Fragile surface zero-energy flat bands in three-dimensional chiral superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Fragile surface zero-energy flat bands in three-dimensional chiral superconductors
Shingo Kobayashi , Yukio Tanaka , and Masatoshi Sato Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: October 28, 2018)We study surface zero-energy flat bands in three-dimensional chiral superconductors with p z ( p x + ip y ) ν -wave pairing symmetry ( ν is a nonzero integer), based on topological arguments and tunnelingconductance. It is shown that the surface flat bands are fragile against (i) the surface misorientationand (ii) the surface Rashba spin-orbit interaction. The fragility of (i) is specific to chiral SCs, whereasthat of (ii) happens for general odd-parity SCs. We demonstrate that these flat band instabilitiesvanish or suppress a zero-bias conductance peak in a normal/insulator/superconductor junction,which behavior is clearly different from high- T c cuprates and noncentrosymmetric superconductors.By calculating the angle resolved conductance, we also discuss a topological surface state associatedwith the coexistence of line and point nodes. I. INTRODUCTION
Gapless phases of matter have received enormous at-tention in recent years. In the context of unconven-tional superconductors (SCs), such gapless phases man-ifest as nodal excitations in superconducting gaps, andthey have played important roles in determination of thepairing symmetry. Recent developments on topologicalclassification have deepened the understanding of thenodal stability. From the topological perspective, thestability of the nodal structure is ensured by topologi-cal numbers, which predict surface zero energy Andreevbound states at the same time, due to the bulk-boundarycorrespondence. In particular, line nodes induce sur-face zero energy flat bands in projected surface Bril-louin zone (BZ) . The existence of the surface zero-energy flat bands results in a zero-bias conductance peak(ZBCP) in the tunneling spectroscopy , which hasbeen observed in high-T c cuprates . In addition, sim-ilar ZBCPs have been anticipated theoretically in time-reversal invariant (TRI) noncentrosymmetric SCs such as CePt Si , CeRh Si , and CeIrSi .Up to this time, line node in TRI SCs have been con-sidered. In TRI SCs, there is chiral symmetry that isobtained as a combination of particle-hole (PHS) andtime-reversal symmetries (TRS) in most topological ar-guments. The chiral symmetry makes it possible to de-fine a one-dimensional (1D) winding number, which en-sures the stability of line nodes and the existence of sur-face flat bands. However, for heavy fermion SCs such asUPt and URu Si , line nodes in time-reversalbreaking gap functions have been also proposed. Thesematerials are candidates of 3D chiral superconductorswith p z ( p x + ip y ) ν pairing symmetry. ν = 1 correspondsto chiral d-wave pairing(URu Si ), and ν = 2 to chi-ral f-wave pairing (B-phase of UPt ). These gap func-tions support a horizontal line node on the k x k y plane,and point nodes in the k z -axis. Interestingly, it has beenshown that the 3D chiral SCs support zero energy surfaceflat bands on a surface perpendicular to the z -axis .(See Fig. 1).In this paper, we address the stability of the line node Line nodePoint nodeZero energy flat band Arc surface stateFermi surface k x k y k z (Chiral linear dispersion) FIG. 1. (color online) Schematic picture of topological surfacestates in 3D chiral SCs with ν = 1, where a spherical Fermisurface is assumed. The chiral SC hosts both line and pointnodes, which induces a zero-energy flat band and an arc sur-face state (chiral linear dispersion) at each projected surface,respectively. and the surface flat band in the 3D chiral SCs, based ontopological arguments and tunneling conductance. Sincethe chiral SCs break time-reversal symmetry, the fore-mentioned chiral symmetry is absent. Nevertheless, byusing a momentum-dependent gauge symmetry, we finda momentum-dependent chiral symmetry specific to thechiral SCs, which allows us to define a similar 1D windingnumber. This winding number explains the existence ofzero energy surface flat bands on a surface normal to the z -axis. In contrast to the original chiral symmetry, themodel-dependent chiral symmetry is fragile, and the sur-face zero energy flat bands disappear when a surface isnot normal to the z -axis. We find that in the latter situa-tion, arc surface states appear, instead. (See Fig. 2). Thesurface arcs connect the projections of the point nodesand the line nodes on the surface BZ. We also reveal thetopological origin of the arc structures. The main pur-pose