II. THE ELIASHBERG EQUATIONS
Eliashberg,12 following what Migdal38 had calculatedin the
normal state, did not “integrate out the phonons,”but instead
adopted the Hamiltonian
H =∑kσ
(�k − µ)c†kσckσ +∑q
h̄ωqa†qaq
+1√N
∑kk′σ
g(k,k′)(ak−k′ + a
†−(k−k′)
)c†k′σckσ .(2)
where ckσ (c†kσ) is the annihilation (creation) operator for
an electron with spin σ and wave vector k, and aq (a†q)
is the annihilation (creation) operator for a phonon withwave
vector q. The electron-phonon coupling function,g(k,k′) is
generally a function of both wave vectors (andnot just their
difference), and in principle is calculablewith the Density
Functional Theory Methods mentionedearlier. Very often models are
adopted based on simple(e.g. tight-binding) considerations.
Eliashberg then applied the apparatus of field theoryto
formulate a pairing theory that accounts for the dy-namics of the
interaction, i.e. for retardation effects.A sketch of the
derivation, taken from Rickayzen47 (seealso Ref. [34]), is provided
in the Appendix. This is myfavourite derivation, as it does not
rely on a formalism(e.g. the Nambu formalism) whose validity
requires anact of faith (or, you simply work through everything
any-ways, to ensure that the formalism “works”).
Following Eliashberg13 with more modernnotation,17,20 the
“normal” self energy Σ(k, iωm) isseparated out into even and odd
(in Matsubara fre-quency) parts, so that two new functions, Z and χ
aredefined:
iωm[1− Z(k, iωm)
]≡ 1
2
[Σ(k, iωm)− Σ(k,−iωm)
]χ(k, iωm) ≡
1
2
[Σ(k, iωm) + Σ(k,−iωm)
].(3)
The equations that emerge are
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3
Z(k, iωm) = 1 +1
Nβ
∑k′,m′
λkk′(iωm − iωm′)g(�F )
(ωm′/ωm
)Z(k′, iωm′)
ω2m′Z2(k′, iωm′) +
(�k′ − µ+ χ(k′, iωm′)
)2+ φ2(k′, iωm′)
(4)
χ(k, iωm) = −1
Nβ
∑k′,m′
λkk′(iωm − iωm′)g(�F )
�k′ − µ+ χ(k′, iωm′)ω2m′Z
2(k′, iωm′) +(�k′ − µ+ χ(k′, iωm′)
)2+ φ2(k′, iωm′)
(5)
along with the equation for the order parameter:
φ(k, iωm) =1
Nβ
∑k′,m′
[λkk′(iωm − iωm′)g(�F )
− Vk,k′] φ(k′, iωm′)ω2m′Z
2(k′, iωm′) +(�k′ − µ+ χ(k′, iωm′)
)2+ φ2(k′, iωm′)
. (6)
These are supplemented with the electron number equation, which
determines the chemical potential, µ:
ne = 1−2
Nβ
∑k′,m′
�k′ − µ+ χ(k′, iωm′)ω2m′Z
2(k′, iωm′) +(�k′ − µ+ χ((k′, iωm′)
)2+ φ2(k′, iωm′)
. (7)
Written in this way both Z and χ are even functions ofiωm (and,
as we’ve assumed from the beginning, they arealso even functions of
k). With electron-phonon pairingthe anomalous self energy, which
determines the anoma-lous pairing amplitude φ(k, iωm), is also an
even func-tion of Matsubara frequency. A generalization of
thislatter result, giving rise to so-called Berezinskii48
“odd-frequency” pairing, is beyond the scope of this review.
Asurvey of Berezinskii pairing is given in Ref. [49].
Other symbols in Eqs. (4-7) are as follows.The numberof lattice
sites is given by N , the parameter β ≡ 1/(kBT ),where kB is the
Boltzmann constant and T is the tem-perature, µ is the chemical
potential, and g(�F ) is theelectronic density of states at the
Fermi level in the band.These equations are generally valid for
multi-band sys-tems, and then the labels k and k′ are to be
understood toinclude band indices. However, we shall proceed for
sim-plicity with the assumption of a single band, with
singleparticle energy �k. Because we are assuming finite
tem-perature right from the start, the equations are writtenon the
imaginary frequency axis, and are functions of theFermion Matsubara
frequencies, iωm ≡ πkBT (2m − 1),with m an integer. Similarly the
Boson Matsubara fre-quencies are given by iνn ≡ 2πkBTn, where n is
an inte-ger. Finally, we have also included a direct Coulomb
re-pulsion in the form of Vk,k′ , which in principle representsthe
full (albeit screened) Coulomb interaction betweentwo
electrons.
The key ingredient of Eliashberg theory (as opposedto BCS
theory) is the presence of the electron-phononpropagator, contained
in
λkk′(z) ≡∫ ∞
0
2να2kk′F (ν)
ν2 − z2dν (8)
with α2kk′F (ν) the spectral function of the phononGreen
function. This function is sometimes written asα2kk′(ν)F (ν) to
emphasize that the coupling part (α
2)can have significant frequency dependence. This
spectralfunction is often called the Eliashberg function. Equa-tion
(8) has been used as a “bosonic glue” to generalize
the application of the Eliashberg/BCS formalism to be-yond that
of phonon exchange. Very often the boson isa collective mode of the
very degrees of freedom that aresuperconducting, i.e. the
conduction electrons. Exam-ples include spin fluctuations or
plasmons, but this workis on more questionable footing.50
A significant anisotropy may exist, specifically throughthe
nature of the coupling in the Eliashberg function.Since the
important physical attribute of the Eliash-berg formalism beyond
BCS is retardation, and there-fore in the frequency domain, we will
nonetheless ne-glect anisotropy in what follows.51 More nuanced
argu-ments for the wave vector dependence of the electron-phonon
coupling are provided in Ref. [20], connected tothe energy scale
hierarchy �F >> νphonon >> Tc, whereνphonon is a
typical phonon energy scale (note that wehave adopted the standard
practice of dropping h̄ andkB , and therefore we refer to
temperatures and phononfrequencies as energies). Indeed very often
the neglect ofanisotropy was justified by the study of so-called
“dirty”superconductors, where the presence of impurities servedto
self-average over anisotropies. We will also drop thewave vector
dependence in the direct Coulomb repul-sion, although this step is
less justified. It means thatthe direct Coulomb repulsion is
represented by a singleparameter, which we will call U , since this
is what wewould obtain by reducing the long-range Coulomb
repul-sion with an on-site Hubbard interaction with strengthgiven
by U . This is one of the weak points of the Eliash-berg
description of superconductivity — an inadequatedescription of
correlations due to Coulomb interactions.In what follows we will
focus more attention on the re-tardation effects, since this is the
part that Eliashbergtheory is best designed to handle properly.
Once we drop the wave vector dependence in the cou-pling
function, all quantities (Z, χ and φ) become inde-pendent of wave
vector. The integration over the FirstBrillouin Zone can then be
performed, although here aseries of approximations are utilized.
