22 Pages·2012·0.66 MB·English

PHYSICAL REVIEW X 2, 041002 (2012) Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of AbelianQuantum Hall States Netanel H. Lindner Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125,USA Department of Physics, California Institute of Technology, Pasadena, California 91125,USA Erez Berg Department of Physics, Harvard University, Cambridge, Massachusetts 02138,USA GilRefael Department of Physics, California Institute of Technology, Pasadena, California 91125,USA Ady Stern Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100,Israel (Received 1 May 2012; published 11 October 2012) Westudythenon-Abelianstatisticscharacterizingsystemswherecounterpropagatinggaplessmodeson the edges of fractional quantum Hall states are gapped by proximity coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electronsofonespindirectionoccupyafractionalquantumHallstateof(cid:1)¼1=m,whileelectronsofthe oppositespinoccupyasimilarstatewith(cid:1)¼(cid:1)1=m.However,wealsoproposeotherexamplesofsuch systems,whichareeasiertorealizeexperimentally.Weﬁndthateachinterfacebetweenaregiononthe edge coupled to a supercondupctﬃoﬃﬃﬃrﬃﬃﬃand a region coupled to a ferromagnet corresponds to a non-Abelian anyonofquantumdimension 2m.Wecalculatetheunitarytransformationsthatareassociatedwiththe braiding of these anyons, and we show that they are able to realize a richer set of non-Abelian representations of the braid group than the set realized by non-Abelian anyons based on Majorana fermions.Wecarryoutthiscalculationbothexplicitlyandbyapplyinggeneralconsiderations.Finally,we show that topological manipulations with these anyonscannot realize universal quantum computation. DOI: 10.1103/PhysRevX.2.041002 Subject Areas: Condensed Matter Physics, Superconductivity, Topological Insulators considerotherquantumHallstates[8,9],spinsystems[10], I. INTRODUCTION p-wave superconductors [11–13], topological insulators Recent years have witnessed an extensive search for coupled in proximity to superconductors [14,15], and electronicsystemsinwhichexcitations(‘‘quasiparticles’’) hybrid systems of superconductors coupled to semicon- follow non-Abelian quantum statistics. In such systems, ductors where spin-orbit coupling is strong [16–22]. the presence of quasiparticles, also known as ‘‘non- Signatures of Majorana zero modes may have been Abeliananyons’’[1–5],makesthegroundstatedegenerate. observed inrecent experiments [23–27]. A mutual adiabatic interchange of the positions of the In the realizations based on superconductors, whether quasiparticles [6] implements a unitary transformation directly or by proximity, the non-Abelian statistics results that operates within the subspace of ground states and from the occurrence of zero-energy Majorana fermions shifts the system from one ground state to another. bound to the cores of vortices or to the ends of one- Remarkably, this unitary transformation depends only on dimensional wires [11–22,28,29]. Majorana-based non- the topology of the interchange and is insensitive to im- Abelian statistics is, on the theory side, the most solid precision and noise. These properties make non-Abelian prediction for the occurrence of non-Abelian statistics, anyons a testing ground for the idea of topological quan- since it is primarily based on the well-tested BCS mean- tumcomputation[7].Thesearchfornon-Abeliansystems ﬁeldtheoryofsuperconductivity.Moreover,ontheexperi- originated from the Moore-Read theory [5] for the (cid:1)¼ mentalside,itistheeasiestrealizationtoobserve[23].The set of unitary transformations that may be carried out on 5=2 fractional quantum Hall (FQH) state and went on to Majorana-based systems is rather limited, however, and does not allow for universal topological quantum compu- tation[30,31]. Published by the American Physical Society under the terms of In this work, we introduce and analyze a non-Abelian the Creative Commons Attribution 3.0 License. Further distri- butionofthisworkmustmaintainattributiontotheauthor(s)and system that is based on proximity coupling to a super- the published article’s title, journal citation, and DOI. conductor but goes beyond the Majorana fermion 2160-3308=12=2(4)=041002(22) 041002-1 Published by the American Physical Society LINDNERet al. PHYS. REV. X 2,041002 (2012) paradigm. The system we analyze is based on the prox- innerdisk,theelectronicspinsarepolarizedparalleltothe imitycouplingoffractionalquantumHallsystemsorfrac- magnetic ﬁeld (spin up); on the annulus, the electronic tional quantum-spin Hall systems [32] to superconductors spins are polarized antiparallel to the magnetic ﬁeld (spin andferromagneticinsulators.(Wewillusetheterm‘‘frac- down). Consequently, two edge modes ﬂow on the two tionaltopologicalinsulator’’[FTI]forafractionalquantum sides of the barrier, with opposite spins and opposite spinHallsystem.)Thestartingpointofourapproachisthe velocities. Such a state may be created under circumstan- following observation, made by Fu and Kane [15] when ceswherethesignofthegfactorismadetovaryacrossthe consideringtheedgestatesof2Dtopologicalinsulatorsof barrier. noninteracting electrons, of which the integer quantum Thethirdsystemisanelectron-holebilayersubjectedto spin Hall state [33,34] is a particular example: In a 2D aperpendicularmagneticﬁeld,inwhichonelayeristuned topological insulator, the gapless edge modes may be to an electron-spin–polarized ﬁlling factor of (cid:1)¼1=m, gapped either by breaking time-reversal symmetry or by and the other to a hole spin-polarized in the (cid:1)¼(cid:1)1=m breaking charge conservation along the edge. The former state.Inparticular,thissystemmayberealizedinamate- may be broken by proximity coupling to a ferromagnet rial with a spectrum that is electron-hole symmetric, such (FM), while the latter may be broken by proximity cou- as graphene. plingtoasuperconductor(SC).Remarkably,theremustbe Inallthreecases,thegaplessedgemodemaybegapped a single Majorana mode localized at each interface be- by proximity coupling either to a superconductor or to a tween a region where the edge modes are gapped by a ferromagnet. We imagine that the edge region is divided superconductor to a region where the edge modes are into2Nsegments,wherethesuperconductingsegmentsare gapped by a ferromagnet. allproximitycoupledtothesamebulksuperconductor,and Our focus is on similar situations in cases where the the ferromagnetic segments are all proximity coupled to gapless edge modes are offractional nature. We ﬁnd that, thesameferromagnet.Thelengthofeachsegmentislarge underthesecircumstances,theMajoranaoperatorscarried compared to the microscopic lengths, so that tunneling bytheinterfacesintheintegercasearereplacedby‘‘frac- between neighboringSC-FM