of this paper is to clarify the disappearance of zeroenergy flat bands and explore the interplay between sur-face states originating from line and point nodes in thehigher order chiral SCs.We also argue the parity-dependence of the stability.For odd ν , the parity of the gap function is even, while foreven ν , the parity is odd. In our previous paper , we haveuncovered that line nodes in odd-parity SCs are topolog-ically unstable, and the corresponding surface states arefragile against the surface Rashba spin-orbit interaction(RSOI). Taking into account the RSOI in the calculationof tunneling conductance, we confirm the flat band insta-bility in the even ν case as the suppression of the ZBCPwith increasing the magnitude of the RSOI.This paper is organized as follows. In Sec. II, we dis-cuss our model and symmetries that the 3D chiral SCshost. In Sec. III, 1D and 2D topological numbers in3D chiral SCs are introduced and their relation to zero-energy Andreev bound states are discussed in subsec-tion III C. In Sec. IV, we show tunneling conductancein a normal/insulator/superconductor junction for chiralSCs. The influence of the misorientation angle and theRSOI on the conductance is discussed in subsection IV Aand IV B, respectively. Finally, we summarize this paperin Sec. V. II. MODEL SYSTEMS
We phenomenologically model 3D chiral SCs with k z ( k x + ik y ) ν -wave pairing symmetry ( ν = 0 , , , · · · )as a single band system described by the Bogoliubov-deGennes (BdG) Hamiltonian H = P k Ψ † k H ( k )Ψ k , withΨ T k = ( c k , ↑ , c k , ↓ , c †− k , ↑ , c †− k , ↓ ) and H ( k ) = (cid:18) E ( k ) ∆( k )∆( k ) † −E T ( − k ) (cid:19) . (1)Here E ( k ) describes a normal dispersion with TRS and∆( k ) takes∆( k ) = ∆ k ν +1 F k z ( k x + ik y ) ν is y ν : odd , ∆ k ν +1 F k z ( k x + ik y ) ν s x ν : even . (2)The parity of the gap function is even (odd) when ν isodd (even). k F is the Fermi wavelength and the direc-tion of d -vector for the odd parity gap function is chosento be parallel to the z direction for convenience. Notethat the k z ( k x + ik y ) ν -wave pairing symmetry is realizedwhen the gap function respects a 2D irreducible represen-tation of point groups; for example, UPt and URu Si correspond to the E u representation of D h and the E g representation of D h , respectively.The BdG Hamiltonian (1) hosts several discrete sym-metries that are relevant to topological numbers dis-cussed in the next section. First of all, the BdG Hamil-tonian hosts PHS: CH ( k ) C − = − H ( − k ) with C = s τ x K , where s i = (1 , σ ) and τ i = (1 , τ ) are the Paulimatrices in spin and Nambu spaces, respectively, and K is the complex conjugation. For ν = 0, the BdG Hamil-tonian also supports TRS: T H ( k ) T − = H ( − k ), where T = is y τ K . We also impose inversion symmetry onthe present model such that P H ( k ) P − = H ( − k ) with P = s τ for even parity and P = s τ z for odd parity.Besides these fundamental symmetries, the systemwith nonzero ν also has an accidental symmetry, whichwe call pseudo TRS. The gap function k z ( k x + ik y ) ν is rewritten as k z ( k x + k y ) ν e iνϕ k with ϕ k = tan − k y k x .Thus, by the local gauge transformation, U ϕ k H ( k ) U † ϕ k with U ϕ k = e − i ν ϕ k s τ z , the gap function becomes ∆ k ν +1 F k z ( k x + k y ) ν , so the system recovers TRS. In otherwords, the BdG Hamiltonian (1) has the following mo-mentum dependent pseudo TRS. U † ϕ k T U ϕ k H ( k ) U † ϕ k T † U ϕ k = H ( − k ) , (3)where U ϕ k = s τ when ϕ k = 0. As we will discuss inSec. IV, the accidental symmetry plays a crucial role in aflat band instability concerning a surface misorientation.Finally, the BdG Hamiltonian possesses spin-rotationsymmetry, [ S z , H ( k )] = 0 with S z = is z τ z . As discussedbelow, the spin-rotation symmetry is essential for the sta-bility of the surface flat bands in the even ν case. III. TOPOLOGICAL NUMBERS AND SURFACESTATES
Topological properties of nodal SCs in 3D are at-tributed to topological numbers of point and line nodes.To characterize the topological numbers in 3D chiral SCsin which point and line nodes exist in the k z axis andon the k x k y plane, we introduce 1D and 2D topologi-cal numbers associated with line and point nodes. Thesetopological numbers are defined within lower dimensionalsubspaces in the BZ. Importantly, the 1D topologicalnumber originates from the interplay between topologyand symmetry and is affected by the pseudo TRS andthe spin-rotation symmetry, as we shall show in the fol-lowing. A. One-dimensional topological number