The result can leadto confusion, so we provide some detail here.
First, once
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4
it is determined that the unknown functions in Eqs. (4-7)do not
depend on wave vector, we can replace the sumover wave vectors in
the first Brillouin zone with an in-tegration over the electronic
density of states,
1
N
∑k
→∫ �max�min
d� g(�), (9)
where g(�) is the single electron density of states and �minand
�max are the minimum and maximum energies of theelectronic band.
Since typically the energy scales are suchthat Tc > Tc. Now the
equationsare
Z(iωm) = 1 +πTcωm
+∞∑m′=−∞
λ(iωm − iωm′)ωm′√
ω2m′ + ∆2(ωm′)
(15)
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5
Z(iωm)∆(iωm) = πTc
+∞∑m′=−∞
[λ(iωm − iωm′)− u∗(ωc)θ(ωc − |ωm′ |)]∆(iωm′)√
ω2m′ + ∆2(ωm′)
, (16)
where u∗(ωc) ≡ g(�F )U∗(ωc). One should note thatU∗(ωc) < U ,
physically corresponding to the fact thatretardation effects allow
two electrons to exchange aphonon with one another while not being
at the sameplace at the same time. This means they do not feel
thefull direct Coulomb interaction with one another.
Thus far we have written the Eliashberg equations asfunctions of
imaginary frequency. As we will see in thenext subsection one can
solve these equations as theyare, to determine many thermodynamic
quantities of in-terest, in particular Tc. However, later we will
extendthese equations to the upper half-plane, and in particu-lar
just above the real axis. This is required for the eval-uation of
dynamic quantities like the tunneling densityof states and the
optical conductivity.17,34,36 In anticipa-tion of these results we
note here that we use functionsZ(z) and φ(z) [and therefore ∆(z)]
with the followingproperties53 as a function of complex frequency
z
Z(z∗) = Z∗(z); Z(−z) = Z(z), (17)φ(z∗) = φ∗(z); φ(−z) = φ(z),
(18)
∆(z∗) = ∆∗(z); ∆(−z) = ∆(z). (19)
A. Results on the imaginary axis: Tc
To compute actual results for Tc, along with the gapfunction
∆(ωm) and the renormalization function Z(ωm),we need to specify α2F
(ν) (now assumed to be isotropic)and u∗(ωc). The latter quantity is
very difficult to com-pute, and the former is more tractable
through Den-sity Functional Theory. Historically it has been
“mea-sured” through tunnelling measurements.18 We use quo-
tation marks around the word “measured” because infact the
current is measured while the spectral func-tion is extracted
through an inversion process that re-quires theoretical input
through the Eliashberg equa-tions themselves.18 We will simply
adopt a model spectralfunction given by
α2F (ν) =λ0ν02π
[�
(ν − ν0)2 + �2− �ν2c + �
2
]θ(νc−|ν−ν0|),
(20)that is, a Lorentzian line shape cut off in such a way
thatthe function goes smoothly to zero in the positive fre-quency
domain. This Lorentzian has a centroid given byν0 and a half-width
given by �. The cutoff frequency pa-rameter νc makes the Lorentzian
go to zero at frequencyν0 + νc and frequency ν0 − νc. For
concreteness we willuse a variety of values of ν0 with � = 0, or �
≈ ν0/10. Thefirst choice results in a δ-function spectrum with
weightsuch that the mass enhancement parameter, λ, definedby
λ ≡ 2∫ ∞
0
dνα2F (ν)
ν(21)
is simply given by λ0. As � increases λ decreases fromλ0;
however in what follows we will adjust λ0 to keep λconstant.54
Since the main focus of Eliashberg theory isthe effect of
retardation, we will often set u∗(ωc) = 0, butwe will nonetheless
note how this quantity affects the gapfunction and Tc.
Superconducting Tc is determined by linearizing thegap
equations, Eqs. (15,16) so that they become
Z(iωm) = 1 +πTcωm
{λ+ 2
m−1∑n=1
λ(iνn)
}. (22)
Z(iωm)∆(iωm) = πTc
+∞∑m′=−∞
[λ(iωm − iωm′)− u∗(ωc)θ(ωc − |ωm′ |)]∆(iωm′)
|ωm′ |. (23)
The latter of these two equations is an eigenvalue equa-tion and
can be solved as such. We use a power methodthat iterates the
eigenvalue and eigenvector simultane-ously by requiring that the
gap function at the lowestMatsubara frequency,55 ∆(iω1), remain at
unity. Thisprocedure tends to converge very quickly for
stronger
coupling, but requires more care for weaker coupling.56
We begin with a standard plot of Tc vs λ in Fig. 1,for a simple
δ-function spectral function with frequencyas shown. There are
scaling relations for Tc with typi-cal phonon frequency, but we
choose to show the resultsexplicitly in real units to make the
result clear. The
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6
0
20
40
60
80
100
120
140
0.0 0.2 0.4 0.6 0.8 1.0
Tc
(K)
λ
U = 0
ε = 0 (δ−function)
ν0 (meV)
100
100
1020
30
4050
Lorentzian, ε = 1 meV
u*(ωc = 1 eV) = 0.1
FIG. 1. The superconducting critical temperature, Tc (K)vs. the
dimensionless mass enhancement parameter, λ, fora variety of
characteristic phonon frequencies, as indicated.The solid curves
are for the Einstein spectrum with U = 0.The square points indicate
how Tc changes (for ν0 = 100 meVonly) when a Lorentzian is used
instead with � = 10 meV andνc = 80 meV, according to Eq. (20). For
the same valueof λ there is only a slight reduction in the value of
Tc. Thepoints marked with asterisks are again for the same
broadenedLorentzian centred at ν0 = 100 meV, but now with u
∗(ωc = 1eV. Note that νc is used as a practical cutoff for the
phononspectrum whereas ωc is used for the Matsubara cutoff for
thedirect Coulomb repulsion. See the discussion in the text forhow
plausible these parameters might or might not be.
possibility of determining an expression for Tc analyt-ically
has been discussed extensively in the literature20
and will not be done here. The trends are clear; higher Tccomes
from higher values of λ and from higher values ofthe characteristic
phonon frequency, which in the presentcase is provided by ν0. The
width of the spectrum playsa minor role, and the Coulomb repulsion
suppresses Tc,as indicated by the marked points. It is well known
thatEliashberg theory predicts that Tc increases with bothfrequency
and coupling strength as Tc ≈ ν0
√λ in the
asymptotic limit.24,57
B. Validity of the Theory
A perhaps more important question is the validity ofthe
parameters used in the calculation. We have shownresults up to a
value of λ = 1. Are higher values al-lowed? In particular, is it
possible for a material to
have a sizeable value of electron-phonon coupling
whilemaintaining a large phonon frequency? As discussed inthe
introduction, this question has been the subject ofprevious
investigation,40–42 although with only qualita-tive conclusions.
