Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions

35 Pages·2011·0.94 MB·English

PHYSICALREVIEWB83,115132(2011)
Projectiveribbonpermutationstatistics:Aremnantofnon-Abelianbraidinginhigherdimensions
MichaelFreedman,1 MatthewB.Hastings,1ChetanNayak,1,2 Xiao-LiangQi,1,3KevinWalker,1andZhenghanWang1
1MicrosoftResearch,StationQ,ElingsHall,UniversityofCalifornia,SantaBarbara,California93106,USA
2DepartmentofPhysics,UniversityofCalifornia,SantaBarbara,California93106,USA
3DepartmentofPhysics,StanfordUniversity,Stanford,California94305,USA
(Received10October2010;revisedmanuscriptreceived11January2011;published23March2011)
Ina recent paper, Teoand Kane [Phys.Rev. Lett.104, 046401 (2010)]proposed athree-dimensional (3D)
modelinwhichthedefectssupportMajoranafermionzeromodes.Theyarguedthatexchangingandtwistingthese
defectswouldimplementasetRofunitarytransformationsonthezero-modeHilbertspacewhichisa“ghostly”
recollectionoftheactionofthebraidgrouponIsinganyonsintwodimensions.Inthispaper,wefindthegroup
T ,whichgovernsthestatisticsofthesedefectsbyanalyzingthetopologyofthespaceK ofconfigurationsof
2n 2n
2ndefectsinaslowlyspatiallyvaryinggappedfree-fermionHamiltonian:T ≡π (K ).Wefindthatthegroup
2n 1 2n
T =Z×Tr,wherethe“ribbonpermutationgroup”Tr isamildenhancementofthepermutationgroupS :
2n 2n 2n 2n
Tr ≡Z ×E((Z )2n(cid:2)S ).Here,E((Z )2n(cid:2)S )isthe“evenpart”of(Z )2n(cid:2)S ,namely,thoseelementsfor
2n 2 2 2n 2 2n 2 2n
whichthetotalparityoftheelementin(Z )2naddedtotheparityofthepermutationiseven.Surprisingly,Ris
2
onlyaprojectiverepresentationofT ,apossibilityproposedbyWilczek[e-printarXiv:hep-th/9806228].Thus,
2n
TeoandKane’sdefectsrealizeprojectiveribbonpermutationstatistics,”whichweshowtobeconsistentwith
locality.Weextendthisphenomenontootherdimensions,codimensions,andsymmetryclasses.Wenotethatour
analysisappliesto3DnetworksofquantumwiressupportingMajoranafermions;thus,thesenetworksarenot
requiredtobeplanar.Becauseitisanessentialinputforourcalculation,wereviewthetopologicalclassification
ofgappedfree-fermionsystemsanditsrelationtoBottperiodicity.
DOI:10.1103/PhysRevB.83.115132 PACSnumber(s): 73.20.−r,73.43.−f,71.10.Pm
I. INTRODUCTION superconductivity, is that they can be understood in a free-
fermionpicture.
In two dimensions (2D), the configuration space of n A collection of 2n Ising anyons has a 2n−1-dimensional
pointlikeparticlesC2Dismultiplyconnected.Itsfirsthomotopy
n Hilbert space (assuming a fixed boundary condition). This
group, or fundamental group, is the n-particle braid group,
canbeunderstoodintermsof2nMajoranafermionoperators
πte1r(cClon2Dck)w=isBene.xTchhaengbersaidσ gorfouthpeBitnhiasndge(nie+rat1e)dthbpyarctoiculnes- γi =γi†, i =1,2,...,n, one associated to each Ising anyon,
i
satisfyingtheanticommutationrules
satisfyingthedefiningrelations
{γ ,γ }=2δ . (2)
i j ij
σ σ =σ σ for|i−j|(cid:2)2,
i j j i The Hilbert space of 2n Ising anyons with a fixed boundary
(1)
σiσi+1σi =σi+1σiσi+1 for1(cid:3)i (cid:3)n−2. condition furnishes a representation of this Clifford algebra;
by restricting to fixed boundary condition, we obtain a
representation of products of an even number of γ matrices,
Thisisaninfinitegroup,evenforonlytwoparticles,because
(σ )m is a nontrivial element of the group for any m>0. In which has a minimal dimension 2n−1. When the ith and
i (i+1)thanyonsareexchangedinacounterclockwisemanner,
fact,evenifweconsiderdistinguishableparticles,theresulting
astateofthesystemistransformedaccordingtotheactionof
group, called the “pure Braid group,” is nontrivial. (For two
particles, the pure braid group consists of all even powers ρ(σi)=eiπ/8e−πγiγi+1/4. (3)
ofσ .)
1
In quantum mechanics, the equation π1(Cn2D)=Bn opens [There is a variant of Ising anyons, associated with SU(2)2
the door to the possibility of anyons.1,2 Higher-dimensional Chern-Simons theory, for which the phase factor eiπ/8 is
representations of the braid group give rise to non-Abelian replaced by e−iπ/8. In the fractional quantum Hall effect,
anyons.3–5 Recently there has been intense effort directed IsinganyonsaretensoredwithAbeliananyonstoformmore
toward observing non-Abelian anyons owing, in part, to complicated models with more particle species; the phase
their potential use for fault-tolerant quantum computation.6,7 factor depends on the model.] A key property, essential for
One of the simplest models of non-Abelian anyons is called applications to quantum computing, is that a pair of Ising
Ising anyons. They arise in theoretical models of the ν = anyons forms a two-state system. The two states correspond
5/2 fractional quantum Hall state8–11 (see also Ref. 12), tothetwoeigenvalues±1ofγiγj.Nolocaldegreeoffreedom
chiralp-wavesuperconductors,13,14asolvablemodelofspins can be associated with each anyon; if we insisted on do√ing
on the honeycomb lattice,15 and interfaces between super- so, we would have to say that each Ising anyon has 2
conductors and either three-dimensional (3D) topological internal states. In superconducting contexts, the γ ’s are the
i
insulators16 or spin-polarized semiconductors with strong Bogoliubov-de Gennes operators for zero-energy modes (or,
spin-orbitcoupling.17AspecialfeatureofIsinganyons,which simply,“zeromodes”)invortexcores;thevorticesthemselves
makes them relatively simple and connects them to BCS are Ising anyons if they possess a single such zero mode γ .
i
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FREEDMAN,HASTINGS,NAYAK,QI,WALKER,ANDWANG PHYSICALREVIEWB83,115132(2011)
Although the Hilbert space is nonlocal in the sense that it Withtheanswertothisquestioninhand,wecouldaddress
cannotbedecomposedintothetensorproductoflocalHilbert questionssuchasthefollowing.Weknowthata3Dincarnation
spaces associated with each anyon, the system is perfectly ofIsinganyonsisonepossiblerepresentationofT ;isa3D
2n
compatiblewithlocalityandarisesinlocallatticemodelsand versionofotheranyonsanotherrepresentationofT ?
