Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states

21 Pages·2013·0.96 MB·English

Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian
quantum Hall states
Netanel H. Lindner∗
Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA and
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
Erez Berg∗
Department of Physics, Harvard University, Cambridge, MA 02138, USA
Gil Refael
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
1
0 Ady Stern
2 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: March 3, 2013)
l
u
We study the non-abelian statistics characterizing systems where counter-propagating gapless
J
modes on the edges of fractional quantum Hall states are gapped by proximity-coupling to su-
5
perconductors and ferromagnets. The most transparent example is that of a fractional quantum

spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of
]
l ν =1/m, while electrons of the opposite spin occupy a similar state with ν =−1/m. However, we
l
a also propose other examples of such systems, which are easier to realize experimentally. We find
h thateachinterfacebetweenaregionontheedgecoupledtoasuperconducto√randaregioncoupled
- toaferromagnetcorrespondstoanon-abeliananyonofquantumdimension 2m. Wecalculatethe
s
unitary transformations that are associated with braiding of these anyons, and show that they are
e
m 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. We carry out this calculation both explicitly
t. andbyapplyinggeneralconsiderations. Finally,weshowthattopologicalmanipulationswiththese
a anyons cannot realize universal quantum computation.
m
-
d I. INTRODUCTION bound to the cores of vortices or to the ends of one di-
n mensional wires10–21,27,28. Majorana-based non-abelian
o
statistics is, from the theory side, the most solid pre-
c Recent years have witnessed an extensive search for
[ electronicsystemsinwhichexcitations(“quasi-particles”) diction for the occurrence of non-abelian statistics, since
it is primarily based on the well tested BCS mean field
follow non-abelian quantum statistics. In such sys-
2
theory of superconductivity. Moreover, from the experi-
v tems,thepresenceofquasi-particles,alsoknownas“non-
mental side it is the easiest realization to observe22. The
3 abelian anyons”1–3, makes the ground state degener-
set of unitary transformations that may be carried out
3 ate. A mutual adiabatic interchange of quasi-particles’
7 positions4 implements a unitary transformation that op- on Majorana-based systems is, however, rather limited,
5 and does not allow for universal topological quantum
erates within the subspace of ground states, and shifts
. computation29,30.
4 the system from one ground state to another. Remark-
0 ably, this unitary transformation depends only on the In this work we introduce and analyze a non-abelian
2 topology of the interchange, and is insensitive to im- system that is based on proximity coupling to a su-
1
precision and noise. These properties make non-abelian perconductor but goes beyond the Majorana fermion
:
v anyons a test-ground for the idea of topological quan- paradigm. The system we analyze is based on the
Xi tum computation5. The search for non-abelian systems proximity-coupling of fractional quantum Hall systems
originated from the Moore-Read theory6 for the ν =5/2 or fractional quantum spin Hall systems31 to supercon-
r
a Fractional Quantum Hall (FQH) state, and went on to ductors and ferromagnetic insulators (we will use the
consider other quantum Hall states7,8, spin systems9, terms “fractional topological insulators” and “fractional
p-wave superconductors10–12, topological insulators in quantum spin Hall states” interchangeably). The start-
proximity coupling to superconductors13,14 and hybrid ing point of our approach is the following observation,
systems of superconductors coupled to semiconductors madebyFuandKane14 whenconsideringtheedgestates
where spin-orbit coupling is strong15–21. Signatures of of two-dimensional (2D) topological insulators of non-
Majorana zero modes may have been observed in recent interacting electrons, of which the integer quantum spin
experiments22–26. Hall state32,33 is a particular example: In a 2D topo-
In the realizations based on superconductors, whether logical insulator, the gapless edge modes may be gapped
directlyorbyproximity,thenon-abelianstatisticsresults eitherbybreakingtimereversalsymmetryorbybreaking
from the occurrence of zero-energy Majorana fermions charge conservation along the edge. The former may be
2
brokenbyproximitycouplingtoaferromagnet,whilethe with a spectrum that is electron-hole symmetric, such as
latter may be broken by proximity coupling to a super- graphene.
conductor. Remarkably,theremustbeasingleMajorana In all these cases, the gapless edge mode may be
mode localized at each interface between a region where gappedeitherbyproximitycouplingtoasuperconductor
the edge modes are gapped by a superconductor to a re- or by proximity coupling to a ferromagnet. We imagine
gionwheretheedgemodesaregappedbyaferromagnet. that the edge region is divided into 2N segments, where
Our focus is on similar situations in cases where the the superconducting segments are all proximity coupled
gapless edge modes are of fractional nature. We find to the same bulk superconductor, and the ferromagnetic
that under these circumstances, the Majorana operators segmentsareallproximitycoupledtothesameferromeg-
carried by the interfaces in the integer case are replaced net. Thelengthofeachsegmentislargecomparedtothe
by “fractional Majorana operators” whose properties we microscopiclengths,sothattunnelingbetweenneighbor-
study. ing SC-FM interfaces is suppressed. We consider the
We consider three types of physical systems. The first proximity interactions of the segments with the super-
(shown schematically in Fig. 1a) is that of a 2D frac- conductor and the ferromegnet to be strong.
tional topological insulator31, that may be viewed as a The questions we ask ourselves are motivated by the
2D system in which electrons of spin-up form an FQH analogy with the non-interacting systems of Majorana
state of a Laughlin34 fraction ν = 1/m , with m being fermions: what is the degeneracy of the ground state?
an odd integer, and electrons of spin-down form an FQH Is this degeneracy topologically protected? What is the
state of a Laughlin fraction ν =−1/m. natureofthedegenerategroundstates? Andhowcanone
The second system (1b) is a Laughlin FQH droplet manipulatethesystemsuchthatitevolves,inaprotected
of ν = 1/m, divided by a thin insulating barrier into way, between different ground states?
an inner disk and an outer annulus. On the inner disk, The structure of the paper is as follows: In Sec. II we
theelectronicspinsarepolarizedparalleltothemagnetic givethephysicalpicturethatwedeveloped,andsumma-
field (spin-up), and on the annulus the electronic spins rize our results. In Sec. III we define the Hamiltonian
are polarized anti-parallel to the magnetic field (spin- of the system. In Sec. IV we calculate the ground state
down). Consequently, two edge modes flow on the two degeneracy. In Sec. V we define the operators that are
sides of the barrier, with opposite spins and opposite ve- localized at the interfaces, and act on the zero energy
locities. Suchastatemaybecreatedundercircumstances subspace. In Sec. VI we calculate in detail the unitary
where the sign of the g-factor is made to vary across the transformationthatcorrespondstoabraidoperation. In
barrier. Sec. VII we show how this transformation may be de-
ducedfromgeneralconsiderations,bypassingtheneedfor
Thethirdsystemisanelectron-holebi-layersubjected
adetailedcalculation. InSec.VIIIwediscussseveralas-
to a perpendicular magnetic field, in which one layer is
pectsofthefractionalizedMajoranaoperators,andtheir
tuned to an electron spin-polarized filling factor of ν =
suitabilityfortopologicalquantumcomputation. Sec.IX
1/m, and the other to a hole spin-polarized ν = 1/m
containssomeconcludingremarks. Thepaperisfollowed
state. In particular, this may be realized in a material
by appendices which discuss several technical details.