The discovery of (very) high tempera-ture superconductivity in the
hydrides58,59 under intensepressure has spurred a reassessment of
this type of anal-ysis, since a part of the community believes that
thesesuperconductors are electron-phonon driven. The mainevidence
has been an observed isotope shift.58 Moreover,the prediction of
superconductivity in some of these com-pounds through density
functional theory calculations60
adds plausibility to this explanation. However, very
highcharacteristic phonon frequencies (60 - 120 meV) andrather
large electron-phonon coupling values (λ ≈ 2 ormore) are required.
The latter is well outside the rangeconsidered reasonable,
especially given that the charac-teristic phonon energy remains so
high. Moreover, the su-perconductivity literature has unfortunately
lapsed intosimply accepting as “standard” or “conventional” a
valuefor the Coulomb pseudopotential u∗ = 0.1, and, espe-cially
given the high value of phonon frequency, the an-ticipated
reduction of the Coulomb interaction throughretardation will be
much lower than previously thought,and the value of the direct
Coulomb repulsion is undoubt-edly higher when the phonon frequency
is so high.
An additional direction of addressing this questioncomes from
microscopic calculations involving Quan-tum Monte Carlo (QMC) and
Exact Diagonalization(ED) techniques, utilizing specific
microscopic models.These methods provide controlled approximations
andare therefore suitable for benchmarking more approxi-mate
theories like Eliashberg theory. By far the mostwork in this
direction has been done on the Holsteinmodel.61 The Holstein model
retains only the short-range(on-site) interaction between local
(Einstein) oscillatorsand the electron charge density. Because it
is a very lo-cal model it is more amenable to the exact or
controlledmethods developed over the past 40 years, and thereforeis
a “favourite” for understanding the electron-phononinteraction,
much like the Hubbard model62 is heavilyused for the study of
electron-electron interactions. Ashort historical account of this
activity is provided in theAppendix of Ref. [34].
Briefly, early Quantum Monte Carlo studies in onedimension63 and
two dimensions64,65 established thatcharge-density-wave (CDW)
correlations dominate athalf-filling and close to half-filling. The
critical questionis whether, sufficiently away from half-filling,
where thesusceptibility for superconductivity is stronger than
thatfor CDW formation, do the “remnant” CDW correla-tions enhance
or suppress suppress superconducting Tc?In Ref. [66] the present
author, based on a comparisonof QMC and Migdal-Eliashberg
calculations on (very!)small lattices, argued that CDW fluctuations
actuallysuppress superconducting Tc. In the so-called renormal-ized
Migdal-Eliashberg calculations a phonon self-energywas included; in
this manner CDW fluctuations impacted
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7
FIG. 2. The singlet pairing susceptibility vs. electron den-sity
for the Holstein model on a 4 × 4 lattice. See Refs. [65and 66] for
pertinent definitions. Here a bare dimensionlesscoupling strength
λ0 = 2 and phonon (Einstein) frequencyωE = 1t are used, and the
susceptibility is plotted for a tem-perature T = t/6, where t is
the nearest neighbour hoppingparameter. In the topmost figure, QMC
results are indicatedwith error bars. The solid curves are the
result for Migdal-Eliashberg theory with a renormalized phonon
propagatorand the dashed curves are the result for the
unrenormalizedcalculations. The renormalized calculations agree
very wellwith the QMC results (both done for a 4 × 4 lattice),
in-dicating that this (combined Migdal-Eliashberg plus
phononself-energy in the bubble RPA approximation) result
accu-rately captures the impact of CDW fluctuations on the
pairingsusceptibility. In the bottom figure the renormalized
(solidcurve) and unrenormalized (dashed curves) are plotted fora
larger system. The renormalized calculations stop at anelectron
density close to n ≈ 0.9 because a CDW instabilityoccurs there. The
unrenormalized calculations carry on tohalf-filling, because they
are oblivious to the CDW instability(and fluctuations). This result
indicates that CDW fluctu-ations at densities less than n = 0.9
suppress pairing, andpresumably Tc, even though λ
eff → ∞. Reproduced fromRef. [66].
the phonon propagator, resulting in softer phonons andan
enhanced coupling constant. Comparisons with theQMC results served
to benchmark the Eliashberg-likecalculations.
Figure 2, reproduced from Ref. [66], illustrates thatthe
so-called renormalized calculations agree with theQMC results.
These calculations (solid curves) includephonon self-energy effects
which are essentially the CDWfluctuations.65 In contrast, the
unrenormalized calcula-tions (dashed curves) are the standard
Migdal-Eliashbergcalculations that omit phonon self-energy effects.
Fig-ure 2(a) illustrates that the renormalized calculationsare more
accurate, and Fig. 2(b) shows that includingCDW fluctuations
suppresses the pairing susceptibility,χSP. We understand these
results to indicate that in thevicinity of a CDW instability, while
the effective couplingconstant (λeff in Ref. [65 and 66])
increases, supercon-ducting Tc actually decreases.
More recently similar calculations have beenperformed67 and
other methodologies have beenemployed.68 In the latter reference
the role of retarda-tion in reducing the direct Coulomb interaction
was alsoaddressed; while they found qualitative agreement withthe
standard arguments, quantitative agreement waslacking, especially
for the expected large values of directCoulomb repulsion. An older
calculation with just twoelectrons,69 based on Exact
Diagonalization studies,also found qualitative support for a
retardation-relatedreduction in the direct Coulomb repulsion. It is
worthmentioning that finding this insensitivity to increasedCoulomb
repulsion, known as the “pseudopotentialeffect,”70,71 has been
looked for in QMC studies, butthese have mostly been unsuccessful.
They may still bethere; part of the problem is that QMC results
becomemore difficult as the electronic and phonon energyscales
differ from one another by a significant amount.Moreover, it may be
that if larger lattices and morerealistic phonon frequencies (i.e.
significantly less thanthe electron hopping parameter, t) are used,
the resultillustrated in Fig. 2 could change qualitatively.
An additional concern has been raised about theelectron-phonon
coupling becoming too strong — thatof polaron collapse.43 Exact
studies in the thermody-namic limit72 have established that a
single electron, in-teracting with Einstein oscillators through the
Holsteinmodel, acquires an additional mass which is modest forλ
-
8
Fermi sea.For example, in Migdal theory, the effective mass
for
electrons near the Fermi energy is m∗/me ≈ 1 + λ, evenfor λ ≈ 2
or more, whereas a single electron with thiscoupling would have an
effective mass many orders ofmagnitude higher. It is important to
note that in thequantum treatment polarons never “self-localize,”
essen-tially because of Bloch’s Theorem. However, with suchlarge
effective mass ratios, any impurities (including sur-faces), would
readily act as localization sites.
There are perhaps a few scenarios to work one’s wayout of this
dilemma. First, as we have already mentioned,perhaps the Holstein
model itself is pathological. Forthis reason it is important to
study other models. Un-fortunately other models are more difficult
to work with,but thus far the conclusions arrived at with the
Holsteinmodel seem to hold for these other models as well. For
ex-ample, in Ref. [76] a variational approach was used withthe
Fröhlich Hamiltonian in the continuum with acous-tic phonons, and
in Refs. [77 and 78] the
Barisić-Labbé-Friedel/Su-Schrieffer-Heeger (BLF/SSH) model was
ex-amined with perturbation theory and the adiabatic
ap-proximation. In either case more definitive results asachieved
with the Holstein model are still lacking, al-though all
indications are that these models have strongpolaronic tendencies
as well.