interfaces is suppressed. We tional Majorana operators’’ whose properties we study. consider the proximity interactions of the segments with We consider three types of physical systems. The ﬁrst the superconductor and the ferromagnet tobe strong. [shownschematicallyinFig.1(a)]isthatofa2Dfractional The questions we ask ourselves are motivated by the topological insulator [32], which may be viewed as a 2D analogy with the noninteracting systems of Majorana fer- mions:Whatisthedegeneracyofthegroundstate?Isthis systeminwhichelectronsofspinupformanFQHstateof a Laughlin [35] fraction (cid:1)¼1=m, where m is an odd degeneracytopologicallyprotected?Whatis thenatureof thedegenerategroundstates?Andhowcanonemanipulate integer, and electrons of spin down form an FQH state of a Laughlinfraction (cid:1)¼(cid:1)1=m. thesystemsuchthatitevolves,inaprotectedway,between different ground states? The second system [shown in Fig. 1(b)] is a Laughlin Thestructureofthepaperisasfollows:InSec.II,wegive FQH droplet of (cid:1)¼1=m, divided by a thin insulating thephysicalpicturethatwehavedevelopedandsummarize barrier into an inner disk and an outer annulus. On the our results. In Sec. III, we deﬁne the Hamiltonian of the system.InSec.IV,wecalculatetheground-statedegener- (a) (b) acy.InSec.V,wedeﬁnetheoperatorsthatarelocalizedat θ the interfaces and act on the zero-energy subspace. In 1 φ θ 1 Sec. VI, we calculate in detail the unitary transformation 3 thatcorrespondstoabraidoperation.InSec.VII,weshow how this transformation may be deduced from general φ θ 2 2 considerations,bypassingtheneedfordetailedcalculation. InSec.VIII,wediscussseveralaspectsofthefractionalized Majorana operators and their suitability for topological quantumcomputation.SectionIXcontainssomeconclud- FIG. 1. Schematicsetup.(a)Afractionaltopologicalinsulator ing remarks. The paper is followed by appendixes that (FTI)realizationofthesystemweconsider.AFTIdropletwith discussseveraltechnicaldetails. an odd ﬁlling factor 1=m is proximity coupled to ferromagnets (FM)andtosuperconductors(SC),whichgapoutitsedgemodes. TheinterfacesbetweentheSCandFMsegmentsontheedgeof II. THE PHYSICAL PICTURE AND SUMMARY the FTI are marked by red stars. (b) A fractional quantum Hall OF THE RESULTS (FQH)realizationofthesystemweconsider.AFQHdropletwith Thesystemsweconsiderhavethreetypesofregions:the ﬁllingfactor1=misseparatedbyathinbarrierintotwopieces:an inner disk and an outer annulus. On either side of the barrier, bulk, the parts of the edge that are proximitycoupled toa there are counterpropagating edge states, which are proximity superconductor,andthepartsoftheedgethatareproximity coupledtosuperconductorsandferromagnets. coupled toa ferromagnet. 041002-2 FRACTIONALIZING MAJORANA FERMIONS:NON-ABELIAN ... PHYS. REV. X 2,041002 (2012) The bulk is either a fractional quantum Hall state or a 2 3 fractional quantum spin Hall state. In both cases, the 1 2 1 2 1 2 bulk is gapped and incompressible, and its elementary excitations are localized quasiparticles whose charge is a 1 4 multipleofe(cid:2) ¼e=melectroncharges.Inouranalysis,we 4 3 4 3 4 3 assumethattheareaenclosedbytheedgemodesencloses n quasiparticles of spin up and n quasiparticles of spin 6 5 " # down. These quasiparticles are assumed to be immobile. In the parts of the edge that are coupled to a supercon- FIG. 2. Braiding process. (a) A FTI disk with six SC-FM ductor, the charge is deﬁned only modulo 2e, because segments. In stages I, II, and III of the braiding process, Cooper pairs may be exchanged with the superconductor. quasiparticle tunneling (represented by blue curves) is turned Thus,theproperoperatortodescribethechargeonaregion on between the SC-FM interfaces. (b) Representation of the of this type is ei(cid:2)Q^i, where Q^i is the charge in the ith braiding procedure, involving interfaces 1, 2, 3, and 4. In the superconductingregion.Sincethesuperconductingregion beginningofeachstage,thetwointerfacesconnectedbyasolid may exchange e(cid:2) charges with the bulk, these operators line are coupled; during that stage, the bond represented by a maytakethevaluesei(cid:2)qi=m,withqianintegerwhosevalue dashed line is adiabatically turned on, and, simultaneously, the solid bond is turned off. By the end of stage III, the system isbetweenzeroand2m(cid:1)1.Thepairinginteractionleads returns to the original conﬁguration. to a ground state that is a spin singlet, and thus the expectationvalueofthespinwithineachsuperconducting regionvanishes.Asweshowbelow,theHamiltonianofthe underthetransformationwheren !n (cid:3)mtogetherwith " " system commutes with the operators ei(cid:2)Q^i in the limit we n# !n#(cid:3)m.Thesesetsmaybespannedbythevalues0(cid:4) consider. For the familiar m¼1 case, these operators n (cid:4)2m(cid:1)1 and 0(cid:4)n (cid:4)m(cid:1)1. " # measure the parity of the number of electrons within The degeneracy of the ground state may be understood each superconducting region. by examining the algebra constructed by the operators The edge regions that are proximity coupled to ferro- ei(cid:2)Q^i and ei(cid:2)S^i. As we show in the next section, the magnets are, insome sense, the dual of the superconduct- operators ei(cid:2)S^i,ei(cid:2)Q^i satisfy ing regions. The ferromagnet introduces backscattering between the two counterpropagating edge modes, leading ½ei(cid:2)Q^i;ei(cid:2)Q^j(cid:5)¼½ei(cid:2)S^i;ei(cid:2)S^j(cid:5)¼0; totheformationofanenergygap.Ifthechemicalpotential (cid:2) (cid:3) (cid:2) (cid:3) YN YN lies within this gap, the region becomes insulating and ei(cid:2)Q^j; ei(cid:2)S^i ¼ ei(cid:2)S^j; ei(cid:2)Q^i ¼0; (2) incompressible. Consequently, the charge in the region i¼1P i¼1 P does not ﬂuctuate, and its value may be deﬁned as zero. The spin, on the other hand, does ﬂuctuate. Since the ei(cid:2)Q^jei(cid:2) lk¼1S^k ¼eið(cid:2)=mÞ(cid:3)jlei(cid:2) lk¼1S^kei(cid:2)Q^j; backscatteringfromspin-upelectrontospin-downelectron where,inthelastequation,1(cid:4)j,l<N.[SeeFig.2(a)for changes the total spin of the region by two (where the theenumerationconvention.]APsmanifestedbyEqs.