a) FM SC b) FM SC II. THE PHYSICAL PICTURE AND SUMMARY
1 FTI 1 1 FQH 1
q OF THE RESULTS
1 f
1
q
3 x = 0 FQH Therearethreetypesofregionsinthesystemswecon-
sider: the bulk, the parts of the edge that are proximity-
f q
2 2 coupled to a superconductor, and the parts of the edge
that are proximity-coupled to a ferromagnet.
SC FM SC FM The bulk is either a fractional quantum Hall state or
2 2 2 2
a fractional quantum spin Hall state. In both cases it
is gapped and incompressible, and its elementary excita-
Figure 1. Schematic setup. (a) A fractional topological
tions are localized quasi-particles whose charge is a mul-
insulator(FTI)realization. AFTIdropletwithanoddfilling
tiple of e∗ = e/m electron charges. In our analysis we
factor 1/m is proximity coupled to ferromagnets (FM) and
willassumethattheareaenclosedbytheedgemodesen-
tosuperconductors(SC),whichgapoutitsedgemodes. The
closesn quasi-particlesofspin-upandn quasi-particles
interfacesbetweentheSCandFMsegmentsontheedgeofthe ↑ ↓
FTI are marked by red stars. (b) A fractional quantum Hall of spin down. These quasi-particles are assumed to be
(FQH) realization. A FQH droplet with filling factor 1/m is immobile.
separated by a thin barrier into two pieces: an inner disk, In the parts of the edge that are coupled to a super-
andanouterannulus. Oneithersideofthebarrier,thereare conductor the charge is defined only modulo 2e, because
counter-propagatingedgestates,whichareproximitycoupled Cooper-pairs may be exchanged with the superconduc-
to superconductors and ferromagnets. tor. Thus, the proper operator to describe the charge on
3
a region of this type is eiπQˆi, with Qˆi being the charge operators eiπSˆi,eiπQˆi satisfy
in i’th superconducting region. Since the superconduct-
ing region may exchange e∗ charges with the bulk, these [eiπQˆi,eiπQˆj]=[eiπSˆi,eiπSˆj]=0,
operatorsmaytakethevalueseiπqi/m, withqi aninteger (cid:34) N (cid:35) (cid:34) N (cid:35)
whose value is between zero and 2m−1. The pairing eiπQˆj,(cid:89)eiπSˆi = eiπSˆj,(cid:89)eiπQˆi =0,
interaction leads to a ground state that is a spin singlet,
i=1 i=1
and thus the expectation value of the spin within each
superconducting region vanishes. As we show below, the eiπQˆjeiπ(cid:80)lk=1Sˆk =eimπδjleiπ(cid:80)lk=1SˆkeiπQˆj, (2)
Hamiltonian of the system commutes with the operators
where, in the last equation, 1 ≤ j,l < N (see Fig 2a for
eiπQˆi in the limit we consider. For the familiar m = 1 theenumerationconvention). AsmanifestedbyEqs. (2),
coafseel,ectthreosnesowpietrhaitnoresacmhesauspuerrecothnedupcatriintgyroefgitohne.number the pairs of operators eiπQˆi,eiπ(cid:80)ik=1Sˆk form N −1 pairs
of degrees of freedom, where members of different pairs
The edge regions that are proximity-coupled to ferro- commute with one another. It is the relation between
magnetsare,insomesense,thedualofthesuperconduct- members of the same pairs, expressed in Eq. (2), from
ing regions. The ferromagnet introduces back-scattering which the ground state degeneracy may be easily read
between the two counter-propagating edge modes, lead- out. As is evident from this equation, if |ψ(cid:105) is a ground
ing to the formation of an energy gap. If the chemical state of the system which is also an eigenstate of eiπQˆj,
pinogteanntdialinliceosmwpirtehsinsibthlei.sgCaopn,stehqeuerengtliyo,ntbheeccohmaersgeinisnultahte- then 2m − 1 additional ground are (cid:16)eiπ(cid:80)ji=1Sˆi(cid:17)k|ψ(cid:105),
region does not fluctuate, and its value may be defined where k isan integerbetween1 and 2m−1. WithN−1
as zero. The spin, on the other hand, does fluctuate. mutuallyindependentpairs,wereachtheconclusionthat
Since the back-scattering from spin up electron to spin the ground state degeneracy, for a given value of n ,n ,
↑ ↓
downelectronchangesthetotalspinoftheregionbytwo is (2m)N−1.
(where the electronic spin is defined as one unit of spin), The operators acting within a sector of given n ,
↑
the operator that may be expected to have an expecta- n of the ground state subspace are represented by
↓
tionvaluewithinthegroundstateiseiπSˆi, withSˆi being (2m)N−1×(2m)N−1 matrices. Theymaybeexpressedin
thetotalspininthei’thferromagnetregion. Again,spins terms of sums and products of the operators appearing
of 1/m may be exchanged with the bulk, and thus these in (2). The physical operations described by the oper-
operatorsmaytaketheeigenvalueseiπsi/m,withsanin- ators eiπSˆi,eiπQˆi can also be read off the relations (2).
teger between zero and 2m−1. The Hamiltonian of the The operator eiπSˆi transfers a quasi-particle of charge
system commutes with the operators eiπSˆi in the limit e/m from the i − 1’th superconductor to the i’th su-
we consider. perconductor. Since the spin within the superconductor
The operators eiπQˆi and eiπSˆi label the different do- vanishes, there is no distinction, within the ground state
manifold, between the possible spin states of the trans-
mainsinthesystem,asindicatedinFig2a. Theysatisfy
a constraint dictated by the state of the bulk, ferred quasi-particle. In contrast, the operator eiπσQˆi
transfers a quasi-particle of spin σ = ±1 across the i’th
superconductor.
N For m = 1 the operators eiπSˆi and eiπQˆi, measuring
(cid:89)eiπQˆi =eiπ(n↑+n↓)/m the parity of the spin and the charge in the i’th ferro-
magnetic and superconducting region, respectively, may
i=1
be expressed in terms of Majorana operators that reside
N
(cid:89)eiπSˆi =eiπ(n↑−n↓)/m (1) at the interfaces bordering that region. A similar repre-
sentation exists also in the case of m(cid:54)=1. Its details are
i=1
given in Section V.
