Dimensional hierarchy of fermionic interacting topological phases

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by  Raquel Queiroz

Dimensionalhierarchyoffermionicinteractingtopologicalphases
Raquel Queiroz,1, Eslam Khalaf,1, and Ady Stern2
∗ †
1Max-Planck-Institutfu¨rFestko¨rperforschung, Heisenbergstrasse1, D-70569Stuttgart, Germany
2DepartmentofCondensedMatterPhysics, WeizmannInstituteofScience, Rehovot76100, Israel
(Dated:November15,2016)
Wepresentadimensionalreductionargumenttoderivetheclassificationreductionoffermionicsymmetry
protectedtopologicalphasesinthepresenceofinteractions. Thedimensionalreductionproceedsbyrelating
the topological character of a d-dimensional system to the number of zero-energy bound states localized at
zero-dimensionaltopologicaldefectspresentatitssurface.Thiscorrespondenceleadstoageneralconditionfor
symmetrypreservinginteractionsthatrenderthesystemtopologicallytrivial,andallowsustoexplicitlywritea
6 quarticinteractiontothisend.Ourreductionshowsthatallphaseswithtopologicalinvariantsmallerthannare
1 topologicallydistinct,therebyreducingthenon-interactingZclassificationtoZ .
n
0
2 PACSnumbers:71.10.Pm,73.20.-r:,03.65.Vf

v
o
Topologicalphasesofmatterarepresentlyoneofthemain tween the topological invariant ν of the d-dimensional bulk
N
researchtopicsincondensedmatterphysics.Thediscoveryof andthenumbern ofzero-energyboundstateslocalizedat
0D
4 time-reversal invariant topological insulators (TIs) [1, 2] and zero-dimensional(0D)topologicaldefectsonitssurface. We
1 superconductors(TSCs)hasled,amongotherresults,toasys- arguethatwhenthezeromodes,localizedinany0Dtopolog-

tematicclassificationoftopologicalphasesofnon-interacting ical defect, are gapped by an interaction, the state becomes
]
l fermions in a general spatial dimension and symmetry class topologically trivial under the same interaction. This consti-
e
- [3–5]. Similar to the quantum Hall states, TIs and TSCs are tutes a general criterion on any interaction that allows for a
r gapped systems hosting gapless modes on their surfaces, in- change in the topological sector. The argument is made ex-
t
s sensitive to small perturbations [6, 7]. However, the robust- plicitbypiercingthesurfacewithalatticeofdefectsandpre-
.
t ness of these surface states relies on the existence of dis- sentingaconcreteexampleofaquarticinteractionthatgapsa
a
m crete antiunitary symmetries, either time-reversal (TRS), , generalsurface.Ouranalysisreproducestheclassificationob-
T
and (or) particle-hole symmetry, C, being denoted symme- tainedin[23],withoutmakinganassumptionontheformof
-
d tryprotectedtopological(SPT)phases. Anaturalquestionis the interaction. Thus, we conclude that lifting the restriction
n whetherSPTphasesarerobustinthepresenceofinteractions ofquarticinteractionsdoesnotaltertheclassification.
o
c thatdonotbreakexplicitlyorspontaneouslytheirprotecting There is an important distinction between classes with Z
[ symmetries. Fidkowski and Kitaev [8] provided the first ex- topologyinevenandodddimensions. Theformerhostchiral
ampleinwhichaninteractioncouldadiabaticallyconnecttwo modes on the boundaries whose gapping is forbidden by the
2
states that are topologically distinct in its absence. This ex-
v conservation of energy; while the latter host nonchiral (heli-
ampleinvolvedonedimensional(1D)spinlessp-wavesuper-
6 cal) boundary modes whose protection depends on the pres-
9 conductorwithTRS,whereitwasshownthatinthepresence ence of chiral symmetry. Following Refs. [12, 13], we con-
5 ofaninteractioninvolvingeightMajoranaoperatorsthenon-
struct nonchiral SPT phases in even dimensions by combin-
1 interactingtopologicalclassificationisreducedfromZtoZ .