(2),the electronicspinisdeﬁnedasoneunitofspin),theoperator that may be expected to have an expectationvalue within pairs of operators ei(cid:2)Q^i, ei(cid:2) ik¼1S^k form N(cid:1)1 pairs of thegroundstateisei(cid:2)S^i,whereS^iisthetotalspinintheith dmeugtreewesitohfofnreeeadnoomth,ewr.hIetriesmtheemreblaetrisoonfbdeitfwfeereenntmpeamirsbecrosmof- ferromagnet region. Again, spins of 1=m may be ex- the same pairs, expressed in Eqs. (2), from which the changed with the bulk, and thus these operators may take ground-state degeneracy may be easily read out. As is theeigenvaluesei(cid:2)si=m,wheresisanintegerbetweenzero evident from this equation, if jci is a ground state of the and 2m(cid:1)1. The Hamiltonian of the system commutes with the operatorsei(cid:2)S^i in the limit we consider. system which is also an eigenstatPe of ei(cid:2)Q^j, then 2m(cid:1)1 Theoperatorsei(cid:2)Q^i andei(cid:2)S^i labelthedifferentdomains additional ground states are ðei(cid:2) ji¼1S^iÞkjci, where k is an integer between 1 and 2m(cid:1)1. With N(cid:1)1 mutually in the system, as indicated in Fig. 2(a). They satisfy a independentpairs,wereachtheconclusionthattheground- constraint dictated by the state of the bulk, statedegeneracy,foragivenvalueofn,n,isð2mÞN(cid:1)1. " # YN YN The operators acting within a sector of given n, n " # ei(cid:2)Q^i ¼ei(cid:2)ðn"þn#Þ=m; ei(cid:2)S^i ¼ei(cid:2)ðn"(cid:1)n#Þ=m: (1) oftheground-statesubspacearerepresentedbyð2mÞN(cid:1)1(cid:6) i¼1 i¼1 ð2mÞN(cid:1)1matrices.Theymaybeexpressedintermsofsums and products of the operators appearing in Eqs. (2). The For the familiar m¼1 case, there are only two possible solutions for these constraints, corresponding to the physicaloperationsdescribed bytheoperatorsei(cid:2)S^i,ei(cid:2)Q^i two right-hand sides of Eq. (1) being both þ1 or both can also be read off the relations (2). The operator ei(cid:2)S^i (cid:1)1. For a general m, the number of topologically distinct transfers a quasiparticle of charge e=m from the constraints is 2m2, since the equations in (1) are invariant (i(cid:1)1)th superconductor to the ith superconductor. Since 041002-3 LINDNERet al. PHYS. REV. X 2,041002 (2012) the spin within the superconductor vanishes, there is no deﬁned in terms of trajectories in parameter space, which distinction, withintheground-statemanifold,between the includes the tunneling amplitudes that are introduced to possible spin states of the transferred quasiparticle. In implementthebraiding.Thebraidingistopologicalinthe contrast, the operator ei(cid:2)(cid:4)Q^i transfers a quasiparticle of sense that it does not depend on the precise details of the spin (cid:4)¼(cid:3)1 across the ith superconductor. trajectorythatimplementsit,aslongasthedegeneracyof Form¼1,theoperatorsei(cid:2)S^i andei(cid:2)Q^i,measuringthe the ground-state manifold does not vary throughout the implementation. Physically, one can imagine realizing parityofthespinandthechargeintheithferromagneticand such operations by changing external gate potentials that superconductingregion,respectively,maybeexpressedin terms of Majorana operators that reside at the interfaces deformtheshapeofthesystem’sedgeadiabatically(similar borderingthatregion.Asimilarrepresentationexistsalsoin totheoperationsproposedfortheMajoranacase[38,40]). thecaseofm(cid:1)1.ItsdetailsaregiveninSec.V. Intheintegerm¼1case,theinterchangeoftwoanyons Westressthattheground-statedegeneracyistopological, positioned at two neighboring interfaces is carried out by in the sense that no measurement of a local operator can subjecting the system to an adiabatically time-dependent determine the state of the system within the ground-state Hamiltonian in which interfaces are coupled to one subspace. For m¼1, this corresponds to the well-known another. When two or three interfaces are coupled to one ‘‘topological protection’’ of the ground-state subspace of another, the degeneracy of the ground state does not de- Majoranafermions[30,36],aslongassingleelectrontun- pendontheprecisevalueofthecouplings,aslongasthey neling is forbidden either between the different Majorana do not all vanish at once. Consequently, one may ‘‘copy’’ modes, or between the Majorana modes and the external anyon a onto anyon c by starting with a situation where world.Inthefractionalcase,thestatesintheground-state corresponding interfaces b and c are tunnel coupled, and manifold can be labeled by the fractional part of the spin then turning on a coupling between a and b while simul- (charge) of the FM (SC) segments, respectively. These taneously turning off the coupling of b to c. Three con- clearly cannot be measured locally. Moreover, they can secutive‘‘copying’’processesthenleadtoaninterchange, changeonlybytunnelingfractionalquasiparticlesbetween and the resulting interchanges generate a non-Abelian differentsegments;tunnelingofelectronsfromtheoutside representation of the braid group. environment cannot split the degeneracy completely, be- In the integer case, only electrons may tunnel between cause such tunnelings change the charge and spin of the twointerfaces,thusallowingustocharacterizethetunnel- systemonlybyintegers. ing term by one tunneling amplitude. In contrast, in the Topological manipulations of non-Abelian anyons con- fractional case, more types of tunneling processes are ﬁned to one dimension are somewhat more complicated possible, corresponding to the tunneling of any number than those carried out in two dimensions. The simplest ofquasiparticlesofchargese=mandspin(cid:3)1=m.Todeﬁne manipulation does not involve any motion of the anyons theeffectiveHamiltoniancouplingtwointerfaces,weneed butratherinvolveseithera2(cid:2)twistoftheorderparameter to specify the amplitudes for all these distinct processes. of the superconductor coupled to one or several super- Asonemayexpect,ifonlyelectronsareallowedtotunnel conducting segments, or a 2(cid:2) rotation of the direction of betweentheinterfaces(asmaybethecaseifthetunneling the magnetization of the ferromagnet coupled to the insu- is constrained to take place through the vacuum), the lating segments [37]. When a vortex encircles the ith m¼1 case is reproduced. When single quasiparticles of superconducting region, it leads to the accumulation of a one spin direction are allowed to tunnel (which is the Berry phase of 2(cid:2) multiplied by the number of Cooper naturalcasefortheFQHErealizationofourmodel),tunnel pairs it encircles. In the problem we consider, this phase couplingbetweeneithertwoorthreeinterfacesreducesthe amounts to ei(cid:2)Q^i, and that is the unitary transformation degeneracyofthegroundstatebyafactorof2m.Thiscase applied by such rotation. As explained above, this then opens the way for interchanges of the positions of transformation transfers a spin of 1=m between the two anyonsbythesamemethodenvisionedfortheintegercase. ferromagnetic regions that the superconductor borders. We analyze these interchangesin detail below. Similarly,arotationofthemagnetizationintheferromag- Our analysis of the unitary transformations that corre- neticregionleadstoatransferofachargeofe=mbetween spondtobraidingschemesfollowstwodifferentroutes.In the two superconductors that the ferromagnet borders. the ﬁrst, detailed in Sec. VI, we explicitly calculate these A more complicated manipulation is that of anyon transformationsforaparticularcaseofanyoninterchange. braiding and its associated non-Abelian statistics. In two In the second, detailed in Sec. VII, we utilize general dimensions, the braiding of anyons is deﬁned in terms of properties of anyons to all non-Abelian representations of worldlinesRðtÞthatbraidoneanotherastimeevolves.On thebraidgroupthatsatisfyconditionsthatweimpose.Itis the other hand, in one dimension—both in the integer natural to expect these conditions from the system we m¼1 and in the fractional case—a braiding operation analyze. Both routes indeed converge to the same result. requirestheintroductionoftunnelingtermsbetweendiffer- While the details of the calculations are given in the ent points along the edge [38,39]. The braiding is then following sections, herewe discuss their results. 