Superconductivity in Weyl metals
SSuperconductivity in Weyl metals
G. Bednik, A.A. Zyuzin, and A.A. Burkov
1, 3 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Department of Physics, University of Basel, Klingelbergstrasse 82, CH-2056 Basel, Switzerland National Research University ITMO, Saint Petersburg 197101, Russia (Dated: August 20, 2018)We report on a study of intrinsic superconductivity in a Weyl metal, i.e. a doped Weyl semimetal.Two distinct superconducting states are possible in this system in principle: a zero-momentumpairing BCS state, with point nodes in the gap function; and a finite-momentum FFLO-like state,with a full nodeless gap. We find that, in an inversion-symmetric Weyl metal the odd-parity BCSstate has a lower energy than the FFLO state, despite the nodes in the gap. The FFLO state, onthe other hand, may have a lower energy in a noncentrosymmetric Weyl metal, in which Weyl nodesof opposite chirality have different energy. However, realizing the FFLO state is in general verydifficult since the paired states are not related by any exact symmetry, which precludes a weak-coupling superconducting instability. We also discuss some of the physical properties of the nodalBCS state, in particular Majorana and Fermi arc surface states.
I. INTRODUCTION
The study of the interplay of superconductivity andnontrivial electronic structure topology has a long his-tory, dating back to the work on superfluid He.
Therecent discovery of topological insulators (TI) has rein-vigorated this subject. Proximity-induced superconduc-tivity on a 3D TI surface has been proposed as a routeto realize Majorana fermions, and bulk topological su-perconductivity has also been studied extensively. Most recently, Weyl, and closely related Diracsemimetals, first discovered theoretically and latelyrealized experimentally, have come to the forefrontof research on topologically-nontrivial phases of mat-ter. Perhaps the most interesting new feature that Weylsemimetals bring to the subject is that they are gap-less. The realization that gapless states of matter maybe topologically nontrivial, just as gapped insulators, isof significant importance.The nontrivial electronic structure topology of Weylsemimetals arises from points of contact of nondegener-ate conduction and valence bands, which act as monopolesources of Berry curvature and thus carry an integertopological charge. A semimetal is realized when theFermi level coincides with the points of contact and thevalence band is filled while the conduction band is empty.This situation, however, is nongeneric and is unstable toimpurities which will always give rise to unintentionaldoping, shifting the Fermi level into the conduction orvalence bands. This motivates the study of
Weyl metals ,i.e. lightly-doped Weyl semimetals. It turns out thatWeyl metals share most of the topologically-nontrivialproperties with undoped Weyl semimetals.
Two fea-tures of the Weyl metals, which are directly related to theWeyl node topology, are very important in this regard.One is that, when the Fermi energy is close enough tothe Weyl nodes, the Fermi surface breaks up into discon-nected sheets, each enclosing one Weyl node. The fluxof the Berry curvature through each such Fermi surfacesheet is equal to the topological charge of the correspond- ing node, which endows each 2D Fermi surface sheet witha nontrivial topological invariant, a Chern number. Thesecond important property of a Weyl metal is the linear-ity of the band dispersion, which necessarily arises in themomentum-space vicinity of each topological charge, asfollows from the so-called Atiyah-Bott-Shapiro construc-tion. This property may be viewed as an emergent low-energy chiral symmetry , which is characteristic of Weylmetals.A nearly universal property of metals is the super-conducting instability, which always exists, at least innonmagnetic metals, at low enough temperature. Fromthis viewpoint, the question of superconductivity in Weylmetals comes up naturally. Moreover, it is certain in thiscase that the superconductivity will be unconventionalsince, at least at weak coupling, the pairing must occurbetween states in a single nondegenerate conduction orvalence band.As has been shown in Refs. 37 and 38, two distinct su-perconducting states are possible in Weyl metals in prin-ciple: a BCS (Bardeen-Cooper-Schrieffer) state, in whichpairing occurs between momentum eigenstates, related toeach other by inversion symmetry [assuming time rever-sal (TR) symmetry is broken, but inversion exists]; andan FFLO (Fulde-Ferrell-Larkin-Ovchinnikov)-like state,with finite-momentum pairs, where the states on the op-posite sides of each Fermi surface sheet, enclosing indi-vidual Weyl nodes, are paired. In the latter case, thestates which are paired are not related to each other byany exact symmetry. What makes such a superconduct-ing state possible in principle, even at weak (but not in-finitesimal) coupling, is the low-energy chiral symmetryof Weyl metals, mentioned above. Moreover, if the inver-sion symmetry is violated, and as a result, Weyl nodesof opposite chirality are shifted away from each other inenergy to a significant degree, the FFLO state becomesthe only superconducting state possible, since the exis-tence of the BCS state relies on inversion symmetry.In this paper we show that, in the presence of inversionsymmetry, the BCS state has a lower energy, at least for a r X i v : . [ c ond - m a t . s t r- e l ] J u l short-range momentum-independent pairing interactionand in the class of Weyl metals, considered in this pa-per. This conclusion differs from that, made in previouswork on this subject, which argued that the FFLOstate has a lower energy. We will explain the origin ofthis disagreement below. For other recent work on Weylsuperconductors, see Refs. 40–42.The rest of the paper is organized as follows. In Sec-tion II we introduce the model of a Weyl metal we willuse, which is based on the TI-NI (normal insulator) mul-tilayer heterostructure, introduced by one of us before. This model has the benefit of being very simple and thusamenable to analytic calculations, yet more realistic thana generic low-energy model of a Weyl metal would be. Inparticular, it does not suffer from spurious exact chiralsymmetry of the simplest “relativistic” low energy mod-els. We also derive in this section the BCS pairing Hamil-tonian, the form of which plays an important role in ouranalysis. In Section III we analyze the BCS Hamiltonian,derived in Section II, using the standard mean-field the-ory and demonstrate explicitly that, in the presence ofinversion symmetry the BCS pairing state has a lowerenergy than the FFLO state. In Section IV we discusssome of the physical properties of the nodal BCS statein more detail, in particular the topological propertiesof the mean-field Hamiltonian and the edge states. Wediscuss the differences between our results and previouswork in Section V and conclude by pointing out that,when the inversion symmetry (in addition to TR) is vio-lated, the BCS state is destroyed while the FFLO statemay survive.
