Mechanism, time-reversal symmetry and topology of superconductivity in noncentrosymmetric systems

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by  Mathias S. Scheurer

Mechanism, time-reversal symmetry and topology of superconductivity in
noncentrosymmetric systems
M.S. Scheurer1
1Institut für Theorie der Kondensierten Materie,
Karlsruher Institut für Technologie, D-76131 Karlsruhe, Germany
(Dated: January 22, 2016)
Weanalyzethepossibleinteraction-inducedsuperconductinginstabilitiesinnoncentrosymmetric
systems based on symmetries of the normal state. It is proven that pure electron-phonon coupling
will always lead to a fully gapped superconductor that does not break time-reversal symmetry and
6 istopologicallytrivial. Weshowthattopologicallynontrivialbehaviorcanbeinducedbymagnetic
1 doping without gapping out the resulting Kramers pair of Majorana edge modes. In case of super-
0
conductivity arising from the particle-hole fluctuations associated with a competing instability, the
2
properties of the condensate crucially depend on the time-reversal behavior of the order parameter

n of the competing instability. When the order parameter preserves time-reversal symmetry, we ob-
a tain exactly the same properties as in case of phonons. If it is odd under time-reversal, the Cooper
J channel of the interaction will be fully repulsive leading to sign changes of the gap and making

0 spontaneous time-reversal symmetry breaking possible. To discuss topological properties, we focus
2 onfullygappedtime-reversalsymmetricsuperconductorsandderiveconstraintsonpossiblepairing
states that yield necessary conditions for the emergence of topologically nontrivial superconductiv-

] ity. These conditions might serve as a tool in the search for topological superconductors. We also
n
discuss implications for oxides heterostructures and single-layer FeSe.
o
c
-
r I. INTRODUCTION of the resulting phase. E.g., in Ref. 7, it has been shown
p
foraspecificmodelofthetwo-dimensional(2D)electron
u
fluid in oxide heterostructures that there is a one-to-one
s In the past few years, topological phases of matter
t. haveattractedconsiderableinterestincondensedmatter correspondence: For conventional, i.e., electron-phonon
a physics1. Acentralconsequenceoftopologicallynontriv- induced, superconductivity, the condensate is trivial,
m whereas in case of an unconventional mechanism, i.e.,
ialbulkstructures,whichcanbeclassifiedbytopological
- invariants in momentum space, is the emergence of zero- superconductivity resulting from electronic particle-hole
d
fluctuations, topologically nontrivial behavior is found.
energy modes localized at the edge of the system. Sev-
n
Thiscanbeusedtoidentifythemicroscopicoriginofthe
o eral different materials have been experimentally identi-
pairing state by determining its topological properties.
c fied as topological insulators (see, e.g., Refs. 2 and 3),
[ whereas unambiguous evidence of their superconducting One promising class of materials for observing intrin-

analogues, topological superconductors, is still lacking sic topological behavior, are superconductors where in-
1
v despite intense research activities4,5. In case of topo- versionsymmetryisalreadybrokeninthenormalstate8.
9 logical superconductors, the edge modes are Majorana This can be the case in the bulk of a three-dimensional
5 bound states (MBS) that are highly sought-after both (3D) material if the crystal structure lacks a center of
4 because of their exotic non-Abelian statistics and poten- inversion as, e.g., for the heavy-fermion superconductor9
5 tial application in topological quantum computation4. CePt3Si, or for 2D superconductivity at interfaces and
0
Concerning the realization of these phases, one has to surfaces. Examples of the latter are given by oxide
.
1 distinguishbetween“intrinsic” and“extrinsic” topological heterostructures10,11 and single-layer12 FeSe on SrTiO3.
0 superconductors. In“extrinsic” systems,superconductiv- Themaindifferenceascomparedtocentrosymmetricsys-
6
ity is induced in a spin-orbit coupled normal conducting temsisthatthebrokeninversionsymmetrytogetherwith
1
material via injection of Cooper pairs from a trivial su- atomicspin-orbitcouplingremovethespindegeneracyof
:
v perconductor and, potentially, additional external fields the Fermi surfaces. The energetic splitting of the Fermi
Xi are applied (see, e.g., Ref. 6). In “intrinsic” topologi- surfacesdefinesanadditionalenergyscaleEso, whiches-
cal superconductors, both superconductivity as well as sentially changes the theoretical description of supercon-
r
a the nontrivial topology arise spontaneously due to the ductivity and has direct consequences for the possible
internal interactions of the system. In particular in the pairing states13–15.
search for the right material for the latter type of topo- In this paper, we address both issues of intrinsic topo-
logical superconductors, guiding principles are required logicalsuperconductorsoutlinedabove: Wederivesimple
that go beyond model studies and depend only on very guiding principles for the search for interaction-induced
few and easily accessible details of the system, such as topological superconductivity and relate the mechanism
symmetries or Fermi surface topologies. driving the superconducting instability to the topology
A related, but more fundamental, question is whether of the corresponding condensate. Our analysis indicates
there is a direct relation between the mechanism that that magnetic fluctuations, either from the proximity to
leadstothesuperconductinginstabilityandthetopology a magnetic instability or due to magnetic impurities, are
2
essential for the formation of a time-reversal symmet- Here “Rashba pair” denotes a pair of Fermi surfaces that
ric topological superconductor. To arrive at these con- mergeintoonedoublydegenerateFermisurfaceuponhy-
clusions, we focus on strongly noncentrosymmetric sys- pothetically switching off the inversion-symmetry break-
tems in the sense that the spin-orbit splitting E ex- ing terms in the Hamiltonian. To discuss the implica-
so
ceedsthetransitiontemperatureT ofsuperconductivity, tions on the topological properties, we focus on fully
c
E (cid:38) T . Our results are based on exact relations fol- gapped (sign changes only between the different Fermi
so c
lowing from symmetries of the system, most notable the surfaces) and time-reversal symmetric superconductors.
time-reversal symmetry (TRS) of the high-temperature The latter assumption is not very restrictive as many
phase, and, thus, do not depend on further microscopic noncentrosymmetric point groups do not allow for spon-
details. To describe unconventional pairing, we apply taneous breaking of TRS by a single superconducting
an effective low-energy approach16 where processes at phase transition14. Most importantly, we find that, for
high energies are assumed to lead to fluctuations in the one-dimensional (1D) and 2D systems, the total num-
particle-hole channel that eventually drive the supercon- beroftime-reversalinvariantmomenta(TRIM)enclosed
ducting instability. by Rashba pairs of Fermi surfaces must be necessarily
More specifically, it is shown that, irrespective of oddforthesuperconductortobetopologicallynontrivial.
whether superconductivity arises from phonons or fluc- Furthermore, in case of a single Rashba pair (two singly-
tuationsofatime-reversalsymmetricorderparameterin degenerate Fermi surfaces) enclosing an odd number of
theparticle-holechannel,theresultingcondensatewillbe TRIM,theresultingsuperconductor,iffullygapped,will
fully gapped, preserve all point symmetries of the high- be automatically topological irrespective of the dimen-
temperaturephaseaswellasTRS.Theinvarianceunder sionalityofthesystem(i.e.,for1D,2Dand3D).Thiscon-
the reversal of the time direction both crucially affects firms and generalizes the correspondence between mech-
the electromagnetic response of the system and deter- anism and topology found in Ref. 7.