041002-4 FRACTIONALIZING MAJORANA FERMIONS:NON-ABELIAN ... PHYS. REV. X 2,041002 (2012) To consider braiding, we imagine that two anyons at ofatransformationei(cid:6)(cid:2)Q^2i thatresultsfromaninterchange the two ends of the ith superconducting region are ofanyons,multipliedbyatransformation eð2(cid:6)(cid:2)i=mÞQ^ik that interchanged. For the m¼1 case, the interchange of two results from a vortex encircling the ith superconducting Majorana fermions correspondto the transformation region 2(cid:6)k=m times. Non-Abelian statistics is the cornerstone of topological p1ﬃ2ﬃ½1(cid:3)iexpði(cid:2)Q^iÞ(cid:5): (3) quantumcomputation[7,30],duetothepossibilityitopens forthe implementation of unitary transformations that are This transformation may be written as exp½i(cid:2)ðQ^ (cid:1)kÞ2(cid:5), topologically protected from decoherence and noise. It is 2 i wherek¼0;1correspondstothe(cid:3) signinEq.(3),oras therefore natural to examine whether the non-Abelian p1ﬃﬃð1(cid:3)(cid:5) (cid:5) Þ, where (cid:5) , (cid:5) are the two localized anyonsthatwestudyallowforuniversalquantumcompu- 2 1 2 1 2 tation, that is, whether any unitary transformation within Majorana modes at the two ends of the superconducting the ground-state subspace may be approximated by topo- region.Thesquareofthetransformationistheparityofthe logical manipulations of the anyons [31]. We ﬁnd that, at chargeinthesuperconductingregion.Thefourthpowerof least forunitary time evolution(i.e., processes that donot the transformation is unity. Note that, in two dimensions, involve measurements), the answer to this question is the two signs in Eq. (3) correspond to anyon exchange in negative, as it is for the integer case. the clockwise and anticlockwise sense. In contrast, inone dimension, the two signs may be realized by different choices of tunneling amplitudes and are not necessarily III. EDGE MODEL associated with a geometric notion. Consistent with the The edge states of a FTI are described by a topological nature of the transformation, a trajectory that hydrodynamic bosonized theory [41,42]. The effective leads to one sign in Eq. (3) cannot be deformed into a Hamiltonian of the edge is writtenas trajectory that corresponds to a different sign without (cid:2) (cid:3) Z passing through a trajectory in which the degeneracy of mu 1 H ¼ dx KðxÞð@ (cid:7)Þ2þ ð@ (cid:8)Þ2 thegroundstatevariesduringtheexecutionofthebraiding. 2(cid:2) x KðxÞ x Guided by this familiar example, we expect that, in the Z fractional case, the unitary transformation corresponding (cid:1) dx½g ðxÞcosð2m(cid:7)Þþg ðxÞcosð2m(cid:8)Þ(cid:5): (6) S F tothisinterchangewilldependonlyonei(cid:2)Q^i.Weexpectto be able towrite it as Here,uistheedge-modevelocity;(cid:7),(cid:8)arebosonicﬁelds satisfying the commutation relation ½(cid:7)ðxÞ;(cid:8)ðx0Þ(cid:5)¼ 2mX(cid:1)1 UðQ^ Þ¼ a expði(cid:2)jQ^ Þ; (4) i(cid:2)(cid:1)ðx0(cid:1)xÞ, where (cid:1) is the Heaviside step function; and i j i m j¼0 gSðxÞ, gFðxÞ describe position-dependent proximity cou- plings to a SC and a FM, which we take to be constant in with some complex coefﬁcients aj, i.e., to be periodic in the SC and FM regions and zero elsewhere, respectively. Q^ ,with the periodbeing2.We expect thevaluesof a to The magnetization of the FM is taken to be in the x i j dependonthetypeoftunnelingamplitudesthatareusedto direction. KðxÞ is a space-dependent Luttinger parameter, implement the braiding. originating from interactions between electrons of Inouranalysis,weﬁndamorecompact,yetequivalent, opposite spins. The charge and spin densities are given form for the transformation U, which is by (cid:9)¼@ (cid:8)=(cid:2) and sz ¼@ (cid:7)=(cid:2), respectively (where x x the spin is measured in units of the electron spin @=2). A UðQ^iÞ¼ei(cid:6)(cid:2)½Q^i(cid:1)ðk=mÞ(cid:5)2: (5) right-orleft-movingelectronisdescribedbytheoperators c ¼eimð(cid:7)(cid:3)(cid:8)Þ. Thevalueof(cid:6)dependsonthetypeofparticlethattunnels (cid:3) Crucially for the arguments below, we assume that the duringtheimplementationofthebraiding,whilethevalue entireedgeisgappedbytheproximitytotheSCandFM, ofkdependsonthevalueofthetunnelingamplitudes.For except (possibly) the SC-FM interface. This can be anelectrontunneling,(cid:6)¼m2.Justasforthem¼1case, 2 achieved, in principle, by making the proximity coupling for this value of (cid:6) the unitary transformation (5) has two tothe SC and FM sufﬁciently strong. possibleeigenvalues,U4 ¼1,anditisperiodicinkwitha period of 2. For braiding carried out by tunneling single IV.GROUND-STATE DEGENERACYOFA DISK quasiparticlesweﬁnd(cid:6)¼m.InthiscaseU4m ¼1,andU 2 WITH 2N SEGMENTS is periodic in k with a period of 2m. Justasinthem¼1case,trajectoriesinparameterspace We consider a disk with 2N FM-SC interfaces on its that differ by their value of k are separated by trajectories boundary [illustrated in Fig. 1(a) for N ¼2]. In order to that involve a variation in the degeneracy of the ground determinethedimensionoftheground-statemanifold,we state.Wenotethat,uptoanunimportantAbelianphase,the constructasetofcommutingoperatorsthatcanbeusedto unitarytransformation(5)maybethoughtofascomposed characterize the ground states. Consider the operators 041002-5 LINDNERet al. PHYS. REV. X 2,041002 (2012) ei(cid:2)Qj (cid:7)eið(cid:8)jþ1(cid:1)(cid:8)jÞ, j¼1;...;N, where (cid:8)j is a (cid:8) ﬁeld thermodynamic limit because the operators ei(cid:2)Sj and evaluated at an arbitrary point near the middle of the jth ei(cid:2)Qj satisfy the commutation relations FM region. The origin (x¼0) is chosen to lie within the ei(cid:2)Siei(cid:2)Qj ¼ei(cid:2)=mð(cid:3)i;jþ1(cid:1)(cid:3)i;jÞei(cid:2)Qjei(cid:2)Si; (7) ﬁrstFMregion[seeFig.1(a)].Theoperator(cid:8) islocated Nþ1 withinthisregion,totheleftoftheorigin(x<0),while(cid:8) whichcanbeveriﬁedbyusingthecommutationrelationof 1 istotherightoftheorigin(x>0).Theﬁelds(cid:8),(cid:7)satisfy the(cid:7)and(cid:8)ﬁelds.Inthestatejfqg;si,thevalueofei(cid:2)Qj is the boundary conditions ei(cid:2)Qtot ¼ei½(cid:8)ðL(cid:1)Þ(cid:1)(cid:8)ð0þÞ(cid:5) and approximately localized near eið(cid:2)=mÞqj. Applying the op- ei(cid:2)Stot ¼ei½(cid:7)ðL(cid:1)Þ(cid:1)(cid:7)ð0þÞ(cid:5), where L is the perimeter of the eratorei(cid:2)Sj tothisstateshiftsei(cid:2)Qj toei(cid:2)½Qjþð1=mÞ(cid:5),ascan system, and Q and S are the total charge and spin on beseenfromEq.