The fate and heir of Majorana zero modes in a quantum wire array
TThe fate and heir of Majorana zero modes in a quantum wire array
Da Wang, Zhoushen Huang, and Congjun Wu
Department of Physics, University of California, San Diego, CA92093
Experimental signatures of Majorana zero modes in a single superconducting quantum wire withspin-orbit coupling have been reported as zero bias peaks in the tunneling spectroscopy. We studywhether these zero modes can persist in an array of coupled wires, and if not, what their remnantcould be. The bulk exhibits topologically distinct gapped phases and an intervening gapless phase.Even though the bulk pairing structure is topological, the interaction between Majorana zero modesand superfluid phases leads to spontaneous time-reversal symmetry breaking. Consequently, edgesupercurrent loops emerge and edge Majorana fermions are in general gapped out except when thenumber of chains is odd, in which case one Majorana fermion survives.
Introduction
Majorana fermions are intriguing ob-jects because they are their own antiparticles. In con-densed matter physics, Majorana fermions arise not aselementary particles, but rather as superpositions of elec-trons and holes forming the zero mode states in topo-logical superconducting states. Majorana fermions werefirst proposed to exist in vortex cores and on boundariesof the p -wave Cooper pairing systems . More recently,they were also predicted in conventional superconduc-tors in the presence of strong spin-orbit (SO) couplingand the Zeeman field . In cold atom physics, SO cou-pling has been realized by using atom-laser coupling .This progress offers an opportunity to realize and manip-ulate Majorana fermions in a highly controllable manner,which has attracted a great deal of attention both theo-retical and experimental .Experimental signatures of Majorana zero modes havebeen reported as zero bias peaks in the tunneling spec-troscopy of a single quantum wire with strong SO cou-pling which is either coupled with an s -wave supercon-ductor through the proximity effect , or, is super-conducting by itself . A further study of an array ofquantum wires is natural , in particular for thepurpose of studying interaction effects among edge Majo-rana zero modes . Topological states in an array ofparallel wires in magnetic fields in the fractional quantumHall regime have been studied recently . Without im-posing self-consistency, flat bands of Majorana zero edgemodes have been found for the uniform pairing as wellas the Fulde-Ferrell-Larkin-Ovchinnikov pairing ,because under time-reversal (TR) symmetry these Majo-rana zero modes do not couple.However, the band flatness of the edge Majorana zeromodes is unstable due to interaction effects. Li and twoof the authors proposed the mechanism of spontaneousTR symmetry breaking for the gap opening in the edgeMajorana flat bands . Even in the simplest case of spin-less fermions without any other interaction channels, thecoupling between Majorana zero modes and the pairingphase spontaneously generates staggered circulating cur-rents near the edge such that Majorana modes can coupleto each other to open the gap due to the breaking of TRsymmetry. Similar results are also obtained recently inRefs. [27 and 48]. The mechanism of gap opening based on spontaneous TR symmetry breaking also occurs inthe helical edge modes of the 2D topological insulatorsunder strong repulsive interactions which leads to edgemagnetism . Main results
In this article, we investigate a coupledarray of s -wave superconducting chains with intra-chainSO coupling and external Zeeman field. We consider boththe proximity-induced superconductivity and the intrin-sic one. For the proximity-induced case, the array isplaced on top of a bulk superconductor, the phase coher-ence induces a nearly uniform pairing distribution in thequantum chains, ∆ r = ∆. The bulk band structure ex-hibits several topologically distinct gapped phases inter-vened by a gapless phase. In the gapless phase, edge Ma-jorana zero modes interpolate between nodes in the bulkenergy spectrum. In the topological gapped phase, theyextend into a flat band across the entire edge Brillouinzone. On the other hand, if either the phase coherence ofthe bulk superconductor is weak, or the superconductiv-ity is intrinsic, such as in the case of Pb nanowires orcold atom systems near Feshbach resonance , then ∆ r has to be solved self-consistently. We find that when thebulk is in the topological gapped phase, the phase dis-tribution of pairing order parameters is inhomogenousalong the edge exhibiting TR symmetry breaking. It in-duces edge currents and gaps out the edge Majorana zeromodes except when the chain number is odd, in whichcase one Majorana zero mode survives. If the bulk is inthe gapless phase, in general TR breaking is also observedbut not always, because Majorana modes associated withopposite winding numbers can coexist on the same edgewhich can be coupled by TR invaraint perturbations. Model of quantum wire array
Consider an array of SOcoupled chains with the proximity effect induced s -wavepairing along the x -direction, which are juxtaposed alongthe y direction. The band Hamiltonian is H = − (cid:88) r σ t (cid:0) c † r σ c r +ˆ x,σ + h.c. (cid:1) − µc † r σ c r σ − (cid:88) r iλ (cid:16) c † r ↑ c r +ˆ x, ↑ − c † r ↓ c r +ˆ x, ↓ (cid:17) + h.c. − (cid:88) r σ t ⊥ (cid:0) c † r σ c r +ˆ y,σ + h.c. (cid:1) , (1) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y FIG. 1. ( a ) Bulk phase diagram of the 2D HamiltonianEqs. 1 and 2 in the µ - B plane with B >
0, and that with
B < B = 0. Theparameter values are t = 1, t ⊥ = 0 . λ = 2, ∆ = 0 .