minesthetopologicalclassificationofthesuperconductor
Our results imply that one should look for supercon-
as well as the structure of the MBS emerging at edges of
ducting systems that are close to a particle-hole insta-
thesystemincaseofatopologicallynontrivialphase1. To
bility with an order parameter that is odd under time-
deducetheassociatedtopologicalinvariant(classDIII17),
reversal(e.g.,aspin-densitywaveinstability)forrealizing
we apply the topological Hamiltonian approach18,19 us-
a noncentrosymmetric topological superconductor. This
ing the full Green’s function obtained from Eliashberg
leads to strong magnetic fluctuations that can drive a
theory20. This captures interaction effects beyond21 the
superconducting instability. Even if the superconduct-
mean-field level as the full frequency dependence of the
ingstateisduetoelectron-phononcoupling,theproxim-
self-energy is taken into account.
ity to a magnetic phase might lead to the spontaneous
We find for phonons or fluctuations of a particle-hole formation22 of local magnetic moments due to initially
order parameter which is even under time-reversal, that nonmagnetic impurities which can induce a transition to
the invariant is generically trivial. Naturally, signifi- a topological phase. Alternatively, intentional magnetic
cantresidualCoulombrepulsioncanleadtoatopological doping can be used to render an electron-phonon super-
phase. We show that also magnetic impurities can drive conductor topological. As the Fermi surface structure is
the system into a topologically nontrivial superconduct- directlyaccessibleexperimentally,e.g.,viaphotoemission
ing state that still preserves TRS. Although the disorder experiments,thenecessaryconditionthatthetotalnum-
requiredtostabilizethetopologicalstructureofthebulk ber of TRIM enclosed by Rashba pairs of Fermi surfaces
system locally breaks TRS, the MBS at the edge of the must be odd for having a 2D topological superconductor
topological domain are shown to be protected if there is can be readily applied for ruling out certain candidate
a residual reflection symmetry at the boundary. systems.
In case of fluctuations of an order parameter that is
We believe that this work will serve as a guiding tool
odd under time-reversal, the situation is completely dif-
in the search for topologically nontrivial superconduct-
ferent: The interaction is fully repulsive within and be-
ing states and, in addition, help determining the pairing
tween all Fermi surfaces such that the resulting super-
mechanism of noncentrosymmetric superconductors.
conducting order parameter will have sign changes, can
break any point symmetry as well as the TRS of the The remainder of the paper is organized as follows.
high-temperaturephase. Weconsiderthelimitwherethe In Sec. II, we introduce the notation used in this work
inversion-symmetrybreakingtermsinduceamomentum- and proof the absence of topological superconductivity
space splitting of the Fermi surfaces that is smaller than in a clean electron-phonon superconductor. To describe
the scale on which the spin-orbit texture varies. This unconventional pairing, the analysis will be extended to
roughly corresponds to E (cid:28) Λ where Λ denotes the general bosonic fluctuations in Sec. III. In Sec. IV, we
so t t
bandwidthofthesystem. Wederiveanasymptoticsym- discuss the modifications when disorder is taken into
metry and show that all possible superconducting order account and show how magnetic impurities can render
parameters can be grouped into two classes: The rela- an electron-phonon superconductor topological. Finally,
tive sign of the order parameter can only be either posi- Sec. V is devoted to illustrating the consequences of our
tive or negative at all “Rashba pairs” of Fermi surfaces. results for two specific materials.
3
II. ELECTRON-PHONON COUPLING ΘˆQˆ Θˆ† = Qˆ , and odd, ΘˆPˆ Θˆ† = −Pˆ , under time-
q −q q −q
reversal, respectively. Using this in Eq. (4), one immedi-
In this section, we will discuss conventional, i.e., ately finds that TRS demands
electron-phonon induced, superconductivity in noncen- (cid:16) (cid:17)∗
trosymmetric systems. Assuming that the normal phase g(l)(k,k(cid:48))=T g(l)(−k,−k(cid:48)) T†. (7)
is time-reversal symmetric, we will proof on a very gen-
eral level that the resulting superconducting state will Finally, the Hamiltonian of the phonons reads
be necessarily topologically trivial in the absence of dis-
orderandadditionalresidualelectronicinteractions. The Hˆph =(cid:88)ˆb†qlˆbqlωql, (8)
inclusionofthelattertwoeffects,whichmaketopological q,l
superconductivity possible, is postponed to Sec. IV.
where the phonon dispersion ω satisfies ω > 0 and
Throughout this work, we consider fermions described ql ql
ω = ω due to stability of the crystal and TRS, re-
by the general noninteracting Hamiltonian ql −ql
spectively.
Hˆ =(cid:88)cˆ† (h ) cˆ , (1)
0 kα k αβ kβ
k
A. Effective electron-electron interaction
where the indices α,β = 1,2,...2N represent all rele-
vant microscopic degrees of freedom, e.g., spin, orbitals Restating the system in the action description and in-
and subbands. Here and in the following we use hats to tegrating out the phonon degrees of freedom yields the
denote operators acting in the many-body Fock space. effective electron-electron interaction
The only symmetry we assume in this section is TRS.
Time-reversalisrepresentedbytheantiunitaryoperators Seff =−(cid:88)(cid:90) ωql g(l)(k +q,k )
Θˆ and Θ in Fock and single-particle space, respectively, int l k1,k2,q Ω2n+ωq2l αβ 1 1 (9)
i.e.
×g(l) (k −q,k ) c¯ c¯ c c .
α(cid:48)β(cid:48) 2 2 k1+qα k2−qα(cid:48) k2β(cid:48) k1β
Θˆcˆ† Θˆ† =cˆ† T , T†T =1, (2)
kα −kβ βα Here c¯ and c are the Grassmann analogues of the
α α
such that ΘˆHˆ0Θˆ† =Hˆ0 is equivalent to fermionic creation and annihilation operato(cid:82)rs cˆ†α and cˆα.
