Nodal topology in d -wave superconducting monolayer FeSe
NNodal topology in d -wave superconducting monolayer FeSe Takeru Nakayama , ∗ Tatsuya Shishidou , and Daniel F. Agterberg The Institute for Solid State Physics, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan. and Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53211, USA. (Dated: December 5, 2019)A nodeless d -wave state is likely in superconducting monolayer FeSe on SrTiO . The lack ofnodes is surprising but has been shown to be a natural consequence of the observed small interbandspin-orbit coupling. Here we examine the evolution from a nodeless state to the nodal state asthis spin-orbit coupling is increased from a topological perspective. We show that this evolutiondepends strongly on the orbital content of the superconducting degrees of freedom. In particular,there are two d -wave solutions, which we call orbitally trivial and orbitally nontrivial. In bothcases, the nodes carry a ± I. INTRODUCTION
Monolayer FeSe grown on SrTiO has generated muchattention due to its high superconducting transition tem-perature T c , which is higher than all the other Fe-basedsuperconductors [1]. Quasiparticle interference [2] exper-iments and scanning tunneling microscopy [1, 3] suggest aplain s -wave pairing state. Angle-resolved photoemissionspectroscopy (ARPES) [4–7] also supports this point ofview by observing a fully gapped superconducting state,although with a nontrivial anisotropy [7]. The appear-ance of an s -wave pairing state in this material seems atodds with the understanding that superconductivity inFe-based materials is due to repulsive electron-electroninteractions and presents a puzzle. Furthermore, mono-layer FeSe lacks the hole pockets about the Γ point ofthe Brillouin zone (BZ) which exist in other iron pnic-tide compounds. This suggests that the usual s ± -wavepairing [8, 9] due to spin fluctuations about a collinear an-tiferromagnetic state with a wave vector that originatesfrom the momentum difference between electron and holepockets is less likely as a pairing mechanism. This hasled to a debate about the nature of the pairing state inmonolayer FeSe. Some proposals include (for a reviewsee Ref. 10) a conventional s -wave pairing state [2, 11],an incipient s -wave pairing state [12], an extended s -wavepairing state [13], a fully gapped spin-triplet pairing state[14], and a nodeless d -wave pairing state [15, 16].Recently, we revisited the nature of the magnetic cor-relations and the pairing state in monolayer FeSe [16, 17].Inelastic neutron scattering in single-crystal FeSe [18]has found that, in addition to collinear antiferromag-netic fluctuations, there are also fluctuations associatedwith translation invariant checkerboard antiferromag-netic (CB-AFM) order. First-principles spin-spiral cal- ∗ Electronic address: [email protected] culations [17] also report the enhanced CB-AFM fluctu-ations in monolayer FeSe, finding that this system sits ata quantum spin-fluctuation-mediated spin paramagneticground state. Motivated by the presence of CB-AFMfluctuations, a symmetry-based k · p theory assuminga single M -point electronic representation was used todescribe fermions coupled to these fluctuations [16, 19].This theory predicts a fully gapped, nodeless d -wave state[16]. Although, typically, symmetry arguments implythat such a d -wave state should be nodal [20], this the-ory reveals that nodal points emerge only if the relevantinterband spin-orbit coupling energy is larger than thesuperconducting gap. This theory thereby naturally ac-counts for the gap minima that are observed along theexpected nodal momentum directions of the d -wave state[7].A natural question is, What is the mechanism thatleads to a nodeless, fully gapped d -wave superconduct-ing state? Indeed, one can ask how such nodeless statesare more generally achieved when symmetry argumentswould dictate nodes. Here we address this questionthrough an examination of the nodal d -wave state. Thisquestion falls naturally into the growing research on topo-logical systems, which originally started with gappedsystems [21] such as quantum Hall systems and topo-logical insulators in which surface states are character-ized by “bulk-edge correspondence.” More recently, thiswas extended to gapless systems such as Weyl and Diracsemimetals [22] and unconventional superconductors [23].In unconventional superconductors that are nodal, thatis, that have momenta with zero gap, it is known thatthe sign change of the pairing potential on the Fermisurface leads to dispersionless Andreev bound states ata surface of the system. These states are characterizedthrough topological arguments [24, 25]. Therefore, stud-ies of nodes in unconventional superconductors are im-portant not only to reveal the pairing mechanism butalso to clarify the topological surface states.Although d -wave superconducting states typically have a r X i v : . [ c ond - m a t . s up r- c on ] D ec topologically protected nodes in one-band systems, thesenodal points can be annihilated in