2n
quantum-fieldtheories. Attempts to generalize the braiding of anyons to higher
In three or more dimensions, the configuration space of dimensions sometimes start with extended objects, whose
n pointlike particles is simply connected if the particles configuration space may have a fundamental group that is
are distinguishable. If the particles are indistinguishable, it richer than the permutation group. Obviously, if one has
is multiply connected, π (C3D)=S . The generators of the linelikedefectsin3Dthatareallorientedinthesamedirection,
1 n n
permutation group satisfy the relations (1) and one more, thenoneisessentiallybacktothe2Dsituationgovernedbythe
σ2 =1. As a result of this last relation, the permutation braid group. This is too trivial, but it is not clear what kinds
i
group is finite. The one-dimensional (1D) representations of extended objects in higher dimensions would be the best
of S correspond to bosons and fermions. One might have startingpoint.Whatisclear,however,isthatTeoandKane’s
n
hoped that higher-dimensional representations of S would topological defects must really be some sort of extended
n
give rise to interesting 3D analogs of non-Abelian anyons. objects. This is clear from the above-noted contradiction
However,thisisnotthecase,asshowninRefs.18and19:Any with the permutation group. It also follows from the “order
higher-dimensionalrepresentationofS ,whichiscompatible parameter”fields,whichmustdeformasthedefectsaremoved,
n
with locality, can be decomposed into the tensor product of aswewilldiscuss.
localHilbertspacesassociatedwitheachparticle.Forinstance, Inthispaper,weshowthatTeoandKane’sdefectsareprop-
suppose we had 2n spin-1/2 particles but ignored their spin erlyviewedaspointlikedefectsconnectedpairwisebyribbons.
values. Then we would have 22n states that would transform Wecalltheresulting2n-particleconfigurationspaceK .We
2n
intoeachotherunderpermutations.Clearly,ifwediscovered compute its fundamental group π (K ), which we denote
1 2n
suchasystem,wewouldsimplyconcludethatweweremissing by T and find that T =Z×Tr. Here, Tr is the “ribbon
2n 2n 2n 2n
some quantum number and set about trying to measure it. permutation group,” defined by Tr ≡Z ×E((Z )2n(cid:2)S ).
2n 2 2 2n
Thiswouldsimplyleadusbacktobosonsandfermionswith The group E((Z )2n(cid:2)S ) is a nonsplit extension of the
2 2n
additionalquantumnumbers.(Thecolorquantumnumberof permutationgroupS byZ2n−1,whichisdefinedasfollows:
2n 2
quarkswasconjecturedbyessentiallythiskindofreasoning.) Itisthesubgroupof(Z )2n(cid:2)S composedofthoseelements
2 2n
Thequantuminformationcontainedinthese22n stateswould for which the total parity of the element in (Z )2n added to
2
nothaveanyspecialprotection. theparityofthepermutationiseven.Theribbonpermutation
The preceding considerations point to the following ten- groupfor2nparticles,byTr,isthefundamentalgroupofthe
2n
sion. The Clifford algebra (2) of Majorana fermion zero reducedspaceof2n-particleconfigurations.
modes is not special to 2D. One could imagine a 3D (or Our analysis relies on the topological classification of
higher)systemwithtopologicaldefectssupportingsuchzero gapped free fermion Hamiltonians26,27—band insulators and
modes. But the Hilbert space of these topological defects superconductors—which is the setting in which Teo and
would be 2n−1 dimensional, which manifestly cannot be Kane’s3Ddefectsandtheirmotionsaredefined.Thestarting
decomposed into the tensor product of local Hilbert spaces point for this classification is reducing the problem from
associated with each particle, seemingly in contradiction classifying gapped Hamiltonians defined on a lattice to
with the results of Refs. 18 and 19 on higher-dimensional classifyingDiracequationswithaspatiallyvaryingmassterm.
representations of the permutation group described above. OnecanmotivatethereductiontoaDiracequationasTeoand
However, as long as no one had a 3D or higher system in Kanedo:TheystartfromalatticeHamiltonianandassumethat
hand with topological defects supporting Majorana fermion the parameters in the Hamiltonian vary smoothly in space,
zero modes, one could, perhaps, sweep this worry under the so that the Hamiltonian can be described as a function of
rug. Recently, however, Teo and Kane20 have shown that a boththemomentumk andthepositionr.Neartheminimum
3D system that is simultaneously a superconductor and a of the band gap, the Hamiltonian can be expanded in a
topologicalinsulator21–24(which,inmanybutnotallexamples, Diracequation,withaposition-dependentmassterm.Infact,
is arranged by forming superconductor-topological insulator Kitaev27 has shown that the reduction to the Dirac equation
heterostructures) supports Majorana zero modes at pointlike withaspatiallyvaryingmasstermcanbederivedmuchmore
topologicaldefects. generally:GappedlatticeHamiltonians,eveniftheparameters
To make matters worse, Teo and Kane20 further showed intheHamiltoniandonotvarysmoothlyinspace,arestably
that exchanging these defects enacts unitary operations on equivalenttoDiracHamiltonianswithaspatiallyvaryingmass
this 2n−1-dimensional Hilbert space, which are essentially term.Here,equivalence oftwoHamiltoniansmeansthatone
equal to (3). But we know that these unitary matrices form can be smoothly deformed into the other while preserving
a representation of the braid group, which is not the relevant locality of interactions and the spectral gap, while stable
groupin3D.Onewouldnaivelyexpectthattherelevantgroup equivalencemeansthatonecanaddadditional“trivial”degrees
is the permutation group, but S has no such representation of freedom (additional sites that have vanishing hopping
n
(and even if it did, its use in this context would contradict matrixelements)totheoriginallatticeHamiltoniantoobtain
locality, according to Refs. 18 and 19 and arguments in a Hamiltonian that is equivalent to a lattice discretization of
Ref. 25). So this begs the question: What is the group T theDiracHamiltonian.
2n
for which Teo and Kane’s unitary transformations form a BecausethisclassificationofDiracHamiltoniansisessen-
representation? tialforthedefinitionofK ,wegiveaself-containedreview,
2n
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following Kitaev’s analysis.27 Our exposition parallels the Theribbonpermutationgroupisaratherweakenhancement
discussion of Bott periodicity in Milnor’s book.28 The basic of the permutation group and, indeed, we conclude that Teo
idea is that each additional discrete symmetry that squares and Kane’s unitary operations are not a representation of the
to −1, which we impose on the system, is encapsulated by ribbon permutation group. However, they are a projective
an antisymmetric matrix that defines a complex structure on representationoftheribbonpermutationgroup.Inaprojective
RN, where N/2 is the number of bands (or, equivalently, representation,thegroupmultiplicationruleisonlyrespected
N is the number of bands of Majorana fermions). For any up to a phase, a possibility allowed in quantum mechanics.
given system, these are chosen and fixed. This leads to a Arepresentationρ (sometimescalledalinearrepresentation)
progressionofsymmetricspacesO(N)→O(N)/U(N/2)→ of some group G is a map from the group to the group of
U(N/2)/Sp(N/4)→··· as the number of such symmetries lineartransformationsofsomevectorspacesuchthatthegroup
is increased. Following Kitaev,27 we view the Hamiltonian multiplicationlawisreproduced:
as a final antisymmetric matrix that must be chosen (and
thus put almost on the same footing as the symmetries);it is ρ(gh)=ρ(g)·ρ(h), (5)
definedbyachoiceofanarbitrarypointinthenextsymmetric
ifg,h∈G.Particlestatisticsarisingasaprojectiverepresen-
space in the progression. The space of such Hamiltonians is
tationofsomegrouprealizesaproposalofWilczek’s,30albeit
topologically equivalent to that symmetric space. However,
fortheribbonpermutationgroupratherthanthepermutation
as the spatial dimension is increased, γ matrices squaring to
+1 must be chosen in order to expand the Hamiltonian in group itself. This difference allows us to sidestep a criticism
ofRead25 basedonlocality,whichTeoandKane’sprojective
the form of the Dirac equation in the vicinity of a minimum
representation respects. The group (Z )2n−1 is generated by
of the band gap. These halve the dimension of subspaces 2
of RN by separating it into their +1 and −1 eigenspaces 2n−1generatorsx1,x2,...,x2n−1,satisfying
and thereby lead to the opposite progression of symmetric
x2 =1,
spaces. Thus, taking into account both the symmetries of i
(6)
the system and the spatial dimension, we conclude that the x x =x x .
space of gapped Hamiltonians with no symmetries in d =3 i j j i
istopologicallyequivalenttoU(N)/O(N).(However,bythe However, the projective representation of (Z )2n−1, which
2
precedingconsiderations,thesamesymmetricspacealso,for gives a subgroup of Teo and Kane’s transformations, is an
instance, classifies systems with time-reversal symmetry in ordinarylinearrepresentationofaZ -centralextension,called
2
d =4.)AllsuchHamiltonianscanbecontinuouslydeformed theextraspecialgroupE1 :
2n−1
into each other without closing the gap, π (U(N)/O(N))=
0
0. However, there are topologically stable pointlike defects x2 =1,
classified by π (U(N)/O(N))=Z . These are the defects i
2 2 x x =x x for |i−j|(cid:2)2, (7)
whose multidefect configuration space we study in order to i j j i
seewhathappenswhentheyareexchanged. xixi+1 =zxi+1xi,
The second key ingredient in our analysis is a 1950’s- z2 =1.
vintagehomotopytheory,whichweusetocomputeπ (K ).