(7).Thisshiftimpliesthattheoverlapof tot tot the edge, respectively. the states jfqg;si and ei(cid:2)Sjjfqg;si decays exponentially Since we are in the gapped phase of the sine Gordon with the system size. model of Eq. (6), we expect in the thermodynamic limit Overall, there are ð2mÞNþ1 distinct approximate eigen- (where the size of all of the segments becomes large) state jfqg;si, corresponding to the 2m allowed values of that the (cid:8) ﬁeld is essentially pinned to the minima of the charges qj of each individual SC segment, and the total cosine potential in the FM regions. (Similar considera- spin s, which can also take 2m values. Not all of these tions hold for the (cid:7) ﬁelds in the SC regions.) In other states are physicPal, however. Labeling the total charge by words, the (cid:8)!(cid:8)þ(cid:2)=m symmetry is spontaneously an integer q¼ Nj¼1qj, we see from Eq. (1) that s and q broken. In this phase, correlations of the ﬂuctuations of must be either both even or both odd, corresponding to a (cid:8) decay exponentially on length scales larger than the totalevenoroddnumberoffractionalquasiparticlesinthe correlation length (cid:10)(cid:8)u=(cid:2) , where (cid:2) is the gap in bulkofthesystem.Becauseofthisconstraint,thenumber F F the FM regions. (See Appendix A for an analysis of of physical states is only 1ð2mÞNþ1. 2 the gapped phase.) Therefore, one can construct approxi- In a given sector with a ﬁxed total charge and total spin, mate ground states that are characterized by thereareN ¼ð2mÞN(cid:1)1groundstates.Form¼1,weobtain GS heið(cid:8)jþ1(cid:1)(cid:8)jÞi¼hei(cid:2)Qji(cid:7)(cid:11)j (cid:1)0, where (cid:11)j ¼j(cid:11)jeði(cid:2)=mÞqj, NGS¼2N(cid:1)1 for each parity sector, as expected for 2N and where q 2f0;...;2m(cid:1)1g can be chosen indepen- MajoranastateslocatedateachoftheFM-SCinterfaces[11]. j Theground-statedegeneracy inthe fractional case sug- dently foreach FM domain.The energy splitting between these ground states is suppressed in the thermodynamic gests that each interface canpbﬃeﬃﬃﬃﬃtﬃﬃhought of as an anyon limit as e(cid:1)R=(cid:10), where R is the length of each region, as whose quantum dimension is 2m. This is reminiscent of recentlyproposedmodelsinwhichdislocationsinAbelian discussed below and in Appendix A. topologicalphasescarryanyonswithquantumdimensions Inaddition,ei(cid:2)Stot commutesbothwiththeHamiltonian that are square roots ofintegers [43–45]. andwithei(cid:2)Qj.Therefore,thegroundstatescanbechosen to be eigenstates of ei(cid:2)Stot, with eigenvalues eið(cid:2)=mÞs, s2 f0;...;2m(cid:1)1g.Welabeltheapproximategroundstatesas V.INTERFACE OPERATORS jfqg;si(cid:7)jq ;...;q ;si, where jfqg;si satisﬁes that 1 N Wenowturntodeﬁnephysicaloperatorsthatactonthe hfqg;sjei(cid:2)Qjjfqg;si¼j(cid:11)jeði(cid:2)=mÞqj. low-energysubspace.Theseoperatorsareanalogoustothe For a large but ﬁnite system, the jfqg;si states are not Majorana operators in the m¼1 case, in the sense that exactly degenerate. There are two effects that lift the theycanbeusedtoexpressanyphysicalobservableinthe degeneracy between them: intrasegment instanton tunnel- low-energysubspace.Theywillbeusefulwhenwediscuss ing events between states with different fqg, and interseg- topological manipulations of the low-energy subspace in ment ‘‘Josephson’’ couplings which make the energy the next section. dependent on the values of fqg. However, both of these Wedeﬁnetheunitaryoperatorsei(cid:7)^i andei(cid:2)Q^j suchthat effects are suppressed exponentially as e(cid:1)R=(cid:10), as they are associated with an action that grows linearly with the ei(cid:2)Q^jjq1;...;qN;si¼ei(cid:2)qj=mjq1;...;qN;si; (8) system size. Therefore, we argue that jfqg;si are approxi- mately degenerate, up to exponentially small corrections, ei(cid:7)^jjq1;...;qN;si¼jq1;...;qjþ1;...;qN;si: (9) for anychoice of the set fqg, s. Similarly, one can deﬁne a set of ‘‘dual’’ operators ei(cid:2)Q^j is a diagonal operator in the jfqg;si basis, whereas ei(cid:2)Sj (cid:7)Qeið(cid:7)j(cid:1)(cid:7)j(cid:1)1Þ, j¼2;...;N, and ei(cid:2)S1 ¼ ei(cid:7)^j shiftsqj byone.Theseoperatorscanbethoughtofas ei(cid:2)Stot Ni¼2e(cid:1)i(cid:2)Si. Although the SC regions are in the projections of the microscopic operators ei(cid:7)j and ei(cid:2)Qj, gappedphase,andtheﬁelds(cid:7)jarepinnedneartheminima introduced in the previous section, onto the low-energy of the corresponding cosine potentials, note that the ap- subspace.Inaddition,wedeﬁnetheoperatorT^ thatshifts s proximate ground states jfqg;si cannot be further distin- the total spinof the system: guishedbytheexpectationvaluesoftheoperatorsei(cid:2)Sj.In fact, these states satisfy hfqg;sjei(cid:2)Sjjfqg;si!0 in the T^sjq1;...;qN;si¼jq1;...;qN;sþ1i: (10) 041002-6 FRACTIONALIZING MAJORANA FERMIONS:NON-ABELIAN ... PHYS. REV. X 2,041002 (2012) Theoperators(9)and(10)willnotbedirectlyusefulto between interfaces essentially couples them, by allowing us, since they cannot be constructed by projecting any quasiparticles to tunnel between them. We shall assume combination of edge quasiparticle operators onto the that only one spin species can tunnel between interfaces. low-energy subspace. To see this, note that they add a The reason for this assumption will become clear in next charge of 1=m and zero spin or spin 1=m with no charge. sections, and we shall explain how it is manifested in As a result, they violate the constraint between the total realizations of the model under consideration. At the end spin and charge, Eq. (1). However, these operators can be of the process, the droplet returns to its original form, but used to construct the combinations the state of the system does not return to the initial state. Yj The adiabatic evolution corresponds to a unitary matrix (cid:12)2j;(cid:4) ¼ei(cid:7)^jðT^sÞ(cid:4) ei(cid:4)(cid:2)Q^i; actingon theground-state manifold. i¼1 (11) Below, we analyze a braid operation between nearest- Yj neighbor interfaces, which we label 3 and 4 (for later (cid:12)2jþ1;(cid:4) ¼ei(cid:7)^jþ1ðT^sÞ(cid:4) ei(cid:4)(cid:2)Q^i; convenience).Theoperationconsistsofthreestages,which i¼1 aredescribedpictoriallyinFig.2(b).Itbeginsbynucleat- where (cid:4)¼(cid:3)1. These combinations, which will be used ing a new, small, segment which is ﬂanked by the inter- below, correspond to projections of local quasiparticle faces1and2.Atthebeginningoftheﬁrststage,thesmall operators onto the low-energy