Westressthatthegroundstatedegeneracyistopologi-
cal, inthesensethatnomeasurementofalocaloperator
For the familiar m = 1 case there are only two possible
can determine the state of the system within the ground
solutions for these constraints, corresponding to the two
state subspace. For m=1, this corresponds to the well-
right hand sides of Eq. (1) being both +1 or both −1.
known “topological protection” of the ground state sub-
For a general m, the number of topologically distinct
constraints is 2m2, since Eqs. (1) are invariant under space of Majorana fermions29,35, as long as single elec-
tron tunneling is forbidden either between the different
the transformation where n → n ± m together with
↑ ↑
Majorana modes, or between the Majorana modes and
n →n ±m. These sets may be spanned by the values
↓ ↓
the external world. In the fractional case the states in
0≤n ≤2m−1 and 0≤n ≤m−1.
↑ ↓
the ground state manifold can be labelled by the frac-
Thedegeneracyofthegroundstatemaybeunderstood tionalpartofthespinorchargeoftheFM/SCsegments,
by examining the algebra constructed by the operators respectively. These clearly cannot be measured locally.
eiπQˆi and eiπSˆi. As we show in the next section, the Moreover, they can only change by tunneling fractional
4
quasi-particles between different segments; even tunnel- two interfaces, thus allowing us to characterize the tun-
ing electrons from the outside environment cannot split neling term by one tunneling amplitude. In contrast, in
the degeneracy completely, because it can only change the fractional case more types of tunneling processes are
the charge and spin of the system by integers. possible, corresponding to the tunneling of any number
Topological manipulations of non-abelian anyons con- of quasi-particles of charges e/m and spin ±1/m. To
fined to one dimension are somewhat more complicated define the effective Hamiltonian coupling two interfaces,
than those carried out in two dimensions. The simplest we need to specify the amplitudes for all these distinct
manipulationdoesnotinvolveanymotionoftheanyons, processes. As one may expect, if only electrons are al-
but rather involves either a 2π twist of the order param- lowed to tunnel between the interfaces (as may be the
eter of the superconductor coupled to one or several su- caseifthetunnelingisconstrainedtotakeplacethrough
perconductingsegments,ora2π rotationofthedirection the vacuum), the m = 1 case is reproduced. When sin-
of the magnetization of the ferromagnet coupled to the gle quasi-particles of one spin direction are allowed to
insulating segments36. When a vortex encircles the i’th tunnel (which is the natural case for the FQHE realiza-
superconducting region it leads to the accumulation of a tion of our model), tunnel coupling between either two
Berry phase of 2π multiplied by the number of Cooper- or three interfaces reduces the degeneracy of the ground
pairs it encircles. In the problem we consider, this phase state by a factor of 2m. This case then opens the way
amountstoeiπQˆi,andthatistheunitarytransformation for interchanges of the positions of anyons by the same
appliedbysuchrotation. Asexplainedabove,thistrans- methodenvisionedfortheintegercase. Weanalyzethese
formation transfers a spin of 1/m between the two fer- interchanges in detail below.
romagnetic regions with which the superconductor bor- Ouranalysisoftheunitarytransformationsthatcorre-
ders. Similarly, a rotation of the magnetization in the spond to braiding schemes goes follows different routes.
ferromagnetic region leads to a transfer of a charge of Inthefirst,detailedinSectionVI,weexplicitlycalculate
e/m between the two superconductors with which the thesetransformationforaparticularcaseofanyonsinter-
ferromagnet borders. change. Inthesecond, detailed inSectionVII,we utilize
generalpropertiesofanyonstofindallnon-abelianrepre-
A more complicated manipulation is that of anyons’
sentationsofthebraidgroupthatsatisfyconditionsthat
braiding,anditsassociatednon-abelianstatistics. While
we impose, which are natural to expect from the system
in two dimensions the braiding of anyons is defined in
we analyze. Both routes indeed converge to the same
terms of world lines R(t) that braid one another as time
result. While the details of the calculations are given in
evolves,inonedimension-bothintheintegerm=1and
the following sections, here we discuss their results.
in the fractional case - a braiding operation requires the
introduction of tunneling terms between different points To consider braiding, we imagine that two anyons at
alongtheedge37,38. Thebraidingisthendefinedinterms the two ends of the i’th superconducting region are in-
of trajectories in parameter space, which includes the terchanged. For the m = 1 case the interchange of two
tunneling amplitudes that are introduced to implement Majorana fermions correspond to the transformation
thebraiding. Thebraidingistopologicalinthesensethat
1 (cid:104) (cid:105)
it does not depend on the precise details of the trajec- √ 1±iexp(iπQˆ ) . (3)
i
tory that implements it, as long as the degeneracy of the 2
ground state manifold does not vary throughout the im-
plementation. Physically, one can imagine realizing such This transformation may be written as exp[iπ2(Qˆi−k)2]
operationsbychangingexternalgatepotentialswhichde- with k = 0,1 corresponding to the ± sign in (3), or
formtheshapeofthesystem’sedgeadiabatically(similar as √1 (1±γ1γ2), with γ1,γ2 the two localized Majorana
2
to the operations proposed for the Majorana case37,39). modesatthetwoendsofthesuperconductingregion. Its
In the integer m = 1 case, the interchange of two square is the parity of the charge in the superconducting
anyonspositionedattwoneighboringinterfacesiscarried region, and its fourth power is unity. Note that in two
outbysubjectingthesystemtoanadiabaticallytimede- dimensions, the two signs in (3) correspond to anyons
pendent Hamiltonian in which interfaces are coupled to exchange in clockwise and anti-clockwise sense. In con-
one another. When two or three interfaces are coupled trast, in one dimension the two signs may be realized
to one another, the degeneracy of the ground state does by different choices of tunneling amplitudes, and are not
not depend on the precise value of the couplings, as long necessarily associated with a geometric notion. Consis-
astheydonotallvanishatonce. Consequently,onemay tent with the topological nature of the transformation,
“copy” an anyon a onto an anyon c by starting with a a trajectory that leads to one sign in (3) cannot be de-
situationwherecorrespondinginterfacesbandcaretun- formed into a trajectory that corresponds to a different
nel coupled, and then turning on a coupling between a sign, without passing through a trajectory in which the
and b while simultaneously turning off the coupling of b degeneracy of the ground state varies during the execu-
to c. Three consecutive “copying” processes then lead to tion of the braiding.