II. MODEL AND DERIVATION OF THE BCSAND FFLO HAMILTONIANS
As mentioned above, we will start from the model of aWeyl metal, based on the TI-NI multilayer heterostruc-ture, introduced by one of us before. The model hasbeen described extensively in the literature and here wewill only mention the most essential points, which arenecessary to understand what follows. The momentumspace Hamiltonian, describing the multilayer structure,is given by H = v F τ z (ˆ z × σ ) · k + ˆ t ( k z ) . (1)Here ˆ z is the growth direction of the heterostructure, σ are Pauli matrices, describing the real spin degree offreedom, while τ is the pseudospin, describing the top(T) and bottom (B) surfaces of the TI layers in the het-erostructure, and ¯ h = 1 units are used here and through-out the paper. The operator ˆ t ( k z ) describes the motionof the electrons in the growth direction of the structure.Explicitly it is given byˆ t ( k z ) = t S τ x + t D (cid:0) τ + e ik z d + h.c. (cid:1) , (2)where t S,D are amplitudes for tunnelling between top andbottom surfaces of the same (S) or neighbouring (D) TI layers and d is the superlattice period. We will take both t S,D to be positive for concreteness, this choice does notaffect any of the physics. This structure describes a Diracsemimetal when t S = t D , with two Weyl fermion com-ponents of opposite chirality residing at the same point(0 , , π/d ) in the first Brillouin zone (BZ). To create aWeyl semimetal, we need to separate the Weyl fermionsof opposite chirality in momentum space. This may beaccomplished by breaking either TR or inversion sym-metries. We choose to break TR and (for now) keep in-version symmetry intact, as this creates the simplest kindof Weyl semimetal state with only two nodes. Breaking ofTR is accomplished by adding a term b σ z to the Hamil-tonian Eq. (1) H = v F τ z (ˆ z × σ ) · k + ˆ t ( k z ) + bσ z . (3)Physically this may arise, for example, from magnetizedtransition-metal impurities, introduced into the sample.Throughout this paper we will assume that undoped het-erostructure is almost a Dirac semimetal, i.e. | t S − t D | issmall.We now introduce the simplest and most natural (atleast for phonon-mediated pairing) kind of pairing inter-action, i.e. an s -wave short-range, and thus necessarilysinglet, pairing term, which in second-quantized notationhas the form H int = − U (cid:90) d r Ψ †↑ ( r )Ψ †↓ ( r )Ψ ↓ ( r )Ψ ↑ ( r ) , (4)where U >
0. We want to eventually rewrite H int in thebasis of the eigenstates of H . To this end, we first writethe electron field operators in the following wayΨ † σ ( r ) = 1 (cid:112) L x L y (cid:88) k iτ φ ∗ iτ ( z ) e − i k · r c † k iστ . (5)Here i labels the unit cells of the TI-NI superlattice in thegrowth direction, σ and τ are the spin and pseudospinlabels, φ iτ ( z ) are Wannier-like exponentially-localizedstates, describing the z -axis behavior of the TI surfacestates in the unit cell i and surface τ , k = ( k x , k y ) is thetransverse momentum, and L x,y are the sample dimen-sions in the x - and y -directions. Due to the exponentiallocalization of the Wannier functions φ iτ ( z ), it is easy tosee that, upon substitution of Eq. (5) into H int , the dom-inant terms will correspond to pairing interaction that islocal in both the i and the τ indices H int = − ˜ UL x L y (cid:88) kk (cid:48) q (cid:88) iτ c † k + q i ↑ τ c † k (cid:48) − q i ↓ τ c k (cid:48) i ↓ τ c k i ↑ τ , (6)where ˜ U = U (cid:90) ∞−∞ dz | φ iτ ( z ) | . (7)Redefining ˜ U d → U and transforming to the crystal mo-mentum basis with respect to the i -index, we then obtain H int = − UV (cid:88) kk (cid:48) q (cid:88) τ c † k + q ↑ τ c † k (cid:48) − q ↓ τ c k (cid:48) ↓ τ c k ↑ τ , (8)where k henceforth means the full 3D crystal momentumand V = L x L y L z is the sample volume.As demonstrated in Refs. 37 and 38, the electron pair-ing will predominantly occur in two distinct channels:BCS and FFLO. Correspondingly, we simplify H int byleaving only the two contributions, H BCSint and H F F LOint ,where H BCSint = − UV (cid:88) kk (cid:48) τ c † k ↑ τ c †− k ↓ τ c − k (cid:48) ↓ τ c k (cid:48) ↑ τ , (9)and H F F LOint = − UV (cid:88) kk (cid:48) Q τ c † Q + k ↑ τ c † Q − k ↓ τ c Q − k (cid:48) ↓ τ c Q + k (cid:48) ↑ τ , (10)Here the momentum vector Q labels the locations of theWeyl nodes, to be specified below. An important prop-erty of Eq. (10), which