To define the 1D topological number associated witha line node, we recall a winding number in TRI SCs(see Refs. 10 and 12 for example). A key ingredient isa so-called chiral operator Γ ≡ − iCT satisfying the anti-commutation relation { Γ , H ( k ) } = 0. Using the chiraloperator and a BdG Hamiltonian, the 1D winding num-ber is normally defined by W ( k k , Γ) ≡ − πi Z ∞−∞ dk ⊥ Tr[Γ H − ( k ) ∂ k ⊥ H ( k )] , (4)which takes an integer under the regularization . k k and k ⊥ are wave vectors parallel and perpendicular to acertain surface, respectively. Although 3D chiral SCs donot have TRS, they have the pseudo TRS (3) instead. k , k , k (a) α=0 (b) 0<α<π/4 (c) α=π/4 (d) π/4 <α<π/2 (e) α=π/2 Fully gaped regionArc surface state by W e and W o Arc surface state by W o Arc surface state by N
Zero-energy flat band Arc surface state Arc surface state Arc surface state Arc surface state
FIG. 2. (color online) Relation between the misorientation angle α and the topological surface states in 3D chiral SCs with aspherical Fermi surface and ν = 0. In the angle 0 < α < π/
4, the zero-energy flat band suddenly disappears and instead anarc surface state appears according to the 1D winding number defined by Eq. (9). When α is greater than π/
4, fully gappedregions exist in the bulk BZ, which allows to exist a nontrivial TKNN number N ( k k , ) defined by Eq. (13). The TKNN numberensures the existence of 2 ν arc surface states, where 2 comes from the spin degeneracy. The combination of the pseudo TRS and PHS also satis-fies the anticommutation relation with the BdG Hamilto-nian, whereby leading to a chiral operator. Furthermore,the existence of inversion symmetry leads to the vanish-ing of the 1D winding number: W ( k k , Γ) = 0 in the even ν case. (See Appendix A for more details). Hence, a linenode in odd-parity SCs is generally unstable under TRSand inversion symmetry and a stable one requires an ad-ditional symmetry. In this paper, in order to acquire astable line node, we use the spin-rotation symmetry pro-vided in the previous section, which makes a line nodestable . In the following, we define a chiral operator andan associated 1D winding number for even and odd paritycases, respectively.In the even-parity case, a proper chiral operator is de-fined by Γ ϕ k ≡ U † ϕ k Γ U ϕ k and the corresponding 1D wind-ing number is given by Eq. (4) replacing Γ with Γ ϕ k : W ( k k , Γ ϕ k ) ≡ W e ( k k ) . (5)Since Γ ϕ k depends on U ϕ k , the quantization of W e re-quires ∂ k ⊥ ϕ k = 0, which is satisfied only for k ⊥ = k z or ϕ k = 0. For this reason, the topological surface statestrongly depends on the relative angle between the linenode and the surface. This is a key feature of 3D chi-ral SCs, resulting in instability of ZBCP in terms of thesurface misorientation.On the other hand, using the local gauge transforma-tion and the spin-rotation symmetry, the chiral operatorin the odd-parity case is defined as Γ s ϕ k ≡ U † ϕ k S z CT U ϕ k .With the chiral operator Γ s ϕ k , we define the 1D topolog-ical number as W ( k k , Γ s ϕ k ) ≡ W o ( k k ) . (6)Here ν = 0 satisfies ϕ k = 0. Thus, W o with ν = 0 iswell-defined at any surface except for the surface beingnormal to the line node. On the other hand, W o with ν = 0 is quantized only for k ⊥ = k z or ϕ k = 0 due to themomentum dependence of the chiral operator. As a re-sult, line-node-induced surface states will be sensitive notonly to a surface misorientation, but also to the RSOI. Assuming that a pairing interaction is weak, ∆( k ) isnegligibly small far way from the Fermi surface. Theweak pairing assumption allows us to simplify the cal-culation of the 1D winding numbers because the maincontribution comes from momentum around the Fermisurface. Substituting Eqs. (1) and (2) into Eqs. (5) and(6), the 1D winding numbers are rewritten in a simpleform . In p z -wave SCs ( ν = 0), it yields for any k ⊥ W o ( k k ) = X E ( k )=0 sgn[ ∂ k ⊥ E ( k )] · sgn[ k z ] , (7)where the summation is taken for k ⊥ satisfying E ( k ) =0. Whereas the 1D winding numbers with ν = 0 aredescribed as follows. When k ⊥ = k z , W e(o) ( k x , k y ) = X E ( k )=0 sgn[ ∂ k z E ( k )] · sgn [ k z ] , (8)and when k ⊥ = k z and k y = 0, W e(o) ( k k , ) = X E ( k )=0 sgn[ ∂ k ⊥ E ( k )] · sgn[ k z k νx ] , ν : odd (even) . (9)Here k ⊥ and k k , lie in the k x k z plane and are orthog-onal to each other. To see the 1D winding numbersconcretely, we shall consider a spherical Fermi surface E ( k ) = ~ m ( k − k F ), where m is the mass of electron,and introduce the misorientation angle between the linenode and the surface as α satisfying (cid:18) k k , k ⊥ (cid:19) = (cid:18) cos α sin α − sin α cos α (cid:19) (cid:18) k x k z (cid:19) . (10)In this case, Eq. (8), i.e., α = 0 leads to 2 at the inside ofthe Fermi surface in the surface BZ. On the other hand,as we change the surface misorientation ( α = 0), Eq. (9)describes the surface state and is evaluated as follows.When 0 < α < π/ W e ( k k , ) = − − cos α < k k , /k F < − sin α, α < k k , /k F < cos α, , (11a) W o ( k k , ) = ( | k k , /k F | < cos α, , (11b)and when π/ < α < π/ W e ( k k , ) = − − sin α < k k , /k F < − cos α, α < k k , /k F < sin α, , (12a) W o ( k k , ) = ( | k k , /k F | < cos α, , (12b)where 2 is due to the spin degeneracy. Equations (11)and (12) show a nontrivial winding number along thedirection k y = 0 for a proper angle. B. Two-dimensional topological number