8 ing two systems with opposite chirality and Chern invariant,
0
. TwogappedquadraticHamiltonianswhosetopologicalinvari- adding a Z2 unitary symmetry R preventing the coupling of
1 antsdifferbyeightcanbeconnectedbyagap-preservingtra-
modes with opposite chirality. The resulting classes, which
0
jectorythatinvolvestheinteraction. Consequently,eightMa-
we refer to as prime () classes, naturally generalize SPT
6 (cid:48)
joranazeromodeslocalizedattheinterfacebetweenthesetwo
1 phases to even dimensions, making our analysis equally ap-
: phases will be gapped by the interaction. Subsequently, the plicabletoalldimensions.
v
full topological classification of interacting 1D SPT phases
i We pursue an analogous path to Ref. [25], in its descrip-
X was obtained [9–11], as well as some examples in two [12–
tionoftheten-foldclassificationofnon-interactingtopologi-
r 14] and three dimensions [15–17]. An exhaustive classifica-
a cal states using a eight- or two-hour “Bott clock”. We relate
tion of SPT phases still remains subject of intense research
the topological character of a SPT class with symmetry s in
[18–24], where methods such as cobordism, group (super)-
dimensiondtoonewiths+1andd+1(Fig.1),andfindthat
cohomology and non-linear sigma model have been recently
in the presence of interactions, Z is reduced to Z , with n
used. n
violatingtheclockperiodicity. Thisisrepresentedasaspiral
Inthiswork,westudythereductionofthetopologicalclas- clockinFigs.1(a)-(c),distinguishedbytheird=1symmetry
sification of interacting fermionic phases in a given dimen- classes: (a)BDI(b)CIIand(c)AIII.Generally,
sion and symmetry class, when this classification in the ab-
sence of interactions is Z. That is, we classify interacting n=2(cid:98)d−21(cid:99)µn0, (1)
fermionic phases which are adiabatically connected to non-
interactingones. Wederiveandemployacorrespondencebe- where n = 4,2,4 for (a) to (c), respectively, and µ = 2
0
{ }
2
forclassesBDI,D andDIII,whentheyarerealizedbytriplet (a) AI0 (b) AI0
(cid:48)
CI BDI CI BDI
superconductors. Forallothercasesµ=1. 32 64
Westartbyreviewinghowthe0Dzeromodesattheendsof 32 d = 14 6(1428) …4 4 8
(8) (16)
1Dsystemsreflecttheirtopologicalinvariants, andhowthey 16 d = 2 32 2
may be gapped by interactions [10, 11, 16]. We then intro- C0 (84)(16248) D0 C0 d = 2 (168) D0
ducethe0Dsurfacedefectsintwoandthreedimensions,and 16 8 (16)8… 322d = 1 16
their reflection of the bulk topological invariant. Following (256)128 16 (32)
CII DIII CII DIII
that, we discuss the possible interaction-induced gapping of
AII0 AII0
zero modes within such a defect, the gapping of a lattice of
such defects and the derivation of the parameter n. Finally, (c) AIII 4 8 8 4 A0
d = 1 d = 3 d = 2
wegeneralizetoanydimension.
Onedimension–WefocusonDiracHamiltonians(DH)as FIG.1. ClassificationofinteractingfermionicSPTphaseswithZ
n
representativesofthedifferenttopologicalsectors [26]. The topology,withngivenbyEq.(1)(blueforµ = 2). Thesymmetry
minimalDHinclassAIIIreads classesarearrangedtoformaeight-ortwo-hour“Bottclock”,start-
ingatthe1Dsymmetryclasses,(a)BDI,(b)CIIand(c)AIII,tracing
(cid:90)
each series clockwise for increasing dimensions. At each even to
= dxc (iσz∂ +m(x)σy)c, c=(c ,c )T, (2)
HAIII † x L R odddimensionstep,nisincreasedbyafactorof2. The8(2)-hour
periodicityofthereal(complex)classesisnotsatisfied,leadingtoan
for c complex fermion operators. Eq. (2) is invariant un- infinitespiral.