an interchange, and the resulting interchanges generate Guidedbythisfamiliarexample,weexpectthatatthe
a non-abelian representation of the braid group. fractional case the unitary transformation corresponding
In the integer case only electrons may tunnel between tothisinterchangewilldependonlyoneiπQˆi. Weexpect
5
to be able to write it as Here, u is the edge mode velocity, φ, θ are bosonic
fieldssatisfyingthecommutationrelation[φ(x),θ(x(cid:48))]=
2m−1
U(Qˆi)= (cid:88) ajexp(cid:16)iπjQˆi(cid:17), (4) imπΘ(x(cid:48)−x) where Θ is the Heaviside step function, and
g (x),g (x)describeposition-dependentproximitycou-
j=0 S F
plings to a SC and a FM, which we take to be constant
with some complex coefficients a , i.e., to be periodic in in the SC/FM regions and zero elsewhere, respectively.
j
Qˆ , with the period being 2. We expect the values of a The magnetization of the FM is taken to be in the x
i j
to depend on the type of tunneling amplitudes that are direction. K(x) is a space-dependent Luttinger param-
used to implement the braiding. eter, 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 ρ = ∂xθ/π and sz = ∂xφ/π, respectively (where the
spin is measured in units of the electron spin (cid:126)/2). A
U(Qˆi)=eiαπ(Qˆi−mk)2. (5) right or left moving electron is described by the opera-
tors ψ =eim(φ±θ).
±
The value of α depends on the type of particle which
Cruciallyfortheargumentsbelow,wewillassumethat
tunnelsduringtheimplementationofthebraiding,while
the entire edge is gapped by the proximity to the SC and
the value of k depends on the value of the tunneling am-
FM,except(possibly)theSC/FMinterface. Thiscanbe
plitudes. For an electron tunneling, α= m2. Just as for achieved, in principle, by making the proximity coupling
2
the m = 1 case, for this value of α the unitary transfor- to the SC and FM sufficiently strong.
mation (5) has two possible eigenvalues, U4 = 1, and it
is periodic in k with a period of 2. For braiding carried
outbytunnelingsinglequasi-particleswefindα= m. In IV. GROUND STATE DEGENERACY OF DISK
2
this case U4m = 1, and U is periodic in k with a period WITH 2N SEGMENTS
of 2m.
Just as in the m = 1 case, trajectories in parameter We consider a disk with 2N FM/SC interfaces on its
spacethatdifferbytheirvalueofk areseparatedbytra- boundary (illustrated in Fig. 1a for N =2). In order to
jectoriesthatinvolveavariationinthedegeneracyofthe determine the dimension of the ground state manifold,
groundstate. Wenotethatuptoanunimportantabelian we construct a set of commuting operators, which can
phase, the unitary transformation (5) may be thought of be used to characterize the ground states. Consider the
ascomposedofatransformationeiαπQˆ2i thatresultsfrom operators: eiπQj ≡ ei(θj+1−θj), j = 1,...,N, where θj is
aninterchangeofanyons,multipliedbyatransformation a θ field evaluated at an arbitrary point near the middle
e2αmπiQˆik that results from a vortex encircling the i’th of the jth FM region. The origin (x=0) is chosen to lie
superconducting region 2αk/m times. within the first FM region (see Fig. 1a). The operator
Non-abelian statistics is the cornerstone of topological θN+1 islocatedwithinthisregion,totheleftoftheorigin
quantum computation5,29, due to possibility it opens for (x < 0), while θ1 is to the right of the origin (x > 0).
the implementation of unitary transformations that are The fields θ, φ satisfy the boundary conditions eiπQtot =
topologicallyprotectedfromdecoherenceandnoise. Itis ei[θ(L−)−θ(0+)] and eiπStot = ei[φ(L−)−φ(0+)], where L is
thennaturaltoexaminewhetherthenon-abeliananyons the perimeter of the system, and Q , S are the total
tot tot
that we study allow for universal quantum computation, charge and spin on the edge, respectively.
that is, whether any unitary transformation within the Since we are in the gapped phase of the sine-Gordon
ground state subspace may be approximated by topo- model of Eq. (6), we expect in the thermodynamic limit
logical manipulations of the anyons30. We find that, at (wherethesizeofall ofthesegmentsbecomeslarge)that
least for unitary time evolution (i.e., processes that do theθfieldisessentiallypinnedtotheminimaofthecosine
notinvolvemeasurements)theanswertothisquestionis potentialintheFMregions. (Similarconsiderationshold
negative, as it is for the integer case. for the φ fields in the SC regions.) In other words, the
θ →θ+π/m symmetry is spontaneously broken. In this
phase,correlationsofthefluctuationsofθdecayexponen-
III. EDGE MODEL tially on length scales larger than the correlation length
ξ ∼ u/∆ , where ∆ is the gap in the FM regions (see
F F
The edge states of a FTI are described by a hydrody- AppendixAforananalysisofthegappedphase). There-
namic bosonized theory40,41. The edge effective Hamil- fore, one can construct approximate ground states which
tonian is written as are characterized by (cid:104)ei(θj+1−θj)(cid:105) = (cid:104)eiπQj(cid:105) ≡ λj (cid:54)= 0,
ˆ where λj = |λ|eimπqj, where qj ∈ {0,...,2m−1} can
(cid:20) (cid:21) be chosen independently for each FM domain. The en-
mu 1
H = 2π dx K(x)(∂xφ)2+ K(x)(∂xθ)2 ergy splitting between these ground states is suppressed
ˆ in the thermodynamic limit as e−R/ξ, where R is the
− dx[g (x)cos(2mφ)+g (x)cos(2mθ)]. (6) length each region, as discussed below and in Appendix
S F
A.
6
Inaddition,eiπStot commutesbothwiththeHamilton- V. INTERFACE OPERATORS
ain and with eiπQj. Therefore the ground states can be
chosentobeeigenstatesofeiπStot,witheigenvalueseimπs, We now turn to define physical operators that act on
s ∈ {0,...,2m−1}. We label the approximate ground the low-energy subspace. These operators are analogous
states as |{q};s(cid:105) ≡ |q1,...,qN;s(cid:105), where |{q};s(cid:105) satis- totheMajoranaoperatorsinthem=1case,inthesense
fies that (cid:104){q};s|eiπQj|{q};s(cid:105)=|λ|eimπqj. that they can be used to express any physical observable
For a large but finite system, the |{q};s(cid:105) states are in the low-energy subspace. They will be useful when
not exactly degenerate. There are two effects that lift we discuss topological manipulations of the low-energy
the degeneracy between them: intra-segment instanton subspace in the next section.
tunneling events between states with different {q}, and We define the unitary operators eiφˆi and eiπQˆj such
inter-segment “Josephson” couplings which make the en-
that
ergy dependent on the values of {q}. However, both of
these effects are suppressed exponentially as e−R/ξ, as
they are associated with an action which grows linearly eiπQˆj|q1,...,qN,s(cid:105)=eiπmqj|q1,...,qN,s(cid:105), (8)
with the system size. Therefore, we argue that |{q};s(cid:105)
are approximately degenerate, up to exponentially small
corrections, for any choice of the set {q},s.