Due to the presence of point nodes ( ν = 0), there existsthe Thouless-Kohmoto-Nightingale-den Nijs (TKNN)number defined at a fixed k k , , which also character-izes the topological structure: N ( k k , ) = i π X n ∈ occ Z BZ dk ⊥ dk k , ǫ ab ∂ k a h u n ( k ) | ∂ k b | u n ( k ) i , (13)where | u n ( k ) i is an eigenstate of H ( k ) and the summa-tion is taken over all of occupied states. Here k k , is adirection orthogonal to k ⊥ and k k , ; k a and k b take k ⊥ and k k , . Using eigenvectors of Eq. (1), we obtain forboth even and odd parity pair potentials N ( k k , ) = 2 ν π Z ∞−∞ Z ∞−∞ dk ⊥ dk k , ǫ ab sin θ k ∂ k a θ k ∂ k b ϕ k , (14)with θ k ≡ tan − [∆ k z ( k x + k y ) ν / ( E ( k ) k ν +1 F )]. As a con-crete model, let us consider a spherical Fermi surfacegiven by ǫ ( k ) = ~ m (cid:20) k − k F + δ (cid:16) k ⊥ + k k , (cid:17) j (cid:21) (2 j >ν ), where the last term is added so as to satisfy the regu-larization at k ⊥ , k k , → ∞ and δ is an infinitesimal coef-ficient. When k k , = k z , i.e., α = π/
2, Eq. (14) is readilycalculated as N ( k z ) = − ν (1 + sgn( k F − k z )) (15)except for k z = 0 (see Fig. 2 (e)). When k k , = k z , N ( k k , ) takes − ν if H ( k ) is fully gapped at a fixed k k , .This condition is satisfied for − sin α < k k , /k F < − cos α and cos α < k k , /k F < sin α within π/ < α < π/ C. Topological surface states
The 1D and 2D topological numbers ensure the exis-tence of zero-energy surface states via the bulk-boundarycorrespondence. We shall consider the semi-infinite su-perconductor on x ⊥ >
0, where x ⊥ is the conjugate co-ordinate of k ⊥ and the surface is at x ⊥ = 0. The BdGequation is given by H ( x ⊥ , k k )Ψ( x ⊥ , k k ) = E ( k k )Ψ( x ⊥ , k k ) , (16)with the boundary condition Ψ(0 , k k ) = 0. The zero-energy surface state satisfies E ( k k ) = 0. We first discussa zero-energy flat band associated with a line node. If achiral symmetry Γ exists in H ( x ⊥ , k k ), it demands thatthe number of positive energy states is equal to that ofnegative energy states for E ( k k ) = 0 due to the anti-commutation relation between Γ and H . Thus, as wedefine the number of zero-energy states with Γ = +( − )as N +0 ( N − ), stable zero energy states exist only when N +0 = N − . It is for the reason that chiral symmetry pre-serving perturbations shift the surface states from zeroenergy within a pair of the zero-energy states with oppo-site chirality. The number of zero-energy states indeedconnects with the winding number (4): W ( k k , Γ) = N − − N +0 . (17)Since the pseudo TRS plays the same role as TRS atthe specific momentum such as k ⊥ = k z and k y = 0,we can apply Eq. (17) to Eqs. (5) and (6) there. In thecase of the spherical Fermi surface, a zero energy flatband will appear at the inside of the Fermi surface in thesurface BZ when the surface is parallel to the line node( α = 0) (see Fig. 2 (a)). In contrast to the zero-energyflat band, as we shift the misorientation angle from 0, themany zero-energy states vanish and instead arc surfacestates emerge in the line k y = 0 as remnants of the zero-energy flat band according to Eqs. (11) and (12). (SeeFig. 2 (b)-(d)). The vanishing of the zero-energy flatband contrasts sharply with that in TRI SCs and is oneof the main results in this paper.Similarly to a line node, a pair of point nodes in-duces an arc surface state terminating the projected pointnodes on the surface BZ (see Fig. 2 (e)). Let us considerthe semi-infinite superconductor ( x >
0) with the spher-ical Fermi surface. From Eq. (15), the TKNN numbertakes − ν within | k z | < k F . Hence, 2 ν arc surface stateswith the chiral linear dispersion appears in the projectedsurface BZ. The same arc surface state has been studiedin chiral SCs without the line node, such as superfluidhelium A phase and Sr RuO . It would benoted that 1D and 2D topological number induced sur-face states coexist in the angle π/ < α < π/
2, togetherwith the emergence of fully gapped regions in the bulkBZ. (See Fig. 2 (d)).