1 2n
WeapplythePontryagin-ThomconstructiontoshowthatK , Here,zgeneratesthecentralextension,whichwemaytaketo
2n
which includes not only the particle locations but also the bez=−1.Theoperationsgeneratedbythexi’sweredubbed
full field configuration around the particles [i.e., the way in “braidlessoperations”byTeoandKane20 becausetheycould
which the gapped free-fermion Hamiltonian of the system beenactedwithoutmovingthedefects.Whiletheseoperations
exploresU(N)/O(N)],istopologicallyequivalenttoamuch formanAbeliansubgroup ofT2n,theirrepresentationonthe
simpler space, namely, pointlike defects connected pairwise Majoranazero-modeHilbertspaceisnotAbelian—twosuch
byribbons.Inordertothencalculateπ1(K2n),werelyonthe operationsthattwistthesamedefectanticommute(e.g.,xiand
longexactsequenceofhomotopygroups, xi+1).
Theremainingsectionsofthispaperwillbeasfollows.In
Sec. II, we rederive Teo and Kane’s zero modes and unitary
···→πi(E)→πi(B)→πi−1(F)→πi−1(E)→···, (4) transformationsbysimplepictorialandcountingargumentsin
a“strong-coupling”limitoftheirmodel.InSec.III,wereview
associatedtoafibrationdefinedbyF →E →B.(Inanexact the topological classification of free-fermion Hamiltonians,
sequence,thekernelofeachmapisequaltotheimageofthe including topological insulators and superconductors. From
previousmap.Iftheexactsequencehasonlyfiveterms,1→ this classification, we obtain the classifying space relevant
A→B →C →1, then it is called a short exact sequence. to Teo and Kane’s model and, in turn, the topological
Ifthesequenceisinfinite,itiscalledalongexactsequence.) classification of defects and their configuration space. In
This exact sequence may be familiar to some readers from Sec. IV, we use a toy model to motivate a simple picture
Mermin’sreviewofthetopologicaltheoryofdefects,29where for the defects used by Teo and Kane and give a heuristic
a symmetry associated with the group G is spontaneously construction of the ribbon permutation group. In Sec. V,
brokentoH,therebyleadingtotopologicaldefectsclassified we give a full homotopy theory calculation. In Sec. VI, we
byhomotopygroupsπ (G/H).Thesecanbecomputedby(4) compare the ribbon permutation group to Teo and Kane’s
n
withE =G,F =H,B =G/H,e.g.,ifπ (G)=π (G)=0, unitary transformations and conclude that the latter form a
1 0
thenπ (G/H)=π (H). projective, rather than a linear, representation of the former.
1 0
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FREEDMAN,HASTINGS,NAYAK,QI,WALKER,ANDWANG PHYSICALREVIEWB83,115132(2011)
Finally,inSec.VII,wereviewanddiscussourresults.Several
appendixescontaintechnicaldetails.
FIG.2. Dimerizationind =1.
II. STRONG-COUPLINGLIMITOFTHE
In general, in d dimensions, we can obtain a Dirac
TEO-KANEMODEL equation with 2d-dimensional γ matrices by the following
Inthissection,wepresentalatticemodelind dimensions iterative procedure. Let the “vertical” direction refer to the
thathas,asitscontinuumlimitind =3,themodeldiscussed directionofthedthbasisvector.Havingconstructedthelattice
by Teo and Kane.20 In the limit that the mass terms in this Hamiltonianind−1dimensions,westacktheseHamiltonians
model are large (which can be viewed as a strong-coupling verticallyontopofeachother,withalternatingsignsineach
limit), a simple picture of topological defects (“hedgehogs”) layer. Then we take all the vertical bonds to be oriented
emerges. Here, a hedgehog is simply a pointlike defect that in the same direction. This Hamiltonian is invariant under
istopologicallystable.Weshowbyacountingargumentthat translationintheverticaldirectionbydistance2.Thus,ifHd−1
hedgehogs possess Majorana zero modes that evolve as the istheHamiltonianind−1dimensions,theHamiltonianHdis
hedgehogs are adiabatically moved. This adiabatic evolution givenby
(cid:2) (cid:3)
isthe3Dnon-Abelianstatistics,whichitisthemainpurpose
ofthispapertoexplain. H = Hd−1 2sin(k/2)I , (8)
Thestrong-coupling limitthatwedescribeisthesimplest d 2sin(k/2)I −Hd−1
way to derive the existence of Majorana zero modes and the
unitarytransformationsoftheirHilbertspacethatresultsfrom where I is the identity matrix and k is the momentum in the
exchangingthem.Thissectiondoesnotrequirethereadertobe verticaldirection.Neark =0,thisis
aucourantwiththetopologicalclassificationofinsulatorsand
superconductors.26,27 (Inthenextsection,wewillreviewthat Hd ≈Hd−1⊗σz+k⊗σx. (9)
classificationinordertomakeourexpositionself-contained.)
This iterative construction corresponds to an iterative
Weuseahypercubiclatticeind dimensions,withasingle
constructionofγ matrices.Havingconstructedd−1different
Musaejaorcahnaaind,eignrdee=of2frweeeduosmeaatsqeuacahresliatett.iTceh,aitnisd,f=or3dw=eu1sweea 2d−1-dimensional γ matrices γ1,...,γd−1, we construct d
different2d-dimensionalγ matrices,γ˜ ,...,γ˜ ,byγ˜ =γ ⊗
cubiclattice,andsoon.Wefirstconstructalatticemodelwhose 1 d i i
σ fori =1,...,d−1,andγ˜ =I ⊗σ .
continuumlimitistheDiracequationwith2d-dimensionalγ z d x
In 1D, dimerization of bonds corresponds to alternately
matrices to reproduce the Dirac equation considered by Teo
strengthening and weakening the bonds as shown in Fig. 2.
and Kane; we then show how to perturb this model to open
In 2D, we can dimerize in either the horizontal or vertical
a mass gap. We begin by considering only nearest-neighbor
directions.Ind dimensions,wehaved differentdirectionsto
couplings.TheHamiltonianH isanantisymmetricHermitian
dimerize.Dimerizinginthe“vertical”directiongives,instead
matrix.Ind =1,wecantakethelinearchaintogivealattice
of(9),theresult
model with the Dirac equation as its continuum limit. That
is, Hj,j+1 =i and Hj+1,j =−i. To describe this state in Hd ≈Hd−1⊗σz+k⊗σx +md ⊗σy, (10)
pictures, we draw these bonds as oriented lines, as shown
in Fig. 1(a), with the orientation indicating the sign of the wherem isthedimerizationstrength.Thiscorrespondstoan
d
bond.ThecontinuumlimitofthisHamiltonianisdescribedby iterativeconstructionofmassmatrices,M ,asfollows.In1D,
i
a Dirac equation with 2D γ matrices. While this system can we have M =iσ . Given d−1 different mass matrices in
1 y
bedescribedbyaunitcellofasinglesite,weinsteadchoose d−1 dimensions, M , we construct M˜ in d dimensions by
i i
todescribeitbyaunitcelloftwositesforconveniencewhen M˜ =M ⊗σ ,fori =1...d−1,andM˜ =iI ⊗σ .