Similarly, one can define a set of “dual” operators eiφˆj|q ,...,q ,s(cid:105)=|q ,...,q +1,...,q ,s(cid:105). (9)
1 N 1 j N
eiπSj ≡ ei(φj−φj−1), j = 2,...,N, and eiπS1 =
eiπStot(cid:81)Ni=2e−iπSi. Although the SC regions are in the eiπQˆj isadiagonaloperatorinthe|{q},s(cid:105)basis,whereas
gimapapoefdthpehacsoer,reasnpdonthdeinfigelcdossiφnjeaproetepnintinaelds,nneoatretthheatmtihne- eiφˆj shifts qj by one. These operators can be thought
of as projections of the “microscopic” operators eiφj and
approximate ground states |{q};s(cid:105) cannot be further
eiπQj, introduced in the previous section, onto the low-
distinguished by the expectation values of the operators energy subspace. In addition, we define the operator Tˆ
eiπSj. In fact, these states satisfy (cid:104){q};s|eiπSj|{q};s(cid:105)→ s
that shifts the total spin of the system:
0 in the thermodynamic limit. That is because the oper-
ators eiπSj and eiπQj satisfy the commutation relations Tˆ |q ,...,q ,s(cid:105)=|q ,...,q ,s+1(cid:105). (10)
s 1 N 1 N
eiπSieiπQj =eimπ(δi,j+1−δi,j)eiπQjeiπSi, (7) Theoperators(9,10)willnotbeusefultous,sincethey
cannot be constructed by projecting any combination of
which can be verified by using the commutation relation
edge quasi-particle operators onto the low energy sub-
of the φ and θ fields. In the state |{q};s(cid:105), the value of
space. To see this, note that they add a charge of 1/m
eiπQj is approximately localized near eimπqj. Applying and zero spin or spin 1/m with no charge. As a result,
theoperatoreiπSj tothisstateshiftseiπQj toeiπ(Qj+m1), they violate the constraint between the total spin and
as can be seen from Eq. (7). This shift implies that
charge, Eq.(1). However, theseoperatorscanbeusedto
the overlap of the states |{q};s(cid:105) and eiπSj|{q};s(cid:105) decays construct the combinations
exponentially with the system size.
Overall,thereare(2m)N+1distinctapproximateeigen- j
state |{q};s(cid:105), corresponding to the 2m allowed values of χ2j,σ =eiφˆj(Tˆs)σ(cid:89)eiσπQˆi,
charges q of each individual SC segment, and the total i=1
j
spin s, which can also take 2m values. Not all of these j
states, however, are physical. Labelling the total charge χ2j+1,σ =eiφˆj+1(Tˆs)σ(cid:89)eiσπQˆi, (11)
byanintegerq =(cid:80)N q ,weseefromEq.(1)thatsand i=1
j=1 j
qmustbeeitherbothevenorbothodd,correspondingto
where σ = ±1. These combinations, which will be used
a total even or odd number of fractional quasi-particles
below, correspond to projections of local quasiparticle
in the bulk of the system. Due to this constraint, the
operators onto the low energy manifold. Indeed, the op-
number of physical states is only 1(2m)N+1.
2 erators χ (σ = ±1) carry a charge of 1/m and a spin
In a given sector with a fixed total charge and total j,σ
of ±1/m (as can be verified by their commutation re-
spin,thereareN =(2m)N−1groundstates. Form=1,
gs lations with the total charge and total spin operators).
we get N = 2N−1 for each parity sector, as expected
gs Therefore, their quantum numbers are identical to those
for 2N Majorana states located at each of the FM/SC
ofasinglefractionalquasi-particlewithspinupordown.
interfaces10.
Moreover, the commutation relations satisfied by χ
i,σ
Thegroundstatedegeneracyinthefractionalcasesug-
and for i<j,
gests that each interface can be thought of as an anyon
√
whosequantumdimensionis 2m. Thisisreminiscentof
recentlyproposedmodelsinwhichdislocationsinabelian χi,σχj,↑ =e−imπχj,↑χi,σ,
topological phases carry anyons with quantum dimen-
sions which are square roots of integers42–44. χi,σχj,↓ =eimπχj,↓χi,σ, (12)
7
S
(a) 2 2 3 (b) TableI.Summaryofthebraidingadiabatictrajectory(shown
also in Fig. 2b). There are three stages, α = I,II,III, along
Q II Q 1 2 1 2 1 2
1 2 eachofwhichtheparameterλ variesfrom0to1. TheHam-
1 I III 4 litonian in each stage is writteαn in the middle column, where
weusethenotationH =−t χ χ† +h.c.(t arecomplex
4 3 4 3 4 3 ij ij j,↑ i,↑ ij
S S parameters). The right column summarizes the symmetry
1 3 (I) (II) (III)
operators which commute with the Hamiltonian throughout
6 5
Q each stage.
3
Stage Hamiltonian Symmetries
Figure 2. Braiding process. (a) An FTI disk with six I (1−λI)H12+λIH23 eiπQˆ3, eiπSˆ3
SC/FM segments. In stages I, II and III of the braiding pro- II (1−λII)H23+λIIH24 eiπQˆ3, e−iπSˆ1
cisestsu,rqnueadsio-pnabrteitcwleeteunntnheelinSgC(/rFepMreisnetnetrefdacbeys.bl(ube)sRoleidprleinseens)- III (1−λIII)H24+λIIIH12 eiπQˆ3, e−iπQˆ2eiπSˆ3
tation of the braiding procedure, involving interfaces 1, 2, 3
and4. Inthebeginningofeachstage,thetwointerfacescon-
nectedbyasolidlinearecoupled;duringthatstage,thebond inal form, but the state of the system does not return to
represented by a dashed line is adiabatically turned on, and the initial state. The adiabatic evolution corresponds to
simultaneously the solid bond is turned off. By the end of
a unitary matrix acting on the ground state manifold.
stage III, the system returns to the original configuration.