FIG. 3. (color online) Normalized conductances for ν = 0(a), ν = 1 (b), and ν = 2 (c) with various barrier potentials Z = 0 , , , and 6. Angle resolved conductances σ S ( eV =0 , θ, φ ) cos θ with Z = 6 for n = 0 , , and 2 are shown inthe pictures (d), (e), and (f), respectively, which indicate theformation of the zero-energy flat band at (001) plane as thefunction of ( k x , k y ). IV. TUNNELING CONDUCTANCE
To test these expectations in the topological argument,we discuss in the following tunneling conductance in a 3Dnormal metal/insulator/chiral superconductor (N/I/S)junction. It is well-known that a surface zero-energy flatband induces a sharp ZBCP for line nodal SCs such ashigh- T c cuprates and noncentrosymmeric SCs. Normally,the ZBCP plays an important role to distinguish a nodalSC from a fully gapped one and is robust against TRSpreserving perturbations, e.g., a nonmagnetic impurity.In contrast, as discussed in the previous section, theZBCP in 3D chiral SCs is expected to be more fragilethan that in TRI SCs. This is because the chiral opera-tor depends on the pseudo TRS and/or the spin-rotationsymmetry. Hence, the breaking of lattice symmetry orspin-rotation symmetry will shift the flat band from zero energy. The purpose of this section is to demonstratethese fragilities of the ZBCP under the effect of the sur-face misorientation and the surface RSOI. These pertur-bations preserve TRS, but breaks each of the acciden-tal symmetries. To see these phenomena, we shall con-sider an N/I/S junction with a flat interface perpendic-ular to the z-axis and the spherical Fermi surface, i.e., E ( k ) = ~ k m − µ , where µ is the chemical potential.In this setup, we calculate tunneling conductance us-ing the Blounder-Thinkham-Klapwijk (BTK) formula ,with taking into account the effect of the surface misori-entation and the surface RSOI. By solving the BdG equa-tion (16) for E ( k k ) = eV and using the quasi-classical ap-proximation, i.e., µ ≫ | E ( k k ) | , | ∆( k ) | , the wave functionansatz for the normal state ( z <
0) and the supercon-ducting state ( z >
0) is given byΨ N σ ( z, k k ) = ψ N e,σ e i k · r + X σ ′ = ± (cid:16) a σ,σ ′ ψ N h,σ ′ e i k · r + b σ,σ ′ ψ N e,σ ′ e i ˜ k · r (cid:17) z < , (18a)Ψ S ( z, k k ) = X σ ′ = ± (cid:16) c σ ′ ψ S e,σ ′ e i k · r + d σ ′ ψ S h,σ ′ e i ˜ k · r (cid:17) z > , (18b)with ψ N e,σ = 1 / σ, − σ, , T , (19a) ψ N h,σ = 1 / , , σ, − σ ] T , (19b) ψ S e,σ = 1 / σ, − σ, (1 − σ ) η Γ + , (1 + σ )Γ + ] T , (19c) ψ S h,σ = 1 / σ )Γ − , (1 − σ ) η Γ − , − σ, σ ] T , (19d)where Γ + ( θ, φ ) = ∆ ∗ ( θ, φ ) / [ E + p E − | ∆( θ, φ ) | ,Γ − ( θ, φ ) = ∆( π − θ, φ ) / [ E + p E − | ∆( π − θ, φ ) | ] and η = − (+) for the even (odd) parity pairing. The super-scripts N and S indicate the normal and superconductingstates and the subscripts e and h describe electron andhole states, respectively. ˜ k = ( k x , k y − k z ) and the alltrajectories of electron and hole have the same incidentangle ( k x , k y , k z ) = k F (sin θ cos φ, sin θ sin φ, cos θ ) sincewe assume that the chemical potential for the normalstate is the same as that for the superconducting state.The coefficients a σ,σ ′ , b σ,σ ′ , c σ , and d σ are determinedby the boundary conditions:Ψ N σ (0 − , k k ) = Ψ S (0 + , k k ) , (20a) d Ψ S dz (cid:12)(cid:12)(cid:12) z =0 + − d Ψ N σ dz (cid:12)(cid:12)(cid:12) z =0 − = 2 mU ~ Ψ S (0 + , k k ) , (20b)where the insulating barrier at z = 0 is modeled as the δ -function V ( z ) = U δ ( z ). Solving Eq. (20), we obtainthe normal reflection and Andreev reflection coefficients b σ,σ ′ and a σ,σ ′ , which give the transmissivity in the N/I/Sjunction as σ S ( eV, θ, φ ) = 1 + 12 X σ,σ ′ = ± [ | a σ,σ ′ | − | b σ,σ ′ | ]= σ N σ N | Γ + | + ( σ N − | Γ + | | Γ − | | σ N − + Γ − | , (21)where σ N = Z and Z = mU ~ k F cos θ ≡ Z cos θ . Integrat-ing σ S ( eV, θ, φ ) with respect to the all incident angles ofinjected electrons, the normalized tunneling conductanceis given by σ ( eV ) = R π dφ R π/ dθ σ S ( eV, θ, φ ) sin θ cos θ R π dφ R π/ dθ σ N sin θ cos θ . (22)We numerically calculate the normalized conductance(22) and the transmissivity (21) for ∆( θ, φ ) = ∆ cos θ ( ν = 0), ∆ cos θ sin θe iφ ( ν = 1), and ∆ cos θ sin θe i φ ( ν = 2) with various barrier potentials, which are shownin Fig. 3. Due to the line node at k z = 0, the zero-energyflat band appears at the (001) plane in all cases with Z = 6 (see Fig. 3 (d), (e), and (f)) and we obtain a sharpZBCP under the high barrier potential (see Fig. 3 (a),(b), and (c)). A. Effect of surface misorientation