i i z d y
addingmasstermslater.Ind =2,wecantakeaπ-fluxstateto Ifthedimerizationisnonzero,andconstant,wecanincrease
obtaintheDiracequationinthecontinuumlimit.Aconvenient thedimerizationstrengthwithoutclosingthegapuntilastrong-
gaugetotaketodescribetheπ-fluxstateisshowninFig.1(b), couplinglimitisreached.In1D,byincreasingthedimerization
with all the vertical bonds having the same orientation, and strength,weeventuallyreachafullydimerizedconfiguration,
theorientationofthehorizontalbondsalternatingfromrowto in which each site has one nonvanishing bond connected to
row.Thecontinuumlimitherehas4Dγ matricesandweuse it. In two or more dimensions, the dimerization can be a
afour-siteunitcell. combinationofdimerizationindifferentdirections.However,
ifthedimerizationiscompletelyinonedirection,forexample
the vertical direction, we increase the dimerization strength
(a) (b)
until the vertical bonds are fully dimerized. Simultaneously,
we reduce the strength of the other bonds to zero without
closing the gap. This is again a fully dimerized state, the
columnarstate,witheachsitehavingonenonvanishingbond.
Any configuration with uniform, small dimerization can be
deformedintothispatternwithoutclosingthegapbyrotating
the direction of dimerization, increasing the strength of
FIG.1. (a)AlatticemodelgivingtheDirac equationin d =1. dimerization,andthensettingthebondsintheotherdirections
(b)Alatticemodelind =2. tozero.
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(a) (b)
(a)
(b)
FIG.4. (a) Defect acting as a source of U(1) flux. Bonds are
oriented from the A to B sublattice. There is a net flux of one
leavingtheregiondefinedbythedashedline.(b)Configurationwith
diagonalbondadded,indicatedbytheundirectedlineconnectingthe
twocircles;eitherorientationofthisline,correspondingtodifferent
choicesofthesignofthetermintheHamiltonian,wouldleadtothe
sameresult.Thereisanetoffluxof2leavingtheregiondefinedby
thedashedline.
FIG.3. (a)A1Dhedgehog.(b)A2Dhedgehog.
everybondhasstrength0or1andeverysitehasexactlyone
bondconnectedtoit,exceptfordefectsites.)Then,thenumber
Itisimportanttounderstandthattheabilitytoreachsucha ofbonds going fromAsitesinthissettoB sitesoutsidethe
strong-couplinglimitsdependsontheperturbationoftheDirac set is exactly equal to the number of bonds going from B
equation that we consider; for dimerization, it is possible to sites in this set to A sites outside the set. On the other hand,
reachastrong-couplinglimit,whileifwehadinsteadchosen if there are defect sites in the set, then this rule is broken.
to open a mass gap by adding, for example, diagonal bonds ConsidertheregiondefinedbythedashedlineinFig.4(a).We
withimaginarycouplingtothe2DDiracequations,wewould define the “flux” crossing, the dashed line, to be the number
openamassgapbyperturbingtheHamiltonianwiththeterm ofbondscrossingthatboundary,whichleavestartingonanA
iγ γ , and such a perturbation cannot be continued to the site,minusthenumberthatleavestartingonaBsite.Theflux
1 2
strong-couplinglimitowingtotopologicalobstruction. is nonzero in this case, but is unchanging as we increase the
Further,ifthedimerizationisnonuniformthenitmaynot size of the region. This flux is the index ν. By the argument
be possible to reach a fully dimerized state without having givenabovefortheexistenceofzeromodes,ν computedfor
defectsites.ConsidertheconfigurationsinFig.3(a)ind =1 any region is equal to the number of Majorana zero modes
andinFig.3(b)ind =2.Thesearethestrong-couplinglimits containedwithintheregion.
ofthehedgehogconfiguration,andeachcontainsazeromode, The index ν can be defined beyond the strong-coupling
asingleunpairedsite.Thisisoneofthecentralresultsofthe limit.Consider,forthesakeofconcreteness,d =3.Thereare
strong-couplinglimit:Topologicaldefectshaveunpairedsites three possible dimerizations, one for each dimension, as we
that,inturn,supportMajoranazeromodes. concludedinEq.(10).Inweakcoupling,thesquareofthegap
Such strong-coupling hedgehog configurations can be isequaltothesumofthesquaresofthedimerizations.Thus,
constructed by the following iterative process in any dimen- if we assume a fixed gap, we can model these dimerizations
sion d. Let x correspond to the coordinate in the vertical byaunitvector.Theintegerindexdiscussedaboveissimply
d
direction. For x (cid:2)0, stack (d−1)-dimensional hedgehog thetotalwindingnumberofthisunitvectorontheboundary
d
configurations.Alongthehalf-linegivenbyx >0andx =0 ofanyregion.
d i
for1(cid:3)i (cid:3)d−1,arrangeverticalbonds,orientedtoconnect However, once diagonal bonds are allowed, the integer
the site with x =2k−1 to that with x =2k, for k (cid:2)1. index ν no longer counts zero modes. Instead, there is a Z
d d 2
Alongthelowerhalf-plane,givenbyx <0,arrangevertical index, equal to ν(mod2), which counts zero modes modulo
d
bonds orientedtoconnect asitewithx =−(2k−1)tothat 2. To see this in the strong-coupling limit, consider the
d
with−2k,fork (cid:2)1.Thisproceduregivesthed =2hedgehog configurationinFig.4(b).Thisisaconfigurationwithν =2
in Fig. 3(b) from the d =1 hedgehog in Fig. 3(a), and gives butnoMajoranazeromodes.However,aν =1configuration
a strong-coupling limit of the Teo-Kane hedgehog in d =3. muststillhaveazeromodeand,thus,anyconfigurationwith
That is, the Teo-Kane hedgehog can be deformed into this oddν musthaveatleastonezeromode.
configuration,withoutclosingthegap. In Fig. 4, we have chosen to orient the bonds from the A
Aslongasweconsideronlynearest-neighborbonds,there toBsublatticetomakeiteasiertocomputeν.However,theν
isanintegerindexνdescribingdifferentdimerizationpatterns anditsresiduemodulo2,definedabove,areindependentofthe
inthestrong-couplinglimit.Thisindex,whichispresentinany orientationofthebonds(thatindicatethesignoftermsinthe
dimension,arisesfromthesublatticesymmetryofthesystem, Hamiltonian) and depend only on which sites are connected
andiscloselyrelatedtotheU(1)symmetryofdimermodelsof by bonds (that indicate which terms in the Hamiltonian are
spinsystems.31LabelthetwosublatticesbyAandB.Consider nonvanishing).
anysetofsites,suchthateverysiteinthatsethasexactlyone Theν(mod2)withdiagonalbondsisthesameasKitaev’s
bondconnectedtoit.(Recallthat,inthestrong-couplinglimit, “Majorananumber.”15 Wecanusethistoshowtheexistence
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the orientation of pairs of neighboring bonds, arriving at the
(a) (b) (c)
configurationinFig.5(c).Finally,werotatebyπ intheplane
containing the other defect site and the last site. This returns
the system to the original configuration, having effected the
desiredoperation.
Becauseweonlyconsideradiabatictransformation,wecan
only perform orthogonal rotations with a unit determinant.