Below, we analyze a braid operation between nearest-
neighborinterfaces,whichwelabel3and4(forlatercon-
coincidewiththoseofquasi-particleoperatorseiφ(x)±iθ(x) venience). The operation consists of three stages, which
are described pictorially in Fig. 2b. It begins by nucle-
localizedattheSC/FMinterfaces(fori=j,[χ ,χ ]=
j,↑ j,↓
ating a new, small, segment which is flanked by the in-
0 if j is odd, and satisfy χ χ = e2iπ/mχ χ if j is
j,↑ j,↓ j,↓ j,↑ terfaces 1 and 2. At the beginning of the first stage, the
even). Note that in our convention, for j odd, χ cor-
j,σ smallsizeofthenewsegmentmeansthatinterfaces1and
responds to the interface between the segments labelled
2 are coupled to each other, and all the other interfaces
by eiπSˆj and eπQˆj, where for j even, between eiπQˆj and
are decoupled. During the first stage, we simultaneously
eiπSˆj+1, see Fig. 2 a. bring interface 3 close to 2, while moving 1 away from
Therefore, the operators χ correspond to quasi-
j,σ both 2 and 3, such that at the end of the process only
particlecreationoperatorsattheSC/FMinterfaces,pro-
2 and 3 are coupled to each other, while 1 is decoupled
jected onto the low-energy subspace. This conclusion is
from them. In the second stage, interface 4 approaches
further supported by calculating directly the matrix ele-
3, and 2 is taken away from 3 and 4. In the final stage,
mentsofthemicroscopicquasi-particleoperatorbetween
we couple 1 to 2 and decouple 4 from 1 and 2, such that
the approximate ground states, in the limit of strong co-
theHamiltonianreturnstoitsinitialform. Inthefollow-
sinepotentials(seeAppendixA).Thiscalculationreveals
ing,weanalyzeanexplicitHamiltonianpathyieldingthis
that the matrix elements of the quasi-particle operators
braid operation, which is summarized in Table I. Later,
withinthelow-energysubspaceareproportionaltothose
we shall discuss the conditions under which the result is
of χ , and that the proportionality constant decays ex-
j,σ independentofthespecificfromoftheHamiltonianpath
ponentially with the distance of the quasi-particle oper-
representing the same Braid operation.
ator from the interface. We note that the commutation
relations of Eq. (12) appear in a one dimensional lattice
model of “parafermions”45,46.
B. Ground state degeneracy
VI. TOPOLOGICAL MANIPULATIONS
To analyze the braiding process, we first need to show
thatitdoesnotchangethegroundstatedegeneracy. We
A. setup
consider a disk with a total of N = 3 segments of each
type. The ground state manifold, without any coupling,
The braiding process is facilitated by deforming the is (2m)2 fold degenerate. We define operators H ,H
12 23
droplet adiabatically, such that different SC/FM inter- and H , the Hamiltonians at the beginning of the three
24
facesarebroughtclosetoeachotherateverystage. Prox- stages I, II, III. These are given by
imity between interfaces essentially couples them, by al-
lowing quasi-particles to tunnel between them. We shall
H =−t χ χ† +h.c.. (13)
assume that only one spin species can tunnel between jk jk j,↑ k,↑
interfaces. The reason for this assumption will become
clearinnextsections,andweshallexplainhowitisman- where the tjk are complex amplitudes.
ifested in realizations of the model under consideration. Consider first the initial Hamiltonian (see Table I),
At the end of the process, the droplet returns to its orig- given by
8
C. Braid matrices from Berry’s phases
(cid:16) (cid:17)
H =−t χ χ† +h.c.=−2|t |cos πQˆ +ϕ .
12 12 2,↑ 1,↑ 12 1 12 The evolution operator corresponding to the braid op-
(14) eration can thus be represented as a block-diagonal uni-
Here, ϕ =arg(t ). It is convenient to work in the ba- tary matrix, in which each (2m)×(2m) block acts on
12 12
sisofeigenstatesoftheoperatorseiπQˆ1,eiπQˆ2,eiπQˆ3,and a separate energy subspace. We are now faced with
eiπSˆtot, which we label by |q1,q2,q3;s(cid:105). The total charge the problem of calculating the evolution operator in the
(cid:80) ground state subspace. Let us denote this operator by
and spin are conserved, and we may set q = 0 and
j j Uˆ , corresponding to a braiding operation of interfaces
s=0. Then,astateinthe(2m)2-dimensionallow-energy 33a4nd 4. The calculation of Uˆ can be done analytically
subspace can be labelled as |q ,q (cid:105), where q is fixed to 34
2 3 1 byusingthesymmetrypropertiesofHamiltonianateach
q =−q −q . TheinitialHamiltonian(14)isdiagonalin
1 2 3 stage of the evolution.
thisbasis,andthereforeitseigen-energiescanbereadoff
easily: E (q ,q ) = −2|t |cos(cid:2)−π (q +q )+ϕ (cid:3). We begin by observing that, since eiπQˆ3 always com-
12 2 3 12 m 2 3 12 mutes with the Hamiltonian, Uˆ and the evolution op-
For generic ϕ , there are 2m ground states. Since this 34
12
erators for each stage are diagonal in the the basis of
FMsegmentisnucleatedinsideaSCregion,itstotalspin
is zero, and the ground states are eiπQˆ3 eigenstates. In every stage, the adiabatic evolu-
tion maps eiπQˆ3 eigenstates between the initial and final
|Ψi(q3)(cid:105)=|q2 =−q3,q3(cid:105), (15) ground state manifolds while preserving the eigenvalue
q , and multiplies by a phase factor that may depend on
3
labelled by a single index q3 =0,...,2m−1. The resid- q3. This is explicitly summarized as
ual 2m-fold ground state degeneracy can be understood
as a result of the symmetries of the Hamiltonian. From
Eq. (14) eiπQˆ3 and eiπSˆ3 commute with H12. The com- Uˆα|Ψαi(q3)(cid:105)=exp(cid:0)iγα(q3)(cid:1)|Ψαf(q3)(cid:105). (18)
mutation relations between eiπQˆ3 and eiπSˆ3 ensures that
the ground state is (at least) 2m-fold degenerate by the Here, Uˆ is the evolution operator of stage α = I,II,III,
α
eignevalues of eiπQˆ3. and |Ψαi(f)(q3)(cid:105) are the ground states of the initial (fi-