follows from momentum conser-vation, is that H F F LOint does not couple different Weylnodes. This will play a significant role in selecting thelowest energy superconducting state.To proceed we diagonalize the noninteracting part ofthe Hamiltonian, H , in order to rewrite the full Hamil-tonian in the basis of the eigenstates of H . Performing acanonical (i.e. commutation relations preserving) trans-formation σ ± → τ z σ ± , τ ± → σ z τ ± , (11)which corresponds to the following unitary transforma-tion of the electron creation operators c † k ↑ T → c † k ↑ T , c † k ↓ T → c † k ↓ T ,c † k ↑ B → c † k ↑ B , c † k ↓ B → − c † k ↓ B , (12)one obtains H = v F (ˆ z × σ ) · k + ˆ m ( k z ) σ z , (13)where ˆ m ( k z ) = b + ˆ t ( k z ). The pairing terms are clearlyunchanged by this transformation.Diagonalizing the 2 × m ( k z ) one obtains thefollowing eigenvalues m r ( k z ) = b + rt ( k z ) , (14)where r = ± and t ( k z ) = (cid:112) t S + t D + 2 t S t D cos( k z d ).The corresponding eigenvectors are | u r ( k z ) (cid:105) = 1 √ (cid:18) , r t S + t D e − ik z d t ( k z ) (cid:19) . (15)Diagonalizing the remaining 2 × H one finallyobtains its eigenvalues (cid:15) sr ( k ) = s(cid:15) r ( k ) = s (cid:113) v F ( k x + k y ) + m r ( k z ) , (16)where s = ± and the corresponding eigenvectors | v sr ( k z ) (cid:105) = 1 √ (cid:32)(cid:115) s m r ( k z ) (cid:15) r ( k z ) , − ise iϕ (cid:115) − s m r ( k z ) (cid:15) r ( k z ) (cid:33) , (17) where e iϕ = k x + ik y √ k x + k y . The full 4-component eigenvectormay be viewed as a tensor product | z sr ( k z ) (cid:105) = | u r ( k z ) (cid:105) ⊗ | v sr ( k z ) (cid:105) . (18)The Weyl nodes correspond to points along the z -axis inmomentum space at which m − ( k z ) = b − t ( k z ) vanishes.Solving the equation b = t ( k z ), one obtains Q = Q ˆ z ,where Q = πd ± d arccos (cid:18) t S + t D − b t S t D (cid:19) ≡ πd ± k . (19)When the Fermi energy is sufficiently close to the Weylnodes, namely when (cid:15) F (cid:28) b , the Fermi level only crossesthe s = + , r = − band, assuming (cid:15) F > c † k στ = z ∗ + − στ ( k ) c † k + − ≡ z ∗ στ ( k ) c † k , (20)i.e. we will omit the explicit s = + , r = − indices fromnow on in all the equations for brevity. Substituting thisinto H BCSint and H F F LOint , we finally obtain the followingprojected low energy BCS and FFLO Hamiltonians H BCS = (cid:88) k ξ ( k ) c † k c k − U V (cid:88) kk (cid:48) f ∗ k f k (cid:48) c † k c †− k c − k (cid:48) c k (cid:48) , (21)and H F F LO = (cid:88) k ξ ( k ) c † k c k − U V (cid:88) kk (cid:48) Q ˜ f ∗ kQ ˜ f k (cid:48) Q c † Q + k c † Q − k c Q − k (cid:48) c Q + k (cid:48) , (22)where in Eq. (22) we assume that Qd (cid:28)
1, i.e. Weylnodes are far away from the BZ boundary. ξ ( k ) = (cid:15) ( k ) − (cid:15) F is the band energy, counted from the Fermi energyand f k = ie iϕ (cid:115) − m ( k z ) (cid:15) ( k ) , (23)while˜ f kQ = ie iϕ (cid:34)(cid:115) m ( Q + k z ) (cid:15) ( Q + k ) (cid:115) − m ( Q − k z ) (cid:15) ( Q − k )+ (cid:115) m ( Q − k z ) (cid:15) ( Q − k ) (cid:115) − m ( Q + k z ) (cid:15) ( Q + k ) (cid:35) . (24)The nontrivial momentum-dependent form-factors f k and ˜ f kQ are a consequence of projection of the pair-ing interaction terms onto a nondegenerate band. Theirproperties play an important role in the physics of theBCS and the FFLO states. It is easy to see that theBCS form-factor vanishes identically everywhere on the k x = k y = 0 line in momentum space. The four points,at which this line intersects the two Fermi surface sheetswill produce four nodes in the BCS gap function, as willbe seen below. On the other hand, the FFLO form-factor˜ f kQ never vanishes, which is directly related to the factthat the function m ( k z ) changes sign at the Weyl nodelocations k x = k y = 0 , k z = Q . This means that theFFLO state is nodeless. It would then seem natural ifthe fully-gapped FFLO state would have a larger conden-sation energy and thus a lower overall energy than thenodal BCS state, with the gap vanishing at four pointsin the BZ. Surprisingly, as we show below, this is not thecase: the BCS state in fact has a lower energy. III. CONDENSATION ENERGIES OF THE BCSAND THE FFLO STATES
In this section we will evaluate and compare the con-densation energies of the BCS and the FFLO states, us-ing the low-energy Hamiltonians, derived in the previoussection.