To show the zero-energy flat band instability in termsof the misorientation angle, we rotate the coordinate ofthe pair potential in the k z k x plane instead of rotating thesurface: ( k x , k y , k z ) → ( k x cos α − k z sin α, k y , k x sin α + k z cos α ), where α is the misorientation angle defined inSec. III A. The pair potential for ν = 0 , θ, φ, α ) = ∆ (sin θ cos φ sin α + cos θ cos α ) , (23)∆( θ, φ, α ) = ∆ (sin θ cos φ sin α + cos θ cos α ) × (sin θ cos φ cos α − cos θ sin α + i sin θ sin φ ) , (24)∆( θ, φ, α ) = ∆ (sin θ cos φ sin α + cos θ cos α ) × (sin θ cos φ cos α − cos θ sin α + i sin θ sin φ ) . (25)Substituting Eqs. (23), (24), and (25) into Eqs. (21) and(22), we numerically calculate the normalized conduc-tance and the angle resolved conductance for each case,where α takes 0 , . π, . π, . π , and 0 . π . The resul-tant conductance is shown in Fig. 4. When α = 0, thesharp ZBCP and the surface zero-energy flat band ap-pear in all cases due to the presence of the line node.As slightly changing α , in the p -wave SC ( ν = 0), theZBCP remains and the zero-energy flat band still formsat the interior of the projected line node as shown inFig. 4 (d-ii). In contrast, the ZBCP and the surfacezero-enrgy flat band in the chiral SCs ( ν = 1 and 2) vanish abruptly (see Figs. 4 (e-ii) and (f-ii)). The rea-son for the disappearance of the flat band is understoodfrom the definition of the 1D wingding number (8) and(9). Hence, the 1D winding number for ν = 1 and 2 isquantized only when ∂ k ⊥ ϕ k = 0, while that for ν = 0is well-defined unless k ⊥ = k x . Therefore, the ZBCPin 3D chiral SCs ( ν = 0) is fragile against the surfacemisorientation and clearly different from the behavior ofthe ZBCP in the p -wave SC. Furthermore, in ν = 1 and ν = 2, an arc surface state emerges for the misorien-tation angle 0 < α ≤ π/ < α < π/
4, the arc surface stateoriginates from the 1D winding number (9). Accordingto Eq. (11), the arc surface state in Fig. 4 (e-ii) is inducedby W e , while that in Fig. 4 (f-ii) comes from W o . As α reaches π/
4, the projected positions of the line and pointnodes overlap, so that the zero-energy states vanish for ν = 1 and that remains for ν = 2 as shown in Fig 4 (e-iii) and (f-iii). When π/ < α < π/
2, the TKNN number N becomes nontrivial because the pair potential is fullygaped in k y k z plane within − sin α < k x /k F < − cos α and cos α < k x /k F < sin α . The TKNN number ensuresthe existence of an arc surface state for ν = 1 (see Fig. 4(e-iv)) and double one for ν = 2 (see Fig. 4 (f-iv)). In-terestingly, a mixed surface state emerges in Fig. 4 (f-iv)with ν = 2, in which the 1D winding number W o inducedsurface state exists at | k x /k F | < cos α and the TKNNnumber N induced one at − cos α < k x /k F < − sin α and sin α < k x /k F < cos α . Finally, achieving α = π/ B. Effect of Rashba spin-orbit interaction
Next, we introduce the surface RSOI in order to in-vestigate the flat band instability caused by the breakingof the spin-rotation symmetry. The interface at z = 0breaks inversion symmetry, which thus gives rise to theRSOI. We take into account this effect in the BTK for-mula based on Refs 54 and 55. Assuming that the RSOIis localized at z = 0, we modify the insulating barrierpotential as V ( z ) = ( U + U ( s × ˆ k ) · ˆ z ) δ ( z ) , (26)where ˆ k and ˆ z mean a unit vector of k and z , respectively.It follows the modified boundary condition:Ψ N σ (0 − , k k ) = Ψ S (0 + , k k ) , (27a) d Ψ S dz (cid:12)(cid:12)(cid:12) z =0 + − d Ψ N σ dz (cid:12)(cid:12)(cid:12) z =0 − = 2 m ~ U U SO U ∗ SO U U − U ∗ SO − U SO U Ψ S | z =0 + , (27b)where U SO = iU sin θe − iφ . By solving the BdG equation FIG. 4. (color online) Normalized conductances with the barrier potential Z = 6 and various misorientation angles α =0 , . π, . π, . π and 0 . π for n = 0 (a), n = 1 (b), and n = 2 (c). Figures (d), (e), and (f) show each angle resolvedconductance σ S ( eV = 0 , θ, φ ) cos θ as varying the misorientation angle α . under the boundary condition (27), a σ,σ ′ and b σ,σ ′ obeythe simultaneous linear equations: iZ − iZ i Γ + i Γ + Z − iZ iZ ∗ iη Γ + Z ∗ iη Γ + Z i Γ − Z − i Γ − Z iZ iZ iη Γ − Z − iη Γ − Z ∗ − iZ ∗ − (2 + iZ ) a σ, + a σ, − b σ, + b σ, − = Γ + (2 − iZ ) − iη Γ + Z ∗ − iZ iZ ∗ if σ = + , − i Γ + Z η Γ + (2 − iZ ) − iZ iZ if σ = − , (28) where Z = i mU sin θe − iφ ~ k F cos θ ≡ Z SO ie − iφ sin θ cos θ . Also, thenormal conductance is given by σ N = 4(4 + | Z | + Z )(4 + | Z | − Z ) + 16 Z . (29)Numerically solving Eq. (28), we determine the transmis-sivity and the normalized conductance. The calculatednormalized conductance and angle resolved conductanceare shown in Fig. 5. As expected, Figs. 5 (a) and (c) showthe suppression of the ZBCP in the normalized conduc-tance with increasing the magnitude of the RSOI, whileFig. 