We use k ≡ (iω ,k), q ≡ (iΩ ,q) with comprising
n n k
Θh Θ† =h . (3) both momentum and Matsubara summation,
−k k
Here, Θ = TK with K denoting complex conjugation. (cid:90) 1 (cid:88)(cid:88)
···= ..., (10)
Since we will focus on spin-1/2 fermions, it holds Θ2 = β
−1 and, hence, TT =−T. k ωn k
The electron-phonon coupling giving rise to supercon- where β denotes the inverse temperature.
ductivity is taken to be of the general form For describing superconducting instabilities, it is very
convenienttoworkintheeigenbasisofthenoninteracting
Hˆ = (cid:88) cˆ† g(l)(k,k(cid:48))cˆ (cid:16)ˆb† +ˆb (cid:17). (4) part h of the high-temperature Hamiltonian. We thus
el-ph kα αβ k(cid:48)β k(cid:48)−kl k−k(cid:48)l k
k,k(cid:48),l write
As mentioned above, further interaction channels are as- c =(ψ ) f , c¯ =f¯ (ψ∗ ) (11)
kα ks α ks kα ks ks α
sumedtobeirrelevantinthissection. InEq.(4),ˆb† and
ql
where ψ denote the eigenstates of h , i.e. h ψ =
ˆb arethecreationandannihilationoperatorsofphonons ks k k ks
ql (cid:15) ψ . If the summation over s includes all 2N values,
of branch l. The associated coupling matrix g(l) can, by ks ks
Eq.(11)willjustconstituteaunitarytransformationand
virtue of spin-orbit interaction, couple states of different
thus be exact. In the following, we will only take into
spin and might have nontrivial structure, e.g., in orbital
account the bands that lead to Fermi surfaces and focus
space. It will not be explicitly specified in this work –
on the degrees of freedom in the energetic vicinity of the
only the constraints resulting from Hermiticity and TRS
chemical potential (−Λ < (cid:15) < Λ). Therefore, Eq. (11)
ks
will be taken into account. The former implies
hastobeunderstoodasalow-energyapproximation. We
(cid:16) (cid:17)∗ will label the states in such a way that, for each s, the
g(l)(k,k(cid:48))= g(l)(k(cid:48),k) . (5)
αβ βα Fermi momenta {k|(cid:15)ks = 0} form a connected set (see
Fig. 1) which we will refer to as Fermi surface s in the
To analyze the consequences of the latter, first note that
remainder of the paper.
Inserting the transformation (11) into Eq. (9) yields
Θˆˆb(†)Θˆ† =ˆb(†) (6)
ql −ql
(cid:90)
sthinecceotnhjeugdaetfeormmoamtieonntuomf tPhˆe∼latit(iˆbc†e−Qˆˆbq ∼)mˆb†−uqst+bˆebqevaennd, Sienfft = k1,k2,qVss31ss42(k1,k2,q)f¯k1+qs1f¯k2−qs2fk2s3fk1(s142)
q q −q
4
In this work we will not have to specify the phases ϕs
k
explicitly, it will only be taken into account that
eiϕs−Kk =−eiϕsk (17)
as a consequence of Θ2 = −1. Using Eqs. (7) and (16),
i.e.theconsequencesofTRSfortheelectron-phononcou-
pling and the wavefunctions of the normal state Hamil-
tonian, it is straightforward to show that
Figure 1. (Color online) Illustration of the parameterization G(sls)(cid:48)(k,k(cid:48))=ei(ϕsk(cid:48)(cid:48)−ϕsk)(cid:16)G(sl)s(cid:48) (−k,−k(cid:48))(cid:17)∗. (18)
K K
of the Fermi surfaces in case of a 2D system. All Fermi sur-
Thisisacentralrelationforouranalysisasitcanbeused
faces are chosen to be connected such that Kramers partner
torewritetheCooperchanneloftheinteraction(12),i.e.,
{(s,Ω),(s ,Ω )} can belong to different Fermi surfaces. All
K K
(distinct) TRIM, defined by k = −k, are indicated as green thescatteringprocessofaKramerspair{s,k;sK,−k}of
dots. quasiparticles into another Kramers pair {s(cid:48),k(cid:48);s(cid:48) ,−k(cid:48)}
K
as depicted in Fig. 2(a). Eq. (18) readily yields for this
type of scattering event
with coupling tensor VssK(cid:48)ss(cid:48)K(k,−k,k(cid:48)−k)=ei(ϕsk−ϕsk(cid:48)(cid:48))Vs(cid:48)s(k(cid:48);k) (19)
Vs1s2(k ,k ,q) (13)
s3s4 1 2 where
=−(cid:88) ωql G(l) (k +q,k )G(l) (k −q,k )
l Ω2n+ωq2l s1s4 1 1 s2s3 2 2 Vs(cid:48)s(k(cid:48);k)
=−(cid:88) ωk(cid:48)−kl (cid:12)(cid:12)G(l)(k(cid:48),k)(cid:12)(cid:12)2 <0. (20)
where we have introduced (ω −ω )2+ω2 (cid:12) s(cid:48)s (cid:12)
l n(cid:48) n k(cid:48)−kl
G(l)(k,k(cid:48))=ψ† g(l)(k,k(cid:48))ψ . (14)
ss(cid:48) ks k(cid:48)s(cid:48) The very same matrix elements also govern the forward
The Hermiticity constraint (5) now becomes scattering processes shown in Fig. 2(b) with amplitude
F (k(cid:48);k) := Vs(cid:48)s(k,k(cid:48),k(cid:48) −k): Using the Hermiticity
(cid:104) (cid:105)∗ s(cid:48)s s(cid:48)s
G(l)(k,k(cid:48))= G(l)(k(cid:48),k) . (15) (5) of the electron-phonon interaction one finds F =V.
ss(cid:48) s(cid:48)s
Consequently, the combination of TRS and the fact
In this paper, we will focus on systems with singly- thattheFermisurfacesaresinglydegeneratehighlycon-
degenerate Fermi surfaces which requires a center of in- straintstheCooperchanneloftheinteraction. Asstated
version to be absent. The combination of broken inver- in Eq. (19), it can be written as the product of the
sion symmetry, e.g., at an interface or in the bulk of a time-reversal phases defined in Eq. (16) and the forward
noncentrosymmetric crystal, and atomic spin-orbit cou- scattering matrix Vs(cid:48)s(k(cid:48);k) which only has negative en-
pling will generally lift the degeneracy of the Fermi sur- tries. We emphasize that this is a very general result
faces. Together with TRS of the normal state Hamil- sincenoadditionalmodelspecificassumptionsotherthan
tonian, Eq. (3), the absence of degeneracy of the Fermi TRSandsingly-degenerateFermisurfaces(suchasnum-
surfaces implies ber/characterofrelevantorbitalsordimensionalityofthe
system) have been taken into account. In the next sub-
ψks =eiϕskΘψ−ksK (16) section, we will analyze the consequences for the result-
ing possible superconducting instabilities using Eliash-
with (k-dependent) phases ϕsk ∈ R that are determined berg theory20.
by how the phases of the eigenstates ψks are chosen Before proceeding, a few remarks are in order: Natu-
(gauge symmetry). Eq. (16), which will be used repeat- rally, the Cooper scattering amplitude (19) is a complex
edly throughout the paper, shows that TRS is more re- number that depends on the phases of the eigenstates
strictive in noncentrosymmetric systems as compared to ψ whereas the forward scattering amplitude V is inde-
ks
the situation with inversion symmetry since the eigen- pendent of the phases as it always involves a wavefunc-
state at k with energy close to the Fermi level fully de- tion and its complex conjugate in pairs [cf. Fig. 2(b)].
terminesthestructureofthewavefunctionatlowenergies Despite the gauge dependence of the Cooper scattering
at momentum −k. Here and in the following sK denotes amplitude, the time-reversal and topological properties
the Fermi surface consisting of the Kramers partners of of the resulting superconducting state are, of course, in-
the momenta of s. Depending on the topology of the dependent of the time-reversal phases ϕs as we will see
k
Fermi surfaces with respect to the TRIM, both sK = s explicitly below.