A. BCS state
We analyze H BCS using the standard mean-field the-ory. The mean-field BCS Hamiltonian has the form H BCS = (cid:88) k (cid:20) ξ ( k ) c † k c k − ∆2 ( f ∗ k c † k c †− k + f k c − k c k ) (cid:21) + V U ∆ , (25)where the pairing amplitude ∆ is given by∆ = UV (cid:88) k f ∗ k (cid:104) c † k c †− k (cid:105) = UV (cid:88) k f k (cid:104) c − k c k (cid:105) . (26)It is important to note that our BCS state has odd parity since f k changes sign under inversion. This is already incontrast to Ref. 37, which claimed an even-parity BCSstate.Diagonalizing H BCS using Bogoliubov transformation,one obtains H BCS = (cid:88) k E ( k ) ψ † k ψ k + 12 (cid:88) k [ ξ ( k ) − E ( k )] + V U ∆ . (27)Here ψ † k are the Bogoliubov quasiparticle creation oper-ators, E ( k ) = (cid:112) ξ ( k ) + | f k | ∆ , (28)is the quasiparticle energy and the pairing amplitude ∆satisfies the standard BCS equation (assuming tempera-ture T = 0) 1 = U V (cid:88) k | f k | E ( k ) . (29) It is clear from Eq. (28) that the quasiparticle energyindeed vanishes at four points in the first BZ, at whichthe form-factor f k vanishes.The T = 0 BCS equation may be easily solved follow-ing the well-known steps. One obtains∆ = 2 ω D e −(cid:104)| f k | ln | f k |(cid:105) / (cid:104)| f k | (cid:105) e − /Ug ( (cid:15) F ) (cid:104)| f k | (cid:105) . (30)Here ω D is the Debye frequency, g ( (cid:15) F ) = (cid:90) d k (2 π ) δ [ (cid:15) ( k ) − (cid:15) F ]= (cid:15) F π v F (cid:90) π/d − π/d dk z Θ[ (cid:15) F − | m ( k z ) | ] , (31)is the density of states at Fermi energy, and (cid:104)| f k | (cid:105) = 1 g ( (cid:15) F ) (cid:90) d k (2 π ) | f k | δ [ (cid:15) ( k ) − (cid:15) F ] , (32)is the Fermi surface average of the BCS gap function | f k | .In the limit (cid:15) F (cid:28) b , i.e. when the Fermi energy isclose to the Weyl nodes, the Fermi surface average maybe easily evaluated explicitly. Indeed, in this case theband dispersion in the z -direction in momentum spacemay be assumed to be linear, which follows from theleading-order expansion of m ( k z ) near the nodes m ( k z ) ≈ m ( Q ) + dm ( k z ) dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q δk z = ± ˜ v F δk z , (33)where˜ v F = d b (cid:112) [ b − ( t S − t D ) ][( t S + t D ) − b ] , (34)is the z -component of the Fermi velocity near the nodes.Then one obtains g ( (cid:15) F ) = (cid:15) F π v F ˜ v F , (35)and (cid:104)| f k | (cid:105) = 14 (cid:20) − (cid:104) m ( k z ) (cid:105) (cid:15) F (cid:21) = 14 (cid:34) − ˜ v F (cid:15) F (cid:90) (cid:15) F / ˜ v F − (cid:15) F / ˜ v F dk z k z (cid:35) = 16 . (36)The pairing amplitude is thus given by∆ ≈ ω D e − /Ug ( (cid:15) F ) . (37)To evaluate the condensation energy we take the ex-pectation value of H BCS at T = 0. In this case there areno quasiparticles and we obtain E BCS V = 12 V (cid:88) k [ ξ ( k ) − E ( k )] + ∆ U . (38)This is again evaluated in the standard way and givesthe following result for the condensation energy, i.e. theenergy gain compared to the normal state energy E E BCS − E V = − g ( (cid:15) F ) (cid:104)| f k | (cid:105) ∆ ≈ − ω D g ( (cid:15) F )24 e − /Ug ( (cid:15) F ) . (39) B. FFLO state
The FFLO state is analyzed in exactly the same wayas above. One important point to note is that the FFLOHamiltonian Eq. (22) clearly does not mix different Weylnodes, i.e. different values of Q , as required by momen-tum conservation. This means that, in mean-field theory,the contributions of different Weyl nodes may be ana-lyzed separately and simply summed when calculatingthe total condensation energy.The mean-field Hamiltonian for a single Weyl node, i.e.with a fixed Q , is given by H Q F F LO = (cid:88) k (cid:20) ξ ( k ) c † k c k − ∆2 (cid:16) ˜ f ∗ kQ c † Q + k c † Q − k + ˜ f kQ c Q − k c Q + k (cid:17)(cid:105) + V U ∆ , (40)where the pairing amplitude is∆ = UV (cid:88) k ˜ f ∗ kQ (cid:104) c † Q + k c † Q − k (cid:105) = UV (cid:88) k ˜ f kQ (cid:104) c Q − k c Q + k (cid:105) . (41)Diagonalizing Eq. (40) one obtains H Q F F LO = (cid:88) k (cid:20) (cid:15) ( Q + k ) − (cid:15) ( Q − k )2 + E ( k ) (cid:21) ψ † k ψ k + 12 (cid:88) k [ ξ ( k ) − E ( k )] + V U ∆ , (42)where E ( k ) = (cid:115)(cid:20) ξ ( Q + k ) + ξ ( Q − k )2 (cid:21) + | ˜ f kQ | ∆ . (43)A crucial difference between the FFLO and the BCSstates is the term [ (cid:15) ( Q + k ) − (cid:15) ( Q − k )] / Q + k and Q − k are not related by any exactsymmetry, unlike k and − k in the BCS case, which arerelated by inversion. An important property of a Weylmetal, however, which may allow the FFLO state to ex-ist in principle, is the low-energy chiral symmetry, whichemerges as (cid:15) F → (cid:15) ( Q + k ) ≈ (cid:15) ( Q − k ) , (44)the equality becoming more and more precise as the en-ergy is lowered. Let us first assume that Eq. (44) holds exactly, as it would in a low-energy model of a Weyl metalwith an exactly linear dispersion. Then we obtain H Q F F LO = (cid:88) k E ( k ) ψ † k ψ k + 12 (cid:88) k [ ξ ( k ) − E ( k )] + V U ∆ . (45)The mean-field equation for the pairing amplitude, againassuming Eq. (44) holds, takes the form, identical to theBCS case 1 = U V (cid:88) k | ˜ f kQ | E ( k ) . (46)To proceed, we now evaluate explicitly the FFLO gapfunction. After elementary algebra, we obtain | ˜ f kQ | = 18 (cid:20) − m ( Q + k z ) m ( Q − k z ) (cid:15) F + (cid:115) − m ( Q + k z ) (cid:15) F (cid:115) − m ( Q − k z ) (cid:15) F (cid:35) . (47)Low energy chiral symmetry implies that m ( Q + k z ) ≈ − m ( Q − k z ) . (48)Assuming this to hold, we finally obtain | ˜ f kQ | = 14 , (49)which is by a factor of 3 / ω D e − /Ug ( (cid:15) F ) (cid:104)| ˜ f kQ | (cid:105) = 4 ω D e − /Ug ( (cid:15) F ) . (50)The extra factor of 2 in the expression above, comparedto the corresponding Eq. (30) in the BCS case, arisesprecisely from the fact that the density of states per Weylnode is g ( (cid:15) F ) / g ( (cid:15) F ) being the total density of states.It is then clear that the magnitude of ∆ in the FFLOstate is exponentially smaller than in the BCS state inthe weak coupling regime U g ( (cid:15) F ) (cid:28) E F F LO − E V = − g ( (cid:15) F )∆ = − ω D g ( (cid:15) F ) e − /Ug ( (cid:15) F ) . (51)Comparing to the corresponding result in the BCS case,Eq. (39), it is clear that the FFLO state condensation en-ergy is exponentially smaller in the weak coupling regime,within our model of a Weyl metal. Since the critical tem-perature T c ∼ ∆ at weak coupling, this also implies that T c of the BCS state is higher than T c of the FFLO stateand we thus do not expect any transitions between themas a function of temperature.In reality the situation is even worse for the FFLOstate, however, since we have so far been assuming ex-act chiral symmetry, expressed by Eq. (44). But chiralsymmetry is not exact, and it is instructive to work outthe consequences of that. While we have already shownthat, in our model, the FFLO state is not favored evenwhen the chiral symmetry exists, this result is based oncomparing energies, and thus may not be universal. Thearguments presented below are of a more general validity.Let us expand (cid:15) ( Q ± k ) in Taylor series with respect tothe deviation from the Weyl node location k , assuming k is small. Let us also take k = k z ˆ z for the sake ofsimplicity. We obtain (cid:15) ( Q + k z ) − (cid:15) ( Q − k z ) ≈ d(cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q + k z + d(cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q − k z + 12 d (cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q + k z − d (cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q − k z + . . . , (52)where we have taken into account the fact that the bandvelocity d(cid:15)/dk z is discontinuous and changes sign at theWeyl node locations. Since d(cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q + = − d(cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q − , (53)the first order term in the expansion above vanishes,which is precisely the expression of the approximate chi-ral symmetry. The quadratic term, however, does notvanish and is given by12 d (cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q + − d (cid:15)dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q − = d m ( k z ) dk z (cid:12)(cid:12)(cid:12)(cid:12) k z = Q = d b − ( t S − t D ) b ∼ d b. (54)We may expect the FFLO state to exist as a local mini-mum of the free energy as long asmax k | (cid:15) ( Q + k ) − (cid:15) ( Q − k ) | < ∼ ∆ , (55)where the maximum is taken over states on the Fermisurface. Taking k z ∼ (cid:15) F / ˜ v F , where ˜ v F ∼ d t S is theFermi velocity at the Weyl nodes, and assuming b (cid:29)| t S − t D | , Eq. (55) becomes b (cid:15) F /t S < ∆ , (56)where t S should be regarded as the highest energy scale inthe problem, of the order of the total bandwidth. UsingEqs. (35) and (50), we may rewrite this inequality as bω D (cid:15) F t S < ∼ exp (cid:18) − π v F ˜ v F U (cid:15) F (cid:19) , (57) Using v F ∼ ˜ v F ∼ t S d , it becomes clear that this in-equality is very hard, if not impossible, to satisfy for anyreasonable values of the relevant parameters.The above arguments lead us to the conclusion that,when the inversion symmetry is present and when thecoupling is weak, nodal odd-parity BCS superconductingstate, which is a close analog of the A phase in He, ismuch more likely to be realized in a Weyl metal. Thisstate is topologically nontrivial and is characterized byan interesting edge state structure, in which Fermi arcsof the “parent” nonsuperconducting Weyl semimetal co-exist with Majorana edge states of the nodal BCS super-conductor, as will be described in detail in the followingsection.
IV. MAJORANA AND FERMI ARC EDGEMODES IN THE NODAL BCS STATE
In this section we will discuss in some detail the non-trivial momentum-space topology of the nodal BCS stateand the corresponding Majorana and Fermi arc edgestates. Some work on this has already been done be-fore in Refs. 40 and 42, and the discussion below mostlyserves to provide a clear connection between these previ-ously published results and place them in the context ofour model.To discuss momentum-space topology, which involvesnot only states near the Fermi energy, it will be neces-sary to consider the BCS Hamiltonian without makingthe projection onto the low-energy states only. We willstill restrict ourselves to the r = − states, since these arethe bands that touch at the Weyl nodes in the nonsuper-conducting state, but will take into account both bandsthat touch. The mean-field BCS Hamiltonian then takesthe form H = (cid:88) k [ v F (ˆ z × σ ) · k + m ( k z ) σ z − (cid:15) F ] c † k c k − (cid:88) k (cid:16) ∆ c † k ↑ c †− k ↓ + ∆ ∗ c − k ↓ c k ↑ (cid:17) , (58)where m ( k z ) ≡ m − ( k z ),∆ = UV (cid:88) k (cid:104) c − k ↓ c k ↑ (cid:105) , (59)and spin indices were suppressed in the first line ofEq. (58) for brevity. To analyze Eq. (58) we introduce aNambu spinor ψ k = ( c k ↑ , c k ↓ , c †− k ↓ , c †− k ↑ ) ≡ ( ψ k ↑ , ψ k ↓ , ψ k ↑ , ψ k ↓ ) . (60)In the Nambu spinor notation, the BCS Hamiltoniantakes the following form H = 12 (cid:88) k [ v F (ˆ z × σ ) · k + m ( k z ) σ z − (cid:15) F κ z ] ψ † k ψ k − (cid:88) k (∆ κ + + ∆ ∗ κ − ) σ z ψ † k ψ k , (61) FIG. 1. (Color online) Numerically calculated eigenstate dis-persions for Eq. (61) along the z -direction in momentum spacefor a sample of finite width in the x -direction. Parameters inEq. (61) are chosen as follows: t S = 1 , t D = 0 . , b = 1 , (cid:15) F =∆ = 0 .
2. Majorana modes are pinned to zero energy and existin intervals of width of order | ∆ | / ˜ v F around each Weyl node.Fermi arc edge modes are split into particle-hole antisymmet-ric and symmetric branches at ± (cid:15) F . where κ is the Nambu pseudospin.It is instructive to start from the limit of (cid:15) F = 0, inwhich case Eq. (61) reduces to the model of Ref. 40. Inthis case it is clear that the Nambu pseudospin block ofthe Hamiltonian may be diagonalized separately and weobtain H = 12 (cid:88) k { v F (ˆ z × σ ) · k + [ m ( k z ) + p | ∆ | / σ z } ψ † k ψ k , (62)where p = ± labels the two eigenvalues ±| ∆ | of the ma-trix (∆ κ + + ∆ ∗ κ − ) /
2. Apart from the factor of 1 / m ( k z ) ± | ∆ | / k z . Sign change of the Dirac massessignals quantum Hall transitions. In particular, when∆ = 0, m ( k z ) changes sign at the locations of the twoWeyl nodes, given by π/d ± k . The topologically non-trivial range of k z , where m ( k z ) >
0, corresponds to therange of momenta in which chiral Fermi arc edge statesexist on any sample surface, not perpendicular to the z -axis. Due to doubling of the number of degrees offreedom in Eq. (62), these chiral Fermi arc edge statescome in degenerate particle-hole symmetric and antisym-metric pairs. Turning on a nonzero ∆, the Dirac massof the particle-hole symmetric states decreases by | ∆ | / δk z = 1 d (cid:20) arccos (cid:18) t S + t D − ( b + | ∆ | / t S t D (cid:19) − arccos (cid:18) t S + t D − ( b − | ∆ | / t S t D (cid:19)(cid:21) ≈ | ∆ | ˜ v F , (63) FIG. 2. (Color online) Numerically calculated eigenstate dis-persions for Eq. (61) along the y -direction in momentum spacefor a sample of finite width in the x -direction, at severalvalues of k z . Parameters in Eq. (62) are chosen as follows: t S = 1 , t D = 0 . , b = 1 , (cid:15) F = 0 . , ∆ = 0 .