manifold. Indeed, the op- size of the new segment means that interfaces 1 and 2 are erators(cid:12) ((cid:4)¼(cid:3)1)carryachargeof1=mandaspinof coupled to each other, and all the other interfaces are j;(cid:4) (cid:3)1=m (as can be veriﬁed by their commutation relations decoupled.Duringtheﬁrststage,wesimultaneouslybring with the total charge and total spin operators). Therefore, interface 3 close to 2, while moving 1 away from both 2 their quantum numbers are identical to those of a single and3,suchthatattheendoftheprocess only2and3are fractional quasiparticle with spin up or down. Moreover, coupledtoeachother,while1isdecoupledfromthem.In the commutationrelations satisﬁed by (cid:12) and for i<j, the second stage, interface 4 approaches 3, and 2 is taken i;(cid:4) awayfrom3and4.Intheﬁnalstage,wecouple1to2and (cid:12) (cid:12) ¼e(cid:1)ið(cid:2)=mÞ(cid:12) (cid:12) ; i;(cid:4) j;" j;" i;(cid:4) decouple4from1and2,suchthattheHamiltonianreturns (12) (cid:12) (cid:12) ¼eið(cid:2)=mÞ(cid:12) (cid:12) ; toits initial form.Inthefollowingdiscussion,we analyze i;(cid:4) j;# j;# i;(cid:4) anexplicitHamiltonianpathyieldingthisbraidoperation, coincide with those of quasiparticle operators ei(cid:7)ðxÞ(cid:3)i(cid:8)ðxÞ which is summarized in Table I. Later, we shall discuss localizedattheSC-FMinterfaces.(Fori¼j,½(cid:12) ;(cid:12) (cid:5)¼ j;" j;# theconditionsunderwhichtheresultisindependentofthe 0ifjisodd,and(cid:12)j;"(cid:12)j;# ¼e2i(cid:2)=m(cid:12)j;#(cid:12)j;"ifjiseven.)Note speciﬁc form of the Hamiltonian path representing the that,inourconvention,(cid:12)2j(cid:1)1;(cid:4)correspondstotheinterface samebraid operation. between the segments labeled by ei(cid:2)S^j and e(cid:2)Q^j, whereas (cid:12)2j;(cid:4) corresponds to the interface between ei(cid:2)Q^j and B. Ground-state degeneracy ei(cid:2)S^jþ1; see Fig. 2(a). To analyze the braiding process, we ﬁrst need to show Therefore,theoperators(cid:12) correspondtoquasiparticle that it does not change the ground-state degeneracy. We j;(cid:4) consider a disk with a total of N ¼3 segments of each creationoperatorsattheSC-FMinterfaces,projectedonto type. The ground-state manifold, without any coupling the low-energy subspace. This conclusion is further sup- between interfaces, is ð2mÞ2-fold degenerate. We deﬁne ported by calculating directly the matrix elements of the operators H , H , and H , the Hamiltonians at the microscopic quasiparticle operator between the approxi- 12 23 24 beginning of the three stages I, II, III. These aregiven by mategroundstates,inthelimitofstrongcosinepotentials (seeAppendixA).Thiscalculationrevealsthatthematrix elements of the quasiparticle operators within the low- energy subspace are proportional to those of (cid:12) , and j;(cid:4) TABLE I. Summary of the braiding adiabatic trajectory that the proportionality constant decays exponentially [shown also in Fig. 2(b)]. There are three stages, (cid:6)¼I, II, III, with the distance of the quasiparticle operator from the along each of which the parameter (cid:11) varies from 0 to 1. The (cid:6) interface. We note that the commutation relations of Hamiltonianineachstageiswritteninthemiddlecolumn,where Eqs. (12) appear in a one-dimensional lattice model of weusethenotationH ¼(cid:1)t (cid:12) (cid:12)y þH:c:(andwheret are ij ij j;" i;" ij ‘‘parafermions’’ [46,47]. complex parameters). The right column summarizes the sym- metryoperatorsthatcommutewiththeHamiltonianthroughout each stage. VI. TOPOLOGICAL MANIPULATIONS Stage Hamiltonian Symmetries A. Setup The braiding process is facilitated by deforming the I ð1(cid:1)(cid:11)IÞH12þ(cid:11)IH23 ei(cid:2)Q^3, ei(cid:2)S^3 darreopblreotuagdhiatbcaltoiscealtloy,esaucchhoththaetrdiaftfeerveenrtySsCta-FgMe. Pinrtoexrifmacietys IIIII ðð11(cid:1)(cid:1)(cid:11)(cid:11)IIIIIÞÞHH2234þþ(cid:11)(cid:11)IIIIIHH2142 ei(cid:2)eQ^i3(cid:2),Q^e3,(cid:1)ei(cid:2)(cid:1)Q^i2(cid:2)eS^i1(cid:2)S^3 041002-7 LINDNERet al. PHYS. REV. X 2,041002 (2012) H ¼(cid:1)t (cid:12) (cid:12)y þH:c:; (13) eachstageoftheevolutionandtheaccidentaldegeneracies jk jk j;" k;" are avoided. where the t are complexamplitudes. jk Consider ﬁrst the initial Hamiltonian (see Table I), C. Braid matrices from Berry’s phases given by The evolution operator corresponding to the braid H ¼(cid:1)t (cid:12) (cid:12)y þH:c:¼(cid:1)2jt jcosð(cid:2)Q^ þ’ Þ: (14) operation can thus be represented as a block-diagonal 12 12 2;" 1;" 12 1 12 unitary matrix, in which each ð2mÞ(cid:6)ð2mÞ block acts Here, ’ ¼argðt Þ. It is convenient towork in the basis on a separate energy subspace. We are now faced with 12 12 of eigenstates of the operators ei(cid:2)Q^1, ei(cid:2)Q^2, ei(cid:2)Q^3, and the problem of calculating the evolution operator in the ei(cid:2)S^tot, which we label by jq1;q2;q3;si. ThPe total charge ground-statesubspace.LetusdenotethisoperatorbyU^34, and spin are conserved, and we may set q ¼0 and correspondingtoabraidingoperationofinterfaces3and4. j j s¼0. Then, a state in the ð2mÞ2-dimensional low-energy The calculation of U^34 can be done analytically by using subspace can be labeled as jq ;q i, where q is ﬁxed to the symmetry properties of the Hamiltonian at each stage 2 3 1 q ¼(cid:1)q (cid:1)q . The initial Hamiltonian (14) is diagonal of the evolution. 1 2 3 in this basis, and therefore its eigenenergies can be read We begin by observing that, since ei(cid:2)Q^3 always com- off easily: E ðq ;q Þ¼(cid:1)2jt jcos½(cid:1)(cid:2)ðq þq Þþ’ (cid:5). muteswiththeHamiltonian,U^ andtheevolutionopera- 12 2 3 12 m 2 3 12 34 For generic ’12, there are 2m ground states. The ground tors for each stage are diagonal in the basis of ei(cid:2)Q^3 statesarenucleatedinsideaSCregion,itstotalspiniszero, eigenstates. In every stage, the adiabatic evolution maps and theground states are ei(cid:2)Q^3 eigenstatesbetweentheinitialandﬁnalground-state manifolds while preserving the eigenvalue q , and multi- j(cid:3)ðq Þi¼jq ¼(cid:1)q ;q i; (15) 3 i 3 2 3 3 plies by a phase factor that may depend on q . This is 3 labeledbyasingleindexq ¼0;...;2m(cid:1)1.Theresidual explicitly summarized as 3 2m-fold ground-state degeneracy can be understood as a U^ j(cid:3)(cid:6)ðq Þi¼exp½i(cid:5) ðq Þ(cid:5)j(cid:3)(cid:6)ðq Þi: (18) resultofthesymmetriesoftheHamiltonian.FromEq.