and sK (cid:54)= s are possible. E.g., for the Fermi surfaces We finally list three properties
shown in Fig. 1, s=1,4 and s=2=3 are examples of
K
the former and latter case, respectively. V (k(cid:48);k)=V (k;k(cid:48)), (21a)
s(cid:48)s ss(cid:48)
5
V (k(cid:48);k)=V (−k(cid:48);−k), (21b) following statements). To begin with charge-conjugation
s(cid:48)s s(cid:48) s
K K
V (cid:0)iω ,k(cid:48);iω ,k(cid:1)=V (cid:0)−iω ,k(cid:48);−iω ,k(cid:1), (21c) symmetry, it holds
s(cid:48)s n(cid:48) n s(cid:48)s n(cid:48) n
ΞG (iω ,−k)Ξ−1 =−G (iω ,k), Ξ=τ K, (24)
whicharereadilyreadofffromEq.(20)andwillbetaken sK n s n 1
into account in the following.
which is just a consequence of the inherent redundancy
of the Nambu Green’s function in Eq. (22).
Secondly, the TRS constraint, which is, in the micro-
B. Eliashberg theory
scopic basis, described by the operator Θ, reads
ingTthheeaNimamobfuElGiarsehebne’rsgfuthnecotiroyn20i,n23tchoenssuisptesrocfoncadluccutliantg- Θ(cid:101)ksGsK(−k)Θ(cid:101)−ks1 =Gs(k), Θ(cid:101)ks =τ3e−iϕskτ3K, (25)
phase. For our purposes, it will be convenient to per- when transformed into the eigenbasis according to
formthecalculationintheeigenbasisofthenormalstate Eq. (11). The phases ϕs enter because of the rela-
k
Hamiltonian h . We thus introduce the Nambu Green’s tion (16) between the wavefunctions of Kramers part-
k
function as ners. Note that the expression for the time-reversal
1 (cid:18) (cid:104)f f¯ (cid:105) (cid:104)f f (cid:105) (cid:19) operator stated above yields Θ(cid:101)2ks = 1 which, at first
Gss(cid:48)(iωn,k):=−β (cid:104)f¯−kkss kf¯sk(cid:48)s(cid:48)(cid:105) (cid:104)f¯−kkss −f−ksk(cid:48)Ks(cid:48) (cid:105) . (22) stiiognhto,fssepeimn-s1/to2fdeirsmagiorenes.wTithhisGcsanbbeiengrecaonGcrieleedn’bsyfunnoct--
K K K
According to this ansatz, all Cooper pairs carry ing that the full time-reversal operator Θ(cid:101)ksI that also
includes the inversion I of momentum indeed satisfies
zero total momentum excluding the formation of
translation-symmetry breaking superconductivity, e.g., (Θ(cid:101)ksI)2 =Θ(cid:101)ksΘ(cid:101)−ksK =−1asaconsequenceofEq.(17).
Fulde-Ferrell-Larkin-Ovchinnikov24,25 states. Note that this subtlety usually does not play any role as
the time-reversal operator in momentum space in many
In addition, we assume that, for determining the su-
cases (e.g. in the microscopic basis as in Eq. (2)) does
perconducting properties, the Green’s function can be
not depend on momentum. It indicates that the prop-
approximated to be diagonal in Fermi-surface space,
erty (17) of the phases ϕs carries the information that
k
G (iω ,k)=δ G (iω ,k), (23) the nondegenerate bands of the system microscopically
ss(cid:48) n s,s(cid:48) s n
arise from spin-1/2 fermions.
which will be referred to as weak-pairing approximation In order to compare our Green’s function approach
in the following. with the mean-field picture, which will be particularly
When the energetic separation E of the Fermi sur- useful when discussing the topological properties below,
so
facesislargerthantheenergyrange2Λofthelow-energy let us consider the generic superconducting mean-field
theory, Eq. (23) is enforced by momentum conservation Hamiltonian
suchthattheweak-pairingapproximationbecomesexact.
Htroixweelveemr,enevtsenoffothreEGsore<en2’Λsf,uEnqc.ti(o2n3)(2c2a)nwbiethusse(cid:54)=dass(cid:48)cmoua-- HˆMF =(cid:88)cˆ†khkcˆk+ 21(cid:88)(cid:18)cˆ†k∆k(cid:16)c†−k(cid:17)T +H.c.(cid:19).
k k
ple single-particle states with energies differing by Eso. (26)
In the calculation of the leading superconducting insta- Performing the transformation into the band basis anal-
bility this will cut off the Cooper logarithms associated ogously to Eq. (11), we get, within the weak-pairing ap-
with these processes unless these integrals are first cut proximation,
offbytemperature. Inotherwords,theweak-pairingap-
proximationisexpectedtobeapplicablefordetermining Hˆ = 1(cid:88)Ψˆ† hBdGΨˆ (27)
thesuperconductingpropertiesaslongasE (cid:38)T . This MF 2 ks ks ks
so c k
criterion agrees with the explicit check of the validity of
the weak-pairing approximation in Ref. 14. withNambuspinorΨˆ† =(cid:16)fˆ† fˆ (cid:17)andBogoliubov-
Physically, the weak-pairing approximation means ks ks −ksK
de Gennes (BdG) Hamiltonian
that the Cooper pairs are made from the same quantum
numbers as the normal state. We emphasize that this is
only a statement about the propagator G and does not hBdG =(cid:18) (cid:15)ks ∆(cid:101)s(k)e−iϕsk(cid:19). (28)
restrict the interaction to be diagonal in Fermi-surface ks ∆(cid:101)∗s(k)eiϕsk −(cid:15)ks
space. On the contrary, interband interactions are even
Here we have introduced the Fermi-surface-diagonal ma-
essential to have a unique superconducting order param-
eter as, otherwise, the free energy would be independent trix elements ∆(cid:101)s(k) = (cid:104)ψks|∆kT†|ψks(cid:105) of the order pa-
rameter.
of the relative phase of the order parameter at different
Demanding that Eq. (26) be time-reversal symmetric,
Fermi surfaces.