A second scenario is that as one assembles a Fermisea of
polarons, they somehow become increasingly un-dressed, presumably
due to some argument involvingPauli blocking. There are no
calculations that we areaware of, however, that provide a
demonstration of this.79
Part of the reason may be psychological; the Migdal
ap-proximation is more often called the Migdal Theorem,and so one
may be inclined to take it for granted thatthis is what will happen
when we assemble a Fermi sea— the “theorem” will be fulfilled.
However, in my opin-ion this is more a belief than an established
argument,as the Migdal approximation does not foresee or accountfor
polaron physics.
This discussion has been a long digression concern-ing the
domain of applicability of Eliashberg theory, andclearly a lot more
investigation is required on this ques-tion. For now we return to
the properties of the solutionsto the Eliashberg equations.
C. Results on the imaginary axis: in thesuperconducting
state
Returning to Eqs. (15,16), or their linearized counter-parts,
Eqs. (22,23), once Tc is determined then the gapfunction can be
determined both at Tc and below Tc.The gap function is a
generalized (frequency-dependent)order parameter. It will grow
continuously from zero atTc to its full value at zero temperature,
but it dependson frequency. In Fig. 3(a) we show the gap functionas
a function of Matsubara frequency for a variety oftemperatures, for
λ = 1 and ν0 = 10 meV. Note that
0.0
0.5
1.0
1.5
2.0
2.5
0 20 40 60 80 100
∆(iω
m)
(meV
)
ωm (meV)
(a)
ν0 = 10 meVε = 0λ = 1U = 0
T/Tc = 0.1 0.5 0.7 0.8 0.9 0.95 0.98 0.99
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100
∆(iω
m)/∆
(iω1)
ωm (meV)
(b)
ν0 = 10 meV
ε = 0λ = 1U = 0
T/Tc = 0.1 0.5 0.7 0.8 0.9 0.95 0.98 0.99 1.00
FIG. 3. (a) The gap function ∆(iωm) vs. Matsubara fre-quency,
ωm, for various temperatures. We have used a δ-function phonon
spectrum (� = 0) with ν0 = 10 meV,electron-phonon coupling
strength, λ = 1 and U = 0. Forthese parameters, Tc = 13.3 K.
Clearly the gap function in-creases in amplitude with decreasing
temperature, and belowabout T/Tc = 0.5 there is very little change
in the ampli-tude and in the frequency dependence. With a
broadenedphonon spectrum there would be only minor changes. Witha
nonzero U , the gap function would have a negative asymp-tote as ωm
→ ∞. In (b), to illustrate that there is very lit-tle change in the
frequency dependence at all temperatureswe show the normalized gap
function, ∆(iωm)/∆(iω1) vs.ωm. Now the results look very similar to
one another, whichmakes convergence from one temperature to the
next rela-tively easy. Also shown is the weak coupling
expectation56 atTc, ∆(iωm)/∆(iω1) = ω
2E/(ω
2E + ω
2m), indicated with a solid
red curve. This result clearly does not resemble the data,
sincewe the numerical results are for λ = 1. However, the
slightlymodified result, ∆(iωm)/∆(iω1) = ω
2E/(ω
2E + ω
2m/(1 + λ)
2),shown as a dashed black curve, is a fairly good fit.
-
9
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
∆(iω
1,T
)/∆
(iω1,
0)
T/Tc
ν0 = 10 meV
ε = 0
λ = 1.0
λ = 0.3
BCS
FIG. 4. The gap function at the first Matsubara
frequency(serving as an order parameter), normalized to the zero
tem-perature gap function at the first Matsubara frequency, vs.
re-duced temperature T/Tc. The blue squares are the results ata few
temperatures for calculations using a phonon δ-functionspectrum
with ν0 = 10 meV, λ = 1, and U = 0. Shown forcomparison is the weak
coupling BCS result (red curve). Thedeviations are very slight.
Also shown for comparison are theEliashberg results for the same
phonon spectrum but withλ = 0.3 (green triangles). These results
fall right on the BCSweak coupling result.
these functions are defined on a discrete set of points(the
Fermion Matsubara frequencies) that become moreclosely spaced as
the temperature is lowered. For temper-atures close to Tc the gap
function diminishes graduallyto zero at all frequencies, while at
the lowest tempera-ture the gap function attains a maximum. In Fig.
3(b)we plot the normalized values, ∆(iωm)/∆(iω1) vs. Mat-subara
frequency, and it is clear that they differ from oneanother by very
little. Returning to Fig. 3(a), the lowestfrequency function value
can be thought of as an orderparameter. In Fig. 4 we show the
lowest frequency gapfunction value, ∆(iω1), now normalized to the
value atT = 0 vs. reduced temperature, T/Tc. These are shownas blue
squares, for about 9 temperatures. Also shown isthe BCS weak
coupling result, given as a red curve. Onecan see that the
differences are small. We have also aplotted many more points
(green asterisks) for a weakercoupling, λ = 0.3 (same phonon
frequency), which fallexactly on the BCS curve. The main point is
that de-viations from the weak coupling BCS result are minor.For
much stronger coupling than given here deviations
are similarly very small, and experiment confirms this tobe the
case.80
Many measurable properties of the superconductingstate can be
calculated from the imaginary axis solu-tions to the gap function.
The renormalization function,Z(iωm), is also required but this does
not change by verymuch in the superconducting state. Examples of
measur-able properties include all thermodynamic quantities likethe
specific heat, and various critical fields. Systematicchanges with
coupling strength, as measured by λ, oralternatively the ratio of
the critical temperature to aparticular phonon frequency moment,
Tc/ωln, have beenreviewed elsewhere,33,34 and will not be repeated
here.
D. Results on the real axis
For any dynamical property (tunneling current, op-tical
response, dynamical penetration depth, etc.), therelevant Green
function (and therefore self-energy) is re-quired as a function of
real frequency. More precisely,for the retarded Green function it
is needed at ω + iδ,i.e. infinitesimally above the real axis. In
the originalliterature12,13,71,81,82 the spectral representation
was in-troduced to replace Matsubara sums with real
frequencyintegrals, and these equations were then solved,
eitheranalytically (with approximations) or numerically. Thiswas a
difficult task (especially with the computers avail-able at that
time), and eventually the procedure wasadopted that first required
a solution on the imaginaryaxis and then analytic continuation (iωm
→ ω + iδ)through some approximate process. For this type of
an-alytic function, the method of Padé approximants wasused,83
although the degree of precision needed for thegap function on the
imaginary axis was very stringent(10−12 for relative errors) in
order to achieve accurateresults on the real axis.