5. (a) Thewhite solid lines enclose the gapless phase and separate therest into two topologically trivial gapped phases and two non-trivial phases, respectively. Inside the gapless phase, states inthe diamond enclosed by the white dashed lines exhibit edgemodes associated with opposite winding numbers, and thoseoutside the diamond only exhibit edge modes associated withsame winding number. Color scale encodes the momentumaveraged winding number r defined in Eq. 20. The gaplessphase is suppressed as decreasing t ⊥ , and it is compressedinto the black dashed line at t ⊥ = 0 (the single chain limit).Points I ∼ IV are used in Fig. 2. (b) W k y v.s. k y and µ are shown along the lines of L ∼ L in ( a ), respectively. Thewhite solid and dashed boundaries of the regions of W k y = ± , k y ) and( π, k y ), respectively. where r is the lattice site index; σ = ↑ , ↓ labels twospin states; t and t ⊥ are intra- and inter-chain nearestneighbor hoppings, respectively, and µ is the chemicalpotential. λ here is the SO coupling, which we choose tolie only in the x -direction. This uni-directional SO cou-pling is a natural setup in cold atom experiments usingatom-laser interaction. The external field part of the Hamiltonian is H ex = (cid:88) r ∆ r ( c † r ↑ c † r ↓ + h.c. ) − B ( c † r ↑ c r ↓ + h.c. ) , (2)The first term accounts for superconducting pairing,where ∆ r is the s -wave pairing on site r , and can beinduced either through proximity effect or intrinsically.For the proximity induced superconductivity, we take∆ r to be spatially uniform, which is a commonly usedapproximation. For intrinsic superconductivity, ∆ r willbe solved self-consistently. The second term arises froman external Zeeman field B , which can also be simulatedusing atom-laser coupling . Uniform pairing
Let us first consider a uniform pair-ing ∆ r = ∆ which can be chosen as real without loss ofgenerality. Under periodic boundary conditions in both x and y -directions, the Hamiltonian Eqs. 1 and 2 can bewritten in momentum space, H = H band + H ex = (cid:88) k ψ † k h k ψ k , (3)where ψ k = [ c k ↑ , c k ↓ , c †− k ↑ , c †− k ↓ ] t , and h k = T k τ + Λ k σ − Bσ τ + ∆ σ τ , (4)The two sets of Pauli matrices σ i and τ i ( i = 1 , ,
3) actin the spin and particle-hole spaces, respectively. T k andΛ k are given by T k = − t cos k x − t ⊥ cos k y − µ, (5)and Λ k = 2 λ sin k x . (6)The energy spectrum of Eq. 4 is E k = T k + Λ k + B + ∆ ± (cid:113) T k Λ k + T k B + B ∆ . (7)Although h k x ,k y does not carry 2D topological indices,nevertheless, we consider the 1D index of h k x ,k y at eachfixed value k y . It is invariant under both particle-hole(Ξ) and TR (Θ) symmetries: defineΞ = τ , Θ = σ τ , (8)then Ξ h k x ,k y Ξ − = − Θ h k x ,k y Θ − = − h ∗− k x ,k y . (9)Here both transformations satisfy Θ = Ξ = 1. Weshould emphasize here that Θ is not the physical timereversal, which should square to − k to − k . Θ and Ξ can be combinedinto a chiral symmetry defined as C = ΞΘ , (10)which gives C h k x ,k y C − = − h k x ,k y . (11)These symmetries put h k x ,k y at fixed k y in the BDI classas pointed out in Ref. [40], which is characterized by a k y -dependent 1D topological index denoted as W k y . Aunitary transformation is performed as U = e i ( π/ σ u e − i ( π/ τ , (12)where u = 12 ( σ + σ ) + 12 τ ( σ − σ ) . (13)It transforms h k into an off-diagonal form U − h k U = (cid:20) A k A † k (cid:21) , (14)where A k = ∆ σ − i ( T k σ + Λ k σ + Bσ ) . (15) W ( k y ) is defined as the winding number of det A k in thecomplex plane as k x sweeps a 2 π cycle, viz. , W k y = − i π π (cid:90) k x =0 d z ( k ) z ( k )= 12 (cid:2) sgn( M + ) − sgn( M − ) (cid:3) sgn( λ ∆) , (16)where z ( k ) = det A k / | det A k | , in whichdet A k = B − T k − (∆ − i Λ k ) , (17) M ± ( k y ) are related to det A k x ,k y as M + ( k y ) = det A k x =0 ,k y , (18) M − ( k y ) = det A k x = π,k y . (19) W k y = ± M + ( k y ) M − ( k y ) < h ( k y ) is topologically nontrivial. W k y changesdiscretely if a gap closing state appears on the line of k y such that M + ( k y ) = 0, or, M − ( k y ) = 0. The momenta ofthese states ( k x , k y ) satisfy that k x = 0 or π , and