Before proceeding with the calculation of the Green’s ΘˆHˆMFΘˆ−1 = HˆMF with Θˆ as defined in Eq. (2), one
function, let us discuss its antiunitary symmetries (see finds that TRS is equivalent to ∆(cid:101)s(k)∈R. Comparison
Appendix A2 for more details on the derivation of the with Eq. (28) shows that TRS on the level of the BdG
6
Φ(cid:101)s(k)=2(cid:88)(cid:90) Vss(cid:48)(k;k(cid:48))DΦ(cid:101)s(cid:48)((kk(cid:48)(cid:48))), (31b)
s(cid:48) k(cid:48) s(cid:48)
δ(cid:15) (k)=−2(cid:88)(cid:90) V (k;k(cid:48))(cid:101)(cid:15)s(cid:48)(k(cid:48)) , (31c)
s ss(cid:48) D (k(cid:48))
s(cid:48) k(cid:48) s(cid:48)
where we have introduced
D (k)=[iω Z (k)]2−(cid:2)(cid:15)2(k)+Φ (k)Φ (k)(cid:3) (32)
s n s (cid:101)s s s
for notational convenience. Here Eqs. (21a) and (21b)
have been taken into account to write the expressions in
more compact form. The additional factor of 2 on the
right-hand sides of Eq. (31) (as compared to the more
Figure2. Tocalculatethesuperconductingpropertiesweonly
frequently encountered form of the Eliashberg equations
needtwospecificchannelsoftheeffectiveelectron-electronin-
teraction: ThescatteringofCooperpairsasshownin(a)and for spinfull fermions) arises since, in the band basis, the
forward scattering (b). Applying the weak-pairing approxi- theory looks as if we were considering spinless particles
mation (23) to Eliashberg theory, the normal and anomalous making more contractions of the interaction vertex pos-
components of the Nambu Green’s function follow from the sible. The time-reversal phases ϕs of the Cooper am-
k
self-consistencyrelationsrepresenteddiagrammaticallyin(c). plitude in Eq. (19) that have been absorbed by defining
Φ(cid:101)s(k) := Φs(k)eiϕsk are reminiscent of the fact that we
are considering not truly spinless particles, but spin-1/2
particles with singly-degenerate bands.
HamiltonianreadsΘ(cid:101)kshB−dkGsKΘ(cid:101)−ks1 =hBksdG whichisjusta In this work, we will focus on the vicinity of the crit-
special case of Eq. (25) restricted to the mean-field level
ical temperature of the superconducting transition and,
where
hence, linearize the Eliashberg equations (31) in Φ. To
G (iω ,k)=(cid:0)iω −hBdG(cid:1)−1. (29) proceed further, let us rewrite the momentum summa-
s n n ks tion as an angular integration over the Fermi surfaces
and an energy integration (momentum perpendicular to
The relation (29) between the mean-field Green’s func-
theFermisurface)subjecttoanenergeticcutoffΛ,which
tionG andtheweak-pairingrepresentationofthegeneral
is a characteristic energy scale of the phonons (e.g., the
multiband mean-field Hamiltonian (26) will be relevant
Debye energy). More explicitly, we replace
in Sec. IIC when discussing topological properties of the
superconducting phase beyond mean-field.
(cid:90) (cid:90) Λ (cid:90)
(cid:88) (cid:88)(cid:88)
Let us next calculate the Green’s function G within ···→β−1 d(cid:15) dΩρ (Ω)..., (33)
s
Eliashberg approximation, i.e., by solving the Dyson s k ωn s −Λ s
equationsfortheelectronicandanomalousGreen’sfunc-
tion represented diagrammatically in Fig. 2(c). The va- where ρ (Ω) > 0 denotes the angle-resolved density of
s
lidity of this approach goes beyond weak coupling. It is states that is taken to be independent of (cid:15). The dimen-
(cid:82)
controlled in the limit m/M (cid:28) 1 with m (M) denot- sionalityof dΩissetbythedimensionalityoftheFermi
s
ing the mass of the electrons (ions) where vertex correc- surface s. For the general purposes of this paper, we do
tions can be neglected according to Migdal’s theorem26. not have to specify any parameterization, we will, as il-
As anticipated by our discussion above, we now see di- lustrated in Fig. 1, only apply the convention that the
rectly that only the forward, Fig. 2(a), and the Cooper, Kramers partner of the state (s,Ω) is given by (s ,Ω ).
K K
Fig. 2(b), scattering amplitudes of the phonon mediated In addition, we take the interaction V, δ(cid:15) and Φ(cid:101) as
interaction enter. well as the quasiparticle residue Z to be only weakly de-
We parameterize the Green’s function according to pendent on the momentum perpendicular to the Fermi
surface (valid for m/M (cid:28)1) and set
(cid:18) (cid:19)
0 Φ (k)
G−1(k)=iω Z (k)τ −(cid:15) (k)τ − s (30)
s n s 0 (cid:101)s 3 Φ (k) 0 V (iω ,k;iω ,k(cid:48))(cid:39)V (iω ,Ω;iω ,Ω(cid:48)),
s ss(cid:48) n n(cid:48) ss(cid:48) n n(cid:48)
(34)
with quasiparticle weight Z (k), (cid:15) (k) = (cid:15) + δ(cid:15) (k), Φ(cid:101)s(iωn,k)(cid:39)Φ(cid:101)s(iωn,Ω)
s (cid:101)s ks s
whereδ(cid:15) (k)isthebandrenormalization,andanomalous
s
self-energiesΦ (k)andΦ (iω ,k)=Φ∗(−iω ,k). These and similarly for δ(cid:15) and Φ(cid:101). With these approximations,
s s n s n
the Eliashberg equations (31) become (for Λ→∞)
quantities, which uniquely determine G, follow from the
self-consistency equations 2 (cid:88)(cid:88)(cid:90)
Z (iω ,Ω)=1− dΩ(cid:48)ρ (Ω(cid:48))
Z (k)=1+ 2 (cid:88)(cid:90) V (k;k(cid:48))iωn(cid:48)Zs(cid:48)(k(cid:48)), (31a) s n 2n+1 n(cid:48) s(cid:48) s(cid:48) s(cid:48) (35a)
s iωn s(cid:48) k(cid:48) ss(cid:48) Ds(cid:48)(k(cid:48)) ×Vss(cid:48)(iωn,Ω;iωn(cid:48),Ω(cid:48))sign(2n(cid:48)+1),
7
(cid:90)
δ (iω ,Ω)=(cid:88)(cid:88) dΩ(cid:48)v (iω ,Ω;iω ,Ω(cid:48)) Eqs. (21c) and (39), we know that δs(−iωn,Ω) is also a
s n ss(cid:48) n n(cid:48)
n(cid:48) s(cid:48) s(cid:48) (35b) solution of Eq. (35b). Due to the absence of degenera-
cies, we conclude that δ (iω ,Ω) = ±δ (−iω ,Ω), i.e.,
×δ (iω ,Ω(cid:48)), s n s n
s(cid:48) n(cid:48) we obtain either an even- or an odd-frequency pairing
δ(cid:15)s(k)=0, (35c) state. As δs(iωn,Ω) > 0 odd-frequency pairing can be
i.e., there is no Fermi velocity correction. In Eq. (35), excluded. In combination with Φ(cid:101)sK(−k)=Φ(cid:101)s(k) follow-
ing from Fermi-Dirac statistics and the property (17) of
the normalized anomalous self-energy
the time-reversal phases, one has
δs(iωn,Ω):=Θs,n(Ω)Φ(cid:101)s(iωn,Ω) (36) Φ(cid:101)s(iωn,Ω)=Φ(cid:101)s(−iωn,Ω)=Φ(cid:101)sK(iωn,ΩK)>0. (40)
with27
Eqs. (39) and (40) constitute the main results of this
(cid:112) subsection.
