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Manipulating Topological Edge Spins in One-Dimensional Optical Lattice Xiong-Jun Liu,1,2 Zheng-Xin Liu,3,4 and Meng Cheng2 1Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 3Institute for Advanced Study, Tsinghua University, Beijing, 100084, P. R. China 4Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: October 8, 2012) We propose to observe and manipulate topological edge spins in 1D optical lattice based on cur- 2 rentlyavailableexperimentalplatforms. Couplingtheatomicspinstatestoalaser-inducedperiodic 1 Zeemanfield, thelatticesystemcanbedrivenintoasymmetryprotectedtopological(SPT)phase, 0 which belongs to the chiral unitary (AIII) class protected by U(1)×ZT symmetry. In free-fermion 2 2 case the SPT phase is classified by a Z invariant which reduces to Z with interactions. The edge 4 t modes of the SPT phase are spin-polarized, with left and right edge spins polarized to opposite di- c rectionsandformingatopologicalspin-qubit(TSQ).Wedemonstrateanovelschemetomanipulate O thespin-polarizedzeromodesandrealizequantum-tunnelingdrivenspinRabi-oscillationsinacon- trollable manner. The manipulation of TSQs has potential applications to quantum computation. 5 PACSnumbers: 71.10Pm,37.10Jk,42.50.Ex,71.70.Ej ] l l a h Therefore, to observe nontrivial topological states with - Introduction.−SincethediscoveryofthequantumHall currentlyavailableexperimentalplatformsisparticularly s e effect in two-dimensional electron gas [1], the search for desired in the field of cold atoms [25]. m nontrivialtopologicalstateshasbecomeanexcitingpur- In this letter, we propose to observe and manipulate . suit in condensed matter physics [2]. The recently ob- topological edge spins in one-dimensional (1D) optical t a served time-reversal (TR) invariant topological insula- lattice with the SO interaction realizable in recent ex- m tors (TIs) have opened a new chapter in the study of periments [11–14]. The predicted SPT phase belongs to - topologicalphases(TPs),attractinggreateffortsinboth AIII class and is protected by U(1) and chiral symme- d theory and experiments [3, 4]. Depending on whether tries,withspin-polarizedzeromodeslocalizedonbound- n o the ground states have long-range or short-range en- aries. Our results not only provide a realistic platform c tanglement, the TPs can be classified into intrinsic or to study SPT states in cold atoms, but also have broad [ symmetry-protected topological (SPT) orders [5–7]. Un- range of applications including realizing single spin con- 2 like conventional quantum phases characterized by bro- trol in optical lattice, which is essential for spin-based v ken symmetries, TPs of quantum matter are classified quantum computation. 0 by topological orders whose emergence does not break Model.−Ourmodelisbasedonquasi-1Dcoldfermions 9 symmetry, and states of different topological orders can- trapped in an optical lattice, with the internal three- 9 not change into each other without closing the bulk gap. levelΛ-typeconfigurationcoupledtoradiation,asshown 2 . Being protected by the bulk gap, the intrinsic TPs are in Fig. 1. The transition |g↑(cid:105) → |e(cid:105) is induced by the 9 robust against any local perturbations, while the SPT laser field with Rabi-frequency Ω (x)=Ω sin(k x), and 0 1 0 0 phases are robust against those respecting given symme- the one |g (cid:105) → |e(cid:105) is driven by another laser Ω (x) = 2 ↓ 2 1 tries [5–8]. This property may be applied to the fault- Ω0cos(k0x). Inthepresenceofalargeone-photondetun- : tolerant quantum computation [9]. ing |∆|(cid:29)Ω and a small two-photon detuning |δ|(cid:28)Ω v 0 0 i WhilethesearchofTRinvariantTIshasbeensuccess- for the transitions (Fig. 1(a)), the Hamiltonian of the X ful in solid-state systems, the recent great advancement light-atom coupling system reads H =H0+H1, with r in realizing effective spin-orbit (SO) interaction in cold a atoms[10–15]opensintriguingnewpossibilitiestoprobe H = (cid:88) (cid:2) p2x +V (x)(cid:3)|g (cid:105)(cid:104)g |+(cid:126)δ|g (cid:105)(cid:104)g |, 0 2m σ σ σ ↓ ↓ SO effects [16] and TPs in a fully controllable fashion. σ=↑,↓ Many theoretical proposals have been introduced in cold H = (cid:126)∆|e(cid:105)(cid:104)e|−(cid:126)(cid:0)Ω |e(cid:105)(cid:104)g |+Ω |e(cid:105)(cid:104)g |+H.c.