An appreciation for the information imbedded in imag-inary axis
solutions can be attained by considering thesimple example of
g(iωm) = sech(ωm/ν0), a very smoothfunction without structure, and
monotonically decreas-ing with frequency on the positive imaginary
axis. Theanalytic continuation can be easily done analytically;
itis g(ω + iδ) = sec[(ω + iδ)/ν0]. This function, in con-trast to
its imaginary axis counterpart, is riddled withdivergences and
discontinuities. And yet, in principle atleast, this information is
embedded in the (smooth) re-sults on the imaginary axis. In
practice, the informationis contained in the 10th significant digit
and beyond.
An alternative, numerically exact procedure was devel-oped in
the late 1980’s.84 Here we simply write down theresulting
expressions, the derivation of which is availablein Ref. [84]. They
are
-
10
φ(ω + iδ) = πT
∞∑m=−∞
[λ(ω − iωm)− u∗(ωc)θ(ωc − |ωm|)
] ∆(iωm)√ω2m + ∆
2(iωm)
+iπ
∫ ∞0
dν α2F (ν)
{[N(ν) + f(ν − ω)
] φ(ω − ν + iδ)√(ω − ν)2Z2(ω − ν + iδ)− φ2(ω − ν + iδ)
+[N(ν) + f(ν + ω)
] φ(ω + ν + iδ)√(ω + ν)2Z2(ω + ν + iδ)− φ2(ω + ν + iδ)
},
(24)
and
Z(ω + iδ) = 1 +iπT
ω
∞∑m=−∞
λ(ω − iωm)ωm√
ω2m + ∆2(iωm)
+iπ
ω
∫ ∞0
dν α2F (ν)
{[N(ν) + f(ν − ω)
] (ω − ν)Z(ω − ν + iδ)√(ω − ν)2Z2(ω − ν + iδ)− φ2(ω − ν +
iδ)
+[N(ν) + f(ν + ω)
] (ω + ν)Z(ω + ν + iδ)√(ω + ν)2Z2(ω + ν + iδ)− φ2(ω + ν +
iδ)
}, (25)
and of course ∆(ω + iδ) ≡ φ(ω + iδ)/Z(ω + iδ). Heref(ω) ≡
1/(exp(βω) + 1) and N(ν) ≡ 1/(exp(βν)− 1) arethe Fermi and Bose
functions respectively. Note that incases where the square–root is
complex, the branch withpositive imaginary part is to be chosen.
The reason forthis can be traced back to Eq. (6) [or Eq. (4)] where
theintegration over �k′ (with the assumptions made there)requires
that the pole (given the same square–root thatappears here) be
above the real axis.
It can easily be verified that substituting ω+ iδ → iωninstantly
recovers the imaginary axis equations (all theFermi and Bose
factors cancel to give zero contribu-tions beyond the initial terms
that require Matsubarasummations). Clearly the inverse is not true
— replac-ing the Matsubara frequency iωm where it appears inEqs.
(15,16) produces the first lines in Eqs. (24,25) involv-ing
Matsubara sums, but leaves out the remaining twolines in each case.
The strategy for solving these equa-tions is straightforward; the
imaginary axis equations[Eqs. (15,16) ] are first solved
self-consistently. Theseare then used in Eqs. (24,25) and these
equations areiterated to convergence. The presence of the first
linesin these equations provides a “driving term” that makesthe
iteration process quite rapid. For example, perform-ing the entire
operation (solution of imaginary axis andreal axis equations) for a
given temperature takes abouta tenth of a second on a laptop.
Moreover, T = 0 is a special case, as is clear fromthese
equations. In fact, this was recognized a long timeago,85 where
they established the following low frequency
behaviour at T = 0,
Re∆(ω + iδ) = c,
Im∆(ω + iδ) = 0,T = 0
ReZ(ω + iδ) = d,
ImZ(ω + iδ) = 0.(26)
where c and d are constants. In contrast, the behaviourat any
non-zero temperature is
Re∆(ω + iδ) ∝ ω2,Im∆(ω + iδ) ∝ ω,
T > 0
ReZ(ω + iδ) = d(T ),
ImZ(ω + iδ) ∝ 1/ω.(27)
For conventional parameter choices this distinction hasvery
little consequence, as the differences are barely dis-cernible. For
example, the expression of the imaginarypart of Z(ω + iδ) in the
normal state is given by
ImZ(ω + iδ) = 2πλν0ω
[N(ν0) + f(ν0)] (28)
for a δ-function spectrum [� → 0 in Eq. (20)] withstrength λ and
central frequency ν0. For ν0 = 10 meVand λ = 1 then Fig. 1
indicates Tc ≈ 10 K, and atT/Tc = 0.1, the exponent in the Bose and
Fermi func-tions makes this part ≈ 10−50. Thus, true to Eq. (27)the
limiting behaviour is ∝ ν0/ω. However, to see thisrequires ω/ν0
< 10
−45, making it unobservable, and in-distinguishable from the T =
0 case.
While this expression is for the normal state, the corre-
sponding one of the superconducting state is even more
-
11
−3
−2
−1
0
1
2
3
4
5
6
0 10 20 30 40 50 60
Re ∆
(ω +
iδ)
(meV
)
ω (meV)
(a)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99
ν0 = 10 meVνc = 8 meVε = 1 meVλ = 1.0 −1
0
1
2
3
4
5
0 10 20 30 40 50 60
Im ∆
(ω +
iδ)
(meV
)
ω (meV)
(b)T/Tc = 0.10
0.50
0.60
0.70
0.80
0.90
0.95
0.99
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 10 20 30 40 50 60
Re Z
(ω +
iδ)
ω (meV)
(c)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 10 20 30 40 50 60
Im Z
(ω +
iδ)
ω (meV)
(d)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.00
0
2
4
6
8
10
0 2 4 6 8 10 12 14
g(ω
)/g(ε
F)
ω (meV)
α2F(ν)
ν (meV)
(e)
0
1
2
0 5 10 15 20
FIG. 5. Frequency dependence of (a) Re ∆(ω+ iδ), (b) Im ∆(ω+
iδ), (c) Re Z(ω+ iδ), (d) Im Z(ω+ iδ), and (e) g(ω)/g(�F ),for
various temperatures in the superconducting state. Features are
discussed in the text. For the spectral function we haveused Eq.
(20) with λ = 1, ν0 = 10 meV, and � = 1 meV. It is displayed in the
inset of (e) and was used for all these figures.The colour coding
in (e) is the same as the others. We used U = 0.
severe, due to the development of the superconductingorder
parameter, which gives rise to a gap in the spec-trum. Again, given
the first two lines in Eq. (27) thereis technically no gap, but in
practice for reasons like wehave just indicated, the practical
results more closely fol-low the behaviour indicated in Eq. (26).
Where the finite
temperature result becomes pronounced and noticeablydifferent
than the zero temperature behaviour is in thestrong coupling
limit,86 but in this case the parametersare not realistic and
undoubtedly beyond the domain ofvalidity of the theory.