ρ (Ω)
s
Θ (Ω)= (37)
s,n (cid:112) (cid:112)
|2n+1| |Z (iω ,Ω)|
s n
C. Topological properties
has been introduced in order to render the kernel
v (iω ,Ω;iω ,Ω(cid:48)) WewillnextdiscusstheconsequencesofEqs.(39)and
ss(cid:48) n n(cid:48)
(38) (40)forthetopologyofthecorrespondingsuperconduct-
:=−2Θ (Ω)V (iω ,Ω;iω ,Ω(cid:48))Θ (Ω(cid:48))
s,n ss(cid:48) n n(cid:48) s(cid:48),n(cid:48) ing phase. For this purpose, we first need to analyze its
antiunitary symmetries. In order to go beyond a mean-
of the gap equation (35b) symmetric.
field description, we have to discuss these symmetries on
Note that, as a consequence of linearizing in Φ(cid:101), the level of Green’s functions.
Eq. (35a) explicitly determines Z (iω ,Ω), i.e., it fol-
s n By design, the Nambu Green’s function defined in
lows without solving a self-consistency equation. We di-
Eq. (22) satisfies the particle-hole symmetry (24). TRS
rectly see that Z (iω ,Ω) ∈ R. In addition, it holds
s n is a much more interesting property of a superconduc-
Z (k)=Z (−k)whichisaconsequenceofitsdefinition
s sK tor in the sense that it can be spontaneously broken by
but can, alternatively, be explicitly seen in Eq. (35a) us-
the formation of the condensate. However, it is straight-
ing Eq. (21b). Together with Z (iω ,Ω) = Z∗(−iω ,Ω)
s n s n forward to check that Eq. (25) is satisfied as a conse-
following from Eq. (21c), we can summarize
quence of Zs(k) and Φ(cid:101)s(k) being real valued and invari-
Z (iω ,Ω)=Z (−iω ,Ω)=Z (iω ,Ω )∈R. (39) ant under in (s,k) → (sK,−k) [cf. Eqs. (39) and (40)]
s n s n sK n K together with (cid:15) =(cid:15) resulting from the TRS of the
ks −ksK
The properties of the superconducting order parameter high-temperaturephase. Therefore,nospontaneousTRS
followfromthesecondEliashbergequation(35b). Asop- breakingispossibleintheweak-pairinglimitifsupercon-
posed to mean-field theory, the temperature dependence ductivity is due to electron-phonon coupling.
ofEq.(35b)ismorecomplicatedandhiddeninthekernel Consequently, the resulting system is invariant both
v definedinEq.(38). However, itcanbeshown(seeAp- underchargeconjugationΞwithΞ2 =1aswellasunder
pendix B) that the leading superconducting instability time-reversal Θ(cid:101)ksI satisfying (Θ(cid:101)ksI)2 = −1 and, thus,
is, as in the mean-field case, determined by the largest belongs to class DIII17. In 1D and 2D, the supercon-
eigenvalueofthe(symmetricandreal)matrixvwhilethe ductor is classified by a Z2 and in 3D by a Z topologi-
order parameter δ (iω ,Ω) belongs to the corresponding cal invariant1. To calculate these invariants, we will use
s n
eigenspace. the topological Hamiltonian approach18,19: For a system
As v only has positive components, we conclude from withafinitegap, thetopologicalpropertiesofthemany-
the Perron-Frobenius theorem28, that the largest eigen- bodysystemdescribedbythefullGreen’sfunctionG are
value of v is nondegenerate with associated eigenvector calculated from the effective mean-field Green’s function
that can be chosen to have purely positive components
Gt(iω ,k)=(cid:0)iω −ht (cid:1)−1 (41)
aswell. Therefore,theleadinginstabilityischaracterized s n n ks
by δs(iωn,Ω) > 0 and, hence, Φ(cid:101)s(iωn,Ω) > 0, i.e., the where the “topological Hamiltonian” is given by
superconductor is fully gapped with the sign of the gap
being the same on all Fermi surfaces. Also as a function ht :=−G−1(iω =0,k). (42)
ks s
of Matsubara frequency, the anomalous self-energy does
not change sign. Due to the absence of any sign change, For the calculation of the topological invariant, we have
the superconducting state cannot break any point-group to go to zero temperature. In Eq. (42), and similarly in
symmetryandmust,therefore,transformunderthetriv- thefollowingexpressions,iω =0hastobeunderstoodas
ialrepresentationofthepointgroup(seeAppendixCfor thelimitT →0ofthefunctionevaluatedattheMatsub-
a proof of this conclusion). ara frequency ω = π/β (or equally well ω = −π/β).
0 −1
Sincev (iω ,Ω;iω ,Ω(cid:48))isinvariantunderasimulta- For this purpose, we assume that no additional topo-
ss(cid:48) n n(cid:48)
neous sign change of ω and ω , which follows from and logical phase transition occurs in the temperature range
n n(cid:48)
8
between the onset of superconductivity and T = 0. In ofTRIMenclosedbyFermisurfacesincaseofa2Dsys-
more mathematical terms, it means that, upon lowing T tem,whereasm =1fora1Dsuperconductor. Again,we
s
to zero, the structure of the solution of the (nonlinear) find,inbothdimensions,atrivialsuperconductor(ν =1)
Eliashberg equations does not change in a way that af- resulting from sign(∆(cid:101)s(ks))=1.
fects the topological invariant. Under this assumption,
the topological properties of the superconducting phase Taken together, superconductivity in noncentrosym-
canbeinferredfromthesolutionofthelinearEliashberg metric systems that arises due to electron-phonon cou-
equations (35). pling alone can neither break TRS nor any point sym-
Due to Φ(cid:101)s(0,Ω) > 0, the superconductor is fully metry of the system and must necessarily be topolog-
gapped and the Green’s function G (iω,k) must be an ically trivial. This has been derived under very gen-
s
analytic function of ω in a finite domain containing the eral assumptions: The inversion-symmetry breaking is
imaginary axis. Consequently, iωZ (iω,k)| = 0 as assumed to be sufficiently strong for the weak-pairing
iωZs(iω,k) is an odd function of ω [scf. Eq. (i3ω9=)]0. There- approximation to be valid (Eso (cid:38) Tc). The Eliashberg
fore, the topological Hamiltonian becomes approach is controlled in the limit of adiabatic ionic mo-
tion (m/M (cid:28) 1), in principle, allowing for arbitrarily
ht =(cid:18) (cid:15)ks Φ(cid:101)s(iω =0,k)e−iϕsk(cid:19), (43) strong interactions V. Also the analysis of topological
ks Φ(cid:101)s(iω =0,k)eiϕsk −(cid:15)ks invariants is performed beyond the mean-field level. We
emphasize that, despite looking deceptively like a mean-
whichismanifestlyHermitian. Furthermore,itisreadily field description, the topological Hamiltonian approach
checked to be particle-hole and time-reversal symmetric we use is equivalent18,30 to the expressions for the topo-
with Ξ and Θ(cid:101)ks as given in Eqs. (24) and (25). logical invariants involving frequency integrals of the full
Theresultingtopologicalpropertiesaremosteasilyin- Green’s functions (see, e.g, Ref. 31). Thus, also inter-
ferredbyreadingtheapproximationofthegeneralmean- action effects without static mean-field counterpart are
field Hamiltonian in Eq. (26) to the weak-pairing de- captured. This is important as dynamical fluctuations
scription (28) in reverse: Comparison of Gt in Eq. (41) can indeed change the topological properties of the sys-
and Eq. (29) shows that ht can be seen as the weak- tem as has been demonstrated in Ref. 21.