(cid:1). 1 1 ↑ 2 ↓ atoms for the study of TIs [17–21] and topological su- perfluids[22–25]. Experimentalobservationsoftheseex- HerethediagonalpotentialsV (x)areusedtoconstruct ↑,↓ otic phases are, however, delicate due to the demanding the 1D optical lattice, and σ are the Pauli matrices y,z conditions such as complicated lattice configurations or in spin space. For |∆| (cid:29) Ω , diagonalizing H yields 0 1 SO interactions. By far the only experimentally realized two nearly degenerate dressed states known as the dark SO interaction [11–15] is the equal Rashba-Dresselhaus- and bright states given by [10] |χ (cid:105) = cos(k x)|g (cid:105) − D 0 ↑ type SO term as theoretically proposed by Liu etal [10]. sin(k x)|g (cid:105)and|χ (cid:105)=sin(k x)|g (cid:105)+cos(k x)|g (cid:105). Pro- 0 ↓ B 0 ↑ 0 ↓ 2 cˆj↓ →eiπxj/acˆj↓, we recast the Hamiltonian into (cid:88) (cid:88) H = −t (cˆ† cˆ −cˆ† cˆ )+ Γ (nˆ −nˆ )+ s i↑ j↑ i↓ j↓ z i↑ i↓ <i,j> i +(cid:2)(cid:88)t(0)(cˆ† cˆ −cˆ† cˆ )+H.c.(cid:3). (2) so j↑ j+1↓ j↑ j−1↓ j WeanalyzethesymmetryoftheaboveHamiltonian. The TRandchargeconjugationoperatorsarerespectivelyde- fined by T = iKσ with K the complex conjugation, y FIG.1: (Coloronline)(a)Coldfermionstrappedin1Doptical and C : (cˆ ,cˆ†) (cid:55)−→ (σ ) (cˆ† ,cˆ ). One can check σ σ z σσ(cid:48) σ(cid:48) σ(cid:48) lattice with internal three-level Λ-type configuration coupled that while both T and C are broken in H, the chiral to radiation. (b) Energy spectra with open boundary condi- symmetry, defined as their product, is respected and tion in the topological (diamond, Γ =0) and trivial (circle, z (CT)H(CT)−1 = H, with (CT)2 = 1. Note the chiral Γ =3t ) phases. The SO coupled hopping t(0) =0.4t . z s so s symmetry is still preserved if a Zeeman term Γ σ along y y y axis is included in H. The complete symmetry group then reads U(1)×ZT, where U(1) gives particle-number jecting H onto the subspace of |χ (cid:105) gives rise to the 2 B,D conservation and the anti-unitary group ZT is formed equalRashba-Dresselhaus-typeSOinteractionwhichhas 2 by {I,CT}. The SPT phase of our free-fermion system beenpredictedintheory[10], andremarkablyrealizedin belongstothechiralunitary(AIII)classandischaracter- the recent experiments [11–15]. In this study, we shall izedbyaZ invariant[5–7]. TheH canberewritteninthe work in the basis of hyperfine states |g↑,↓(cid:105). The effect of k-space H = −(cid:80) cˆ† [d (k)σ + d (k)σ ] cˆ , thesmalltwo-photondetuningisequivalenttoaZeeman k,σσ(cid:48) k,σ z z y y σ,σ(cid:48) k,σ(cid:48) field along z axis Γz =(cid:126)δ/2, which in experiment can be with dy =2t(s0o)sin(ka) and dz =−Γz+2tscos(ka). This preciselycontrolledwithacoustic-opticmodulator. Elim- Hamiltonian describes a nontrivial topological insulator yiniealtdinsgthteheeffeexcctiitveedHsatmatieltobnyia|ne(cid:105) ≈ ∆1(Ω∗1|g↑(cid:105) + Ω∗2|g↓(cid:105)) fbourlk|Γgza|p<E2gts=anmdino{th|2etrsw−ise|Γaz|t|r,i2v|ita(s0ol)i|n}su(Flaitgo.r1, w(bit)h).thIne particular,whenΓ =0andt =|t(0)|,ourmodelgives y,z s so H = p2x + (cid:88) (cid:2)VLatt(x)+Γ σ (cid:3)|g (cid:105)(cid:104)g |− rise to a flat band with nontrivial topology. eff 2m σ z z σ σ Edge states.−The nontrivial topology can support de- σ=1,2 generateboundarymodes. Consideringhardwallbound- (cid:2) (cid:3) − M(x)|g↑(cid:105)(cid:104)g↓|+H.c. , (1) aries located at x = 0,L, respectively [27] and diag- (cid:80) onalizing H in position space H = H(x ) with where M(x) = M0sin(2k0x) with M0 = (cid:126)2Ω∆20 represents H(xi)=−(tsσz+it(s0o)σy)cˆ†xicˆxi+a+Γzσzcˆ†xxiicˆxi+hi.c.,we an additional laser-induced Zeeman field. To form a 1D obtaintheedgestatelocalizedonleftboundaryx=0as lattice,weconsidertheopticaldipoletrappingpotentials 1 V↑Latt(x) = V↓Latt(x) = −V0cos2(2k0x) with the lattice ψL(xi)= √N[(λ+)xi/a−(λ−)xi/a]|χ+(cid:105), (3) trapping frequency ω = (8V k2/m)1/2 [26]. This effec- 0 0 and accordingly the one on x = L by ψ (x ) = tive model describes a system of spin-1/2 fermions expe- R i riencing a periodic transverse Zeeman