Returning to T = 0, for cases like the present where
-
12
the phonon spectrum has a gap there is a special
sim-plification. Basically, no iteration is required — thelow
frequency gap and renormalization functions comeentirely from the
first lines of Eqs. (24,25), and thesecan be constructed explicitly
from the imaginary axis re-sults. For a δ-function phonon spectrum,
however, onehas to be careful to convert the Matsubara summationto
an actual integral, as a discontinuity will occur at thephonon
frequency (at non-zero temperature this discon-tinuity is broadened
into a gradual drop). Once the lowfrequency gap and renormalization
functions are so con-structed, higher frequency values require the
Matsubarasum and real axis values of these functions at lower
fre-quencies only. Eventually the entire functional forms areso
constructed, without the need for iteration. As we willsee, low
temperature results converge quite rapidly to thezero temperature
result, so this non-iterative method canbe used as an alternative,
for the lowest temperatures.Nonetheless, we will proceed with fully
converged (itera-tive) finite temperature results, since these
require suchfew iterations anyways.
To show real axis results, we utilize a phonon spectrumas in Eq.
(20) with ν0 = 10 meV, νc = 8 meV, and� = 1 eV, and λ = 1. Results
with a δ-function spectrumtend to have a series of singularities,
that are anywaysartifacts of the singular spectrum, so we prefer to
showresults corresponding to this model spectrum. A series ofsuch
plots was also shown in Ref. [34] for the tunneling-derived Pb
spectrum, with a much larger value of λ, andmany more such results
have been shown in the literature,often using this same method.87
In Fig. 5 we show (a) thereal part of the gap function, (b) the
imaginary part ofthe gap function, (c) the real part of the
renormalizationfunction, (d) the imaginary part of the
renormalizationfunction, and (e) the tunneling density of
states,
g(ω)
g(�F )= Re
ω√ω2 −∆2(ω + iδ)
, (29)
which is measurable in single-particle tunneling experi-ments.
The first observation, difficult to make with justthese results, is
that an image of the α2F (ν) spectrumis contained in both the real
and imaginary parts of thegap function. Here it is the peak
structure clearly evi-dent in (a) centred around 10 meV for the
highest tem-perature shown. As the temperature decreases this
peakshifts to higher frequency, roughly by an amount equalto the
value of the gap function at low frequency (about2 meV in the
present case). Experimentation with differ-ent spectral functions
makes this observation more self-evident. See, for example, the
distinctive spectrum forPb in Fig. 4.35(a) of Ref. [34].
Both functions in (a) and (b) go to zero as the criti-cal
temperature is approached from below. As discussedearlier, they
both go to zero at zero frequency at all tem-peratures shown,
according to Eqs. (27), although onecannot see this on the scale
shown. Even for the highesttemperatures shown this behaviour can
barely be seen,but becomes evident when one expands the low
frequency
scale. For the lowest temperatures shown even expandingthe
frequency scale by a few orders of magnitude is notenough to reveal
the low-frequency behaviour indicatedby Eqs. (27). For this reason
one cannot use ∆(ω + iδ)as an order parameter at any frequency;
either one hasto revert to φ(ω + iδ) at zero frequency, or one can
usethe imaginary axis result for ∆(iωm), as we did in Fig. 4.Also
note that these functions approach zero at high fre-quency. If a
Coulomb repulsion is included then the realpart of ∆(ω+ iδ)
approaches a negative constant at highfrequency.
In contrast the real and imaginary parts of Z(ω + iδ)plotted in
(c) and (d) have changed very little in thesuperconducting state,
and remain non-zero at the su-perconducting critical temperature,
as indicated by theblack curve. An image of α2F (ν) is present in
this func-tion as well, particularly in the imaginary part (see
alsoFig. 4.35 (c) and (d) in Ref. [34]). Finally, the tunnel-ing
density of states is shown in Fig. 5(e), and revealsa “gap” that
opens from zero at T = Tc rather quicklyand then saturates to a low
temperature value as indi-cated. In fact a plot of this “gap” vs.
temperature wouldfollow the result displayed in Fig. 4 very
closely. How-ever, as first pointed out by Karakozov et al.85 there
isno “gap” (hence the parentheses) and in fact this is evi-dent in
Fig. 5(e), where there is a noticeable rounding ofthe curves at
frequencies below the sharp peak at almostall temperatures shown.
The peak is a remnant of thesquare-root singularity known from BCS
theory, whichis evident from Eq. (29) if a constant gap function
isused, ∆(ω+ iδ) = ∆0. In fact Eliashberg theory predictssmearing
of this singularity simply due to the presenceof imaginary
components of all the functions involved inEqs. (27) for all finite
temperatures. It is also worthpointing out that the BCS limit of
Eliashberg theory isnot achieved by setting the gap function to a
constant,∆(ω+ iδ)→ ∆0, but in fact the gap function is a decay-ing
function of frequency in this limit.85 This frequencydependence and
its implications for the weak couplinglimit has been further
explored in Refs. [56, 88, and 89].
Finally, though not so evident in Fig. 5(e), there are“ripples”
in the density of states beyond the “gap” region,caused by the
coupling of electrons to phonons. The pres-ence of these ripples in
experiments (see, in particular,Refs. [18, 90, and 91]) is perhaps
the strongest evidenceof the validity of Eliashberg theory. In fact
the most in-tense scrutiny has been superconducting Pb, where
theelectron-phonon coupling is particularly strong, λ ≈ 1.5,with a
value well beyond the expected domain of validity.These
experiments, coupled with an inversion techniquethat use the
Eliashberg equations themselves, result in aconsistent description
of the superconducting state for Pband other so-called “strong
coupling” superconductors.
Many systematic deviations from BCS theory havebeen
characterized, for example the gap ratio,2∆0/(kBTc),
92 the specific heat jump, and many otherdimensionless
ratios93,94 These have all been reviewed inRef. [33], and show very
systematic behaviour as a func-
-
13
tion of the strong coupling index, Tc/ωln. On the otherhand,
when systematics are examined with purely exper-imental parameters,
the picture is not so clear.95
III. SUMMARY AND OUTLOOK
I have provided just a sketch of what we consider theessence of
Eliashberg theory — retardation effects, in thecontext of a single
featureless band. The generalizationof these types of calculations
to more complicated scenar-ios is well documented in a number of
places, and havenot been reviewed here. These include order
parame-ter anisotropy, multi-band superconductivity,
Berezinskii“odd-frequency” pairing, sharply varying electronic
den-sity of state, impurity effects, and so on. These addi-tional
complications are increasingly taken into accountto understand new
classes of compounds that exhibit su-perconductivity, such as the
hydrides, MgB2, and thepnictides. In some cases, these additional
effects havebeen invoked to explain higher critical temperatures
aswell, but for the most part they are motivated by match-ing
theory to experiment.
In its bare form, Fig. 1 presents the possibilities for
Tcprovided by Eliashberg theory. The conscious decisionwas made to
extend the domain of coupling strength tounity only and not beyond,
because there are reasons tobelieve that going beyond this regime
is not viable. Atthe same time, large values of the characteristic
phononfrequency have been used, and this is why the plot ex-tends
to beyond ≈ 50 K for the vertical axis, Tc. Arethese values of
frequency, together with large values ofλ ≈ 1 viable? Probably not,
but given these sorts ofparameter values, this is what Eliashberg
in its standardform predicts.