L,R
der the chiral symmetry defined by S−1cL,RS = c†R,L and
1i = i. A 0D edge can be implemented by forc-
−
S S −
ing the mass m(x) to change sign at x = 0. Choosing d = 2 does not increase n. Class D(cid:48) can be constructed by
m(x)=tanhxwithoutlossofgenerality,weobtainthezero addingtwocopiesof2Dp-wavesuperconductorswithbroken
energy localized operator ψ = (cid:82)dx(c +c )sech x, obey- TRS,bearingelectronsofoppositespindirectionandgapless
L R
ing 1ψ = ψ . For a set of AIII chains there is one edge modes of opposite chirality. The resulting system has
− †
zeroSmodeSat the end of each chain, so that n0D = ν, i.e., TRS with T2 = −1, but an additional Z2 unitary symme-
the number of zero energy end modes equals the topological try, R, distinguishes it from the class DIII, in which an even
invariant. Inter-chaincouplingisrestrictedbythesymmetry. number of pairs of counter-propagating edge modes may be
MasstermsMijψi†ψj areforbiddensincehermiticityrequires gappedwithoutviolatingTRS[27]. ThesymmetryRcanbe
MQuiajrt=ictMerjm∗i,s,wohniltehechoitrhaelrsyhmanmd,eatrlylorweqfuoirreasfuMlliyjs=ym−mMetrj∗iic. puhloys2ic(aclolynsiemrvpalteimonenotfedtheeitphaerrityas(a c1o)Nns↑e)rvsaattiisofnyionfgSRz2m=od1-
−
interactionbetweenthechains and ,R = 0[12,13]orasamirrorsymmetrysatisfying
{T }
R2 = 1and[ ,R] = 0[12]inwhichcaseD corresponds
(cid:48)
− T
Hint =Vψ1†ψ2ψ3†ψ4+h.c.. (3) tothecrystallinephaseDIII+R.Inbothoptions,thecombined
operator ˜ = Risantiunitaryandsatisfies ˜2 = 1. Here,
Herethesubscriptenumeratesthechains. Theinteraction(3) we choosTe theTfirst R realization. The D syTstem has zero
(cid:48)
hasaunique -symmetricgroundstateseparatedbyagapV ChernnumberN +N =0butanon-zerospinChernnum-
S
fromtheremainingstatesgivenby 0101 1010 ,with0,1 ber 1(N N ),↑result↓inginaZtopologicalclassificationat
theeigenvaluesofψi†ψi. Duetoth|eabse(cid:105)n−ce|ofed(cid:105)gemodes, then2on-i↑n−terac↓tinglevel.
weconcludethata1DAIIIsystemwithν =4becomestopo-
TheHamiltonianforνpairsofMajorana1Dedgemodesis
logically trivial once the interaction (3) is included. Thus,
n=4ford=1andclassAIII. (cid:90) ν
(cid:88)
Analogously, we can describe a chain in class BDI by HD(cid:48) =i dx ηa†σz∂xηa, η =(η↑,η↓)T, (4)
Eq.(2),wherethefermionicoperatorscarereplacedbyMa- a=1
joranaoperatorsη. Inthisbasis,chargeconjugationC isthe with and R defined by 1(η ,η ) = (η , η ) and
−
identity, and = . The edge Majorana bound states are R 1(Tη ,η )R=(η , η ).TWerel↑ate↓itTtotheBD↓I−Ma↑jorana
formedbyγ =T(cid:82)dxS(ηL+ηR)sechx,localizedatx=0.Pair- ch−ain s↑tud↓ied above↑b−y a↓dding the mass term m(x)η†σyη at
ingtheoperatorsγiintocomplexfermionsψi =γ2i 1+iγ2i, the edge [13]. Locally, the mass term breaks both and R,
we see that, Eq. (3) satisfies the symmetries of c−lass BDI but preserves the combination ˜ = R. ConsequTently, the
and can gap out groups of 8 Majoranas, reducing Z Z8 resulting edge Hamiltonian hasTthe aTntiunitary symmetry ˜
(n = 8). Similarly, we find n = 2 in class CII, whe→re the squaring to +1, representing spinless fermions (class BDIT).