1. (a) k z is outsidethe Fermi surface. Two pairs (corresponding to two surfaces)of chiral Fermi arc modes, crossing at ± (cid:15) F at k y = 0 arevisible. (b) k z is just inside the Fermi surface. Two chiralMajorana modes are visible now, crossing at k y = 0 at zeroenergy. (c) k z is closer to a Weyl node location. Majoranamodes directly connect with the Fermi arcs. in which only the particle-hole antisymmetric states aretopologically nontrivial (the last equality holds assuming | ∆ | (cid:28) b ). These momentum intervals give rise to chi-ral Majorana edge modes, which do not have particle-hole symmetric partners, unlike the “ordinary” Fermiarc edge states. The Majorana edge states terminateat points at which m ( k z ) = ±| ∆ | /
2. These are pointnodes in the spectrum of the Bogoliubov quasiparticles.Each Weyl node in the normal state thus splits intotwo Bogoliubov-Weyl nodes in the superconducting state,which inherit the chirality of the parent Weyl node, asseen from Eq. (62).When (cid:15) F >
0, Eq. (61) no longer has the simple formof two independent massive Dirac Hamiltonians, but theeigenstate spectrum is still easily found. We obtain E ± ( k ) = (cid:15) ( k ) + | ∆ | (cid:15) F ± (cid:113) (cid:15) ( k ) (cid:15) F + | ∆ | m ( k z ) . (64)The node locations are now given by the solutions of theequation | m ( k z ) | = (cid:114) (cid:15) F + | ∆ | . (65)The edge state spectrum may be easily found numeri-cally, as shown in Fig. 1. The degeneracy of the Fermi arcdoublet is split, with the particle-hole symmetric branchmoving down to − (cid:15) F in energy, while the particle-holeantisymmetric branch moves up to (cid:15) F . The Majoranastates, which extend between the nodes, remain pinnedat zero energy (note that zero is at the Fermi energyhere).In the limit | ∆ | (cid:28) (cid:15) F , we may obtain a simple pic-ture of the Majorana states from the BCS Hamiltonian,projected onto the low-energy s = + states H = 12 (cid:88) k (cid:20) ξ ( k ) κ z −
12 (∆ f ∗ k κ + + ∆ ∗ f k κ − ) (cid:21) ψ † k ψ k , (66)where ψ k = ( c k , c †− k ) is the projected Nambu spinor and ξ ( k ) = (cid:113) v F ( k x + k y ) + m ( k z ) − (cid:15) F . This again has theform of the Hamiltonian of a 2D Dirac fermion, with themass ξ (0 , , k z ), which depends on k z as a parameter.The mass changes sign at points, satisfying the equation | m ( k z ) | = (cid:15) F , which coincides with the locations of thenodes of the superconducting gap function f k . The topo-logically nontrivial momentum range, which in this casecorresponds to negative Dirac mass, coincides with therange of momenta inside the two Fermi surface sheets,enclosing the Weyl nodes. There are thus Majorana zero-energy edge states, which exist on arcs, connecting thegap function nodes of the opposite sides of each Fermisurface sheet. For sample surfaces perpendicular, say,to the x -direction, the dispersion of the Majorana edgemodes in the y -direction is chiral and exist for a given k z within the energy interval ±| f k ∆ | . Near the center ofeach Fermi surface sheet, i.e. close to the locations of theWeyl nodes along the z axis, the superconducting gap islargest in magnitude and the Majorana edge state dis-persion extends all the way to the Fermi surface, whereit gets reconnected with the Fermi arc states, as shownin Fig. 2.However, there is a subtlety here, since the projectedlow-energy Hamiltonian Eq. (66) does not actually givethe correct Majorana edge mode dispersion in the y -direction. The reason is that the two Bogoliubov-Weyl FIG. 3. Same parameters as in Fig. 2, but values of k z aretaken near (a) and beyond (b) the Weyl node location. Dis-apperance of the Fermi arcs necessitates a “flat band” atthe transition point. The chirality of the Majorana modeschanges sign at the transition. The transition itself occursexactly at the Weyl node location in the limit ∆ /(cid:15) F →
0, butis shifted slightly away for any finite ∆. nodes on each Fermi surface sheet have the same chiral-ity, inherited from the Weyl node, enclosed by the Fermisurface sheet, as discussed above. This can not be de-duced from Eq. (66) and requires analysis of the full un-projected Hamiltonian Eq. (61). Another way to see thisis to notice that the Majorana modes must connect withthe Fermi arcs, which are not low-energy modes and re-quire the full Hamiltonian Eq. (61) to describe. An in-teresting consequence of this, first described in Ref. 42,is a topological transition that happens when one crossesthe Weyl node locations along the z -axis. The Fermiarcs must disappear on approaching Weyl nodes, whichmeans that the finite- k z crossing points in the edge statedispersions in Fig. 2 must disappear. This leads unavoid-ably to the appearance of a dispersionless “flat band” atthe value of k z at which the Fermi arcs disappear, asshown in Fig. 3(a). As k z is further increased, the Majo-rana dispersion again becomes chiral, but with chiralityof opposite sign, as seen in Fig. 3(b). V. DISCUSSION AND CONCLUSIONS