(14), (cid:6) i 3 (cid:6) 3 f 3 ei(cid:2)Q^3 andei(cid:2)S^3 commutewithH12.Thecommutationrela- Here, U^ is the evolution operator of stage (cid:6)¼I, II, III, tionsbetweenei(cid:2)Q^3 andei(cid:2)S^3 ensurethatthegroundstateis and j(cid:3)(cid:6)(cid:6) ðq Þi are the ground states of the initial (ﬁnal) (atleast)2m-folddegeneratebytheeigenvaluesofei(cid:2)Q^3. HamiltoiðnfiÞan3instage(cid:6),respectively,whicharelabeledby theSimgriolaurndc-osntasitdeerdateigoennsercaacnybetharopupgliheoduitn tohrederbrtaoidﬁinndg their ei(cid:2)Q^3 eigenvalues. Likewise, (cid:5)(cid:6)ðq3Þ are the phases accumulated in each of the stages. operation. The operator ei(cid:2)Q^3 always commutes with the In order to determine (cid:5) ðq Þ, we use the additional (cid:6) 3 Hamiltonian,atanystage.Thiscanbeseeneasilyfromthe symmetry operator (cid:4) for each stage, as indicated in (cid:6) factthatthesegmentlabeledbyei(cid:2)Q^3 nevercouplestoany Table I. This symmetry commutes with the Hamiltonian, other segment at any stage [see Fig. 2(a)]. Using the andthereforealsowiththeevolutionoperatorforthisstage deﬁnition of the (cid:12)i(cid:4) operators in Eqs. (11), one ﬁnds that ½(cid:4)(cid:6);U^(cid:6)(cid:5)¼0. Acting with (cid:4)(cid:6) on both sides of (18), we ﬁndthat H ¼(cid:1)2jt jcosð(cid:2)S^ þ’ Þ; (16) 23 23 2 23 and U^ (cid:6)(cid:4)(cid:6)j(cid:3)(cid:6)i ðq3Þi¼ei(cid:5)(cid:6)ðq3Þ(cid:4)(cid:6)j(cid:3)(cid:6)fðq3Þi: (19) H24 ¼(cid:1)2jt24jcos½(cid:2)ðS^2þQ^2Þþ’24(cid:5): (17) Furthermore, the relation ei(cid:2)Q^3(cid:4)(cid:6) ¼eið(cid:2)=mÞ(cid:4)(cid:6)ei(cid:2)Q^3 impliesthatthe operator (cid:4)(cid:6) advances ei(cid:2)Q^3 by one incre- Ineachstage,(cid:6)¼I,II,III,thereisasymmetryoperator ment, and therefore for both the initial and ﬁnal stage at (cid:4) that commutes with the Hamiltonian and satisﬁes (cid:6) each stage, we have (cid:4)(cid:6)ei(cid:2)Q^3 ¼e(cid:1)ið(cid:2)=mÞei(cid:2)Q^3(cid:4)(cid:6).Wespecify(cid:4)(cid:6)foreachstage (cid:4) j(cid:3)(cid:6) ðq Þi¼exp½i(cid:3)(cid:6) ðq Þ(cid:5)j(cid:3)(cid:6) ðq þ1Þi; (20) in the right column of Table I, and the aforementioned (cid:6) iðfÞ 3 iðfÞ 3 iðfÞ 3 relationcanbeveriﬁedusingEqs.(2).Thiscombinationof where (cid:3)(cid:6) ðq Þ are phases that depend on gauge choices symmetries dictates that every state is at least 2m-fold iðfÞ 3 degenerate, where each degenerate subspace can be for the different eigenstates, to be determined below. labeledbyq .Assumingthatthespecialvalues’ ,’ ¼ Inserting Eq. (20) into Eq. (19), we obtain the recursion 3 12 23 (cid:2)ð2lþ1Þ=ð2mÞ and ’ ¼(cid:2)l=m (l integer) are avoided, relation 24 theground state is exactly 2m-fold degenerate throughout (cid:5) ðq þ1Þ¼(cid:5) ðq Þþ(cid:3)(cid:6)ðq Þ(cid:1)(cid:3)(cid:6)ðq Þ: (21) thebraidingprocess.(Thespecialvaluesforthe’ givean (cid:6) 3 (cid:6) 3 f 3 i 3 ij additional twofold degeneracy.) Note that these conclu- Note that, while the phase accumulation at each point sions hold for any trajectory in Hamiltonian space, as along the path depends on gauge choices, the total phase long as the appropriate symmetries are maintained in accumulated along a cycle does not. It is convenient to 041002-8 FRACTIONALIZING MAJORANA FERMIONS:NON-ABELIAN ... PHYS. REV. X 2,041002 (2012) choose a continuous gauge, for which the total phases are given by (cid:4) (cid:5) X = U^ j(cid:3)(cid:6)ðq Þi¼exp i (cid:5) ðq Þ j(cid:3)(cid:6)ðq Þi: (22) 34 i 3 (cid:6) 3 i 3 (cid:6) A continuous gauge requires j(cid:3)(cid:6)ðn Þi¼j(cid:3)(cid:6)þ1ðn Þi. f 3 i 3 Therefore, the values of the phases (cid:3)(cid:6) depend only on iðfÞ three gauge choices. These are the gauge choices for the 11 2 3 1 2 3 eigenstates of the Hamiltonians H , H , and H , which 12 23 24 constitutetheinitialHamiltonianatthebeginningofstages FIG. 3. DiagrammaticrepresentationoftheYang-Baxterequa- IthroughIII,aswellastheﬁnalHamiltonianforstageIII. tions[Eq.(27)].Threeinterfaces1,2,3arebraidedintwodistinct Making the necessary gauge choice allows us to solve sequences. The Yang-Baxter equations state that the results of Eq.(21)for(cid:5) ðq Þ,yieldingthetotalphase(thedetailsof (cid:6) 3 thesetwosequencesofbraidingoperationsarethesame. the calculation aregiven in AppendixB): (cid:2) representationofthebraidgroup,itisnecessaryandsufﬁ- (cid:5)ðq Þ¼ ðq (cid:1)kÞ2þ(cid:5) ; (23) 3 2m 3 0 cient that theysatisfy where (cid:5)0 is an overall ( q3-independent) phase that con- ½U^ðkiÞ ;U^ðkjÞ (cid:5)¼0 ðji(cid:1)jj>1Þ; (26) tains both a dynamical phase and a Berry’s phase. We i;iþ1 j;jþ1 cannot compute (cid:5) with the present approach, and, more- 0 over, the dynamical phase depends on the details of U^ðk1Þ U^ðk2Þ U^ðk1Þ ¼U^ðk2Þ U^ðk1Þ U^ðk2Þ : (27) the path. Note, however, that the differences between the j;jþ1 jþ1;jþ2 j;jþ1 jþ1;jþ2 j;jþ1 jþ1;jþ2 phases of states with different q are independent of 3 Equation (26) clearly holds because the spin or charge this dynamical phase. In the following discussion, we set operators of non-nearest-neighbor segments commute. (cid:5) ¼0 for convenience. The integer k depends on the 0 UsingEq.(25),itisnotdifﬁculttoshowthatEq.(27)holds choice for the phases ’ . Recall that the Hamiltonians ij as well. [See Appendix D.] Equation (27) is depicted in Hij,Eqs.(14), (16), and (17), havean additionaldegener- Fig.3.Therefore,U^ formarepresentationofthebraid i;iþ1 acyforadiscretechoiceofthe’ .Anytwochoicesforthe ij group. For the actual evolution operators (for which the ’ that can be deformed to each other without crossing a ij phase(cid:5) ispathdependent),Eq.(27)canholdonlyuptoan 0 degeneracy point yield the same k. overallphase,andthebraidingoperationsformaprojective The evolution operator for the braiding path can be representationofthebraidgroup.Inthatrespect,oursystem written explicitly by its application on the eigenstates exhibits a form of non-Abelian statistics. By combining a of the Hamiltonian in the beginning of the cycle, sequenceof nearest-neighborexchanges,an exchange op- U^ð3k4Þj(cid:3)iðq3Þi¼eið(cid:2)=2mÞðq3(cid:1)kÞ2j(cid:3)iðq3Þi. Since, by Eq. (15), erationofarbitrarilyfarinterfacescanbedeﬁned. the ground states of the initial Hamiltonian satisfy In anyphysical realization, we do not expect to control q ¼(cid:1)q , the evolution operator can be written in a the precise form of the Hamiltonian in each stage. It is 2 3 basis-independent form in terms of the operator ei(cid:2)Q^2. thereforeimportanttodiscusstheextenttowhichtheresult Loosely speaking, U^ can bewritten as of the braiding process depends on the details of the 34 (cid:2) (cid:4) (cid:5) (cid:3) Hamiltonian along the path. We argue that the braiding is i(cid:2)m k 2 U^ ðkÞ ¼exp Q^ þ : (24) ‘‘topological,’’ in the sense that it is, to a large degree, 34 2 2 m independent of these precise details. To see this, one needs to note that the braiding unitary Alternatively, using the identity [48] eið(cid:2)=2mÞq2 ¼ qﬃﬃﬃﬃﬃ matrix was derived above without referring to the precise P 1 2m(cid:1)1eið(cid:2)=mÞ½pq(cid:1)ðp2=2Þ(cid:5)þið(cid:2)=4Þ, one can write adiabatic path in Hamiltonian space. All we use are the 2m p¼0 symmetry properties of the Hamiltonian in each stage sﬃﬃﬃﬃﬃﬃﬃ 1 2mX(cid:1)1 (Table I). These symmetries depend, not on the precise