anothercondition T k + ∆ − B = 0 which determines k y .Based on W k y ’s behavior over the range of [ − π, π ), weplot the bulk phase diagram for the 2D Hamiltonian Eqs.1 and 2 in the parameter plane µ - B shown in Fig. 1( a ).The gapped phases are characterized by k y -independentvalues of W : two phases with W = ± W = 0 aretrivial pairing states. For the gapless phase, a momentumaveraged topological number is defined as r = (cid:90) d k y π W k y . (20)The values of W k y v.s. µ and k y are depicted in Fig. 1( b )along the line cuts L ∼ L in Fig. 1 (a). Usually, W k y only changes the value by 1 at one step as varying k y , −1 0 1−0.500.5 k y / π E (a) −1 0 1−0.500.5 k y / π E (b)−1 0 1−0.500.5 k y / π E (c) −1 0 1−0.500.5 k y / π E (d) FIG. 2. Edge spectra with the open and periodical boundaryconditions along the x and y -directions, respectively. ( a ), ( b ),( c ), and ( d ) correspond to points I, II, III, and IV marked inFig. 1 ( a ), respectively. ( a ) gapped trivial phase; ( b ) gap-less phase with edge modes associated with the same windingnumber; ( c ) gapped weak topological phase; ( d ) gapless phasewith edge modes associated with opposite winding numbers.Parameters used are the same as those in Fig. 1 ( a ). but along line L , W k y can directly change between 1and -1 without passing 0, which means two Dirac points(0 , k y ) and ( π, k y ) appear at the same value of k y . Notethat the SO coupling λ is related to W k y only throughits sign ( c.f. Eq. 16), therefore the phase diagram Fig. 1is independent of λ (up to an overall sign flip). Next we discuss edge spectra in the above differentphases. The open boundary condition is applied alongthe x -direction. In the topological trivial phase shown inFig. 2 ( a ), the zero energy edge modes are absent, whilethey appear and run across the entire 1D edge Brillouinzone in the gapped weak topological pairing phase shownin Fig. 2 ( b ). In the gapless phase, flat Majorana edgemodes appear in the regimes with W ( k y ) = ± These flat Ma-jorana edge modes are lower dimensional Majorana ana-logues of Fermi arcs in 3D Weyl semi-metals . Thisanalogy goes further as in both cases: the gapless phaseintervenes topologically distinct gapped phases.The flat edge Majorana modes in the gapless phasecan behave differently. In Fig. 2 ( b ), all the edge flatMajorana modes are associated with the same value of W k y . In this case, these Majorana modes on the sameedge are robust at the zero energy if TR symmetry ispreserved, which means that they do not couple. Nev-ertheless, TR symmetry may be spontaneously brokento gap out these zero modes . On the other hand, forstates inside the white dashed diamond in Fig. 1 ( a ),edge Majorana modes appear with both possibilities of W k y = ±
1. In particular, in the case of µ = 0, the re-lation W ( k y ) = − W ( k y + π ) holds for edge Majoranamodes as shown in Fig. 2 ( d ). Majorana modes withopposite winding numbers on the same edge can couple (b) J rr’ (a) ∆ r yx FIG. 3. Self-consistent solutions for ∆ r ( a ) and supercurrent J rr (cid:48) ( b ). Parameter values are L x = 120, L y = 8, B = 1 . t = 1, t ⊥ = 0 . µ = − λ = 2, g = 5. Open and periodicalboundary conditions are used along the x -direction (vertical)and y -direction (horizontal), respectively. Only the first 10sites from the upper edge are plotted. The distributions of∆ r and J rr (cid:48) are reflection symmetric with respect to the cen-ter line of the system. ( a ) Direction and length of each arrowrepresent the phase and amplitude of ∆ r on site r . Its dis-tribution is nearly uniform in the bulk but exhibits spatialvariations near the edge. ( b ) Each arrow represents J rr (cid:48) onbond rr (cid:48) , which is prominent near the edge but vanishes inthe bulk. to each other even without TR breaking, and thus arenot topologically stable. r represents the net density ofstates of zero modes in the edge Brillouin zone which arestable under TR-conserved perturbations. Self-consistent solution