Thus, any transformation that swaps two defects and returns
thebondstotheiroriginalconfigurationmustchangethesign
of one of the zero modes: γ →γ ,γ →−γ . Indeed, any
i j j i
FIG.5. (a)Pairsofdefectsconnectedbyastring.(b)Firstrotation orthogonal transformation with a determinant equal to −1
appliedtotheconfigurationin(a).Theopencirclereplacesthefilled would change the sign of the fermion parity in the system,
circle to indicate a sign change of the Majorana mode on the site. asthefermionparityoperatorisequaltotheproductoftheγ
i
(c) After rotating along the string. (d) Rotating the last site and operators.
restoringthestringtoitsoriginalconfiguration. We used the ability to change the orientation of a pair of
bondsinthisconstruction.Thefactthatonecanonlychange
of zero modes in the Teo-Kane hedgehog even outside the the orientation of bonds in pairs, and not the orientation
strong-coupling limit. Consider a hedgehog configuration. of a single bond, is related to a global Z invariant: The
2
OutsidesomelargedistanceRfromthecenterofthehedgehog, Hamiltonian is an antisymmetric matrix and the sign of its
deform to the strong-coupling limit without closing the gap. Pfaffiancannotbechangedwithoutclosingthegap.Changing
Then,outsideadistanceR,wecancountν(mod2)bycounting thedirectionofasinglebondchangesthesignofthisPfaffian
bonds leaving the region, and we find a nonvanishing result andsoisnotpossible.
relative to a reference configuration: If there is an even The above discussion left open the question of which
number of sites in the region then there is an odd number zero changes its sign, i.e., is the effect of the exchange
of bonds leaving in a hedgehog configuration, and if there γ →γ ,γ →−γ or γ →−γ ,γ →γ ? The answer is
i j j i i j j i
is an odd number of sites then there is an even number of that it depends on how the bonds are returned to their
bondsleaving.However,becausethisimpliesanonvanishing originalconfigurationaftertheexchangeiscompleted(which
Majorananumber,theremustbeazeromodeinsidetheregion, is a clue that the defects must be understood as extended
regardless of what the Hamiltonian inside is. We note that objects, not pointlike ones). For the bonds to be restored,
this is a highly nontrivial result in the weak-coupling limit, one of the defects must be rotated by 2π; the corresponding
where the addition of weak diagonal bonds, all oriented the zero mode acquires a minus sign. We will discuss this in
same direction, to the configuration of Fig. 1(b) corresponds greater detail in a later section. The salient point here is
toaddingthetermiγ1γ2 totheHamiltonianind =2.Bythe that the effect of an exchange is a unitary transformation
argumentgivenabove,eventhisHamiltonianhasazeromode generated by the operator e±πγiγj/4. This is reminiscent of
inthepresenceofadefectwithnonzeroν(mod2). the representation of braid group generators for non-Abelian
Given any two zero modes, corresponding to defect sites quasiparticles in the quantum Hall effect9 and vortices in
in the strong-coupling limit, we can identify a string of sites chiral p-wave superconductors,14 namely, the braid group
connectingthem.Ifwehaveapairofdefectsitesonopposite representation realized by Ising anyons.7 But, of course, in
sublattices, corresponding to opposite hedgehogs, then one 3Dthebraidgroupisnotrelevant,andthepermutationgroup,
particular string corresponds to the north pole of the order whichisassociatedwithpointlikeparticlesind >2,doesnot
parameter, as in Fig. 5(a). However, we can simply choose havenontrivialhigher-dimensionalrepresentationsconsistent
anyarbitrarystring.Letγi,γj betheMajoranaoperatorsatthe with locality.18,19 As noted in the Introduction, this begs
two defect sites. The operation γi →−γi,γj →−γj can be the question: What group are the unitary matrices e±πγiγj/4
implementedasfollows.Webeginwithanadiabaticoperation representing?
on one of the defect sites and the nearest two sites on the
line.TheHamiltonianonthosethreesitesisanantisymmetric,
Hermitianmatrix.Thatis,itcorrespondstoanorientedplane
in 3D. We can adiabatically perform orthogonal rotations of
III. TOPOLOGICALCLASSIFICATIONOFGAPPED
thisplane.Thus,byrotatingbyπ intheplanecorresponding
FREE-FERMIONHAMILTONIANS
tothedefectsiteandthefirstsiteonthestring,wecanchange
the sign of the mode on the defect and the orientation of A. Setupoftheproblem
the bond, as shown in Fig. 5(b). This rotation is an adiabatic In this section, we will briefly review the topological
transformationofthethree-siteHamiltonian, classification of translationally invariant or slowly spatially
⎛ ⎞
varyingfree-fermionHamiltoniansfollowingKitaev’sanaly-
0 0 isin(θ)
⎜ ⎟ sisinRef.27.(Foradifferentperspective,seeSchnyderetal.’s
⎝ 0 0 icos(θ)⎠, (11) approach in Ref. 26.) The 3D Hamiltonian of the previous
−isin(θ) −icos(θ) 0 section is a specific example that fits within the general
scheme and, by implication, the 3D non-Abelian statistics
along the path θ =0→π. We then perform rotations on that we derived at the end of the previous section also holds
consecutivetriplesofsitesalongthedefectline,whichchanges for an entire class of models into which it can be deformed
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withoutclosingthegap.Ourdiscussionwillfollowthelogic Clearly,wecancontinuouslydeformA withoutclosingthe
ij
ofMilnor’streatmentoftheBottperiodicityinRef.28. gapsothatλ =(cid:9)foralli.(Thisisusuallycalled“spectrum
i
Consider a system of N flavors of electrons c (k) in d flattening.”)Then,wecanwrite
j
dimensions. The flavor index j accounts for spin as well as
A=(cid:9)·OTJO, (15)
the possibility of multiple bands. Because we will not be
assumingchargeconservation,itisconvenienttoexpressthe where
⎛ ⎞
complex fermion operators cj(k) in terms of real fermionic 0 −1
operators (Majorana fermions), cj(k)=[a2j−1(k)+ ⎜ ⎟
ia2j(k)]/2 (the index j now runs from 1 to 2N). The ⎜⎜1 0 ⎟⎟
momentum k takes values in the Brillouin zone, which has J =⎜⎜ 0 −1 ⎟⎟. (16)
thetopologyofthed-dimensionaltorusTd.TheHamiltonian ⎜ ⎟
⎜ 1 0 ⎟
maybewrittenintheform ⎝ ⎠
H =(cid:10)iA (p)a (p)a (−p), (12) ...
ij i j
i,j,p The possible choices of Aij correspond to the possible
where, by Fermi statistics, A (p)=−A (−p). Let us sup- choicesofO ∈O(2N),moduloO,whichcommutewiththe
ij ji matrix J. But the set of O ∈O(2N) satisfying OTJO =
posethattheHamiltonian(12)hasanenergygap2(cid:9),bywhich
J is U(N)⊂O(2N). Thus, the space of all possible 0D
wemeanthatitseigenvaluesE (p)(αisanindexlabelingthe
α
eigenvaluesofH)satisfy|E (p)|(cid:2)(cid:9).Thebasicquestionthat free-fermionic Hamiltonians with N single-particle energy
α
levels is topologically equivalent to the symmetric space
weaddressinthissectionisthefollowing:Whattopological
O(2N)/U(N).
obstructionspreventusfromcontinuouslydeformingonesuch
Thiscanberestatedinmoregeometricaltermsasfollows.
gappedHamiltonianintoanother?