field M(x) which √1 [(λ+)(L−xi)/a−(λ−)(L−xi)/a]|χ−(cid:105). HereN isthenor- N is commensurate with the 1D optical lattice [Fig. 1(a)]. malization factor, |χ (cid:105) = [1,±1]T are spin eigensates of ± (cid:113) Wenextderivethetight-bindingmodelbytakingthat σ , and λ =(Γ ± Γ2−4t2+4|t(0)|2)/(2t +2|t(0)|). x ± z z s so s so the fermions occupy the lowest s-orbitals φsσ (σ =↑,↓). Therefore the two edge states are polarized to the oppo- Fromtheevenparityofφsσ,onecanverifythattheperi- site ±x directions. Note the edge modes ψL,R span the odictermM(x)doesnotcoupletheintrasiteorbitalsφs↑ complete Hilbert space of one single 1/2-spin or spin- ańd φs↓, but leads to a spin-flip hopping given by tisjo = qubit. Each edge state equals one-half of a single spin, dxφ(i)(x)M(x)φ(j)(x), asaconsequenceoftheinduced similar to the relation between a Majorana fermion and s↑ s↓ SO interaction. The spin-conserved hopping is given by a complex fermion in topological superconductors. As a ´ ts = dxφ(sjσ)(x)(cid:2)2pm2x +V(cid:3)φ(sjσ+1)(x). Bearing these re- result,weexpecttherobustnessofthezeromodestoany sults in mind we write down the effective Hamiltonian localoperationswithoutbreakingtheU(1)andCT sym- in the tight-binding form: H = −t (cid:80) cˆ† cˆ + metries. These properties of the topological spin qubit s <i,j>,σ iσ jσ (cid:80) Γ (nˆ −nˆ )+(cid:2)(cid:80) tijcˆ† cˆ +H.c.], with nˆ = (TSQ) may be applicable to the fault-tolerant quantum i z i↑ i↓ <i,j> so i↑ j↓ iσ computation [9]. Moreover, the Z classification implies cˆ† cˆ . It can be verified that tj,j±1 = ±(−1)jt(0), iσ iσ ´ so so that single-particle couplings respecting U(1) and CT where t(s0o) = Ω∆20 dxφs(x)sin(2k0x)φs(x − a) with a cannotgapouttheedgemodesinarbitraryN-chainsys- the lattice constant. Redefining the spin-down operator temof1Dlattices,while,interestingly,wehaveconfirmed 3 that weak interactions can break Z down to Z classifi- with a charge gap. The spin sector remains at the fixed 4 cation [28]. This result suggests an interesting platform point with K∗ = 1, described by SU(2) Wess-Zumino- σ 1 to study classification of SPT phases with interactions. Witten model [31]. When u,w (cid:54)= 0, such symmetry is reduced to the U(1) subgroup. For the large-U limit, the charge boson φ is pinned √ ρ to its classical minima cos2 2φ = 1, and the system ρ is a Mott insulator, regardless of the mass terms. In the weak-couplingregime,theinteractioneffectsonthephase boundary can be studied with RG method up to one- loop order [2]. The RG flow equations for the coupling constants u,v,g ,g are given by [28]: ρ σ FIG. 2: (Color online) (a) Wave functions for zero modes du 3−K g u g u |ψ (cid:105); (b) 1/2-particle fractionalization (seen by ∆N ) for = ρu− ρ + σ , L,R 1,2 dl 2 4πv 4πv zeromodes. Theparameterst(0) =0.4t andΓ =0.3t ,with F F so s z s dw 3−K g w g w which the localization length of bound modes ξ0=2.36a. = ρw+ ρ + σ , (4) dl 2 4πv 4πv F F Existenceofzeromodesleadstoparticlefractionaliza- dgρ = gρ2 , dgσ = gσ2 . tion, which is another direct observable in experiment. dl πv dl πv F F A zero mode is contributed half from the valence band Herelisthelogarithmofthelengthscale. Therenormal- and half from the conduction band. Therefore, an edge izationofLuttingerparametersK hasbeenneglectedas ν statecarries+1/2(−1/2)particleifitisoccupied(unoc- it is a higher order correction. For U >0, g marginally σ cupied) [28]. This result is confirmed by numerical sim- flows to zero so we drop it off below. g is marginally ρ ú0lxaitdioxn(cid:48)Nsh1,o2w(xn(cid:48))in−Fxigi/.a2,awthhearlef-wfilelincagl,cuwliatthe ∆N1N(21),2(x(x)i)th=e sretilteuvtainntgatnhdiscraensublteinsotolvRedGbeyqugaρ(tlio)n=s oπfπvuFvF−aggnρρ(d(00))wl.ySieuldbs- density of fermions when the left (right) edge mode is filled. The fermion number carried by an occupied (un- afterintegrationu(l)=u(0)[1−gπρv(0F)l]41e(3−Kρ)l/2,w(l)= wocicthupξied(cid:29)) eξdg.eHmeoredeξis=the−nag/ilvne|nλby| ins1(t2h)e=lo∆caNli1z(a2t)i(oξn) wpu(l0s)i[v1e+intgπeρv(r0Fa)clt]i41oen(3(−gKρ>)l/02). sTuhpeprpehsysessicSsOisicnldeaurc:edthmearses- 0 0 + ρ length of ψ . The 1/2-fractionalization is clearly seen term u while enhances the trivial mass term w. The fate L,R whenξisseveraltimesgreaterthanξ (Fig.2(b)). Being of the system depends on which of u and w reaches the 0 a topological invariant, the 1/2-fractionalization can be strong-couplingregimefirst. Assuming|g (0)l|(cid:28)v ,we ρ F confirmed to be robust against weak disorder scatterings find the TP transition occurs at without breaking the given symmetries. Correlation effects.