We now impose self-consistency on the pairing order parameter ∆ r ,which is necessary for the case of intrinsic pairings. Thepairing interaction is modeled as H ∆ = − g (cid:88) r n r , ↑ n r , ↓ , (21)and the self-consistent equation is∆ r = − g (cid:104) G | c r ↓ c r ↑ | G (cid:105) , (22)where (cid:104) G | ... | G (cid:105) means the ground state average. We haveverified numerically that ∆ r is nearly uniform inside thebulk. Thus the bulk shares a similar phase diagram tothe case of uniform pairing ( cf. Fig. 1), except that thevalues of ∆ should be self-consistently determined.Nevertheless, near edges ∆ r varies spatially in the self-consistent solutions. If the bulk is in the topologicalgapped phase, the edge Majorana zero modes can couplewith each other by breaking TR symmetry spontaneouslyas shown in Ref. [26]. Because of the band flatness, thiseffect is non-perturbative. This will gap out the zeroMajorana modes and lower the edge energy. The systemconverges to an inhomogeneous distribution of arg[∆ r ]near the edges as shown in Fig. 3 ( a ), even if this costsenergy by disturbing the Cooper pairing . This edgeinhomogeneity in the pairing phase leads to an emergentcurrent pattern as depicted in Fig. 3 ( b ). B (a) J max ∆ bulk E B (b)-0.300.3 0.8 1 1.2 1.4 1.6 1.8 Ly = 7 FIG. 4. Self-consistent solutions for coupled chains with vary-ing B -field. Open and periodical boundary conditions areused along the x and y -directions, respectively. Parametersare L x = 120, L y = 8, t = 1, t ⊥ = 0 . µ = − λ = 2, and g = 5. In both ( a ) and ( b ), the bulk gapless phase is markedas the shaded region, which separates the topologically trivial(on its left) and nontrivial (on its right) gapped phases. ( a )The bulk pairing ∆ bulk and the characteristic edge currentmagnitude J max extracted as the maximal current in the sys-tem. ( b ) The energy spectra close to E = 0. The inset of ( b ) isfor the case of L y = 7. TR symmetry is spontaneously brokenbetween the two dashed red lines as evidenced by J max (cid:54) = 0.Please note that: At large values of B , the edge current van-ishes which is an artifact due to the finite length of L x . Thedecaying lengths of edge Majorana modes are at the order ofthe superconducting coherence length which is long due to thesuppression of the pairing gap. As a result, Majorana modeson opposite edges can hybridize and are gapped out withoutbreaking TR symmetry. A natural question is under what conditions TR sym-metry is spontaneously broken near edges. We havecarried out extensive numerical studies and results of µ = − a ) and ( b ). TR sym-metry is always broken in the topological gapped phasesuch that Majorana edge fermions are pushed to midgapenergies, while TR symmetry remains unbroken in thetrivial gapped phase. The latter is easy to understandbecause there are no Majorana fermions to begin with.If the bulk is in the gapless phase (shaded area in Fig. 4),the situation is more complicated. TR symmetry break-ing solutions are found in most part of the gapless phase.In this regime, | r | < L y . These modes are associated with the samevalue of W k y , and thus TR symmetry breaking is neededto gap out these edge modes. There exists a small re-gion inside the gapless phase in which TR symmetry isunbroken in Fig. 4 ( a ), which is largely due to the fi-nite value of L y . We have tested that as increasing L y the TR breaking regime is extended, and thus we expectthat it will cover the entire gapless phase in the thermo-dynamic limit. On the other hand, for the case of thegapless phase with µ = 0 in which r = 0 for even valuesof L y , our calculations show that all the Majorana modesare gapped out without developing currents. Instead abond-wave order appears at the wavevector of k y = π along the edge, which is consistent with the fact thatTR invariant perturbations can destroy Majorana zeroenergy modes at r = 0. In general, we expect that TRsymmetry is spontaneously broken in the case of r (cid:54) = 0in the thermodynamic limit.However, not all Majorana edge modes have to begapped out in the topological gapped phase. As shownin Fig. 4 ( b ), for the case of L y = 8, all the edge modesbecome gapped due to TR symmetry breaking, whereasfor L y = 7, one Majorana mode survives at zero energy. The reason is that breaking TR brings the system fromclass BDI to class D , and the latter is characterized bya Z index. Physically it is because (in the infinite chainlength limit) only the Majorana modes on the same edgecan be paired and gapped out, thus beginning with L y Majorana fermions per edge, for odd L y , one of them willalways remain unpaired. In short, if TR is spontaneously broken, only L y mod Discussion