Let us here and henceforth take units in which (cid:9)=1. Then
Such an analysis can apply, as we will see, not only
theeigenvaluesofAare±i.Ifweviewthe2N ×2N matrix
to free-fermion Hamiltonians, but also to those interacting
AasalineartransformationonR2N,thenitdefinesacomplex
fermion Hamiltonians that, deep within ordered phases, are
structure. Consequently, we can view R2N as CN because
well approximated by free-fermion Hamiltonians. (This can
multiplicationofv(cid:9)∈R2N byacomplexscalarcanbedefined
include rather nontrivial phases such as Ising anyons, but
as (a+ib)v(cid:9)≡av(cid:9)+bAv(cid:9). The set of complex structures on
notFibonaccianyons.)Insuchsettings,thefermionsmaybe
R2N is given by performing an arbitrary O(2N) rotation on
emergentfermionicquasiparticles;iftheinteractionsbetween
a fixed complex structure, modulo the rotations of CN that
thesequasiparticlesareirrelevantintherenormalization-group
respect the complex structure, namely, U(N). Thus, once
sense, then an analysis of free-fermion Hamiltonians can
again, we conclude that the desired space of Hamiltonians
shed light on the phase diagrams of such systems. Thus,
istopologicallyequivalenttoO(2N)/U(N).
the analysis of free-fermion Hamiltonians is equivalent to
What are the consequences of this equivalence? Consider
theanalysisofinteractingfermiongroundstateswhoselow-
thesimplestcase,N =1.Thenthespaceof0DHamiltonians
energyquasiparticleexcitationsarefreefermions.
is topologically equivalent to O(2)/U(1)=Z : There are
Let us begin by considering a few concrete examples, in 2
two classes of Hamiltonians, those in which the single
orderofincreasingcomplexity.
fermionic level is unoccupied in the ground state, c†c=
(1+ia a )/2=0, and those in which it is occupied. For
1 2
B. Zero-dimensionalsystems
larger N, O(2N)/U(N) is a more complicated space, but it
First, we analyze a zero-dimensional (0D) system that stillhastwoconnectedcomponents,π (O(2N)/U(N))=Z ,
0 2
we will not assume to have any special symmetry. The so that there are two classes of free-fermion Hamiltonians,
Hamiltonian(12)takesthesimplerform correspondingtoevenoroddnumbersofoccupiedfermionic
(cid:10)
levelsinthegroundstate.
H = iA a a , (13)
ij i j Suppose now that we restrict ourselves to time-reversal-
i,j invariant systems and, furthermore, to those time-reversal-
whereA isa2N ×2N antisymmetricmatrix,A =−A . invariant systems that satisfy T2 =−1, where T is the
ij ij ji
Anyrealantisymmetricmatrixcanbewrittenintheform antiunitaryoperatorgeneratingtimereversal.Then,following
⎛ ⎞ Ref. 27, we write Ta T−1 =(J ) a . The matrix J is
0 −λ1 antisymmetric and satisifies J2 =1−ij1.jT invariance of1the
⎜ ⎟ 1
⎜λ 0 ⎟ Hamiltonianrequires
⎜ 1 ⎟
A=OT ⎜⎜ 0 −λ2 ⎟⎟O, (14) J A=−AJ . (17)
⎜ ⎟ 1 1
⎜ λ 0 ⎟
⎝ 2 ⎠ BecauseJ isantisymmetricandsatisfiesJ2 =−1,itseigen-
1 1
... values are ±i. Therefore, J1 defines a complex structure on
R2N thatmay,consequently,beviewedasCN.Nowconsider
where O is an orthogonal matrix and the λ ’s are positive. A, which is also antisymmetric and satisfies A2 =−1, in
i
The eigenvalues of A come in pairs ±iλ ; thus, the absence addition to anticommuting with J . It defines a quaternionic
i 1
ofchargeconservationcanalsobeviewedasthepresenceof structure onCN that may, consequently, be viewed asHN/2.
a particle-hole symmetry. By assumption, λ (cid:2)(cid:9) for all i. Multiplication of v(cid:9)∈R2N by a quaternion can be defined
i
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as (a+bi+cj +dk)v(cid:9)≡av(cid:9)+bJ v(cid:9)+cAv(cid:9)+dJ Av(cid:9). The (which breaks time-reversal symmetry) and charge-density-
1 1
possible choices of A can be obtained from a fixed one waveorder(CDW).Thesetaketheform
by performing rotations of CN, modulo those rotations that
respectthequaternionicstructure,namely,Sp(N/2).Thus,the Maipby =(cid:9)ipyiτyδab (22)
setoftime-reversal-invariant0Dfree-fermionicHamiltonians
and
withT2 =−1istopologicallyequivalenttoU(N)/Sp(N/2). (cid:11) (cid:12)
Because π0(U(N)/Sp(N/2))=0, any such Hamiltonian can MaCbDW =ρ2kFτy cosθμzab+sinθμxab , (23)
be continuously deformed into any other. This can be under-
whereμx,z arePaulimatricesandθ isanarbitraryangle.For
stoodasaresultofKramersdoubling:Theremustbeaneven
an analysis of the possible mass terms in the more complex
numberoffermionsinthegroundstatesothedivisionintotwo
situationofgraphenelikesystems,see,forinstance,Ref.32.
classesofthepreviouscasedoesnotexisthere.
Letusconsiderthespaceofmasstermswithafixedenergy
gap(cid:9),whichisthesameforallfouroftheMajoranafermions
C. 2Dsystems:T-breakingsuperconductors in the model (i.e., a flattened mass spectrum). An arbitrary
Now,letusconsidersystemsinmorethan0D.Onceagain, gapped Hamiltonian can be continuously deformed to one
walesowaisllsuamsseumtheatthtiamtec-hraevrgeersaisl nsyomt mcoentsryervisedn,oatnpdrewseervweidll. tρh2akFt s=ati0sfioersρt2hkiFs=co(cid:9)nd,it(cid:9)ioipny.=Th0en(inwethcealnathtearveca(cid:9)sei,pyar=bit±ra(cid:9)ry,
For the sake of concreteness, let us consider a single band θ is allowed). If both order parameters are present, then the
ofspin-polarizedelectronsona2Dlattice.Letussupposethat energy gap is not the same for all fermions. It is not that
the electrons condense into a (fully spin-polarized) p -wave thereisanythingwrongwithsuchaHamiltonian—indeed,one
x
superconductor.Forafixedsuperconductingorderparameter, canimagineasystemdevelopingbothkindsoforder.Rather,
the low-energy theory is a free-fermion Hamiltonian for it is that such a Hamiltonian can be continuously deformed
gaplessfermionicexcitationsatthenodalpoints±k(cid:9)F ≡(0,± to one with either (cid:9)ipy =0 or ρ2kF =0 without closing the
gap. For instance, if (cid:9) >ρ , then the Hamiltonian can
pF). We now ask the question, what other order parameters be continuously deformipeyd to o2knFe with ρ =0. (However,
could develop that would fully gap the fermions? For fixed if we try to deform it to a Hamiltonian 2wkFith (cid:9) =0, the
values of these order parameters, we have a free-fermion gap will close at (cid:9) =ρ .) Hence, we concluipdye that the
Hamiltonian. Thus, these different possible order parame- ipy 2kF
spaceofpossiblemasstermsistopologicallyequivalenttothe
ters correspond to different possible gapped free-fermion
disjoint union U(1)∪Z : a single one-parameter family and
Hamiltonians. 2
twodisjointpoints.