Hamiltonianactsonspinfulfermionswithanintrinsicdouble Asintheprevioussection,wechoosem(x)tochangesignat
degeneracy. x = 0toforma0Dmassdefectandfindν Majoranabound
From one to two dimensions – In 2D we focus on class states at x = 0. Thus, in the noninteracting limit, the 1D
D. The analysis for A and C proceeds similarly. In all BDIsystemconstructedattheboundaryinheritsthetopology
(cid:48) (cid:48) (cid:48)
cases we find n = ν, such that the step from d = 1 to of the bulk 2D system of class D. If the mass oscillates in
0D (cid:48)
3
(a) D D+III� (b) D0 (c) D0 � x y iivnn+gde=oxn.(ηW1±,e0ic,sa0gn,iv0the)enannecxdoavnc−tcllyu=dbey(0tEh,qa0.t,(o40n),1wth)i.ethTn±ohner-Heinpatlmearciailcntotginntighaenlesavpceitln-,
⌘
+ the 2D boundary D system inherits the topology of the 3D
(cid:48)
D � + DIII DIII DIIIsystemwithaninvariantν/2ratherthanν. Addinganx-
DIII dependentmassm(x)=tanhx,the1Dedgemodebecomes
⌘ ⌘ BDI gappedanda0Ddefectisintroducedwithν/2zeromodes.
�
The surface Hamiltonian (5) may be equivalently written
FIG.2.Illustrationofthereductionschemefroma3DDIIIsystemto intermsofDiracmatrices, tobeconvenientlygeneralizedto
a1DBDIsystem,hostingzeroenergymodesγ: (a)Iso-spinmulti- higher dimensions. We construct the DH using Γ1,...,5 ma-
pletχa,Eq.(6),ofgappedsurfacemodes(lightgrey)withprotected trices satisfying the usual Clifford algebra Γi,Γj = 2δij.
1Dcounter-propagatingchiralmodesη±(blueandredlines),local- The matrices are chosen to be symmetric an{d antis}ymmetric
izedaty =0. (b)EffectiveD(cid:48)systemwithhelicalmodeη,Eq.(4).
for odd and even i, respectively. The kinetic part of Eq. (6)
(c)Gapped1DBDIsystem(darkergray)withzeromode(blueand
becomes Γ ∂ + Γ ∂ , and TRS acts as = Γ with
reddots)atx=y=0. 1 x 3 y T 2K
2 = 1. Byaddingthemasstermm(x)Γ +m(y)Γ ,we
2 4
T −
obtain a 0D defect localized at points where both m(x) and
m(y) change sign. The mass term breaks of the original
spacem(x) = m cos(qx), alatticeof0Ddefectsisformed T
0 Hamiltonian but leads to an emergent antiunitary symmetry
with a separation of 2π/q. When each defect contains eight
given by ˜ = Γ with ˜2 = 1. That is, the new Hamil-
zero modes, the interaction (3) between the zero modes may T 5K T
tonian is in class BDI, hosting Majorana bound states at the
be used to induce the topological transition from m > 0 to
0 0D topological defect [25]. For m(x) = ( 1)sxtanhx and
m <0withoutclosingtheenergygap[13]leadington=8. −
0 m(y) = ( 1)sytanhy, the zero modes are given explicitly
From two to three dimensions – We now follow the same byγ =(cid:82)d−2rsechxsechy vχ,withvthenon-zeroeigenvec-
csyosntsetmrucctliaosnsDto(cid:48)raenladtetociltass0sDDIsIuIrfiance3Ddetfoecittss.dInestcheinsdsatenpt 2wDe tio(ro1f)sthyΓe3pΓro4j)e.ctHioenreopsexr,astoyraPre=int41e(g1er+s.i(T−h1e)ssxigΓn1Γch2)a(n1ge+s
find a doubling of the ratio ν/n , giving rise to a doubling −
0D they introduce emerge naturally when the mass oscillates in
of n. Class DIII represents superconductors with 2 = 1.
space to introduce a lattice of defects, for example using the
Atthefreefermionlevel,DIIIhasaZclassificatioTnwhe−rea
massfunctionsm(x)=cosqxandm(y)=cosqy.