We will start this section by discussing differences be-tween our results and the ones obtained previously inRefs. 37 and 38, which both claimed that the FFLO statehas a lower energy. In Ref. 38 a specific low-energy linear-dispersion model for a Weyl metal was used, in which theBCS interaction, Eq. (9) vanished identically. This is cer-tainly possible in some realizations of Weyl metals, butcan not be a general feature. Moreover, such an exactcancellation would presumably not happen even withinthe model of Ref. 38, if chiral symmetry violating correc-tions to the low energy model, which are always present,as discussed above, were included. However, in agree-ment with our results, all superconducting states, foundin Ref. 38, had odd parity.Ref. 37 considered a model, quite similar to ours as faras its symmetry properties are concerned, also assuminga strictly linear band dispersion in all calculations. How-ever, the BCS state, found in this reference, was claimedto have even parity, which disagrees with our results. TheFFLO state in this reference also appears to be different,with a single order parameter for both nodes, while in ourcase there are two independent (in mean-field theory) or-der parameters. In fact, a single order parameter wouldbe impossible in our case: this would imply internodepair scattering already at the level of mean-field theory,which violates momentum conservation, as clearly seenfrom Eq. (10).We will conclude by pointing out one possible situ-ation in which the FFLO, rather than the nodal BCSstate, may be realized. FFLO state may happen to bethe ground state if the inversion symmetry, assumed tobe present in the calculations above, is violated, in ad-dition to TR symmetry. The presence of at least oneof those symmetries, i.e. TR or inversion, is necessaryfor the existence of the BCS superconducting state, sinceonly those symmetries guarantee that the band eigen-states at momenta k and − k have the same energy. In aWeyl metal, either TR or inversion must be violated, toremove the two-fold Kramers degeneracy. In the modelthat we have discussed above, TR was violated from thestart. This already puts BCS-type superconductivity un-der some strain, which is manifest in the gap functionhaving nodes. When inversion symmetry is violated aswell, the Weyl nodes will generally be shifted to differentenergies, which implies that the states with momenta k and − k will have different energies. Once the inversionbreaking is strong enough, such that the energy differ-ence between the Weyl nodes is comparable to the BCSpairing amplitude ∆, given by Eq. (37), the BCS statewill be destroyed (possibly going through an intermediate“ordinary” small-wavevector FFLO state, which we will not discuss here). However, the FFLO state is largely un-affected by this, since its existence relies not on the exactmicroscopic inversion symmetry, but rather on the low-energy chiral symmetry, which is unaffected by the bro-ken inversion symmetry, unless inversion breaking is sostrong that the energy difference between the Weyl nodesbecomes comparable to the Fermi energy. Thus, providedthe inequality Eq. (57) may be satisfied (which, as dis-cussed above, appears to be hard in the weak couplingregime, but may be possible at intermediate coupling),the FFLO state will be realized. However, it is impor-tant to keep in mind that since nonlinear chiral symmetryviolating corrections to the band dispersion are alwayspresent in a real Weyl metal, realizing the FFLO statecertainly requires a finite pairing interaction strength, asthe logarithmic divergence of the FFLO pairing suscep-tibility will always be cut off by the nonlinearity. Fromthis viewpoint, a Weyl metal, in which both TR and in-version are violated, is an example of a metal withoutany weak-coupling BCS instability and thus with a trueFermi liquid ground state at T = 0. Another closely re-lated and interesting possibility, worthy of further, moredetailed study, is a state, in which the superconductingorder parameter is nonzero on one of the Fermi surfacesheets, but is zero on the other one, since the inequalityEq. (57) may be satisfied for one of the sheets, but notthe other, when the inversion symmetry is absent. Thiswould be a helical superconductor , coexisting with anormal Fermi liquid. Helical superconductors have beenstudied extensively in the general context of supercon-ductivity in noncentrosymmetric materials, and theirrealization in Weyl metals would be of significant inter-est.Our results may be used as a starting point for furtherstudies of superconductivity in Weyl metals. In particu-lar, since the nodal BCS state is topologically nontrivial,as discussed in Section IV and in Refs. 40 and 42, thequestion of its electromagnetic response seems to be ofinterest. ACKNOWLEDGMENTS
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