Thelow-energyHamiltonianofafullyspin-polarizedp -
x Because π (U(1)∪Z )=Z , there are three distinct
wavesuperconductorcanbewrittenintheform 0 2 3
classesofquadraticHamiltoniansforfourflavorsofMajorana
H =ψ†(iv ∂ τ +iv ∂ τ )ψ, (18) fermions in 2D. The one-parameter family of CDW-ordered
(cid:9) x x F y z
Hamiltonians counts as a single class because they can be
where v , v are, respectively, the Fermi velocity and slope
F (cid:9) continuously deformed into each other. The parameter θ is
of the gap near the node. The Pauli matrices τ act in the
thephaseoftheCDW,whichdetermineswhetherthedensity
particle-holespace:
is maximum at the sites, the midpoints of the bonds, or
(cid:2) (cid:3)
somewhere in between. It is important to keep in mind,
ψ(k)≡ ck(cid:9)F+k(cid:9) . (19) however, that, although there is no topological obstruction
†
c−k(cid:9)F+k(cid:9) to continuously deforming one θ into another, there may be
an energetic penalty that makes it costly. For instance, the
This Hamiltonian is invariant under the U(1): ψ →eiθψ,
couplingofthesystemtothelatticemayprefersomeparticular
which corresponds to conservation of momentum in the p
y valueofθ.Theclassificationdiscussedhereaccountsonlyfor
direction (not to charge conservation). Because we will be
topological obstructions; the possibility of energetic barriers
consideringperturbationsthatdonotrespectthissymmetry,it
mustbeanalyzedbydifferentmethods.
isconvenienttointroduceMajoranafermionsχ ,χ according
1 2 We can restate the preceding analysis in the following,
toψ =χ +iχ .Then
1 2 more abstract language. This formulation will make it clear
that we have not overlooked a possible mass term and will
H =iχ (v ∂ τ +v ∂ τ )χ , (20)
a (cid:9) x x F y z a generalize to more complicated free-fermion models. Let us
witha =1,2.Notethatwehavesuppressedtheparticle-hole writeγ1 =τxδab,γ2 =τzδab.Then
index on which the Pauli matrices τ act. Because χ , χ are
1 2 {γ ,γ }=2δ . (24)
each a two-component real spinor, this model has four real i j ij
Majoranafields. The Dirac Hamiltonian for N =4 Majorana fermion fields
Wenowconsiderthepossiblemasstermsthatwecouldadd takestheform
tomakethisHamiltonianfullygapped:
H =iχ(γ ∂ +M)χ. (25)
i i
H =iχ (v ∂ τ +v ∂ τ )χ +iχ M χ . (21)
a (cid:9) x x F y z a a ab b
The matrix M plays the role that the matrix A did in the 0D
If we consider the possible order parameters that could case.Asinthatcase,weassumeaflattenedspectrumthathere
develop in this system, it is clear that there are only two meansthateachMajoranafermionfieldhasthesamegapand
choices: an imaginary superconducting order parameter ip thatthisgapisequalto1.(Itdoesnotmeanthattheenergyis
y
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independentofthemomentumk.)Inordertosatisfythis,we O(N)/O(k)×O(N −k), i.e., we can take the restriction of
mustrequirethat γ1M toX+tobeoftheform
⎛ ⎞
1
{γi,M}=0 and M2 =−1. (26) ⎜⎜ ... ⎟⎟
⎜ ⎟
⎜ 1 ⎟
NotethatitiscustomarytowritetheDiracHamiltonianin γ1M =OT ⎜⎜ −1 ⎟⎟O, (30)
⎜ ⎟
aslightlydifferentform,
⎝ ... ⎠
−1
H =ψ(iγ ∂ +m)ψ, (27)
i i
with k diagonal entries equal to +1 and N −k entries equal
whichcanbemassagedintotheformof(25)usingψ =ψ†γ : to−1.Thus,thespaceofHamiltonians forN flavorsoffree
0 Majoranafermionsistopologicallyequivalentto
H =ψ†(iγ γ ∂ +mγ )ψ (cid:17)N
=ψ†(iα0∂i+i mβ)ψ0 M2N = O(N)/[O(k)×O(N −k)]. (31)
=iψ†(αi∂i −imβ)ψ, (28) k=0
i i
However, because π (O(N)/[O(k)×O(N −k)])=0, inde-
0
pendent of k (note that 0 is the group with a single element,
where α =γ γ and β =γ . Thus, if we write γ ≡α and
M ≡−iimβ a0ndiconsider M0ajorana fermions (or diecomipose nottheemptyset∅),π0(M2N)=ZN+1.
In the model analyzed above, we had only a single spin-
DiracfermionsintoMajoranas),werecover(25).Wehaveused
polarizedbandofelectrons.Byincreasingthenumberofbands
theform(25)sothatitisanalogousto(13),with(γ ∂ +M)
i i andallowingbothspins,wecanincreasethenumberofflavors
replacing A and the i pulled out front. Then, the matrix M
ij ofMajoranafermions.Inprinciple,thenumberofbandsina
determines the gaps of the various modes in the same way
solidisinfinity.Usually,wecanintroduceacutoffandrestrict
as A does in the 0D case. Similarly, assuming a “flattened”
attentiontoafewbandsneartheFermienergy.However,for
spectrumleadstotheconditionM2 =−1.
a purely topological classification, we can ignore energetics
HowmanywayscanwechoosesuchanM?Becauseγ2 =
2 and consider all bands on equal footing. Then we can take
1,itseigenvaluesare±1.Hence,viewedasalinearmapfrom
R4 to itself, this matrix divides R4 into two 2D subspaces N →∞,sothatπ0(M∞)=Z.Thisclassificationpermitsus
todeformHamiltoniansintoeachotheraslongasthereisno
R4 =X+⊕X− with eigenvalue ±1 under γ2, respectively. topological obstruction, with no regard to how energetically
Forγ =τ δ ,thisistrivial:
2 z ab costlythedeformationmaybe.Thus,the2N =4classification
(cid:13)(cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15)(cid:16) thatwediscussedabovecanperhapsbeviewedasa“hybrid”
X+ =span 1 ⊗ 1 , 1 ⊗ 0 , (29) classification, which looks for topological obstructions in a
0 0 0 1 class of models with a fixed set of bands close to the Fermi
energy.
where τ acts on the first spinor and the second spinor is But even this point of view is not really natural. The
z
indexed by a =1,2, i.e., is acted on by the Pauli matrices discussionabovetookasitsstartingpointanexpansionabout
μx,z in (23). This construction generalizes straightforwardly apx superconductor;thepx superconductingorderparameter
to arbitrary numbers N of Majorana fermions, which is why was assumed to be large and fixed while the ipy and CDW
weuseitnow. order parameters were assumed to be small. In other words,
Now γ M commutes with γ and satisfies (γ M)2 =1. weassumedthatthesystemwasatapointinparameterspace
1 2 1
Thus, it maps X+ to itself and defines subspaces X+1,X+2 atwhichthegap,thoughnonzero,wassmallattwopointsin
with eigenvalue ±1 under γ1M (and equivalently for X−). the Brillouin zone (the intersection points of the nodal line
X+ can decomposed into X+1 ⊕X+2 =X+. Choosing M is in the px superconducting order parameter with the Fermi
thusequivalenttochoosingalinearsubspaceX+1 ofX+. surface). This allowed us to expand the Hamiltonian about
Thiscanbedividedintothreecases.Ifγ Mhasonepositive these points in the Brillouin zone and write it in Dirac form.
1
eigenvalueandonenegativeonewhenactingonX+ thenthe And this may, indeed, be reasonable in a system in which
space of possible choices of γ1M is equal to the space of px superconducting order is strong (i.e., highly energetically
1D linear subspaces of R2, which is simply U(1). If, on the favored) and other orders are weak. However, a topological
other hand, γ M has two positive eigenvalues, then there is classificationshouldallowustotakethesystemintoregimesin
1
auniquechoice,whichissimplyM =γ1γ2.Ifγ1M hastwo whichpxsuperconductivityissmallandotherordersarelarge.
negativeeigenvalues,thenthereisagainauniquechoice,M = Suppose,forinstance,thatwetookourmodelofspin-polarized
−γ γ . Therefore, the space of possible M’s is topologically electrons(whichweassume,forsimplicity,tobeathalf-filling
1 2
equivalenttoU(1)∪Z . on the square lattice) and went into a regime in which there
2
Now, suppose that we have 2N Majorana fermions. was a large dx2−y2 density-wave (or “staggered flux”) order
Then γ2 defines N-dimensional eigenspaces X+,X− such parameter33(cid:14)ck†+Qck(cid:15)=i(cid:17)(coskxa−coskya),whereaisthe
that R2N =X+⊕X− and γ1M defines eigenspaces of X+: latticeconstantand(cid:17)isthemagnitudeoftheorderparameter.