Before closing, a few remarks are in or-der. (1) The phenomenon of spontaneous TR symme-try breaking in topological superconductors has previ-ously been found in a spinless p -wave superconductor inRef. 26. Our work extends this observation in three ways:(a) Our results confirm that spontaneous TR breakingalso occurs in a different setup with SO coupling and s-wave pairing, which is more relevant to experiments. (b)Our model hosts a gapless phase, wherein spontaneousTR breaking may also occur. (c) We also found a param-eter regime where Majorana modes with opposite wind-ing numbers can coexist. This provides another routeto gap out the Majorana modes without invoking TRbreaking. (2) In this work, we only considered SO cou-pling in the x direction, which can be exactly simulatedin cold atom systems. However, in solid state physics,both Rashba and Dresselhaus SO couplings will involveSO coupling along the y direction as well (unless Rashbaand Dresselhaus are of equal strength, in which case SOcoupling along y will vanish). This will break TR sym-metry (as defined in Eq. 8, which is not the usual physi-cal TR symmetry) and bring the system from class BDI to D. In the presence of a y -direction SO coupling term( ∼ sin( k y ) σ τ ), the Majorana flat bands will developdispersion, either connecting upper and lower bulk bandsor forming isolated mid-gap states which may cross zeroat k y = 0 or π , consistent with a Z description. (3)Disorders such as spatial variations of chemical potential( ∼ σ τ ) and Cooper pairing amplitude ( ∼ σ τ ) can beadded without changing any of our conclusions (providedthe disorder is not strong enough to close the bulk gap).This is because these two terms are invariant under bothparticle-hole(Ξ) and TR(Θ) symmetries, hence the sys-tem still belongs to the BDI class. (4) Finally, althoughwe modeled the constituent nanowires each as a 1D lat-tice, switching to a continuum formulation in the chaindirection should not affect the formation of edge Majo-rana modes (that is, before they couple and gap out). Thus we expect the edge physics obtained here to be in-sensitive to how the bulk of the chains is formulated interms of continuum vs. lattice.
Summary
We have studied quantum wire arrays withSO coupling and s -wave superconductivity in an exter-nal Zeeman field. The relation between edge Majoranazero modes and the bulk band structure is investigatedin both topologically nontrivial gapped phase and thegapless phase. The coupling between Majorana modesand superfluid phases leads to spontaneous TR symme-try breaking. Our results have several experimental bear-ings. For proximity effect induced superconductivity, thenumber of edge Majorana fermions in the gapless phasecan be tuned by the Zeeman field from zero all the wayup to the number of chains. This could be detected asa prominent change in the height of zero bias peaks intunneling spectroscopy experiments. For the intrinsic su-perconductivity, edge supercurrent loops resulting fromspontaneous TR breaking will induce small magnetic mo-ments, which can be detected using magnetically sensi-tive experiments such as nuclear magnetic resonance orneutron scattering. The fluctuation in the number ofpersisting Majorana mode between 1 and 0, in the TR-broken topological gapped phase, may also show up intunneling spectroscopy. Acknowledgments
We thank Yi Li for early collab-orations, Hui Hu and D. P. Arovas for helpful discus-sions, and D. P. Arovas for comments after readinga draft of this paper. DW and CW are supportedby the NSF DMR-1105945 and AFOSR FA9550-11-1-0067(YIP). ZSH is supported by NSF through grantDMR-1007028. CW acknowledges the support from theNSF of China under Grant No. 11328403.
Note added
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