multiband supercon-ductors [26, 27]. Indeed, it has been pointed out that themerging nodal points near the Γ point have winding num-bers of opposite-sign in Fe-based superconductors [28]. Inaddition, a nodeless d -wave superconductor has also beendiscussed in the context of cuprates [29]. These works didnot include spin-orbit coupling, which is essential in ourtheory. Our work highlights the annihilation of nodessolely due to spin-orbit coupling and demonstrates thatthe nodal charge is protected by a chiral symmetry thatis the product of time-reversal and particle-hole symme-tries. Furthermore, we find that the nodal annihilationdepends upon the orbital structure of the d -wave gap.In particular, we find two types of d -wave pairing: (a)orbitally trivial usual d -wave anisotropy with a k x k y mo-mentum dependence and (b) orbitally nontrivial with nomomentum dependence. For the latter case, nodal anni-hilation arises in a natural and straightforward manner,while for the orbitally trivial case, the annihilation ismuch less straightforward, proceeding initially throughthe creation of additional nodes which then annihilatewith the original nodes as the interband spin-orbit cou-pling is decreased.The remainder of this paper is organized as follows. InSec. II, we introduce the symmetry-based effective modelthat describes the electronic excitations that stem froma single M point representation of the BZ; these repre-sentations are fourfold degenerate and thus lead to twobands. We then briefly review the emergence of nodalpoints due to interband spin-orbit coupling. In Sec. III,we give the topological charges for these nodal points asa 2 Z invariant and show that there are topologically dis-tinguished phases which manifest themselves through thepresence of dispersionless Andreev surface states. Theresults are summarized in Sec. IV. II. MODEL
In this section, we present a brief review of the low-energy symmetry-based k · p -like theory that describesthe electronic states of monolayer FeSe in the vicinity ofthe Fermi level [16]. Density functional theory calcula-tions show that two states, which are k -dependent linearcombinations of Fe { xz, yz } and x − y orbitals, whichare the two electronic M -point representations M and M using the nomenclature of Ref. [19], are dominant atthe Fermi level around the M point. These states can bedescribed as originating from a single M -point four-foldelectronic representation (with two orbital and two spindegrees of freedom) through an effective k · p theory. Thesimplicity of this model allows insight into the underlyingphysics that cannot be found using a theoretical modelsimply based on ten orbital and two spin degrees of free-dom. In addition, it captures the relevant physics of thesuperconducting state that appears in theories of mono-layer FeSe that include two M -point representations [14]. - - - - k x k y - - - - k x k y FIG. 1: Fermi surfaces in normal states (a) without spin-orbitcoupling and (b) with spin-orbit coupling v so = 12 meV ˚A.The units of horizontal and vertical axes are ˚A − . The otherparameters are given in the main text. In this theory, the normal-state Hamiltonian is H ( k ) = (cid:15) τ σ + γ xy τ z σ + τ x [ γ x σ y + γ y σ x ] , (1)where k = ( k x , k y ) is the momentum measured fromthe M -point of the BZ and the τ i ( σ i ) matrices de-scribe the two orbital (spin) degrees of freedom. The τ x term is the interband spin-orbit coupling that playsan essential role in the d -wave superconducting state.This term has a magnitude that is related to the on-site spin-orbit coupling but is also determined by otherfactors and can be small even if the on-site spin-orbitcoupling is substantial. The Fermi surface, as observedby ARPES, is reasonably described when we chose (cid:15) = (cid:15) ( k ) = ( k x + k y ) / m − µ , γ xy = γ xy ( k ) = ak x k y , γ x = γ x ( k ) = v so k x , γ y = γ y ( k ) = v so k y and pa-rameters as µ = 55 meV, 1 / (2 m ) = 1375 meV ˚A , a = 600 meV ˚A and | v so | ≤
15 meV ˚A . The normalstate dispersions are given by ξ ± = (cid:15) ± (cid:113) γ x + γ y + γ xy ,which have positive helicity and negative helicity, respec-tively. Figures 1(a) and 1(b) show the Fermi surfaceswithout spin-orbit coupling and with spin-orbit coupling v so = 12 meV ˚A, respectively.Superconducting pairing is assumed to be inducedby the fluctuations associated with translation-invariantCB-AFM. This yields a d xy -like pairing state. Impor-tantly, for this paper, there are two such pairing statesthat are described in more detail below. The Hamiltonianis given by the following in the Bogoliubov-de Gennesform: H ( k ) = Γ z ( (cid:15) τ σ + γ xy τ z σ + γ x τ x σ y )+ γ y Γ τ x σ x + i Γ y (∆ d, τ + ∆ d,z τ z ) iσ y , (2)where the Γ i matrices describe the particle-hole degreeof freedom, ∆ d, = ∆ d, ( k ) = ∆ k x k y /k , ∆ d,z = ∆ d,z ( k ) = ∆ , (3)and we take the typical Fermi wave vector k = 0 . − .The two gap functions ∆ d, and ∆ d,z are the two d xy pairing degrees of freedom mentioned above. The pair-ing term ∆ d, τ represents an orbitally trivial and usual d xy pairing with a k x k y momentum dependence. ∆ d,z τ z represents an orbitally nontrivial pairing state with nomomentum dependence; it also has d xy pairing symme- try due to the τ z orbital dependence and the differentsymmetries of the two orbitals that give rise to this gapfunction. In general, since both ∆ d, and ∆ d,z channelshave the same symmetry, the gap function will be a linearcombination of both these pairing channels.In order to gain a deeper understanding of these two types of d xy order, it is convenient to change basis from theorbital basis to the band basis. The Hamiltonian in (2) can be written in block diagonal form with two 4 × (cid:15) + γ xy γ y − iγ x d, + ∆ d,z γ y + iγ x (cid:15) − γ xy − ∆ d, + ∆ d,z − ∆ d, + ∆ d,z − (cid:15) + γ xy γ y + iγ x ∆ d, + ∆ d,z γ y − iγ x − (cid:15) − γ xy , (4)while the other matrix is given by transforming ∆ i → − ∆ i and γ x → − γ x . Performing a unitary transformation thatdiagonalizes the normal part of the Hamiltonian we obtain in the band basis, we find (cid:15) + (cid:113) γ x + γ y + γ xy ∆ d, + ∆ d,z γ xy √ γ x + γ y + γ xy ∆ d,z ( γ y − iγ x ) √ γ x + γ y + γ xy ∆ d, + ∆ d,z γ xy √ γ x + γ y + γ xy − (cid:15) − (cid:113) γ x + γ y + γ xy ∆ d,z ( γ y − iγ x ) √ γ x + γ y + γ xy ∆ d,z ( γ y + iγ x ) √ γ x + γ y + γ xy (cid:15) − (cid:113) γ x + γ y + γ xy ∆ d, − ∆ d,z γ xy √ γ x + γ y + γ xy ∆ d,z ( γ y + iγ x ) √ γ x + γ y + γ xy d, − ∆ d,z γ xy √ γ x + γ y + γ xy − (cid:15) + (cid:113) γ x + γ y + γ xy . (5)This band basis clarifies that the Hamiltonian has both intraband and interband pairings, as is the case in otherproposals for nodeless d -wave superconductors [27]. The interband pairing arises only from the orbitally nontrivial∆ d,z (in combination with the interband spin-orbit coupling). The intraband pairing contains both pairing channels.In this case, the orbitally nontrivial ∆ d,z channel explicitly gains d -wave momentum anisotropy through the γ xy normal state term. Figure 2 shows the pairing anisotropy in the case of only orbitally trivial pairing [Fig. 2(a)] andthe orbitally nontrivial one in the band basis [Fig. 2(b)]. Note that here only spin-singlet pairing is considered. Ingeneral, there can be mixing of spin-singlet and -triplet pairings due to the interband spin-orbit coupling.The interband pairing in the band basis is essential to generate a gapless superconducting d xy state, provided theinterband spin-orbit coupling is sufficiently small. To understand how a large interband spin-orbit coupling gives riseto nodal points, it is useful to consider the quasiparticle dispersion for Hamiltonian (2). This is given by E ± ( k ) = (cid:115) (cid:15) + γ xy + γ x + γ y + ∆ d, + ∆ d,z ± (cid:114) ( (cid:15) γ xy + ∆ d, ∆ d,z ) + (cid:0) γ x + γ y (cid:1) (cid:16) (cid:15) + ∆ d,z (cid:17) . (6)Notice that there are also two negative quasiparticle dis-persion − E ± ( k ) due to chiral symmetry. Along the nodaldirection k y = 0, so that γ xy = γ y = ∆ d, = 0, yielding E ± ( k ) = (cid:12)(cid:12)(cid:12)(cid:113) (cid:15) + ∆ d,z ± | γ x | (cid:12)(cid:12)(cid:12) . Therefore, the followingequation must be satisfied at the nodal points (labeled k ∗ ): (cid:15) = γ x − ∆ d,z . (7)This means that once the interband spin-orbit couplingsatisfies | γ x | > ∆ d,z , nodal points exist. As the interbandspin-orbit coupling is reduced, there is consequently atransition from a nodal d xy state to a fully gapped d xy state, which is the focus of the remainder of this paper.Note that a generic consequence of this theory is thatgap minima in the fully gapped state are along the nodal directions; this agrees with what is observed in ARPESmeasurements. III. NODAL TOPOLOGICAL CHARGES ANDANDREEV FLAT BAND STATESA. Nodal topological charges
Now we examine how the fully gapped d xy state ap-pears as the interband spin-orbit coupling is reduced. Inparticular, for sufficiently large interband spin-orbit cou-pling we have a nodal d xy state, and we examine thetopological charge of the nodal points. We show thattopological charge at the nodal points can be defined as a k x k y k x k y FIG. 2: Pairing anisotropy and topological charges in (a) or-bitally trivial pairing and (b) orbitally nontrivial pairing inthe band basis with only intraband pairing. The solid linesrepresent the Fermi surface in normal states. The circles rep-resent ± Z invariant. The key symmetries in defining this chargeare time reversal (with operator T ) and particle-hole con-jugation (with operator C ). These act on H ( k ) as T H ( k ) T − = H ( − k ) , (8) CH ( k ) C − = − H ( − k ) , (9)where T = K Γ τ ( iσ y ), C = K Γ x τ σ , and K is thecomplex conjugate operator. Since T = − C =1, this Hamiltonian belongs to Altland-Zirnbauer classDIII [30]. Furthermore, we define a chiral operator S , S = − iT C = Γ x τ σ y . (10)Since chiral symmetry is preserved and S anticommuteswith H ( k ), H ( k ) can be written in block off-diagonalform using the basis in which S is diagonal: H ( k ) → V H ( k ) V † = (cid:20) q ( k ) q † ( k ) 0 (cid:21) , (11)where q ( k ) = (cid:15) τ σ + γ xy τ z σ + γ x τ x σ y + γ y τ x σ x + i (∆ d, τ + ∆ d,z τ z ) σ (12)and V = 1 √ (cid:20) I − τ σ y I τ σ y (cid:21) , (13)where I = τ σ is a 4 × q ( k ∗ ) = 0 because of the nodal condition E − ( k ∗ ) =0. In addition, given that chiral symmetry leads tothe topological protection discussed here, we mentionphysically relevant perturbations that preserve and breakthis symmetry. In particular, the mirror glide planesymmetry-breaking term M I = λ I (cid:0) k x − k y (cid:1) τ x σ andnematic order η Q τ z σ preserve chiral symmetry, but aZeeman field h τ · σ does not. In class DIII, a topological charge can be defined bythe winding number [31], which is given by W L = 12 πi (cid:73) L dk l Tr (cid:2) q − ( k ) ∂ k l q ( k ) (cid:3) , (14)where the contour L is a loop around the nodal point.This charge is an integer Z invariant. In the problemwe are considering, we also have parity symmetry, whichensures a twofold degeneracy of the nodal point. Conse-quently, the nodes have a 2 Z topological charge [32]. Wefind that the orbitally trivial and orbitally nontrivial gapfunctions exhibit different nodal charge distributions inmomentum space and that a topological transition existsbetween these two cases.To understand the different nodal charge distributionsbetween the orbitally trivial and nontrivial cases (seeFig. 2), it is useful to consider the limit in which theinterband pairing can be ignored. This can be achievedin the orbitally trivial case by setting ∆ d,z = 0 and inthe orbitally nontrivial case by setting ∆ d, = 0 and alsorequiring that the interband spin-orbit coupling satisfy | γ i | (cid:28) | γ xy | . When the interband pairing can be ignored,we can consider the nodal points in each band indepen-dently. In this case, following Ref.s [24, 25], Eq. (14) canbe simplified to W L ± = − (cid:88) k ∈ S L± sgn (cid:16) ∂ k l ξ ± k (cid:12)(cid:12) k = k (cid:17) sgn (cid:0) ∆ ± k (cid:1) , (15)where ξ ± = (cid:15) ± (cid:113) γ x + γ y + γ xy , ∆ ± k is the supercon-ducting gap of positive and negative helicity, and thesum is over the set of points S L ± given by the intersec-tion of positive- and negative-helicity Fermi surfaces withthe one-dimensional contour L ± . We consider explic-itly the topological charges of the adjacent pair of nodalpoints in the k x ( >
0) direction, ( k ∗− x ,
0) and ( k ∗ + x , ± k of each band is ∆ ± k = − ∆ d, . Therefore, two nodalpoints will have same-sign topological charge, which wecall same-sign pair states. On the other hand, for theorbitally nontrivial case, ∆ ± k ∼ ∓ γ xy ∆ d,z , so that thetwo nodal points have opposite-sign topological charges,which we call opposite-sign pair states. In general, thepairing state will be a linear combination of the orbitallytrivial and orbitally nontrivial gap functions, but it isintuitively clear that the nodes can still be classified assame-sign pair or opposite-sign pair states and a tran-sition between these two topological states can occur.Furthermore, in both cases, as the spin-orbit couplingis decreased, a gapped d xy superconducting state mustarise (assuming that ∆ d,z (cid:54) = 0). The development of thisgapped state for opposite-sign pair states is intuitivelyclear, but this is not the case for same-sign pair states.To gain a deeper understanding of the physics dis-cussed above, we consider a more general treatment ofthe topological charge. In particular, the topologicalcharge (14) can be cast in the following form: W L = 1 π (cid:73) L dk l ∂ k tan − (cid:34) (cid:15) ∆ d, − γ xy ∆ d,z ) (cid:15) − γ x − γ y − γ xy − ∆ d, + ∆ d,z (cid:35) . (16)This can be understood as the winding number of thevector ( (cid:15) − γ x − γ y − γ xy − ∆ d, +∆ d,z , (cid:15) ∆ d, − γ xy ∆ d,z )rotating around the nodal point. The crucial term whichdetermines whether same- or opposite-sign pairs appearis the numerator (cid:15) ∆ d, − γ xy ∆ d,z (the denominator (cid:15) − γ x − γ y − γ xy − ∆ d, +∆ d,z behaves similarly for both sameand opposite-sign pairs). Substituting detailed forms (3),the numerator is given by (cid:15) ∆ d, − γ xy ∆ d,z = − ak x k y ∆ ∆ = 0 , k x k y k ∆ (cid:104) (cid:15) − ak
20 ∆ ∆ (cid:105) ∆ (cid:54) = 0 . (17)If ∆ = 0, the sign of the numerator is the same betweenthe two nodal points k ∗− and k ∗ + , leading to topolog-ical charges of opposite signs at the two nodal points,that is, opposite-sign pair states. However, if ∆ (cid:54) = 0and the sign of (cid:15) − ak ∆ / ∆ changes between the twonodal point k ∗− and k ∗ + , the topological charges havethe same sign at the two nodal points, leading to same-sign pair states. In order to develop an analytic conditionto distinguish these two cases, we consider the k y = 0 di-rection and set ˜ k x as (cid:15) (˜ k x ) − ak ∆ / ∆ = 0. In thecase of same-sign pair states, k ∗− x < ˜ k x < k ∗ + x , this is notsatisfied for opposite-sign pair states. With the nodalcondition (7), we get the following inequality:2 mv − m (cid:114) µm v − ∆ m + v < a ∆ ∆ k < mv + m (cid:114) µm v − ∆ m + v . (18)As an example, if we take the values ∆ = 11 meV and∆ = − . v so = 80 meV ˚A,then the topological character of nodal points is classifiedas opposite-sign pair states.Now we turn to the development of the gapless d xy state due to the merging and annihilation of nodal points.It is worth emphasizing that this has been studied inDirac and Weyl semimetals [22] and also in s and d -wavesuperconductors [28] in a framework different from oursin which spin-orbit coupling is not an essential interac-tion. In the case of opposite-sign pair states, the nodalpoints can merge and are annihilated as the interbandspin-orbit coupling decreases because they have oppositetopological charges. However, in the case of same-signpair states, merging and annihilation of nodal points can-not occur directly. We find that this annihilation occurs opposite sign pair states same sign pair statesnodeless states FIG. 3: Schematic picture of transition to nodeless states from opposite - (left) and same-sign pair states (right). The arrowsrepresent that two nodal points merge with each other. Insame-sign pair states, each inner nodal point splits into threenodal points (surrounded by a dashed line) in transition tonodeless states. through an involved mechanism. Indeed, as the inter-band spin-orbit coupling is decreased from the same-signpair state (which we take to be positive for both in thedescription that follows), a new pair of opposite-chargenodal points is created near the nodal point at k ∗− . Asthe interband spin-orbit coupling is further decreased,the negatively charged nodal point stays near k ∗− , whilethe two positively charged nodal points move off the k x (or k y ) axis. The positively charged nodal points thatmove off the k x axis eventually merge with similarlyformed negatively charged nodal points that have movedoff the k y axis. This leaves an opposite-sign pair state,for which the nodes merge and annihilate as before whenthe interband spin-orbit coupling is further decreased (seeFig. 3). B. Andreev flat-band states
We find that, typically, either same-sign pair states oropposite-sign pair states occur when the superconduct-ing state is nodal. In particular, the state we find abovewith 16 nodal points exists only in a narrow range ofparameters, so we do not consider it further here. Itwould be of interest to be able to experimentally iden-tify whether same-sign or opposite-sign pair states exist.As we show below, this can be done through an exami-nation of edge states. Prior to discussing this, we notethat the values of the spin-orbit coupling used below arelarger than those observed in monolayer FeSe grown onSrTiO . Consequently, we do not predict flat-band en-ergy states for this material (however, there still existin-gap edge states that are not topologically protected).In this context we note that the spin-orbit coupling maybe larger when monolayer FeSe is grown on a differentsubstrate or if it is doped, for example, with Te, whichmay allow for the flat-band edge states to be observed.The nontrivial topological charges at nodal points im-ply the existence of dispersionless Andreev band states orAndreev flat band states as edge states. The number ofAndreev flat-band states is related to a one-dimensional FIG. 4: Schematic pictures of the relation between W L (left)and N ( k y ) (right) in the case of (a) opposite-sign pair and (b)same-sign pair states. Red and blue points indicate W L = +2and −
2, respectively. (1D) winding number N ( k (cid:107) )[25, 33], which is given by N (cid:0) k (cid:107) (cid:1) = (cid:90) d k ⊥ Tr (cid:2) q − ( k ) ∂ k ⊥ q ( k ) (cid:3) , (19)where k (cid:107) ( k ⊥ ) is the bulk momentum parallel (perpendic-ular) to the surface. We consider edges running along the y direction and take k (cid:107) ( k ⊥ ) as k y ( k x ). Figure 4 showsthe relation between the 1D winding number N ( k y ) andthe topological charge W L . Figure 4(a) shows the 1Dwinding number is nonzero between nodal points whichhave opposite-sign topological charges but is zero at theorigin in the case of opposite-sign pair states. On theother hand, the 1D winding number is nonzero for allmomenta between the outer nodal points in the case ofsame-sign pair states [Fig. 4(b)].In order to investigate the edge states further, we in-troduce a lattice model which corresponds to Eq. (2)(see Appendix A). We suppose that the system has twoedges at i x = 1 and N x in the x direction and takethe boundary condition in the y direction to be peri-odic. Then, we examine the edge states by numericallyobtaining the energy spectrum as a function of the mo-mentum k y . We set N x = 10000. Figure 5 shows theenergy spectra for no nodal points [Fig. 5(a)], opposite-sign pair states [Fig. 5(a) and Fig. 5(c)], and same-signpair states [Fig. 5(d)]. Indeed, with no nodal points wedo not have Andreev flat-band states, and once the nodalpoints appear with increasing interband spin-orbit cou-pling, flat-band states appear. In the cases of opposite-sign pair states, the flat-band states exist