bulk topological invariant ν is associated with ν helical gap-
A local interaction that renders the system topologically
lesssurfacemodes. ThesurfaceHamiltonianreads
trivial must guarantee that the 0D defect zero modes are
(cid:90) ν gappedbyactingintheprojectedspaceofzeromodes. This
(cid:88)
HDIII =i d2r ηa†[σz∂x+σx∂y]ηa, (5) is required since the interaction matrix elements which cou-
a=1 ple the zero and high energy modes can be made arbitrarily
small by an appropriate choice of m, and the elements cou-
withνtakentobeeveninthediscussionbelow.Anaiveintro-
ductionofasurfacemasstermoftheformm(r)η σyηresults pling zero modes and bulk states vanish by locality. Hence,
†
we conclude that when the defect contains less than 8 zero
inagappedsurfaceofclassD,withνchiraledgemodesalong
modes the topological sector is stable to any interaction. In
a 1D defect (a line along which m = 0), which cannot be
thecasewherea0Ddefectcannotbeconstructed, forexam-
gapped. Gapping requires two time-reversed copies of class
pleforoddvaluesofν,itisalwayspossibletoreducethe2D
D at the surface, obtained by grouping the ν surface modes
intopairsdefininganisospindoubletχ = (ηT ,ηT ),and gaplessmodestoasingle1Dgaplessmodecoupledtoanum-
a 2a 1 2a
adding the mass term mχ σy τzχ, with τi t−he Pauli ma- berof0Dzeromodes. The1Dmodecanneverbegappedout
†
⊗ by interactions, guaranteeing the stability of the topological
trices in isospin space. Taking the mass to have an opposite
sector. Put together, these considerations imply the stability
signforthetwoisospindirectionsleadstoaclassD surface
(cid:48)
ofthe2DDIIIsurfacewithν <16toanyinteraction.
with counter-propagating chiral modes. Combining with
T
anisospinflipresultsinamodifiedTRS(Fig.2(a)). The2D For ν = 16, 0D defects containing 8 zero modes can be
surfaceHamiltonianreads, constructedandgappedbyinteractions.Wecanexplicitlyfind
one interaction that gaps the full surface by combining pairs
(cid:90) (cid:88)ν/2 of Majorana surface states into four complex fermions ψi =
H= d2r χ†a[i(σz∂x+σx∂y)⊗τ0+m(r)σy⊗τz]χa. χ2i 1+iχ2iinanalogytothe0Dcase,
a=1 −
(6) (cid:90)
HisinvariantunderthemodifiedTRST = iσyτxKandun- Hint = d2rV(ψ1†Γ5ψ2)(ψ3†Γ5ψ4), (7)
der R = τz satisfying 2 = 1, R2 = 1 and ,R = 0,
T − {T }
whichimplementstheD systemdescribedabove. Choosing with Γ = Γ Γ Γ Γ . This interaction reduces to the inter-
(cid:48) 5 1 2 3 4
m(r) to change sign along the y-direction, m(y) = tanhy, action (3) upon projecting to the space of zero modes in the
we find the 1D gapless modes η = (cid:82) dy sech y v χ with defect. Itrespectsboth and ˜ symmetries.
± ± T T
4
The gapping procedure of the 2D surface is similar to the thereforeforany0Ddefectinthelattice)reads
one employed to gap the 1D surface of the class D system
(cid:48) (cid:90)
a0tDdde=fec2t, t[u1n3n].elcInoutphleinagbbseentwceeeonfzienrtoermacotdioensiwnintheiinghebaocrh- Hint = dd−1rV(ψ1†Γ2d−1ψ2)(ψ3†Γ2d−1ψ4), (9)
ing defects creates a spectrum that is gapless at zero energy.
