Towards a Complete Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions

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by  Meng Cheng

Towards a Complete Classification of Symmetry-Protected Phases for Interacting
Fermions in Two Dimensions
Meng Cheng,1 Zhen Bi,2 Yi-Zhuang You,2 and Zheng-Cheng Gu3
1Station Q, Microsoft Research, Santa Barbara, CA 93106, USA
2Department of physics, University of California, Santa Barbara, CA 93106, USA
3Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
(Dated: January 19, 2015)
Recently,it has been shown that two-dimensional bosonic symmetry-protected topological(SPT)
5 phaseswithon-siteunitarysymmetryGcanbecompletelyclassifiedbythegroupcohomologyclass
1 H3(G,U(1)). Later, group super-cohomology class was proposed as a partial classification for SPT
0 phases of interacting fermions. In this work, we revisit this problem based on the mathematical
2 framework of G-extension of unitary braided tensor category(UBTC) theory. We first reproduce
the partial classifications given by group super-cohomology, then we show that with an additional
n
a H1(G,Z2)structure,acompleteclassificationofSPTphasesfortwo-dimensionalinteractingfermion
J systems for a total symmetry group G×Zf2 can be achieved. We also discuss the classification of
6 interacting fermionic SPT phases protected bytime reversal symmetry.
1

] Introduction — Due to the Pauli exclusion princi- fluxes into the fSPT state. The classificationis obtained
l
e ple,non-interactingfermionshaverichstructuresintheir bystudyingthetopologicalpropertiesofthedefects,such
- ground state wavefunction and a complete classification astheirfusionrulesandbraidingstatistics. Similarideas
r
t of symmetry protected topological(SPT) phases of free have proven to be quite successful in classifying bosonic
s
. fermions has been achievedby using ideas like Anderson SPT phases in 2D [27–30]. The mathematical objects
t
a localization [1] and K theory [2]. It is then of no sur- thatclassify2DfSPTphasescanbesummarizedasthree
m prise that the experimentalobservationofthese nontriv- group cohomologies of the symmetry group: H1(G,Z ),
2
- ialstatesisrelativelyeasyandavarietyofmaterialshave BH2(G,Z2) and H3(G,U(1)). H1(G,Z2) which one-
d been discoveredto be time-reversal-invarianttopological to-one corresponds to the Z subgroups of G, classifies
n 2
insulators/superconductors[3–6]. Ontheotherhand,re- fSPT phases with Majorana edge modes. BH2(G,Z )
o 2
c centstudies showthat SPT phases alsoexist ininteract- has been derivedin the groupsuper-cohomologyclassifi-
[ ingbosonsystemsandcanbesystematicallyclassifiedby cation[24]andweclarifyitsphysicalmeaningasthepro-