between twonodal points that have opposite topological charges andthe number of the flat-band states is two for each edge.On the other hand, in the cases of same-sign pair states[Fig. 5(d)], the flat-band states exist at k y = 0, and the FIG. 5: Energy spectra for (a) no nodal pointsand (b) opposite-sign pair, (c) opposite-sign pair, and(d) same-sign pair states. We set the parameters( v so [meV˚A] , ∆ [meV] , ∆ [meV]) as (a) (50 , , − . , (b)(60 , , − . , (c) (70 , , − . , , − t = (2 m ) − . number of the flat-band states across k y = 0 and be-tween two nodal points in positive k y is four and twofor each edge, respectively. In these cases, the numberof flat-band states has a one to one correspondence with | N ( k y ) | , which is shown in Fig. 4. Note that in Fig. 5(d)the finite-size effect creates a gap at k y = 0. We haveconfirmed that there is no gap at k y = 0 by using therecursive Green’s function method (see Appendix B). Inaddition to the flat-band edge states that appear whennodes exist in the bulk spectrum, note that we find edgestates within the gap, although not at zero energy, evenin the fully gapped case. These can be attributed to signchanges in the gap that still appear in a fully gapped d xy superconductor.In actual experiments misalignments would appear,and it is worth mentioning the consequences of this on thedistinct topological phases and the resultant anisotropyof the number of Andeev flat bound states. The one-to-one correspondence between the number of flat-bandstates and (cid:12)(cid:12) N (cid:0) k (cid:107) (cid:1)(cid:12)(cid:12) is also useful for the edge in other di-rections. For instance, consider the edges running alongthe (1 ,
1) direction and denote the wave-vector compo-nent k (cid:107) parallel to the edges. Figures 6(a) and 6(b) showthe 1D winding number (cid:12)(cid:12) N (cid:0) k (cid:107) (cid:1)(cid:12)(cid:12) and the topologicalcharge W L for the cases of opposite-sign pair and same-sign pair states, respectively. For both cases (cid:12)(cid:12) N (cid:0) k (cid:107) (cid:1)(cid:12)(cid:12) = 0for any k (cid:107) ; therefore, there are no Andreev flat-bandstates.Finally, we note that the examination of the Andreevbound state spectra should take into account interac-tion effects. It has been pointed out that due to thelarge density of states intrinsic to flat-bands, they aresusceptible to surface instabilities [33, 34]. The most FIG. 6: Schematic pictures of the relation between W L (left)and N ( k (cid:107) ) (right) in the case of (a) opposite-sign pair and (b)same-sign pair states. We consider the edges running alongthe (1 ,
1) direction. Red and blue points indicate W L = +2and −
2, respectively. likely candidate is edge ferromagnetism that spits theflat-bands [34]. Such a surface instability is seen in tun-neling spectroscopy experiments on the cuprate super-conductor YBa CU O where the zero-bias conductancepeak is seen to split into two below an edge transitiontemperature that is approximately 0 . T c [35]. We leavethe study of possible edge instabilities of Andreev flat-band states due to interactions in the context of the mod-els examined here to future work. IV. CONCLUSION
We have studied nodal topological charges in d -wavesuperconducting monolayer FeSe to help understand theorigin of a fully gapped d -wave state. The nodal pointsthat arise when interband spin-orbit coupling is suffi-ciently strong have 2 Z topological charges that give riseto zero-energy dispersionless Andreev edge bound states.The momentum space distribution of the nodal chargesdepends strongly on the orbital character of the super-conducting state, allowing this to be probed through theobservation of Andreev bound states. Acknowledgments
We thank Philip Brydon, Andrey Chubukov, HirokazuTsunetsugu, and Michael Weinert for useful discussions.Our numerical calculations were partly carried out atthe Supercomputer Center, The Institute for Solid StatePhysics,The University of Tokyo. T. N was supported byJapan Society 433 for the Promotion of Science throughProgram for Leading Graduate Schools (MERIT).
Appendix A: lattice model
In order to obtain the lattice model which correspondsto Eq. (2), we replace k i → sin k i and (cid:0) k x + k y (cid:1) / (2 m ) →− t (cos k x + cos k y ) + 4 t where t − = 2 m in Eq. (2) (the lattice constant is unity). We use A i σ and B i σ , whichare annihilation operators of two orbital, spin σ = ↑ and ↓ electrons at i , and we divide H into H , H SOC , and H ∆ . They are given by H = − t (cid:88) (cid:104) i , j (cid:105) ,σ (cid:104) A † i σ A j σ + B † i σ B j σ (cid:105) − ( µ − t ) (cid:88) i ,σ (cid:104) A † i σ A i σ + B † i σ B i σ (cid:105) + a (cid:88) i ,σ (cid:104) A † i σ A i + x + y σ + A † i + x + y σ A i σ − (cid:16) A † i σ A i + x − y σ + A † i + x − y σ A i σ (cid:17)(cid:105) − a (cid:88) i ,σ (cid:104) B † i σ B i + x + y σ + B † i + x + y σ B i σ − (cid:16) B † i σ B i + x − y σ + B † i + x − y σ B i σ (cid:17)(cid:105) , (A1) H SOC = − v so (cid:88) i (cid:104)(cid:110) A † i ↑ B i + x ↓ − A † i + x ↑ B i ↓ (cid:111) − (cid:110) A † i ↓ B i + x ↑ − A † i + x ↓ B i ↑ (cid:111) + (cid:110) B † i ↑ A i + x ↓ − B † i + x ↑ A i ↓ (cid:111) − (cid:110) B † i ↓ A i + x ↑ − B † i + x ↓ A i ↑ (cid:111)(cid:105) + v so i (cid:88) i (cid:104)(cid:110) A † i ↑ B i + y ↓ − A † i + y ↑ B i ↓ (cid:111) + (cid:110) A † i ↓ B i + y ↑ − A † i + y ↓ B i ↑ (cid:111) + (cid:110) B † i ↑ A i + y ↓ − B † i + y ↑ A i ↓ (cid:111) + (cid:110) B † i ↓ A i + y ↑ − B † i + y ↓ A i ↑ (cid:111)(cid:105) , (A2) H ∆ = − ∆ k (cid:88) i (cid:104) A † i ↑ A † i − x − y ↓ + A † i ↑ A † i + x + y ↓ − (cid:16) A † i ↑ A † i − x + y ↓ + A † i ↑ A † i + x − y ↓ (cid:17) − (cid:110) A † i ↓ A † i − x − y ↑ + A † i ↓ A † i + x + y ↑ − (cid:16) A † i ↓ A † i − x + y ↑ + A † i ↓ A † i + x − y ↑ (cid:17)(cid:111)(cid:105) − ∆ k (cid:88) i (cid:104) B † i ↑ B † i − x − y ↓ + B † i ↑ B † i + x + y ↓ − (cid:16) B † i ↑ B † i − x + y ↓ + B † i ↑ B † i + x − y ↓ (cid:17) − (cid:110) B † i ↓ B † i − x − y ↑ + B † i ↓ B † i + x + y ↑ − (cid:16) B † i ↓ B † i − x + y ↑ + B † i ↓ B † i + x − y ↑ (cid:17)(cid:111)(cid:105) +∆ (cid:88) i (cid:104) A † i ↑ A † i ↓ − A † i ↓ A † i ↑ − (cid:16) B † i ↑ B † i ↓ − B † i ↓ B † i ↑ (cid:17)(cid:105) + H . c . (A3) Appendix B: Energy spectrum using the Green’sfunction method
Our Hamiltonian matrix of the edge problem has asimple band form, H = A B · · B † A B · · B † A B · · B † A B · · ·· · · , (B1)where A and B are small square matrices of order 8 (or4 in the reduced block form). L´opez Sancho et al. [36]developed a highly convergent iterative scheme to cal-culate the surface and bulk Green’s functions ( G and G ∞∞ , respectively) for this form of Hamiltonian. At the i th iteration, the (renormalized) G is given in terms ofeffective interaction with the 2 i th layer:( ωI − (cid:15) s i ) G = I + α i G i , (B2)and other elements are given by( ωI − (cid:15) i ) G i n, = β i G i ( n − , + α i G i ( n +1) , , (B3)( ωI − (cid:15) i ) G i n, i n = I + β i G i ( n − , i n + α i G i ( n +1) , i n , (B4)where ω is an energy with a small imaginary part iη and ( ω -dependent) energy matrices (cid:15) s i , (cid:15) i , α i , and β i aredetermined recursively starting from (cid:15) s0 = (cid:15) = A , α = B , and β = B † . As the iteration proceeds, the effectiveinteractions α i and β i decay quickly. We take η/t = 10 − ,and the iteration is truncated when | α i /t | , | β i /t | < − .The required number of iterations is at most 20.Figure 7 shows k y -resolved spectral functions obtainedwith this method, N n ( k y , E ) = − π Im Tr G nn ( k y , E + iη ) , (B5) with n = 0 (edge) and n = ∞ (bulk), for the four param-eter sets used in Figs. 5(a)-5(d). A blowup of spectralfunctions near k y ∼ −0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 160 180 −0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70−0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 160 180 −0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70−0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 160 180 −0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70−0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 160 180 −0.01−0.005 0 0.005 0.01 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70 FIG. 7: Momentum-resolved spectral function calculated by the Green’s function method. Left (right) panels provide the localdensity of states at the bulk (edge). The dark blue area represents a no-state region. (a) Full gap, (b) and (c) opposite-signpairs of nodal points, and (d) same-sign pair of nodal points. The energy is given in units of t . -0.0020.0020 0 0.01 0.02 0.03 0.17 0.18 0.23 FIG. 8: Blowup of the spectral function (edge+bulk) of the parameter set in Fig. 5(d) around gapless regions with fineresolution in energy and momentum. [1] Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S.Zhang, W. Li, H. Ding, Y.-B. Ou, P. Deng, K. Chang,J. Wen, C.-L. Song, K. He, J.-F. Jia, S.-H. Ji, Y.-Y.Wang, L.-L. Wang, X. Chen, X.-C. Ma, and Q.-K. Xue,Interface induced high temperature superconductivity insingle unit-cell FeSe films on SrTiO , Chin. Phys. Lett. , 037402 (2012).[2] Q. Fan, W. H. Zhang, X. Liu, Y. J. Yan, M. Q. Ren, R.Peng, H. C. Xu, B. P. Xie, J. P. Hu, T. Zhang, andD. L. Feng, Plain s -wave superconductivity in single-layer FeSe on SrTiO probed by scanning tunneling mi-croscopy, Nat. Phys. , 946 (2015).[3] Z. Li, J.-P. Peng, H.-M. Zhang, W.-H. Zhang, H. Ding,P. Deng, K. Chang, C.-L. Song, S.-H. Ji, L. Wang, K.He, X. Chen, Q.-K. Xue, and X.-C. Ma, Molecular beamepitaxy growth and post-growth annealing of FeSe filmson SrTiO : A scanning tunneling microscopy study, J.Phys.: Condens. Matter , 265002 (2014).[4] D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang,Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C.Chen, L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z.Xu, J. Hu, X. Chen, X. Ma, Q. Xue, and X. J. Zhou,Electronic origin of high-temperature superconductivityin single-layer FeSe superconductor, Nat. Commun. ,931 (2012).[5] S. He, J. He, W. Zhang, L. Zhao, D. Liu, X. Liu, D. Mou,Y.