I have also tried to touch on aspects of the theory wheremore
critical scrutiny is possible, by comparing results tothose
obtained with microscopic models, such as the Hol-stein model. We
believe there are significant difficultiesthat arise when these
comparisons are made. One re-action is to dismiss such comparisons,
as the Holsteinmodel (or the Hubbard model, for that matter) may
beregarded as “toy models,” possibly fraught with patholo-gies.
However, if the Holstein model is lacking in someway, it is
important to know why, and what other aspectof the electron-phonon
interaction (wave-vector depen-dence?) is essential to the success
of Eliashberg theory.For example, if the super-high-Tc of the
hydrides is con-firmed to originate in the electron-phonon
interaction,then clearly one or more missing gaps in our
understand-ing of how this happens needs to be filled.
Moreover, as presented here, Eliashberg theory fo-cusses on the
superconducting instability, and does notconsider other, possibly
competing, or potentially en-hancing, instabilities. This
possibility has come up asmore and more phase diagrams of families
of materials ex-hibit a nearby antiferromagnetic or
charge-density-waveinstability, as a function of some tuning
parameter (dop-
ing, pressure, etc.). It would be desirable to
generalizeEliashberg theory to be more “self-regulating,” and
havethe theory itself indicate when a competing instability
islimiting superconducting Tc, for example.
The other aspect that goes hand in hand with theelectron-phonon
coupling is the direct Coulomb interac-tion. We cannot claim to
understand superconductivityto the point of having predictive power
until we under-stand the role of Coulomb correlations, and their
detri-mental (or perhaps favourable?) effects on pairing. Akey
advancement has to come in further understandingthe role that
competitive tendencies or instabilities playin superconductivity.
Many of the modern-day meth-ods (Dynamical Mean Field Theory, for
example) seek toaddress the question of competing interactions.
Studieswith Quantum Monte Carlo methods, like the ones men-tioned
here, will also aid in furthering our understandingof interacting
electrons, and similar studies with the now-accessible much larger
lattice sizes would be welcome.
ACKNOWLEDGMENTS
This work was supported in part by the NaturalSciences and
Engineering Research Council of Canada(NSERC). I would like to
thank the many students andpostdoctoral fellows in my own group who
have con-tributed to the work described here. I also want tothank
in particular Jules Carbotte, who first taught meabout Eliashberg
theory, and Ewald Schachinger, whowas instrumental in teaching me
about programming theEliashberg imaginary axis equations. I also
want to thankSasha Alexandrov, who over the years continued to
ques-tion the validity of MIgdal-Eliashberg theory. In thesame way,
Jorge Hirsch, with whom I have worked agreat deal on other matters,
has been instrumental indiscussions concerning the validity of the
work reviewedhere. I also want to thank him for critical comments
con-cerning parts of this review. Finally, Jules sadly passedaway
earlier this year (April 5, 2019), and this review isdedicated to
his memory. He was a wonderful man, and Ifeel extremely fortunate
to have first entered the physicsworld under his guidance.
APPENDIX: Derivation of Eliashberg Theory
In this Appendix, we will first outline a derivation
ofEliashberg theory, based on a weak coupling approach.Our primary
source for this derivation is Ref. [47]. Migdaltheory of the normal
state follows by simply dropping theanomalous amplitudes in what
follows.
If we know the many-body wave function of system,we can
calculate the expectation value for any observ-able. However,
usually this is something we do not know,and instead we calculate
multi-electron Green functions,which are themselves related to
observables. The Greenfunctions are necessarily almost always
approximate, andthose computed in Eliashberg theory are no
exception. In
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14
fact, Eliashberg theory is essentially a mean-field
theory,though because of the inherent frequency dependence inthe
self energy, it is in many ways a precursor to Dynam-ical Mean
Field Theory.96
We begin with the definition of the one-electron Greenfunction,
defined in momentum space,97 as a function ofimaginary time, τ
,
G(k, τ − τ ′) ≡ − < Tτ ckσ(τ)c†kσ(τ′) >, (30)
where k is the momentum and σ is the spin. The an-gular brackets
denote a thermodynamic average. Wecan Fourier-expand this Green
function in imaginary fre-quency:
G(k, τ) =1
β
∞∑m=−∞
e−iωmτG(k, iωm)
G(k, iωm) =
∫ β0
dτG(k, τ)eiωmτ . (31)
The frequencies iωm are the Fermion Matsubara frequen-cies,
given by iωm = iπT (2m − 1), m = 0,±1,±2, ...,where T is the
temperature. Because the c’s are Fermionoperators, the Matsubara
frequencies are odd multiplesof iπT. The imaginary time τ takes on
values from 0 toβ ≡ 1/(kBT ).
A similar definition holds for the phonon Green func-tion,
D(q, τ − τ ′) ≡ − < TτAq(τ)A−q(τ ′) >, (32)
where
Aq(τ) ≡ aq(τ) + a†−q(τ). (33)
The Fourier transform is similar to that given in (31)except
that the Matsubara frequencies are iνn ≡ iπT2n,n = 0,±1,±2, ... and
occur at even multiples of iπT .These are the Boson Matsubara
frequencies.
To derive the Eliashberg equations, we follow Ref. [47],and use
the equation-of-motion method. The startingpoint is the time
derivative of Eq. (30),
∂
∂τG(k, τ) = −δ(τ) − < Tτ
[H−µN, ckσ(τ)
]c†kσ(0) >,
(34)where we have put τ ′ = 0, without loss of gener-ality. We
use the Hamiltonian (2), and assume, forthe Coulomb interaction,
the simple Hubbard model,HCoul = U
∑i ni↑ni↓. Including the Coulomb repulsion,
the result is repeated here,
H =∑kσ
�kc†kσckσ
+∑q
h̄ωqa†qaq
+1√N
∑kk′σ
g(k,k′)(ak−k′ + a
†−(k−k′)
)c†k′σckσ
+U
N
∑k,k′,q
c†k↑c†−k+q↓c−k′+q↓ck′↑, (35)
where the various symbols have already been defined inthe text.
We consider only the Green function with σ =↑;the commutator in Eq.
(34) is straightforward and weobtain(
∂
∂τ+ �k
)G↑(k, τ) =
−δ(τ)− 1√N
∑k′
gkk′ < TτAk−k′(τ)ck′↑(τ)c†k↑(0) >
+U
N
∑pp′
< Tτ c†p′−k+p↓(τ)cp′↓(τ)cp↑(τ)c
†k↑(0) > . (36)
Various higher order propagators now appear; to deter-mine them
another equation of motion can be written,which would, in turn,
generate even higher order prop-agators, and this eventually leads
to an infinite set ofequations with hierarchical structure. This
infinite se-ries is normally truncated at some point by the
processof decoupling, which is simply an approximation proce-dure.