This spectrum is not identical to the one obtained in the ab- whereΓ2d 1 = (cid:81)di=−11αiβi. Forvaluesofν < n,atopolog-
icallyprote−cted0Ddefectcannotbeformed,orwillhostless
senceofthemassterms,butitslowenergycharacteristicsare
than four zero modes that cannot be gapped by any interac-
identical. The zero modes within each defect all share the
tion. Thus,anyinteractioninvolvinglessthatnfermionswill
samevaluesofs ,s andthustheinteraction(7)respectsthe
x y
beeitherprojectedtoonethatcannotlocallygapthesurface,
symmetry. Whentheinteractionisstrongerthanthehopping
oronethatisnotrelevantinthelowenergysubspace.
terms between defects, the surface is gapped. This proce-
dure bears similarity to the proliferation of monopoles used Theanalogousderivationofnfortherealclassesfindsthe
inRefs.[24,28]. same doubling of n0D whenever d is increased by two, as
described here for the complex classes. As explained below
Higher dimensions — We now generalize beyond d =
Eq. (1),however,thethreeZ-classseriesdifferinn inthe
1,2,3toobtainthecompleteclassificationoffermionicSPTs 0D
d=1case,andthisdifferencecarriesovertoalllargerdimen-
,assummarizedinFig.1andEq.(1). Westartwiththecom-
sions. Furthermore,thevalueofndependsalsoonthenature
plexseries(Fig.1(c)), andcommentontheextensiontoreal
ofthezeromodesbeingcomplexorMajoranafermions.
classes. We first identify the correspondence between ν and
n , and then introduce an interaction that gaps the surface Conclusion—Weshowedthatunderinteractionstheclassi-
0D
once it is pierced with a lattice of 0D defects. The (d 1)- ficationoftopologicalphaseswithchiralsymmetryisreduced
dimensionalsurfaceisdescribedby − fromZtoZn,andreproducedthevalueofnderivedin[23],
summarizedinEq.(1)andFig.1.Inourapproach,thegapless
(cid:90)
surfaceseparatingtopologicallydistinctphasesisreplacedby
= dd 1rχ (iα ∇+β M)χ. (8)
H − † · · agaplesslatticeofcoupled0Ddefects. Thesedefectsenclose
zeromodes,whosenumbern isdeterminedbydimension,
0D
Here, α = (α1,...,αd 1), β = (β1,...,βd 1) are Dirac bulksymmetriesandbulktopologicalinvariantν.Weidentify
matrices satisfying the C−lifford algebra. In d s−patial dimen- therelationbetweenν andn andfindittodoublewithan
0D
sions,theminimalmasslessDHwithchiralsymmetryhasdi-
increaseofthedimensionbytwo. Thenumberofzeromodes
mension 2(cid:98)d2(cid:99). This can be understood from the doubling of ford=1dependsonthesymmetryoftheproblem,leadingto
thebulkDHdimensionfromoddtoevend,duetotheintrin- adifferencebetweenthetheseries(a) (c)(Fig.1).Ourcon-
sicdoublingofclassA,andthefactthatthesurfaceandbulk −
(cid:48) struction establishes the stability of topological phases with
DHsdifferindimensionbyafactoroftwo.
ν <nforanysymmetry-preservinginteraction,andprovides
AgaplessedgeHamiltonianofasystemwithatopological a necessary and sufficient condition for ones that gap the n-
invariantνmaybeconstructedbyνcopiesoftheedgeHamil- boundary modes. It turns out it is always possible to find a
tonian (8) with β = 0, enlarging the size of the matrix by a quarticinteractionsatisfyingthisconditionforanarbitrarydi-
factorν,givingthecombineddimensionν2(cid:98)d2(cid:99). Weseekthe mensionandsymmetryclass,Eq.(9).
valueν forwhichamasstermthatallowstheformationofa WeacknowledgestimulatingdiscussionswithA.P.Schny-
0Ddefectmaybeintroduced. der,J.S.HofmannandB.A.Bernevig.Weacknowledgefinan-
A zero mode is an operator that commutes with (8). For cialsupportbytheEuropeanResearchCouncilundertheEu-
a single 0D defect, the i-th component of the mass vector is ropean Unions Seventh Framework Programme (FP7/2007-
chosentosatisfyMi =( 1)sitanhri.Thezeromodeisthen 2013) / ERC Project MUNATOP, Microsoft Station Q, Min-
γ = (cid:82)dd−1r(cid:81)di=−11sech−ri vχ, where v corresponds to the ervafoundation,andtheU.S.-IsraelBSF.
non-zeroeigenvectorofthed 1commutingprojectionoper-
atorsPisi = 21(1+i(−1)siαi−βi). Toobtainasinglenon-zero
eigenvaluetothissetofoperators,weneedanHamiltonianof
dimension2d 1. Theratioofthisdimensiontothedimension
−
ν2(cid:98)d2(cid:99) fixes n0D(ν) to be (cid:98)ν21−d2(cid:99) for even d and (cid:98)ν21−2d(cid:99) ∗† re..qkuheailraofz@@fkfkf.fm.mppgg.d.dee
foroddd.