ks
pairing approximation of some mean-field Hamiltonian
of the form of Eq. (26) with the property Note that these conclusions are not altered when elec-
tronic renormalization effects of the phononic dispersion
∆(cid:101)s(k)≡(cid:104)ψks|∆kT†|ψks(cid:105)=Φ(cid:101)s(iω =0,k). (44) aretakenintoaccountsinceωql inEq.(8)canalreadybe
regarded as the fully renormalized spectrum. In Sec. III
InRef.29,ithasbeenshownthatthetopologicalclass- wewillshow,usingexactrelationsderivedfromthespec-
DIII invariant ν of a mean-field Hamiltonian of the form tralrepresentation,thatthesameholdseveniffrequency-
of Eq. (26) is, within the weak-pairing limit, determined dependentcorrectionstothephononpropagatorarecon-
bythesignoftheorder-parametermatrixelements∆(cid:101)s(k) sidered.
on the different Fermi surfaces of the system. More ex-
plicitly, in the 3D case, it holds Itisinstructivetocomparethisresultvalidfornoncen-
trosymmetric systems with the situation where inversion
ν = 21(cid:88) sign(cid:16)∆(cid:101)s(ks)(cid:17)C1s, (45) sayremdmoeutbrlyyidsepgerneseerravteeda.ndInthtehissupcaesrec,onadlluFcteirnmgistsautrefaccaens
s
only be either singlet or triplet. As has been shown in
whereksisanarbitrarypointonandC1sdenotesthefirst Ref.32usingmean-fieldtheory, singletandtripletstates
Chern number of the Fermi surface s (for the definition will be degenerate if the electron-phonon coupling satis-
of C1s we refer to Eq. (E1)). Due to Eqs. (40) and (44), fiescertainsymmetriessuchthatalreadyaninfinitesimal
we find amount of residual Coulomb repulsion favors the triplet
1(cid:88) state that breaks inversion symmetry and has nontriv-
ν = 2 C1s =0, (46) ial topological properties33. In our case, there are two
s main differences: Firstly, the absence of inversion sym-
metry generally mixes singlet and triplet components.
i.e.,atopologicallytrivialsuperconductor. Inthesecond
Secondly, e.g., for a Fermi surface enclosing the Γ-point,
equality of Eq. (46), we have used that the total Chern
number of all Fermi surfaces vanishes29. breaking point symmetries necessarily implies the pres-
ence of nodes.
In lower dimensions, the expression (45) for the topo-
logical invariant assumes the form
In the remainder of the paper, we will discuss sev-
(cid:89)(cid:104) (cid:16) (cid:17)(cid:105)ms eralgeneralizationsoftheconsiderationspresentedabove
ν = sign ∆(cid:101)s(ks) (47) including coupling to generic collective bosonic modes
s
(Sec.III),residualCoulombinteractionsaswellasdisor-
as has been in shown in Ref. 29 by means of dimen- der(Sec.IV)renderingtopologicallynontrivialproperties
sional reduction. In Eq. (47), m denotes the number possible in the weak-pairing limit.
s
9
and all other residual electron-electron interactions will
beneglectedsince,byassumption,thechanneldescribed
by the collective mode φˆ is dominant. In Eq. (49), the
qj
matrices {λ(j)} have to be Hermitian and satisfy
Θλ(j)Θ† =tλ(j) (50)
resulting from φˆ being real and Eq. (48), respectively.
In case of a model with only a single orbital where α
just refers to the spin of the electrons, one could have
{λ(j)} = {σ }, N = 1, in case of t = + and {λ(j)} =
0 B
{σ ,σ ,σ }, N = 3, for t = −. Here σ , j = 0,1,2,3,
1 2 3 B j
Figure 3. In (a), the energy scales of the effective low-energy denote the Pauli matrices in spin space. For simplicity
approach weusetodescribeunconventionalpairingareillus- of the presentation of the results, the discussion of more
trated. Part (b) and (c) show self-energy corrections of the
complex fermion-boson couplings will be postponed to
bosonic propagator due to normal conducting electrons and
Sec. IIIC.
thesuperconductingorderparameter,respectively. Thelatter
The dynamics of the bosons will be described by the
type of corrections are at least of quadratic order in Φ.
action
S = 1(cid:90) φ (cid:0)χ−1(iΩ ,q)(cid:1) φ , (51)
col 2 qj 0 n jj(cid:48) −qj(cid:48)
q
III. UNCONVENTIONAL
SUPERCONDUCTIVITY whereφisthefieldvariablecorrespondingtotheoperator
φˆ and χ (iΩ ,q) is the bare susceptibility with respect
0 n
In this section, we will extend the analysis to uncon- totheorderparameterofthecompetingparticle-holein-
ventional superconductors, i.e., systems where supercon- stability the system is close to. The full susceptibility
ductivityisnotbasedonelectron-phononinteractionbut χ(iΩn,q), renormalized by particle-hole fluctuations as
arisesfromapurelyelectronicmechanism. Eventually,it shown in Fig. 3(b), is more important since it is experi-
is the Coulomb interaction which, strongly renormalized mentally accessible, e.g., via neutron scattering or NMR
dependingonthemicroscopicdetailsofthesystem,gives relaxationrate16, andbecauseitwillenterthesupercon-
risetothesuperconductinginstability. Herewewillfully ductingself-consistencyequationsdiscussedinSec.IIIA.
neglect the electron-phonon interaction and treat the in- As shown in Appendix A1, χ has to satisfy (the same
teracting electron problem in the following low-energy holds for χ0) the exact relations
approach16: We are not interested in the behavior of
χ(iΩ ,q)=χT(−iΩ ,−q), (52a)
the system at high energies, e.g., of the order of the n n
bandwidth Λt, but only focus on the physics for energies χ(iΩn,q)=χ†(−iΩn,q), (52b)
smaller than some cutoff Λ < Λt. As shown schemati- χ(iΩn,q)=χ(−iΩn,q). (52c)
callyinFig.3(a),itisassumedthatprocessesatenergies
The first identity is just a consequence of χ being a
between Λ and Λ drives the system close to some insta-
bilitythatwedesctribebythecollectivereal(φˆ† =φˆ ) correlator of twice the same operator φˆ evaluated at q
qj −qj and −q, whereas thesecond line isbased onHermiticity,
bosonic mode φˆ , j = 1,2,...,N . For simplicity, we
qj B φˆ† =φˆ . Finally, the third relation follows from TRS
will first assume that the associated order parameter is qj −qj
of the system.
either even (t = +) or odd (t = −) under time-reversal,
Being Hermitian, χ(iΩ ,q) has real eigenvalues all of
which means mathematically n
which have to be positive as required by stability: By
Θˆφˆ Θˆ† =tφˆ . (48) assumption,thecompetinginstabilitywillnotoccurand,
qj −qj
hence, the bosons have to have a finite mass.