For example, in (36) the Coulomb term is normallydecoupled at this
point and becomes
< Tτ c†p′−k+p↓(τ)cp′↓(τ)cp↑(τ)c
†k↑(0) >→
< Tτ c†p′−k+p↓(τ)cp′↓(τ) >< Tτ cp↑(τ)c
†k↑(0) >→
−δkpG↓(p′, 0)G↑(k, τ). (37)
The case of the electron–phonon term is more difficult;in this
case we define a ‘hybrid’ electron/phonon Greenfunction,
G2(k,k′, τ, τ1) ≡< TτAk−k′(τ)ck′↑(τ1)c†k↑(0) >, (38)
and write out an equation of motion for it. We simplyget
∂
∂τG2(k,k
′, τ, τ1) = −ωk−k′ < TτPk−k′(τ)ck′↑(τ1)c†k↑(0) >,(39)
where Pq(τ) = aq(τ)− a−q(τ). Taking a second deriva-tive
yields[∂2
∂τ2− ωk−k′
]G2(k,k
′, τ, τ1) =∑k′′σ
2ωk−k′gk−k′ < Tτ c†k′′−k+k′σ(τ)ck′′σ(τ)ck′↑(τ1)c
†k↑(0) > .
(40)
At this point we do not simply decouple the last line ofEq.
(40). We first need to take the phonon propaga-tor into account,
and the standard procedure is to usethe “non-interacting” phonon
propagator. The adjective“non-interacting” is in quotes because
part of the phi-losophy of proceeding in this way was a desire to
notcompute corrections to the phonon propagator, becausethe
information going into this part of the calculations(e.g. the
phonon spectral function) was going to comefrom experiment. Coming
from experiment means that
-
15
nature “had already done the calculation,” and we didnot want to
double count. Clearly, if the purpose of thiscalculation is to
compare to Quantum Monte Carlo cal-culations where this is not the
case, then something dif-ferent should be done, and this is what
motivated therenormalized Migdal-Eliashberg calculations of Refs.
[65and 66].
For now, we proceed with the standard Eliashberg cal-culations.
The equation of motion for the non-interactingphonon propagator
is(
∂2
∂τ2− ω2q
)D(q, τ − τ ′) = 2ωqδ(τ − τ ′). (41)
Inserting this expression into Eq. (40) then yields
G2(k,k′, τ, τ) =
1
N
∑k′′σ
∫ β0
dτ ′gk′′,k′′+k−k′D(k− k′, τ − τ ′)
× < Tτ c†k′′σ(τ′)ck′′+k−k′σ(τ
′)ck′↑(τ)c†k↑(0) >, (42)
where now τ1 has been set equal to τ as is requiredin (36). It
is important that this be done only afterapplying Eq. (41). The
result can be substituted intoEq. (36), and then Fourier
transformed (from imaginarytime to imaginary frequency). Before
doing this however,we recall that Gor’kov11 realized the important
role ofthe so-called Gor’kov anomalous amplitude, in the
Wickdecomposition97 of the various two–particle Green func-tions
encountered above. We therefore have to accountfor these in
addition to the pairing of fermion operatorsused in Eq. (37).
The anomalous amplitudes are defined to be
F (k, τ) ≡ − < Tτ ck↑(τ)c−k↓(0) > (43)
and
F̄ (k, τ) ≡ − < Tτ c†−k↓(τ)c†k↑(0) > . (44)
Now we need to repeat the same steps as above with Fand F̄ as we
did with G. Skipping the intermediate steps,
the result is an equation analogous to Eq. (36)
(∂
∂τ− �k
)F̄ (k, τ) =
− 1√N
∑k′
g−k′,−k < TτAk−k′(τ)c†−k′↓(τ)c
†k↑(0) >
+U
N
∑k′′,q
< Tτ c†k′′↑(τ)c
†−k′′+q↓(τ)ck+q↑(τ)c
†k↑(0) >,
(45)
and similarly for the function F . This leads to the needfor
another ‘hybrid’ electron/phonon anomalous Greenfunction,
F̄2(k,k′, τ, τ1) ≡< TτAk−k′(τ)c†−k′↓(τ1)c
†k↑(0) >, (46)
and, following the same process as for the regular
Greenfunction, we find
F̄2(k,k′, τ, τ) =
1
N
∑k′′σ
∫ β0
dτ ′gk′′,k′′+k−k′D(k− k′, τ − τ ′)
× < Tτ c†k′′σ(τ′)ck′′+k−k′σ(τ
′)c†−k′↓(τ)c†k↑(0) >, (47)
where again τ1 has been set equal to τ after applyingEq.
(41).
The Fourier definitions of the anomalous Green func-tion are the
same as Eq. (31):
F̄ (k, τ) =1
β
∞∑m=−∞
e−iωmτ F̄ (k, iωm)
F̄ (k, iωm) =
∫ β0
dτF̄ (k, τ)eiωmτ , (48)
and similarly for F . In frequency space one finds thattwo self
energies naturally arise,
Σ(k, iωm) = −1
Nβ
∑k′,m′
gkk′gk′kD(k− k′, iωm − iωm′)G(k′, iωm′), (49)
φ(k, iωm) = −1
Nβ
∑k′,m′
gk′kg−k′−kD(k− k′, iωm − iωm′)F (k′, iωm′). (50)
One can show that gk′k = g∗kk′ ; normally one expects a similar
relation with negative wave vectors, and we assume it
in what follows. These equations are then written
self-consistently and lead to Eqs. (4-7) once Eq. (3) is used.
1 See, for example, Superconducting Quantum Levitation ona 3π
Möbius Strip.
2 See, for example, Applications with Superconductivity.
https://www.youtube.com/watch?v=Vxror-fnOL4https://www.youtube.com/watch?v=Vxror-fnOL4http://www.supraconductivite.fr/en
-
16
3 R.D. Parks, in the Preface of Superconductivity, edited byR.D.
Parks (Marcel Dekker, Inc., New York, 1969) page v.
4 This phrase was first used by an anonymous referee in
ap-praising a colleague’s understanding of the subject matterof his
grant proposal (in 1990). I have plagiarized it eversince.
5 See the various articles in the Special Issue of PhysicaC,
‘Superconducting Materials: Conventional, Unconven-tional and
Undetermined’, edited by J.E. Hirsch, M.B.Maple, and F. Marsiglio,
Physica C 514, 1-444 (2015).
6 See, for example, P. P. Kong, V. S. Minkov, M. A. Ku-zovnikov,
S. P. Besedin, A. P. Drozdov, S. Mozaffari,L. Balicas, F.F.
Balakirev, V. B. Prakapenka, E. Green-berg, D. A. Knyazev, M. I.
Eremets, ‘Superconductivityup to 243 K in yttrium hydrides under
high pressure,’ inhttps://arxiv.org/abs/1909..
7 Danfeng Li, Kyuho Lee, Bai Yang Wang, Motoki Osada,Samuel
Crossley, Hye Ryoung Lee, Yi Cui, Yasuyuki Hikitaand Harold Y.
Hwang, ‘Superconductivity in an infinite-layer nickelate’, Nature
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