X+1 ⊕X+2 =X+.Ifγ1MhaskpositiveeigenvaluesandN −k With nearest-neighbor hopping only, the energy spectrum
negative ones, then the space of possible choices of γ M is is E2 =(2t)2(cosk a+cosk a)2+(cid:17)2(cosk a−cosk a)2.
1 k x y x y
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Thus, the gap vanishes at four points, (±π/2,±π/2) and closing. But any gapped Hamiltonian can be continuously
(∓π/2,±π/2). The Hamiltonian can be linearized in the deformed so that the gap becomes small at some points in
vicinityofthesepoints: the Brillouin zone. Thus, the problem of classifying gapped
free-fermion Hamiltonians in d dimensions is equivalent to
H =ψ†(iv ∂ τ +iv ∂ τ )ψ
1 (cid:9) x x F y z 1 the problem of classifying possible mass terms in a generic
+ψ†(iv ∂ τ +iv ∂ τ )ψ , (32) d-dimensional Dirac Hamiltonian as long as the number of
2 (cid:9) y x F x z 2 bandsissufficientlylarge.27Thisstatementcanbemademore
where vF, v(cid:9) are, respectively, the Fermi velocity and slope preciseandputonmoresolidmathematicalfootingbyusing
of the gap near the nodes; the subscripts 1,2 refer to the two ideasthatwediscussinAppendixB.
sets of nodes (±π/2,±π/2) and (∓π/2,±π/2); and ψ ,
A
A=1,2aredefinedby
(cid:14) (cid:15) D. Classificationoftopologicaldefects
ψ1,2(k)≡ cc(−(ππ//22,±,∓ππ//22)+)+k(cid:9)k(cid:9) . (33) onlTyhfeortcolpaoslsoegsiocfaltrcalnasslsaitfiiocnaatilolynindveascriraibnetdHaamboilvteonhiaonldsssuncoht
IfweintroduceMajoranafermionsψA =χA1+iχA2,thenwe as(25),butalsofortopologicaldefectswithinaclass.Suppose,
canwritethisHamiltonianwithpossiblemasstermsas for instance, that we consider (25) with a mass that varies
slowlyastheoriginisencircledatagreatdistance.Wecanask
H =iχ (v ∂ τ +v ∂ τ )χ
1a (cid:9) x x F y z 1a whether such a Hamiltonian can be continuously deformed
+iχ (v ∂ τ +v ∂ τ )χ into a uniform one. In a system in which the mass term is
2a (cid:9) y x F x z 2a
+iχ M χ . (34) understood as arising as a result of some kind of underlying
Aa Aa,Bb Bb
ordering such as superconductivity or CDW order, we are
We have suppressed the spinor indices (e.g., χ11 is a two- simplytalkingabouttopologicaldefectsinanorderedmedia,
component spinor); with these indices included, MAa,Bb is but with the caveat that the order parameter is allowed to
an 8×8 matrix. However, in order for the gap to be the explore a very large space that may include many physically
same for all flavors, the mass matrix must anticommute distinctorunnaturalorders,subjectonlytotheconditionthat
withτx,z.Thus,MAa,Bb =τyM˜Aa,Bb.ThematrixM˜Aa,Bb can thegapdoesnotclose.
have 0,1,2,3, or 4 eigenvalues equal to +1 (with the rest Let us, for the sake of concreteness, assume that we have
being −1). The spaces of such mass terms are, respectively, a mass term with N/2 positive eigenvalues when restricted
0,O(4)/[O(1)×O(3)],O(4)/[O(2)×O(2)],O(4)/[O(3)× to the +1 eigenspace of γ . [For N large, the answer
2
O(1)], and 0. Mass terms with 0 or 4 eigenvalues equal obviouslycannotdependonthenumberofpositiveeigenvalues
to +1 correspond physically to ±idxy density-wave order, k so long as k scales with N. Thus, we will denote the
(cid:14)c† c (cid:15)=±sink asink a.Masstermswith2eigenvalues space M defined in Eq. (31) by Z×O(N)/[O(N/2)×
k+Q k x y 2N
equal to +1 correspond physically to superconductivity, to O(N/2)], where the integers in Z correspond to the number
Q(cid:17) =(π,0)CDWorder,andtolinearcombinationsofthetwo. of positive eigenvalues of the mass term when restricted
Mass terms with 1 or 3 eigenvalues equal to +1 correspond to the +1 eigenspace of γ .] Then M(r =∞,θ) defines a
2
to (physically unlikely) hybrid orders with, for instance, loop in O(N)/[O(N/2)×O(N/2)] that cannot be continu-
superconductivityat(±π/2,±π/2)and±id density-wave ously unwound if it corresponds to a nontrivial element of
xy
order at (±π/2,∓π/2). Clearly, this is the 2N =8 case of π (O(N)/[O(N/2)×O(N/2)]).
1
the general classification discussed above. Thus, the same To compute π (O(N)/[O(N/2)×O(N/2)]), we
1
underlying physical degrees of freedom—a single band of parametrize O(N)/[O(N/2)×O(N/2)] by symmetric
spin-polarized electrons on a square lattice—can correspond matrices K that satisfy K2 =1 and tr(K)=0. [Such
toeither2N =4or2N =8,dependingonwherethesystem matrices decompose RN into their +1 and −1 eigenspaces:
isinparameterspace.Onecanimagineregionsofparameter RN =V+⊕V−.K canbewrittenintheformK =OTK0O,
spacewherethegapissmallatanarbitrarynumberNofpoints. where K has N/2 diagonal entries equal to +1 and N/2
0
Thus, if we restrict ourselves to systems with a single band, equal to −1, i.e., K =diag(1,...,1,−1,...,−1).] Note
0
then different regions of the parameter space (with different thatanysuchK isitselfanorthogonalmatrix,i.e.,anelement
numbers of points at which the gap is small) will have very of O(N); thus O(N)/[O(N/2)×O(N/2)] can be viewed
differenttopologies.Althoughsuchaclassificationmaybea as a submanifold of O(N) in a canonical way. Consider a
necessaryevilinsomecontexts,itisfarpreferable,giventhe curve L(λ) in O(N)/[O(N/2)×O(N/2)] with L(0)=K
choice, to allow topology to work unfettered by energetics. and L(π)=−K. We will parametrize it as L(λ)=KeλA,
Then we can consider a large number n of bands. Suppose whereAisintheLiealgebraofO(N).Inorderforthisloopto
thatthegapbecomessmallatr pointsintheBrillouinzonein remaininO(N)/[O(N/2)×O(N/2)],weneed(KeλA)2 =1.
eachband.Then,thelow-energyHamiltoniantakestheDirac Because(KeλA)2 =KeλAKeiλA =eiλKAKeλA,thiscondition
form for 2N =2rn Majorana fermion fields. As we will see implies that KA=−AK. In order to have L(π)=−K, we
below,ifN issufficientlylarge,thetopologyofthespaceof need A2 =−1. Such a curve is, in fact, a minimal geodesic
possiblemasstermswillbeindependentofN.Consequently, fromK to−K.Eachsuchgeodesiccanberepresentedbyits
for n sufficiently large, the topology of the space of possible midpoint L(π/2)=KA, so the space of such geodesics is
mass terms will be independent of r. In other words, we are equivalenttothespaceofmatricesAsatisfyingA2 =−1and
in the good situation in which the topology of the space of KA=−AK. As discussed in Ref. 28, the space of minimal
Hamiltonians will be the same in the vicinity of any gap geodesicsisagoodenoughapproximationtotheentirespace
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