[1] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann,
As before, a group of four complex fermions in a single
L. W. Molenkamp, X.-l. Qi, and S.-C. Zhang, Science 318,
defect may be gapped by interactions. The surface may be 766(2007).
gappedbypiercingitbyalatticeof0Ddefects,eachofwhich [2] D.Hsieh,D.Qian,L.A.Wray,Y.Xia,Y.S.Hor,R.J.Cava,
containingfourzeromodes,hencereducingtheZclassifica- andM.Z.Hasan,Nature452,970(2008).
tiontoZn,withn = 2d+22 forevendandn = 2d+23 forodd [[34]] AA..KP.itSacehv,niyndAerI,PSC.oRnyfue,reAnc.eFPurruoscaekeid,inagnsd(AAI.PW,2.0W09.)Lpu.d2w2.ig,
d.
PhysicalReviewB78,195125(2008).
Whentherearefourzeromodesinasingle0Ddefect,one [5] A.P.Schnyder,S.Ryu,A.Furusaki, andA.W.W.Ludwig,in
interactionthatreducestoEq.(3)foranychoiceofsi’s(and AIPConferenceProceedings(AIP,2009)p.10.
5
[6] M.Z.HasanandC.L.Kane,ReviewsofModernPhysics82, ofHighEnergyPhysics2015,52(2015).
3045(2010). [20] Z.C.GuandX.G.Wen,PhysicalReviewB90,115141(2014).
[7] X.-l.QiandS.-C.Zhang,ReviewsofModernPhysics83,1057 [21] Z. Bi and C. Xu, Physical Review B 91, 184404 (2015),
(2011). arXiv:1501.02271v1.
[8] L. Fidkowski and A. Kitaev, Physical Review B 81, 134509 [22] J.C.Wang,Z.-C.Gu, andX.-G.Wen,PhysicalReviewLetters
(2010). 114,031601(2015).
[9] A.M.Turner,F.Pollmann, andE.Berg,PhysicalReviewB83, [23] T.Morimoto,A.Furusaki, andC.Mudry,PhysicalReviewB
075102(2011). 92,125104(2015).
[10] L. Fidkowski and A. Kitaev, Physical Review B 83, 075103 [24] Y.-Z.YouandC.Xu,PhysicalReviewB90,245120(2014).
(2011). [25] J. C. Y. Teo and C. L. Kane, Physical Review Letters 104,
[11] E.TangandX.G.Wen,PhysicalReviewLetters109,096403 046401(2010).
(2012). [26] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig,
[12] H.YaoandS.Ryu,PhysicalReviewB88,064507(2013). NewJournalofPhysics12,065010(2010).
[13] X.L.Qi,NewJournalofPhysics15,65002(2013). [27] InthegeneralizationtoclassA(cid:48),werecoverchiralsymmetryby
[14] Z.C.GuandM.Levin,PhysicalReviewB89,201113(2014). imposing an additional Z symmetry (e.g. S mod 2). This
2 z
[15] C.WangandT.Senthil,PhysicalReviewB89,195124(2014). excludestheS conservingspinquantumHalleffect,forwhich
z
[16] T.Senthil,AnnualReviewofCondensedMatterPhysics6,299 theedgestatescannotbegappedbyaninteraction[29].
(2015). [28] Y. You, Y. BenTov, and C. Xu, arXiv preprint 1402.4151
[17] A. Vishwanath and T. Senthil, Physical Review X 3, 011016 (2014).
(2013). [29] M. Levin and A. Stern, Physical Review Letters 103, 196803
[18] T.Senthil,AnnualReviewofCondensedMatterPhysics6,299 (2009).
(2015).
[19] A.Kapustin,R.Thorngren,A.Turzillo, andZ.Wang,Journal