Similar considerations can be applied in order to find nal) Hamiltonian in stage α, respectively, which are la-
tehraetgiorno.unTdhsetaotpeedreagtoenreeriaπcQˆy3tahlrwoauygshocuotmtmheutbersaiwdiinthgothpe- btheellepdhabsyesthacecirumeiuπQlˆa3teedigiennevaacluheos.f tLhiekeswtaigsee,s.γα(q3) are
Hamiltonian, at any stage. This can be seen easily from In order to determine γ (q ), we use the additional
α 3
thefactthatthesegmentlabelledbyeiπQˆ3 nevercouples symmetry operator Σ for each stage, as indicated in
α
to any other segment at any stage (see Fig. 2a). Using TableI.ThissymmetrycommuteswiththeHamiltonian,
the definition of the χiσ operators, Eq. (11), one finds and therefore also with the evolution operator for this
that stage [Σ ,Uˆ ] = 0. Acting with Σ on both sides of
α α α
(cid:16) (cid:17) (18), we get that
H =−2|t |cos πSˆ +ϕ , (16)
23 23 2 23
and Uˆ Σ |Ψα(q )(cid:105)=eiγα(q3)Σ |Ψα(q )(cid:105). (19)
α α i 3 α f 3
(cid:104) (cid:16) (cid:17) (cid:105)
H =−2|t |cos π Sˆ +Qˆ +ϕ . (17)
24 24 2 2 24 Furthermore, the relation eiπQˆ3Σα = eimπΣαeiπQˆ3 im-
In each stage, α = I,II,III, there is a symmetry op- plies that the operator Σα advances eiπQˆ3 by one incre-
ment,andthereforeforboththeinitialandfinalstageat
erator Σ that commutes with the Hamiltonian, and
α each stage we have,
satisfies ΣαeiπQˆ3 = e−imπeiπQˆ3Σα. We specify Σα for
each stage in the right column of Table I, and the afore-
mentioned relation can be verified using Eq. (2). This (cid:16) (cid:17)
Σ |Ψα (q )(cid:105)=exp iδα (q ) |Ψα (q +1)(cid:105), (20)
combinationofsymmetriesdictatesthatevery state is at α i(f) 3 i(f) 3 i(f) 3
least2mfolddegenerate,whereeachdegeneratesubspace
can be labelled by q3. Assuming that the special values where δiα(f)(q3) are phases which depends on gauge
ϕ ,ϕ =π(2l+1)/(2m)andϕ =πl/m(linteger)are choicesforthedifferenteigenstates,tobedeterminedbe-
12 23 24
avoided, the ground state is exactly 2m-fold degenerate low. Inserting(20)into(19),wegettherecursionrelation
throughout the braiding process (the special values for
the ϕ give an additional two fold degeneracy). Note
ij
that these conclusions hold for any trajectory in Hamil- γ (q +1)=γ (q )+δα(q )−δα(q ). (21)
α 3 α 3 f 3 i 3
tonian space, as long as the appropriate symmetries are
maintained in each stage of the evolution, and the acci- Note that while the phase accumulation at each point
dental degeneracies are avoided. alongthepathdependsongaugechoices,thetotalBerry
9
phase accumulated along a cycle does not. It is conve-
nient to choose a continuous gauge, for which the total
Berry’s phases are given by
=
Uˆ |Ψα(q )(cid:105)=exp(cid:0)i(cid:88)γ (q )(cid:1)|Ψα(q )(cid:105). (22)
34 i 3 α 3 i 3
α
A continuous gauge requires |Ψα(n )(cid:105) = |Ψα+1(n )(cid:105).
f 3 i 3
Therefore, the values of the phases δα depend only on
i(f)
three gauge choices. These are the gauge choices eigen-
1 2 3 1 2 3
statesoftheHamiltoniansH ,H ,andH ,whichcon- Q S Q S
12 23 24
stitute the initial Hamiltonian at the beginning of stages
I-III, as well as the final Hamiltonian for stage III.
Making the necessary gauge choice, allows us to solve Figure 3. Diagrammatic representation of the Yang-
Eq. (21) for γ (n ), yielding the total Berry phase (the Baxter equations (Eq. D2). Three interfaces 1,2,3 are
α 3
braided in two distinct sequences. The Yang-Baxter equa-
details of the calculation are given in Appendix B)
tionsstatethattheresultsofthesetwosequencesofbraiding
π operations are the same.
γ(q )= (q −k)2. (23)
3 2m 3
The integer k depends on the choice for the phases ϕij. Equation(26)clearlyholdsbecausethespinorchargeop-
RecallthattheHamiltoniansHij,Eqs.(14),(16–17),have erators of non-nearest neighbor segments commute. Us-
an additional degeneracy for a discrete choice of the ϕij. ing(25), itisnotdifficulttoshowthat(27)holdsaswell
Anytwochoicesfortheϕij thatcanbedeformedtoeach (see Appendix D. Eq. (27) is depicted in Fig. 3). There-
otherwithoutcrossingadegeneracypointyieldthesame fore, Uˆ form a representation of the braid group. In
i,i+1
k. that respect, our system exhibits a form of non-abelian
The evolution operator for the braiding path can statistics. By combining a sequence of nearest-neighbor
be written explicitly by its application on the eigen- exchanges, an exchange operation of arbitrarily far seg-
states of the Hamiltonian in the beginning of the cycle, ments can be defined.
Uˆ3(4k)|Ψi(q3)(cid:105) = ei2πm(q3−k)2|Ψi(q3)(cid:105). Since by Eq. (15), In any physical realization, we do not expect to con-
the ground states of the initial Hamiltonian satisfy q2 = trol the precise form of the Hamiltonian in each stage.
−q3, this can be written in a basis-independent form in It is therefore important to discuss the extent to which
terms of the operator eiπQˆ2. Loosely speaking, Uˆ34 can the result of the braiding process depends on the details
be written as of the Hamiltonian along the path. We argue that the
braidingis“topological”,inthesensethatitis,toalarge
(cid:32)iπm(cid:18) k (cid:19)2(cid:33) degree, independent of these precise details.
Uˆ(k) =exp Qˆ − . (24)
34 2 2 m Toseethis,oneneedstonotethatthebraidingunitary
matrixwasderivedabovewithoutreferringtotheprecise
Alternatively, using the identity47 ei2πmq2 = adiabatic path in Hamiltonian space. All we used were
(cid:113) 1 (cid:80)2m−1eimπ(cid:16)pq−p22(cid:17)+iπ4, one can write t(hTeabsylemIm).eTtrhyesperosypemrtmieestroiefsthdeoHnoatmdiletpoenniadnoinntehaechprsetcaigsee
2m p=0
details of the intermediate Hamiltonian, but only on the
(cid:114) 2m−1 overallconfiguration,e.g.,whichinterfacesareallowedto
Uˆ3(4k) = 21m (cid:88) e−2imπ(p+k)2+iπ4 (cid:16)eiπQˆ2(cid:17)p. (25) couple in each stage.