−A particular advantage in cold u(0)=(cid:2)w(0)(cid:3)γ, γ ≈1− gρ(0) . (5) 4πv (3−K ) atomsisthatonecaninvestigatecorrelationeffectsonthe F ρ predictedSPTphasebypreciselycontrollingtheinterac- This gives the scaling law at the phase boundary with tion. For spin-1/2 cold fermions the onsite Hubbard in- interaction. Note γ <1 for U >0. The above scaling re- (cid:80) teractionU n n canbewellcontrolledbyFeshbach lation implies that a repulsive interaction suppresses the i i↑ i↓ resonance [26]. We adopt here the Abelian bosonization SPT phase. Accordingly, if initially the noninteracting approach combined with renormalization group (RG) system is topologically nontrivial with u(0) > w(0) > 0, analysis to probe interaction effects in single-chain sys- increasing U to the regime u(0)<[w(0)]γ drives the system at critical point. With low energy approximation tem into a trivial phase. we linearize the fermion dispersion near Fermi points Single spin control.−Now we study an interesting ap- kF = 2πa by cˆs(x)≈(cid:80)r,s=±1ψrseirkFx, where r/s is the plication of the present results to realizing single spin chiral/spin index. Using the standard bosonization for- control. Besides the edge modes localized on the ends, mula [29]: ψrs = √1 e−√i2[rφρ−θρ+s(rφσ−θσ)], we reach TSQs can also be obtained in the middle areas by cre- 2πa (cid:80) ating mass domains in the lattice. This can be achieved thebosonizedHamiltoniandensityH= H +H ν=ρ,σ ν√ 1 by applying a local Zeeman term Γ or Γ . For exam- withHν = v2ν[Kν√(∂xθν)2+K√ν−1(∂xφν)2]+(2√2πgaν)2 cos2√2φν ple, we consider Γz = 0 everywherey, butzΓy = Γ0 for and H1 = πuasin 2φρcos 2θσ + πwacos 2φρsin 2θσ. x1 < x < x2 and Γy = 0 otherwise. The local Γy can Here gρ,σ arise from umklapp scattering and spin be generated by applying another two lasers which cross backscattering, respectively. The bare values are vν = with the 1D lattice and couple the atoms in the area v K ,g =−g =U,u=2t(0),w =Γ in units of t . At x < x < x to induce a local resonant Raman coupling F ν ρ σ so y s 1 2 half-fillingHrespectsO(4)∼=SU(2)×SU(2)symmetryif between|g (cid:105)and|g (cid:105)(Fig.3(a)). Employingaπ/2-phase ↑ ↓ u=w =0 [30]. For U >0, the umklapp scattering term offset in the Rabi-frequencies of the two lasers, the Ra- √ cos2 2φ drivesthesystemintoaMottinsulatingphase man coupling takes the form Γ σ , with Γ controlled ρ 0 y 0 4 by the laser strength. When |Γ | > 2|t(0)| a mass do- 0 so main is created, associated with two midgap spin states |ψ (cid:105) respectively localized around x = x (Fig. 3(a)). ± 1,2 The width ∆x = x −x and height of the domain are 2 1 respectively adjusted by the waist size and strength of the two laser beams. Due to the nonlocality of the TSQ, creation of a single qubit here is not restricted by the size of the laser beams. This is a fundamental differ- ence from creating conventional single qubit by optical dipole trapping which requires tiny-sized laser beams to reachaverysmalltrappingvolume[33]. Noteinrealistic case the laser induced Γ may vary fast but not in the y form of step functions around x = x , which, however, 1,2 does not affect the main results presented here. Cou- pling between |ψ (cid:105) and |ψ (cid:105) results in an energy split- + − ting 2E ∝ e−(|Γ0|−2|t(s0o)|)∆x/(2ats), which is controlled by Γ0 and ∆x (Fig. 3(a), lower panel). In the limit (|Γ0|− FIG.3: (Coloronline)SpinRabi-oscillationswiththeparam- 2|t(s0o)|)∆x/(2ats) (cid:29) 1, such coupling is negligible, and eters ts = 3.15kHz, t(s0o) = 0.4ts, and ∆x = 10a ∼ 4µm. (a) thetwozeromodesconsistofasinglespinqubitwhichis Mass domain created by setting |Γ |>2t(0) for x <x<x 0 so 1 2 topologically stable. Let |ψ (cid:105) be initially occupied while which localizes a spin-qubit composed