5 (b) maintains the ZBCP under the RSOI. Alongwith the suppression of the ZBCP, the disappearance ofthe zero-energy flat band is verified in Fig. 5 (d) and (f)via the angle resolved conductance in eV = 0. Thus,the obtained results agree well with the topological ar- FIG. 5. (color online) Normalized conductances for n = 0 (a), n = 1 (b), and n = 2 (c) with the barrier potential Z = 6,taking into account the RSOI Z SO = 0 , , , σ S ( eV = 0 , θ, φ ) cos θ with Z = 6 and Z SO = 8 for n = 0 , , and 2 are shown in the pictures (d),(e), and (f), respectively. A large region of the zero-energysurface states disappears due to the RSOI when ν is even. guments. Hence, in even ν , the RSOI breaks the spin-rotation symmetry and leads to the flat band instability,while the suppression of the ZBCP does not occur in odd ν because the pseudo TRS only stabilize the flat bandand is still preserved under the RSOI. V. SUMMARY
In summary, we have discussed surface states in 3D chi-ral SCs with p z ( p x + ip y ) ν -wave pairing symmetry andfound fragility of surface zero-energy flat bands againstthe surface misorientation and the surface RSOI. Theseinstabilities are due to the breaking of the protectingsymmetries: the pseudo TRS and the spin-rotation sym-metry. Using the 1D winding number, we have shown that zero-energy flat bands in the 3D chiral SCs are pro-tected by the accidental chiral symmetry consisting of thepseudo TRS and/or the spin-rotation symmetry. Thus,the resulting flat bands are more fragile than is normallyunderstood. We have demonstrated the suppression ofthe ZBCP in the N/I/S junction by numerically calcu-lating tunnel conductance, with taking into account theeffect of the surface misorientation and the surface RSOI.As a result, we found the suppression of the ZBCP interms of the surface misorientation in all of 3D chiralSCs ( ν = 0) because the pseudo TRS is sensitive to thebreaking of lattice symmetry. Also, including the RSOIbreaks the spin-rotation symmetry, resulting in the de-crease of the ZBCP with increase in the magnitude ofthe RSOI in odd-parity SCs. We summarized the chiraloperators discussed in this paper and the flat band in-stabilities in Table I, in which TRI noncentrosymmetricand TRI even-parity SCs are included. It is noteworthythat similar flat band instabilities associated with thebreaking of chiral symmetry have been also discussed in2D systems in the disordered limit . Proximity effectof the flat bands into diffusive normal metal is also aninteresting problem .Finally, we mention the implication of our results forheavy fermion compounds UPt , URu Si , andSrPtAs . In this paper, we have mainly focused onTRS breaking SCs with p z ( p x + ip y ) ν -wave pairing sym-metry, assuming that the normal Hamiltonian possessesTRS. In the heavy fermion systems, however, the gapfunctions are representation of a point group and theycan be more complicated. Nevertheless, as long as theytake the form p z ( p x + ip y ) ν near the Fermi surface, our re-sult is applicable even for heavy fermion SCs. In our cal-culation, we do not determine the spatial dependence ofthe pair potential . As far as we are considering zeroenergy states, the obtained results will not be changedeven if the spatial depletion of the pair potential nearthe surface is taken into account . VI. ACKNOWLEDGMENTS
We thank K. Yada and A. Yamakage for valuablediscussions. This work was supported in part by theTopological Quantum Phenomena Grant-in Aid for Sci-entific Research on Innovative Areas from the MEXT ofJapan (No. 22103005), the Topological Materials Sci-ence Grant-in Aid for Scientific Research on Innova-tive Areas from the MEXT of Japan (No. 15H05853,15H05855), a Grant-in-Aid for Scientific Research B(Grant No. 15H03686) (YT), a Grant-in-Aid for Chal-lenging Exploratory Research (Grant No. 15K13498)(YT), a Grant-in-aid for JSPS Fellows (No. 256466)(SK) and a Grant-in-Aid for Scientific Research B (No.25287085) (MS).