-B. Ou, Q.-Y. Wang, Z. Li, L. Wang, Y. Peng, Y. Liu,C. Chen, L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z.Xu, X. Chen, X. Ma, Q. Xue, and X. J. Zhou, Phase di-agram and electronic indication of high-temperature su-perconductivity at 65 K in single-layer FeSe films, Nat.Mater. , 605 (2013).[6] S. Tan, Y. Zhang, M. Xia, Z. Ye, F. Chen, X. Xie, R.Peng, D. Xu, Q. Fan, H. Xu, J. Jiang, T. Zhang, X.Lai, T. Xiang, J. Hu, B. Xie, and D. Feng, Interface-induced superconductivity and strain-dependent spindensity waves in FeSe/SrTiO thin films, Nat. Mater. , 634 (2013).[7] Y. Zhang, J. J. Lee, R. G. Moore, W. Li, M. Yi, M.Hashimoto, D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen, Superconducting gap anisotropy in monolayerFeSe thin film, Phys. Rev. Lett. , 117001 (2016).[8] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du,Unconventional Superconductivity with a Sign Reversalin the Order Parameter of LaFeAsO − x F x , Phys. Rev.Lett. , 057003 (2008).[9] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H.Kontani, and H. Aoki, Unconventional Pairing Originat-ing from the Disconnected Fermi Surfaces of Supercon-ducting LaFeAsO − x F x , Phys. Rev. Lett. , 087004(2008).[10] D. Huang and J. E. Hoffman, Monolayer FeSe on SrTiO ,Annu Rev. Condens. Matter Phys. , 311 (2017).[11] S. Coh, M. L. Cohen, and S. G. Louie, Large electron-phonon interactions from FeSe phonons in a monolayer,New Journal of Physics , 073027 (2015).[12] X. Chen, S. Maiti, A. Linscheid, and P. J. Hirschfeld,Electron pairing in the presence of incipient bands iniron-based superconductors, Phys. Rev. B , 224514(2015).[13] I. I. Mazin, Symmetry analysis of possible superconduct- ing states in K x Fe y Se superconductors, Phys. Rev, B , 024529 (2011).[14] P. M. Eugenio and O. Vafek, Classification of symmetryderived pairing at the M point in FeSe, Phys. Rev. B ,014503 (2018).[15] Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee, What makesthe T c of monolayer FeSe on SrTiO so high: A sign-freequantum Monte Carlo study, Sci. Bull. , 925 (2016).[16] D. F. Agterberg, T. Shishidou, J. OHalloran, P. M.R. Brydon, and M. Weinert, Resilient Nodeless d -WaveSuperconductivity in Monolayer FeSe, Phys. Rev. Lett. , 267001 (2017).[17] T. Shishidou, D. F. Agterberg, and M. Weinert, Magneticfluctuations in single-layer FeSe, Commun. Physics, , 8(2018).[18] Q. Wang, Y. Shen, B. Pan, X. Zhang, K. Ikeuchi, K.Iida, A. D. Christianson, H. C. Walker, D. T. Adroja, M.Abdel- Hafiez, X. Chen, D. A. Chareev, A. N. Vasiliev,and J. Zhao, Magnetic ground state of FeSe, NatureCommun. , 12182 (2016).[19] V. Cvetkovic and O. Vafek, Space group symmetry, spin-orbit coupling, and the low-energy effective Hamiltonianfor iron-based superconductors, Phys. Rev. B , 134510(2013).[20] M. Sigrist and K. Ueda, Phenomenological theory of un-conventional superconductivity, Rev. Mod. Phys. , 239(1991).[21] M. Z. Hasan and C. L. Kane, Colloquium: Topologicalinsulators, Rev. Mod. Phys. , 3045 (2010).[22] O. Vafek and A. Vishwanath, Dirac Fermions in Solids:From High- T c Cuprates and Graphene to TopologicalInsulators and Weyl Semimetals, Annu. Rev. Condens.Matter Phys. , 83 (2014).[23] A. P. Schnyder and P. M. R. Brydon, Topological sur-face states in nodal superconductors, J. Phys.: Condens.Matter , 024522 (2012).[25] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Topol-ogy of Andreev bound states with flat dispersion, Phys.Rev. B , 224511 (2011).[26] A. V. Chubukov, O. Vafek, and R. M. Fernandes, Dis-placement and annihilation of Dirac gap nodes in d -waveiron-based superconductors, Phys. Rev. B , 174518(2016).[27] E. M. Nica, R. Yu, and Q. Si, Orbital-selective pairingand superconductivity in iron selenides, npj QuantumMater. , 24 (2017).[28] D. V. Chichinadze and A. V. Chubukov, Winding num-bers of nodal points in Fe-based superconductors, Phys.Rev. B , 094501 (2018).[29] G.-Y. Zhu, F.-C. Zhang, and G.-M. Zhang, Proximity-induced superconductivity in monolayer CuO oncuprate substrates, Phys. Rev. B , 174501 (2016).[30] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Classification of topological insulators and super-conductors in three spatial dimensions, Phys. Rev. B ,195125 (2008).[31] B. B´eri, Topologically stable gapless phases of time-reversal-invariant superconductors, Phys. Rev. B , , 155105 (2017).[33] A. P. Schnyder and S. Ryu, Topological phases and sur-face flat bands in superconductors without inversion sym-metry, Phys. Rev. B , 060504(R) (2011).[34] A. C. Potter and P. A. Lee, Edge ferromagnetism fromMajorana flat bands: application to split tunneling-conductance peaks in high- T c cuprate superconductors, Phys. Rev. Lett , 117002 (2014).[35] M. Covington, M. Aprili, E. Paraoanu, L. H. Greene, F.Xu, J. Zhu, and C. A. Mirkin, Observation of surface-induced broken time-reversal symmetry in YBa Cu O tunnel junctions, Phys. Rev. Lett. , 277 (1997).[36] M. P. L Sancho, J. M. L. Sancho, and J Rubio, Highlyconvergent schemes for the calculation of bulk and sur-face Green functions, J. Phys. F14