of two bound modes + |ψ−(cid:105) be left vacancy. Reducing |Γ0| smoothly can open |ψ±(cid:105) on x=x1,x2, respectively; (b) Spin Rabi-oscillation by the coupling in |ψ±(cid:105) and lead to spin state evolving as smoothlyreducing|Γ0|from8t(s0o)to2.5t(s0o);(c)Themasscur- [34] |ψ(t)(cid:105)=α(t)ϕ (x−x )|χ (cid:105)+β(t)ϕ (x−x )|χ (cid:105), rentJm(t)andexpectationvaluesofparticlenumbers(cid:104)n±(t)(cid:105) − 1 − + 2 + in states |ψ (cid:105); (d) Spin-flip operation by controlling that with α(0)=0,β(0)=1, and ϕ the spatial parts of the ± ± γ(t ) = (2m+1)π with m = 1. The initial spin state |χ (cid:105) bound state wave-functions. The spin-polarization den- 3 + (Points A) flips to be |χ (cid:105) (Points B). − sities are given by sx,y,z(x,t) = (cid:104)ψ(x)|σx´,y,z|ψ(x)(cid:105), and the spin expectation values S (t) = dxs (x,t). x,y,z x,y,z It can be verified that S (t)=0, and y 5.984ms(b-c). Notethequantumstateofthespincanbe S (t)=|α(t)|2−|β(t)|2, precisely controlled by properly manipulating γ(t). For x ˆ example, in Fig. 3 (d) we demonstrate the spin-flip op- (6) S (t)=2Re(cid:2)α(t)β∗(t) dxϕ∗(x)ϕ (x)(cid:3). eration |χ (cid:105) → |χ (cid:105) by requiring γ(t ) = (2m + 1)π. z + − + − 3 Here m ∈ Z and in (d) we take m = 1. Note one may Thisphenomenonisanalogousto´spinprecessionwiththe integrate multiple TSQs with e.g. atom-chip technology rotatingangleyieldingγ(t)=2 tdt(cid:48)E(t(cid:48)). Wehavethen and individually control them by creating multiple mass 0 α = cosγ(t) and β =´sinγ(t). The amplitude of Sz(t) domains in the 1D lattice. The precise manipulation of is given by Smax = | dxϕ∗(x)ϕ (x)|, which measures such integrated TSQs may have interesting applications z + − the overlapping integral of ϕ . Accordingly, if we ap- in developing scalable spin-based quantum computers. ± ply the local Zeeman field along z rather than y axis, we Before conclusion we estimate the parameter values shall obtain that the spin evolves in the x-y plane. Note for realistic experimental observations. For example, in the spin Rabi-oscillation is induced by quantum tunnel- 40K atoms we have the recoil energy E /(cid:126)=(cid:126)k2/2m= R ing. Therefore it is associated with a tunneling current 48kHz using red-detuned lasers of wavelength 780nm to given by J (t) = −∆xE∂ |α(t)|2 between x and x . In form the optical lattice. Taking that V = 5E and m 2π(cid:126) t 1 2 0 R experimenttheinternalstatesofasingleatomcanbede- M = 2E , we have that the lattice trapping frequency 0 R tected without energy transfer [35], which is applicable ω = 214kHz, and hopping coefficients t /(cid:126) (cid:39) 3.15kHz s toobservethespinRabi-oscillations,whiletheoscillation and t(0)/(cid:126) (cid:39) 1.3kHz. Then the bulk gap equals E /(cid:126) = so g of Sx(t) can be more conveniently observed by measur- 2.6kHz for Γz = 0, indicating a temperature T = 19nK ingthenumberoffermions(cid:104)n±(t)(cid:105)localizedonx1,2,and for the experimental observation. J (t)canbedetectedbymeasuringthechangeratewith m Conclusions.−In conclusion, we have proposed to ob- time of such fermion numbers. serve and manipulate the topological edge spins in 1D We show in Fig. 3 (b-d) the numerical simulation opticallatticewithcurrentlyavailableexperimentalplat- for single spin control with the parameter regime that forms. In the free-fermion case the SPT phase belongs ts = 3.15kHz, t(s0o) = 0.4ts, and ∆x = 10a. For t < 0, to the AIII class characterized by a Z invariant which Γ0 = 8t(s0o) and the coupling in |ψ±(cid:105) is negligible. Re- breaks down to Z4 with interactions. The spin-polarized ducing Γ0 at t > 0 leads to spin evolution and by fixing zeroedgemodesinthenontrivialSPTphaseformTSQs. Γ =2.5t(0) fort>t thespinoscillateswithaperiodof Our results are of broad range of interests and applica- 0 so 1 5 tions including the manipulation of TSQs which may be Martin-Delgado, M. Lewenstein, I. B. Spielman, Phys. applicable in developing spin-based quantum computers. Rev. Lett. 105, 255302 (2010); N. Goldman, J. Beugnon, We thank X. G. Wen, X. Chen, Vincent W. Liu, and and F. Gerbier, ibid 108, 255303 (2012). [20] X. Li, E. Zhao, and W. V. Liu, arXiv:1205.0254 (2012). Cenke Xu for helpful discussions. We acknowledge the [21] G. Liu, S. -L. Zhu, S. Jiang, F. Sun, and W. M. 