p=0 InAppendixC1,westatemoreformallytheconditions
under which the result of the braiding is independent
In the case m=1, Uˆ reduces to the braiding rule of of details. Special care must be taken in stage III of
34
Ising anyons10–12. the braiding, in which quasi-particles of only one spin
Following a similar procedure, one can construct the species, e.g. spin up, must be allowed to tunnel between
operator representing the exchange of any pair of neigh- interfaces 2 and 4. We elaborate on the significance of
boring interfaces: Uˆ(k) = eiπ2m(Sˆj−k/m)2, Uˆ(k) = this requirement and the ways to meet it in the various
2j−1,2j 2j,2j+1
physical realizations in Appendix C2.
eiπ2m(Qˆj+1−k/m)2. In order for these operations to form
a representation of the braid group, it is necessary and
sufficient that they satisfy VII. BRAIDING AND TOPOLOGICAL SPIN OF
BOUNDARY ANYONS
(cid:104) (cid:105)
Uˆ(ki) ,Uˆ(kj) =0 (|i−j|>1), (26)
i,i+1 j,j+1
In the previous section, we derived the unitary matrix
Uˆ(k1) Uˆ(k2) Uˆ(k1) =Uˆ(k2) Uˆ(k1) Uˆ(k2) . (27)
j,j+1 j+1,j+2 j,j+1 j+1,j+2 j,j+1 j+1,j+2 representing braid operations by an explicit calculation.
10
Inthefollowing,weshalltrytoshedlightonthephysical ber. This therefore suggests the following fusion rules
picture behind these representations. To do so, we show
X×X =0+1+...+2m−1
that the results of the previous section can be derived
almost painlessly, just by assuming that the representa- q1×q2 =(q1+q2) mod 2m (28)
tionofthebraidshavepropertieswhichareanalogousto
Wenotethatthelabellingq =0,1...,2m−1doesnotde-
thoseofanyonsintwodimensions. Thefirstandmostba- 1
pend on the gauge choices in the definition of the opera-
sicassumption,isverynatural: thereexistsatopological
torsexp(iπQ). Equation(28)suggeststhatthelabelling
operation in the system which corresponds to a braid of
can be defined by the addition law for charges, in which
two interfaces, in that the unitary matrices representing
each type of charge plays a different role. Indeed, this
this operation obey the Yang-Baxter equation.
addition rule has a measurable physical content which
The operations we consider braid two neighboring in- does not depend on any gauge choices.
terfaces, butdo not changethetotalcharge(spin)inthe In two dimensional theories of anyons, it is convenient
segment between them. This results from the general to think about particles moving in the two dimensional
form of the braid operations - to exchange two interfaces plane, and consider topological properties of their world
flankingaSC(FM)segment,weusecouplingstoanaux- lines (such as braiding). In this paper, we have de-
iliarysegment,ofthesametype. Therefore,charge(spin) finedbraidingbyconsideringtrajectoriesinHamiltonian
canonlybeexchangedwiththeauxiliarysegment. Since space. In the following, we represent these Hamiltonian
the auxiliary segment has zero charge (spin) at the be- trajectoriesasworld-linesoftherespective“particles” in-
ginning and end of the operation, the charge of the main volved, keeping in mind that they do not correspond to
segment cannot change by the operation. Indeed, this motion of objects in real space.
can be seen explicitly in the analysis presented in the We are now ready to define the TS in our system. In
previous section. As a result, the unitary matrix repre- short, the TS of a particle is a phase factor associated to
senting the braid operation is diagonal with respect to the world line appearing in Fig 4(a). For interfaces, it is
the charge (spin) of the segment. concretely defined by the phase acquired by the system
by the following sequence of operations, as illustrated in
Thederivationnowproceedsbyconsideringaproperty
Fig 4(e): (i) nucleation of a segment to the right (by
ofanyonscalledthetopologicalspin(TS).Intwodimen-
convention) of the interface X (note that the notation
sions, the topological spin gives the phase acquired by a 1
X corresponds to particle X at coordinate r ). The
2π rotation of an anyon. For fermions and bosons, the 1 1
total spin or charge of this segment is zero (the nucle-
topological spin is the familiar −1 and +1 respectively
ation does not add total charge to the system). The
(corresponding to half-odd or integer spins). There is a
couplings between X and X flanking the new segment
closeconnectionbetweenthebraidmatrixforanyonsand 2 3
is taken to zero, increasing the ground state degeneracy
theirtopologicalspin. Intwodimensions, theserelations
by a factor of 2m. (ii) A right handed braid operation
have been considered by various authors9,48. The sys-
is performed between X and X . (iii) The total charge
tem under consideration is one dimensional, and there- 1 2
q of the segment between X and X is measured, and
foreseeminglydoesnotallowa2π“rotation” ofaparticle. 2 3
we consider (post-select) only the outcomes correspond-
However, as we shall explain below, the TS of a particle
ing to zero charge. Therefore, the system ends up in
canbedefinedinoursystemusingtherelationsoftheTS
the same state (no charges have been changed anywhere
to the braid matrix. We shall then see how to use these
in the system), up to a phase factor. Importantly, this
relationstoderivethepossibleunitaryrepresentationsof
phase factor does not depend on the state of the system,
the braid operations in the system at point.
since the operation does not change the total charge in
In our one dimensional system, we consider the TS the segment of X and X (see Appendix E for a more
1 2
of two different kinds of objects (particles) - interfaces, detailed discussion). We can therefore define this phase
which we denote by X, and the charge (or spin) of a factor as θ , the topological spin of particle type X.
X
segment, which we shall label by q = 0,1,..2m−1. In In order to define θ , the topological spin of a charge
q
what follows, we need to know how to compose, or fuse q, we first need to define the operation corresponding to
different objects in our system. As we saw above, two an exchange of two charges. Consider the sequence of 4
interfaces yield a quantum number exp(iπQ) (exp(iπS)) righthandedexchangesoftheinterfaces, asinFig.4(c).
which is the total charge (spin) in the segment between The figure suggests that this sequence should yield an
them, respectively. Suppose we consider two neighbor- exchange of the fusion charges q and q of the two pairs
1 2
ing SC segments with quantum numbers exp(iπQ ) and of X particles, as would indeed be the case for anyons
1
exp(iπQ ), and we “fuse” them by shrinking the FM re- in two dimensions (see Appendix. E for more details).
2
gion which lies between them. This results in tunneling Our second assumption is that this is indeed the case.
of fractional quasi-particles between the two SC regions, Since by Eq. (28) there is only one fusion channel for
and energetically favors a specific value for exp(iπS) in the q ’s, the state is multiplied by a phase factor which
i
the FM region. The two SC segments are for all pur- depends only on q and q - the charges q are abelian.
1 2 i
poses one, where clearly, in the absence of other cou- It is straightforward to check that exchanges of charges
plings,exp(iπ(Q +Q ))remainsagoodquantumnum- satisfy the Yang-Baxter equation.
1 2