TABLE I. Stability of surface flat bands and types of the chiral operators for several superconducting phases with a line node.The second column shows the definition of the chiral operators for each phase, where S ( S = −
1) is an additional symmetrysuch as spin-rotation symmetry and mirror-reflection symmetry. The third and fourth columns describe the influence of thesurface misorientation and the surface RSOI on the surface zero-energy flat bands, respectively. Here X ( × ) indicates (un)stableflat bands. Systems Chiral operator Misorientation RSOITRI noncentrosymmetric SCs − iCT X X
TRI even-parity SCs − iCT X X
TRI odd-parity SCs S CT X × Chiral even-parity SCs − iU † ϕ k CT U ϕ k × X Chiral odd-parity SCs U † ϕ k S CT U ϕ k × × Appendix A: Vanishing of 1D winding number inTRI odd-parity superconductors
In the presence of TRS and PHS, we always have thechiral operator Γ = − iCT , and then it is possible todefine the 1D winding number W ( k k , Γ) by Eq. (4). Inwhat follows, we show that the 1D winding number W vanishes in the case of an odd-parity pair potential owingto inversion symmetry. We start with the general BdGHamiltonian described by H = 12 X k ,α,α ′ (cid:16) c † k α , c − k α (cid:17) H ( k ) c k α ′ c †− k α ′ ! , (A1)where H ( k ) is given by H ( k ) = E ( k ) αα ′ ∆( k ) αα ′ ∆( k ) † αα ′ −E ( − k ) Tαα ′ ! . (A2) c † k α ( c k α ′ ) represents the creation (annihilation) operatorof an electron with momentum k . The suffix α representsother degrees of freedom such as spin, orbital, and sub-lattice indices. ǫ ( k ) αα ′ and ∆( k ) αα ′ are the Hamiltonianin the normal state and gap function, respectively. TheTRI BdG Hamiltonian possesses TRS:Θ H ( k )Θ − = H ( − k ) ∗ , Θ = U αα U ∗ αα ′ ! , (A3)where U satisfies U E ( k ) U ∗ = E ( − k ) ∗ and U ∆( k ) U T =∆( − k ) ∗ , and PHS: CH ( k ) C − = − H ( − k ) ∗ , C = δ αα ′ δ αα ′ ! . (A4)Thus, the combination of Eqs. (A4) and (A3) satisfies { CT, H ( k ) } = 0, which gives the chiral operatorΓ = − iCT = − iU ∗ αα ′ − iU αα ′ ! . (A5) For convenience, we choose the basis with the chiral op-erator being diagonal such that U † Γ Γ U Γ = δ αα ′ − δ αα ′ ! , U Γ = δ αα ′ − iδ αα ′ − iU αα ′ U αα ′ ! . (A6)In this basis, H ( k ) becomes off-diagonal form U † Γ H ( k ) U Γ = q ( k ) q ( k ) † ! , (A7)with q ( k ) ≡ i E ( k ) + ∆( k ) U. (A8)With this basis, the 1D winding number is rewritten as W ( k k , Γ) = i π Z ∞−∞ dk ⊥ Tr[Γ H − ( k ) ∂ k ⊥ H ( k )]= 12 π Im (cid:20)Z ∞−∞ dk ⊥ ∂ k ⊥ ln det q ( k ) (cid:21) , (A9)where k ⊥ is momenta perpendicular to the surface. In theTRI odd-parity SC, the BdG Hamiltonian hosts inversionsymmetry in addition to the chiral symmetry:˜ P H ( k ) ˜ P − = H ( − k ) , ˜ P = P αα ′ − P αα ′ ! , (A10)where P is a real matrix and satisfies P = 1. Under theunitary transformation U Γ , the inversion operator alsotransforms into U † Γ ˜ P U Γ = − iP αα ′ iP αα ′ ! , (A11)where [ U, P ] = 0 is assumed because P always acts triv-ially on the spin space. From Eqs. (A7) and (A11), theaddition constraint is added on q ( k ): P q † ( k ) P = − q ( − k ) . (A12)0Furthermore, U and q ( k ) satisfies the relation: U q ( k ) U † = q ( − k ) T . (A13)Combining Eqs. (A12) and (A13), we obtain U P q ( k ) † P U † = − q ( k ) T . (A14)Finally, we take the determinant for the both side ofEq.(A14), det q ( k ) † = det( −
1) det q ( k ) T = det q ( k ) T , (A15) where det( −
1) = 1 due to the spin degrees of freedom.Therefore, det q ( k ) is a real function of k , which imme-diately proves W ( k k , Γ) = 0 for any k k . G. E. Volovik,
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