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The full symmetry of y y z z y y so z z s the system is U(1)×(CT), where the phase transformation operator U(1), charge conjugation operator C and time reversal operator T are defined as: U(θ)cÛ(θ)−1 =eiθcˆ, CcˆC−1 =σ cˆ†; z TcˆT−1 =−iσ Kcˆ, (CT)cˆ(CT)−1 =σ cˆ†. y x The following commutation relations can be checked: C2 = −T2 = (CT)2 = 1,{C,T} = 0, and [CT,U(1)] = 0. The subgroup {I,CT} is anti-unitary and can be denoted as ZT, and the complete symmetry group can also be written 2 as U(1)×ZT. Owning to this symmetry group, the free fermion system belongs to chiral unitary (AIII) class and is 2 characterized by a Z invariant in the noninteracting case. If the Zeeman field along y axis vanishes, i.e. Γ = 0, alternatively the symmetry group can be chosen as U(1)(cid:111) y ×(C ×T) where U(θ)cÛ(θ)−1 = eiθcˆ, CcˆC−1 = σ cˆ†, TcˆT−1 = Kcˆ, with C2 = T2 = (CT)2 = 1, and [C,T] = x 1,TU(θ)=U(−θ)T,CU(θ)=U(−θ)C. In this case both T and C are symmetries of the Hamiltonian and the system then belongs to the BDI class which is also classified by a Z invariant in the non-interacting case. 2. Interacting regime ForaN-chainsystemunderthehalf-fillingcondition,thetotalground-statedegeneracyisCN withoutinteractions. 2N Let cˆ and cˆ† be the annihilation and creation operators of the left/right edge mode for the m-th chain, m,L/R m,L/R √ √ respectively, where cˆ = (cˆ +cˆ )/ 2 and cˆ = (cˆ −cˆ )/ 2. The edge states of the m-th chain can be written as L ↑ ↓ R ↑ ↓ cˆ† |0(cid:105) and c† |0(cid:105) , where |0(cid:105) is the ground state for the bulk. m,L m m,R m m Aninterestingquestioniswhathappensifweturnoninteractions. Itturnsoutthatinthepresenceofinteractions, the Z classification breaks down to Z . To confirm this result, we will study the ground states of a N-chain system 4 step by step. First, for N =2, a generic U(1)×(CT)-symmetric interaction between the edge zero modes reads H = (cid:88) (cid:104)V(4)cˆ† cˆ cˆ† cˆ +(V(4))∗cˆ cˆ† cˆ cˆ† (cid:105) 12 12 1,s 1,s 2,s 2,s 12 1,s 1,s 2,s 2,s s=L,R = (cid:88)V(4)(2cˆ† cˆ cˆ† cˆ −cˆ† cˆ −cˆ† cˆ +1). (A2) 12 1,s 1,s 2,s 2,s 1,s 1,s 2,s 2,s s With above interaction the minimum degeneracy of the 2-chain system is two-fold, which is obtained when V(4) = 12 (V(4))∗ = −V < 0. It is straightforward to check that the states |01(cid:105) |01(cid:105) and |10(cid:105) |10(cid:105) have energy −2V while 12 0 1 2 1 2 0 the states |10(cid:105) |01(cid:105) , |01(cid:105) |10(cid:105) ,|00(cid:105) |11(cid:105) , and |11(cid:105) |00(cid:105) have a higher energy 0. Then the ground state is two-fold 1 2 1 2 1 2 1 2 degenerate, as given by |01(cid:105) |01(cid:105) and |10(cid:105) |10(cid:105) . 1 2 1 2 7 Second, for N = 3, the ground state degeneracy C3 = 20 can be reduced to 2 by two-chain interactions according 6 to (A2): H =H +H . For instance, if V(4) =V(4) =−V <0, the two-fold degenerate ground states are int 12 23 12 23 0 |01(cid:105) |01(cid:105) |01(cid:105) , |10(cid:105) |10(cid:105) |10(cid:105) ; (A3) 1 2 3 1 2 3 and if V =−V =−V <0, the ground states read 12 23 0 |01(cid:105) |01(cid:105) |10(cid:105) , |10(cid:105) |10(cid:105) |01(cid:105) . (A4) 1 2 3 1 2 3 However, for the 3-chain system, one should also consider three-body interactions. One of the possible 3-chain interactions reads H =(cid:88)(V(6)cˆ† cˆ cˆ† cˆ cˆ† cˆ +V(6)cˆ cˆ† cˆ cˆ† cˆ cˆ† ), (A5) 123 123 1,s 1,s 2,s 2,s 3,s 3,s 123 1,s 1,s 2,s 2,s 3,s 3,s s where V(6) = (V(6))∗. It is straightforward to see that above interaction is identical to a summation of two-body 123 123 interactions and can not split the degeneracy of the ground states. Other possible 3-chain interactions include H(cid:48) = (cid:88)V(4)(cid:104)(cid:16)c† c† c c +c c c† c† (cid:17)+h.c.