U^ðkÞ ¼ e(cid:1)ði(cid:2)=2mÞðp(cid:1)kÞ2þið(cid:2)=4Þðei(cid:2)Q^2Þp: (25) details of the intermediate Hamiltonian, but only on the 34 2m p¼0 overallconﬁguration,e.g.,whichinterfacesareallowedto couple ineach stage. In the case m¼1, U^ reduces to the braiding rule of 34 InAppendixC1,westatemoreformallytheconditions Ising anyons [11–13]. under which the result of the braiding is independent of Following a similar procedure, one can construct the details.SpecialcaremustbetakeninstageIIIofthebraid- operator representing the exchange of any pair of neigh- ing,inwhichquasiparticlesofonlyonespinspecies,e.g., boring interfaces: U^ð2kjÞ(cid:1)1;2j ¼eði(cid:2)m=2ÞðQ^jþk=mÞ2, U^ð2kjÞ;2jþ1 ¼ spin up, must be allowed to tunnel between interfaces 2 eði(cid:2)m=2ÞðS^jþ1þk=mÞ2. In order for these operations to form a and4.Weelaborateonthesigniﬁcanceofthisrequirement 041002-9 LINDNERet al. PHYS. REV. X 2,041002 (2012) andthewaystomeetitinthevariousphysicalrealizationsin respectively.SupposeweconsidertwoneighboringSCseg- AppendixC2. ments with quantum numbers expði(cid:2)Q^ Þ and expði(cid:2)Q^ Þ, 1 2 and we fuse them by shrinking the FM region that lies betweenthem.Thisfusingresultsintunnelingoffractional VII. BRAIDING AND TOPOLOGICALSPIN quasiparticlesbetweenthetwoSCregionsandenergetically OF BOUNDARYANYONS favorsaspeciﬁcvalueforexpði(cid:2)S^ÞintheFMregion.The Intheprevioussection,wederivedtheunitarymatrices twoSCsegmentsareforallpurposesone,whereclearly,in representingbraidoperationsbyanexplicitcalculation.In theabsenceofothercouplings,exp½i(cid:2)ðQ^ þQ^ Þ(cid:5)remainsa 1 2 the following discussion, we shall try to shed light on the goodquantumnumber.Thisthereforesuggeststhefollow- physicalpicturebehindtheserepresentations.Todoso,we ingfusionrules: showthattheresultsoftheprevioussectioncanbederived almostpainlessly,justbyassumingthattherepresentation ofthebraidoperationshaspropertiesthatareanalogousto X(cid:6)X ¼0þ1þ(cid:9)(cid:9)(cid:9)þ2m(cid:1)1; (28) thoseofanyonsintwodimensions.Theﬁrstandmostbasic q (cid:6)q ¼ðq þq Þmod2m: assumption is very natural: There exists a topological 1 2 1 2 operation in the system that corresponds to a braid of two interfaces, in that the unitary matrices representing We note that the labeling q¼0;1;...;2m(cid:1)1 does not this operationobey the Yang-Baxter equation[49]. dependonthegaugechoicesinthedeﬁnitionoftheopera- Theoperationsweconsiderbraidtwoneighboringinter- torsexpði(cid:2)Q^Þ.Equations(28)suggestthatthelabelingcan faces,butdonotchangethetotalcharge(spin)intheseg- be deﬁned by the addition law for charges, in which each mentbetweenthem.Thisresultsfromthegeneralformof type of charge plays a different role. Indeed, this addition thebraidoperations—Toexchangetwointerfacesﬂanking rulehasameasurablephysicalcontentthatdoesnotdepend aSC(FM)segment,weusecouplingstoanauxiliaryseg- onanygaugechoices. mentofthesametype.Therefore,charge(spin)canonlybe In two-dimensional theories of anyons, it is convenient exchangedwiththeauxiliarysegment.Sincetheauxiliary to think about particles moving in the two-dimensional segmenthaszerocharge(spin)atthebeginningandendof planeandtoconsidertopologicalpropertiesoftheirworld the operation, the charge (spin) of the original segment lines (such as braiding). In this paper, we have deﬁned cannot change by the operation. Indeed, this can be seen braiding by considering trajectories in Hamiltonian explicitlyintheanalysispresentedintheprevioussection. space. In the following discussion, we represent these Asaresult,theunitarymatrixrepresentingthebraidopera- Hamiltonian trajectories as world lines of the respective tion is diagonal with respect to the charge (spin) of the ‘‘particles’’ involved, keeping in mind that they do not segmentﬂankedbythebraidedinterfaces. correspondto motionof objectsin real space. Thederivationnowproceedsbyconsideringapropertyof We are now ready to deﬁne the TS in our system. In anyonscalledthetopologicalspin(TS).Intwodimensions, short,theTSofaparticleisaphasefactorassociatedwith the topological spin gives the phase acquired by a 2(cid:2) the world line appearing in Fig. 4(a). For interfaces, it is rotation of an anyon. For fermions and bosons, the topo- concretelydeﬁnedbythephaseacquiredbythesystemby logicalspinisthefamiliar(cid:1)1andþ1,respectively(corre- the following sequence of operations, as illustrated in sponding to half-odd or integer spins). There is a close Fig. 4(e): (i) nucleation of a segment to the right (by connection between the braid matrix for anyons and their convention) of the interface X . (Note that the notation 1 topological spins. In two dimensions, these relations have X corresponds to particle X at coordinate r .) The total 1 1 been considered by various authors [10,50]. The system spin or charge of this segment is zero (i.e., the nucleation under consideration is one dimensional, and therefore it does not add total charge to the system). The couplings seemingly does not allow a 2(cid:2) ‘‘rotation’’ of a particle. betweenX and X ﬂankingthe newsegment aretaken to 2 3 However,asweshallexplainbelow,theTSofaparticlecan zero,increasingtheground-statedegeneracybyafactorof bedeﬁnedinoursystemusingtherelationsoftheTStothe 2m. (ii) A right-handed braid operation is performed be- braidmatrix.Weshallthenseehowtousetheserelationsto tween X and X . (iii) The total charge q of the segment 1 2 derive the possible unitary representations of the braid between X and X is measured, and we consider (post- 2 3 operationsinthesystematpoint. select) only the outcomes corresponding to zero charge. In our one-dimensional system, we consider the TS of Therefore, the system ends up in the same state (i.e., no twodifferentkindsofobjects(particles):interfaces,which chargeshavebeenchangedanywhereinthesystem),upto we denote by X, and the charge (or spin) of a segment, a phase factor. Importantly, this phase factor does not which we label by q¼0;1;...;2m(cid:1)1. In what follows, depend on the state of the system, since the operation we need to know how to compose or ‘‘fuse’’ different does not change the total charge in the segment of X 1 objects in our system. As we saw above, two interfaces and X . (See Appendix E for a more detailed discussion.) 2 yield a quantum number expði(cid:2)Q^Þ [expði(cid:2)S^Þ], which is We can therefore deﬁne this phase factor as (cid:8) , the topo- X the total charge [spin] in the segment between them, logical spin of particle type X. 041002-10

the topology of the interchange and is insensitive to im- precision and noise. journal citation, and DOI. PHYSICAL REVIEW X 2, 041002 (2012) .. fractional case, more types of tunneling processes are possible, corresponding to

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