The proximity to, e.g., (real) charge-density or spin- Inthefollowing,wewillproceedinamannerverysim-
density wave order correspond to time-reversal even ilar to Sec. II: Writing the entire model in the field inte-
(TRE), t = +1, or time-reversal odd (TRO), t = −1, gral representation and integrating out the bosons leads
fluctuations, respectively. toaneffectiveelectron-electroninteractionoftheformof
Furthermore,weassumethattheinteractionprocesses Eq. (12) with
atenergieslargerthanΛneitherdestroytheFermi-liquid
Vs1s2(k ,k ,q)
behaviorofthefermionsnorleadtoTRSbreaking. Con- s3s4 1 2
sequently,thenoninteractingpartofthefermionicHamil- =−1ΛT (k −q,k )χ(q)Λ (k +q,k ), (53)
tonian is still of the form of Eq. (1) satisfying Eq. (3). 2 s2s3 2 2 s1s4 1 1
The fermions are coupled to the bosons via where we have introduced the N -component vector of
B
Hˆ =(cid:88)cˆ† λ(j)cˆ φˆ (49) matrix elements
int k+qα αβ kβ qj Λ (k,k(cid:48))=ψ† λψ (54)
k,q ss(cid:48) ks k(cid:48)s(cid:48)
10
inanalogytoGinEq.(14). Notethatweuseχinsteadof the gap equation (35b), i.e., v is replaced by tv. Note
thebareχ suchthattheinteractionvertexinEq.(53)is that the renormalized propagator χ is taken into ac-
0
already fully renormalized by particle-hole fluctuations. count which, diagrammatically, corresponds to replacing
Due to Eqs. (16) and (50), TRS of the system implies the bare bosonic line in Fig. 2(c) by the full line (see
Fig. 3(b)). We emphasize that, for the linearized Eliash-
Λss(cid:48)(k,k(cid:48))=tei(ϕsk(cid:48)(cid:48)−ϕsk)(cid:0)Λs s(cid:48) (−k,−k(cid:48))(cid:1)∗, (55) berg equations, there are no anomalous propagators en-
K K teringthefullbosonicline: Anyterminthebosonicself-
which constitutes the obvious generalization of Eq. (18) energyinvolvingtheanomalousself-energyΦ,suchasthe
includingnotonlyTRE(suchasphonons)butalsoTRO oneshowninFig.3(c),isatleastofquadraticorderinΦ
fluctuations. and,hence,doesnotcontribute. Therefore,wecansafely
We have seen in Sec. IIB that, in the weak-pairing usetheTRSconstraint(52c)nearthetransitionwithout
approximation, the superconducting properties are fully a priori knowledge about the time-reversal properties of
determined by the Cooper and the forward scattering the superconducting condensate.
channel shown in Fig. 2(a) and (b). As before, we still RepeatingtheargumentspresentedinSec.IIB,wedi-
find that these two interaction channels are determined rectly conclude that Eq. (39) is still valid. The kernel tv
by the same matrix elements, ofthegapequationissymmetric[cf.Eq.(21a)],realand,
hence, diagonalizable. As shown in Appendix B2, the
VssK(cid:48)ss(cid:48)K(k,−k,k(cid:48)−k)=tei(ϕsk−ϕsk(cid:48)(cid:48))Vs(cid:48)s(k(cid:48);k), (56a) leading superconducting instability is again determined
F (k(cid:48),k)=V (k(cid:48);k) (56b) by its largest eigenvalue.
s(cid:48)s s(cid:48)s
TobeginwithTREfluctuations,t=+,thekernelhas,
with exactly as V in Eq. (57), only positive components, such
thatthePerron-Frobeniustheoremcanbeapplied. Itfol-
1
Vs(cid:48)s(k(cid:48);k)=−2Λ†s(cid:48)s(k(cid:48),k)χ(k(cid:48)−k)Λs(cid:48)s(k(cid:48),k). (57) lows that the resulting superconducting order parameter
satisfies δ >0 and, thus, preserves TRS and has no sign
To show this, TRS (55) and Hermiticity, λ† = λ, have changes, neither on a given Fermi surface nor between
been taken advantage of. different Fermi surfaces. It must transform under the
Recalling that stability forces χ(q) to be positive defi- trivial representation of the point group. Again Eq. (40)
nite, we conclude that V <0. We see that the forward issatisfiedand,accordingtoouranalysisofSec.IIC,the
s(cid:48)s
scattering amplitude F is, exactly as in case of phonons, associatedstateistopologicallytrivial–exactlyasinthe
negative forallstatesonthe Fermisurfaces, whereas the case of electron-phonon coupling.
global sign of the Cooper channel is reversed in case of For TRO fluctuations, we have t=− such that δ now
TROfluctuationsascomparedtophonons(orTREelec- belongs to the eigenspace of v with the smallest eigen-
tronicfluctuations). Notethat,ingeneral,thisonlyholds value. This has two crucial consequences. Firstly, we
for the renormalized electron-electron interaction since cannot generically exclude spontaneous TRS breaking
only χ and not the bare χ has to be positive definite. since it is no longer guaranteed that this eigenspace is
0
one-dimensional. Although all eigenvectors of the real
matrix v can always be chosen to be real valued, the
A. Superconducting instability superconducting order parameter can be a complex su-
perposition of the degenerate eigenvectors which makes
Let us next analyze the consequences for the possi- TRSbreakingpossible. Notethat, apartfromaccidental
ble superconducting phases. As before, we apply Eliash- degeneracies which we will neglect here, these degenera-
berg theory that is frequently used for studying super- cies can be enforced by symmetry if the point group of
conductivity caused by collective bosonic modes other the system allows for multidmensional or complex irre-
than phonons (see, e.g., Ref. 34 and references therein). ducible representations14,37. Secondly, the eigenvectors
As in this case, m/M (cid:28) 1 will not hold in general, one with minimal eigenvalue can have many sign changes
expects this approach only to be applicable in the weak- which, depending on the form of the Fermi surfaces, can
coupling regime. However, in the limit of large numbers break any point symmetry of the system and lead to
of fermion flavors, neglecting vertex corrections is also nodal points.
justified in the strong-coupling case34. While there are To proceed, we will assume that the resulting super-
complications with this large-N theory in 2D35, some ef- conducting state preserves TRS and, thus, belongs to
forts have been made to develop controlled approaches class DIII. This is not very restrictive since it has been
for this case as well36. shown14 that, for 2D systems, spontaneous TRS break-
Using Hermiticity of λ, Eqs. (55) and (52c), it is ing can only occur in the weak-pairing limit if there is a
straightforward to check that the three properties (21) threefold rotation symmetry perpendicular to the plane.
of the vertex function V are still satisfied. Consequently, Furthermore, as we want to discuss topological proper-
the linearized Eliashberg equations are again of the form ties of superconductors, we will focus on fully gapped
of Eq. (35) with V (k(cid:48);k) now given by Eq. (57) and systems, where the sign changes take place between dif-
s(cid:48)s
an additional prefactor of t on the right-hand side of ferent Fermi surfaces. Being fully gapped, we can apply