(cid:105) (A6) 123 123 1,s 2,s 2,s 3,s 1,s 2,s 2,s 3,s s = V(4)(2c† c −1)(c† c +c† c ), (A7) 123 2,s 2,s 1,s 3,s 3,s 1,s It can be easily checked that in the case (A3) the perturbation H(cid:48) have zero matrix elements in the ground-state 123 subspaceandhencecannotspiltthedegeneracy. Theconditionissimilarinanothercase(A4). Thedifferenceisthat, in case (A4), H(cid:48) mixes the ground states with higher energy states. For instance, the state |01(cid:105) |01(cid:105) |10(cid:105) is mixed 123 1 2 3 with |11(cid:105) |01(cid:105) |00(cid:105) to lower its energy. At the same time the state |10(cid:105) |10(cid:105) |01(cid:105) is mixed with |00(cid:105) |10(cid:105) |11(cid:105) to 1 2 3 1 2 3 1 2 3 lowertheenergywiththesameamount. Asaresult,thetwonewstates,asthenewgroundstates,arestilldegenerate. Therefore, thetopologicalpropertiesof3chainsarestableagainstinteractionsrespectingthesymmetry, andweneed to investigate the 4-chain system. Finally,forN =4,itturnsoutthatwecanfindapathtosmoothlyreducethegroundstatedegeneracyto1. Itiseasy toverifythatdegeneracyofthegroundstatescanbereducedto4undertwo-bodyinteractionH =H +H +H int 12 23 34 with V =−V =V =−V <0. These four ground states are given by 12 23 34 0 |01(cid:105) |01(cid:105) |10(cid:105) |10(cid:105) , |10(cid:105) |10(cid:105) |01(cid:105) |01(cid:105) , 1 2 3 4 1 2 3 4 |11(cid:105) |11(cid:105) |00(cid:105) |00(cid:105) , |00(cid:105) |00(cid:105) |11(cid:105) |11(cid:105) , (A8) 1 2 3 4 1 2 3 4 with the energy equal to −4V . However, the above ground states can be further gapped out by taking into account 0 the following interactions: H =(cid:88)(V(4) c† c† c c +V(4) c c c† c† ). (A9) 1234 1234 1,s 2,s 3,s 4,s 1234 1,s 2,s 3,s 4,s s In the 4-dimensional Hilbert space spanned by the ground states in Eq. (A8), the above interaction can be written in the matrix form   0 0 1 1 H1234 =V1(243)401 01 10 10, (A10) 1 1 0 0 whose eigenvalues are −2V(4) ,0,0, and 2V(4) . We therefore obtain the single non-degenerate ground state with the 1234 1234 energy−2|V(4) |−4V . Thisimpliesthatunderinteractionthe4-chainsystemcanbesmoothlyconnectedtoatrivial 1234 0 phasewithoutclosingthebulkgap, andwethereforecompletetheproofthattheZ classificationcanbebrokendown to Z under interactions. 4 8 Appendix B: Particle fractionalization We prove in this section that each edge state leads to 1/2-fractionalization. A convenient way is to consider the semi-infinite geometry which has the open boundary at x = 0. We then calculate the particle number of the zero mode localized on this boundary. Note the total number of quantum states in the system is given by (cid:88) (cid:88) N = (cid:104)ψ |nˆ |ψ (cid:105)+ (cid:104)ψ |nˆ |ψ (cid:105)+(cid:104)ψ |nˆ |ψ (cid:105), (B1) E E E E E E 0 E 0 E<0 E>0 wherewedenotebynˆ =Ithestatenumberoperatorand|ψ (cid:105)istheeigenstatewithenergyE. SincetheHamiltonian E E (cid:80) (cid:80) satisfies {H,σ } = 0, the energy spectrum is symmetric. We have then (cid:104)ψ |nˆ |ψ (cid:105) = (cid:104)ψ |nˆ |ψ (cid:105). It x E<0 E E E E>0 E E E follows that (cid:88) 1 (cid:104)ψ |nˆ |ψ (cid:105)= [N −(cid:104)ψ |nˆ |ψ (cid:105)]. (B2) E E E 2 0 E 0 E<0 Theparticlenumberofthezeromodedependsonitsoccupation. Ifthezeromodeisunoccupied,theparticlenumber of it is given by (cid:88)(cid:2) (cid:3) n = (cid:104)ψ |nˆ |ψ (cid:105) −(cid:104)ψ |nˆ |ψ (cid:105) . (B3) 0 E E E 1 E E E 0 E<0 Here (cid:104)(cid:105) and (cid:104)(cid:105) represents the cases with one (topological phase) and zero (trivial phase) bound modes, respectively. 1 0 Using the Eq. (B2) one finds directly 1 1 n =− (cid:104)ψ |nˆ |ψ (cid:105)=− . (B4) 0 2 0 E 0 2 Similarly, if the zero mode is occupied, the particle number is n = 1/2. It is trivial to know that this result can 0 be applied to the case with two boundaries located far away from each other, say respectively at x = 0 and x = L. Since the two zero modes are obtained independently, each of them carries +1/2 (−1/2) particle if it is occupied (unoccupied). Appendix C: Derivation of the RG equations In this section we provide the derivation of the one-loop RG equations. We consider that in Hamiltonian (A1) only one Zeeman term, e.g. Γ is nonzero (The case with Γ (cid:54)=0 can be studied in the similar way). We find it convenient y z toredefinethatcˆj↓ →eiπxj/acˆj↓ andrewritetheHamiltonian(A1)intheform: H =−ts(cid:80)<i,j>,σcˆ†iσcˆjσ+HSO+HZ, whereH =t(0)(cid:80) (−1)j(c† c −c† c +h.c.)andH =Γ (cid:80) (−1)j(ic† c −h.c.). Thelow-energyphysics SO so j j↑ j+1,↓ j↑ j−1,↓ Z y j j↓ j↑ can be well captured by the continuum approximation (x=ja): √ c ≈ a[ψ (x)eikFx+ψ (x)e−ikFx], (C1) jσ Rσ Lσ with k =π/2. The continuum representation of the two mass terms are given by: F ˆ ˆ H ≈iu dx(ψ†σxψ −ψ†σxψ ), H =w dx(ψ†σyψ +ψ†σyψ ). (C2) SO L R R L Z R L L R Here u = 2tso,w = Γy. Using the standard bosonization formula ψrs = √1 e−√i2[rφρ−θρ+s(rφσ−θσ)] with r = R,L 2πa and s=↑,↓, we reach that u √ √ w √ √ H = sin 2φ cos 2θ , H = cos 2φ sin 2θ . (C3) SO πa ρ σ Z πa ρ σ The allowed four-fermion interactions in the (unperturbed) Hubbard model at half-filling is highly constrained by the SU(2)×SU(2) symmetry. It is convinient to define current operators: 1 J =ψ† ψ ,J(cid:126) = ψ† ψ ,I =ψ (cid:15) ψ . (C4) r rα rα r 2 rααβ rβ r rα αβ rβ 9 Here (cid:15) is the fully anti-symmetric tensor. The general form of four-fermion interaction is given by g H =2g J(cid:126) ·J(cid:126) + ρ(I†I +h.c.). (C5) int σ R L 4 R L We then derive the RG flow equations to understand the fate of the topological phase transition driven by the competition of the staggered Zeeman term and the spin-orbit coupling term. The tree level term can be easily read off from the scaling dimensions of H and H in their bosonization form (C3), both of which are (1+K )/2. Thus SO Z ρ it is necessary to go to the next order in the perturbative expansion. The one-loop order RG flow can be most easily derived by calculating the operator algebra of the various operators [1, 2]. To derive the RG equation, we consider the partition function in the Euclidean functional integral representation ˆ Z = DψDψe−S. (C6) ´ TheEuclideanactionS = dxdτ(H +H )whereH istheunperturbedGaussianpartandtheH =H +H isthe 0 1 0 1 SO Z (cid:80) perturbation. To perform the RG, we expand the exponential to the second order in H . Let us write H = g O . 1 1 i i Then the second order term is given by ˆ ˆ 1g g (cid:88) (cid:104)O (z)O (w)(cid:105)(cid:39) 1g g (cid:88) cijk O . (C7) 2 i j i j 2 i j 4π2|z−w|2 k i,j z,w i,j z,w We have introduced complex coordinates z,w where z = vτ −ix. Here we assume the following operator product expansion (OPE) c O (z)O (0)∼ ijk O +regular terms, (C8) i j 4π2|z|2 k which is sufficient for our purpose. The OPEs are valid when two points z and 0 are brought close together, as replacement within correlation functions. At this point the cutoff prescription needs to be carefully specified. We will choose a short-distance cutoff a in space, but none in imaginary time. For a rescaling factor b, we must then perform the integral ˆ ˆ ∞ 1 2π I = dx dτ = lnb. (C9) v2τ2+x2 v a<|x|<ba −∞ Eq. (C7) then becomes ˆ c ijkg g lnb O . (C10) 4πv i j k Upon re-exponentiating we obtain the one-loop RG equation dgk =− 1 (cid:88)c g g . (C11) dl 4πv ijk i j ij Here l=lnb. Now let us be more specifit. Besides the current operators defined in (C), we also need to define M =ψ†σiψ , i=x,y. (C12) i R L The fermionic field operators satisfy the following OPEs: δ δ ψ (z)ψ† (0)∼ αβ, ψ (z)ψ† (0)∼ αβ . (C13) Rα Rβ 2πz Lα Lβ 2πz∗ The OPEs of the currents and the mass M can be calculated from (C13) using Wick’s theorem. Those between the i currents are standard and can be found in [2] which we do not duplicate. Below are the needed ones: 1 1 M (z)J(cid:126) ·J(cid:126) (0)∼ M (z), M†(z)J(cid:126) ·J(cid:126) (0)∼ M†(z) i R L 16π2|z|2 i i R L 16π2|z|2 i (C14) 2 2 M (z)[I I† +h.c.](0)∼− M†(z), M†(z)[I I† +h.c.](0)∼− M (z). i R L 4π2|z|2 i i R L 4π2|z|2 i 10 Applying the above formalism to the model at hand, we find du 3−K g u g u = ρu− ρ + σ , dl 2 4πv 4πv F F dw 3−K g u g u = ρw+ ρ + σ , (C15) dl 2 4πv 4πv F F dg 1 dg 1 σ = g2, ρ = g2. dl πv σ dl πv ρ F F (cid:80) If we take the interaction in the lattice model to be an on-site Hubbard form: H =U n n , the bare values of U i i↑ i↓ the coupling coefficients are g =U,g =−U. (C16) ρ σ Then for the repulsive interaction U > 0, the coupling parameter g of the spin sector is marginally flows to zero, σ while g is marginally relevant. ρ [1] J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, 1996) [2] L. Balents and M. P. A. Fisher, Phys. Rev. B 53, 12133 (1996).