(generalized) group cohomology theory [7–9]. Neverthe- jective representations carried by fermion-parity fluxes.
2
v less, bosonic SPT phases remainquite elusive since their H3(G,U(1)) is the well-known classification of bosonic
3 occurrence requires strong interactions (weakly interact- SPT phases [7].
1 ing bosons tend to condense and become a superfluid). Generalities — First of all, let us clarify the meaning
3
Despite the remarkably successful classifications of of the symmetry group G in a fermionic system. A fun-
1
0 non-interactingbandHamiltonians,thenon-perturbative damental symmetry of fermionic systems that can never
. effects of interactions in fermionic SPTs still remain an be brokenis the conservationoftotal fermion parity,de-
1
important theoretical question. In many cases, the free- noted by Zf. In addition to this symmetry, we assume
0 2
5 fermion classifications are shown to be stable against the system has an on-site symmetry group G. The total
1 interaction effects, e.g. the Z classification of time- symmetry group of the system is actually Zf G [48].
: reversal-invariant topological in2sulators [10]. A break- Formostofourpaper weassumeGisunitary2a×ndfinite.
v
i through by Fidkowski and Kitaev [11] demonstrated We will consider anti-unitary time-reversal symmetry in
X
that in one-dimensional fermionic systems with time- the end.
r reversal symmetry T2 = 1, the non-interacting Z clas- Ourapproachtotheproblemisbasedonthealgebraic
a
sification breaks down to Z8 when strong interactions theory of two-dimensional gapped quantum phases [31].
are present [11, 12]. The result was then generalized to Given a gapped phase in two dimensions, one can clas-
twodimensionswithanon-siteZ2 symmetry[13–18]and sifythelow-energylocalizedquasiparticleexcitationsinto
three-dimensional time-reversal-invarianttopological su- superselection sectors (topological charges). Different
perconductors [19–23]. Recently several general classi- topological charges can not be transformed into each
fication schemes have been proposed for fermionic SPT other by applying local operators. Fusion and braiding
phases, including group super-cohomology [24], cobor- of the topological charges are described by the math-
dism [25] and invertible topological field theories [26]. ematical framework of unitary braided tensor category
In this work we pursue an alternative route to clas- (UBTC)[31]. Inafermionicsystem,stateswithevenand
sify fermionic SPT(fSPT) phasesin 2D. Followingprevi- odd numbers of fermions belong to different superselec-
ous works on bosonic SPT phases, to characterize fSPT tion sectors. Therefore we model a gapped fermion SPT
phaseswe introduce extrinsic defects carryingsymmetry phaseabstractlybythe(premodular)UBTC = I,ψ ,
f
C { }
2
whereψ representsasinglefermionicexcitation. Thefu- construct the fusion outcome of σ σ , we im-
g h gh
× ∈ C
sion rule is obviously ψ ψ =I. mediately see that the defects in the gh sector must be
×
To classify fSPT phases with a unitary symmetry Abelianjusttomatchthequantumdimension. Thereare
G, we exploit the idea of weakly gauging the symme- still three possibilities: σ σ = 2σ+ ,σ σ = 2σ−
g× h gh g × h gh
try [17, 27, 28]. To be more concrete, we introduce and σ σ = σ+ +σ− . The former two are impossi-
intothesystemnon-dynamicalpoint-likedefectscarrying ble forgt×he hfollowginhg reagshon: Assuming σ σ = 2σ+ .
symmetry fluxes. In other words, these defects are en- g × h gh
Using the symmetry of the fusion coefficients, we must
tqiurealsyipeaxrttricinlesiecx.cWitaetitohnesntoenilnacrlguedeouthreaslegedberfeacictst.hDeoerfeycotsf have σg+h × σg−1 = 2σh. The left-hand side has di-
mension √2 while the right-hand already has dimension
byconstructioncarrygrouplabels,buttheycanalsohave
2√2, which is clearly impossible. So we conclude that
their own topological charge labels. We therefore collect
σ σ =σ+ +σ− .
alldefects labeledbythe samegroupelementg G into g× h gh gh
∈ On the other hand, if the g sector has a non-Abelian
ag-sector , anddefine the so-calledG-extension as
Cg CG defect but the h sector has Abelian ones, the only avail-
= . Notice that the g = 1 sector is just
CthGeoriLginga∈lGthCegoryCf. Similartoanyonmodels,Ct1hemost adbefleectfuisniotnheruglhesiesctσogr×is aσlh±so=nonσ-gAhb,eilmiapnl.yiOngbvtiohuastlyt,hief
fundamental property of defects is their fusion rules, i.e.
both g and h sectors have only Abelian defects, the gh
how the topological charges are combined. But because
sector only has Abelian ones, too.
defectsalsocarrygrouplabels,theirfusionrulesmustbe
What we have just established is that whether the g-
G-graded, namely for a ,b , we have
g g h h
∈C ∈C sectorisnon-Abelianornotgivesahomomorphismfrom
a b = Ncgh c , (1) G to Z2. The inverse statement is quite obvious. Given
g× h X agbh gh anysuchhomomorphism,wecanwritedownfusionrules
cgh accordingly. Physically,a non-Abeliandefect with quan-
where Ncgh are non-negative integers indicating the tum dimension d = √2 is associated with an odd num-
agbh berofMajoranazeromodeslocalizedatthedefectwhich
numberofwaysdefectsa andb cancombinetoproduce
g h
impliesatopologicaldegeneracywhentherearemultiple
charge c .
gh
defects.
InordertocompletelydefinetheG-extension,weneed
We also need to determine the fusion rules of the
tostudymoresubtlestructures,suchastheassociativity
Abelian sectors G consisting of all Z -even group ele-
of defect fusion and braiding transformations of defects. e 2
ments. For any g,h G , we need to specify whether
These are captured in a mathematical formalism called σ+ σ+ isσ+ orσ− ∈=ψe σ+ . Wecangenerallywrite
G-crossedbraidedfusioncategory,andwerefertheread- g × h gh gh × gh
ers to Ref. [29] for a complete treatment. In our case,
we follow a more physically intuitive argument to avoid σg+×σh+ =σgωh(g,h) =ψn(g,h)×σg+h, (2)
solving complicated algebraic equations. In particular,
wetakeadvantageofaspecial“invertibility”propertyof whereω(g,h)= 1andn(g,h)= 1−ω(g,h). Sincefusion
± 2
SPT phases: for each SPT state, there is a unique “con- mustbeassociative,comparing(σ+ σ+) σ+ andσ+
g × h × k g ×
jugate” state such that by stacking them up one obtain (σ+ σ+) yields
h × k
thetrivialstate. Furthermore,giventwoSPTstates,one
canstackthemtogethertogetanotherSPTstate,which ω(g,h)ω(gh,k)=ω(g,hk)ω(h,k). (3)
is defined as their sum [49]. In other words, there is a
Formally, this means that ω(g,h) is a Z -valued 2-
natural Abelian group structure on SPT states. We will 2
cocycle: ω Z2(G ,Z ). However, we must remem-
denote the Abelian group of fSPT phases with a given ∈ e 2
symmetry group G by G. ber that the definition of σg+ is completely arbitraryand
has no physical meaning. One can always swap σ+ and
Classifying fusion rules — The G-grading structure g
σ− for each individual g, or more explicitly, we redefine
of the fusion rules of defects has a profound conse- g
σ˜+ =ψmg σ+. In terms of σ˜±, we find
quence : one can show that all sectors g have the g × g g
C
same total quantum dimensions: 2 = 2 = 2 where
Dg D1 ( 1)mg( 1)mh
2 = d2 [29, 32]. In our case, 2 = 2 leaves ω˜(g,h)= − − ω(g,h). (4)
Dg Pag∈Cg ag Dg ( 1)mgh
us with onlytwopossibilities: (a)There aretwoAbelian −
tdheefemctbsyinσCg±g.a(nbd)tThheeyrdeiiffsearbsiynfgulesinngonw-Aithbeψli.anWdeedfeecntoitne ωeqgu,hivsarleelnatt.edThbeyesquuchivarleednecfiencitlaiosnsesssohfou2l-dcobceyccloesns[iωd]eraerde
with quantum dimension √2. We denote it by σ . classified by the second group cohomology H2(G ,Z ).
g g e 2
C
First we show that the possible non-Abelian fusion Classifying fSPT phases — As we mentioned before,
ruleshaveone-to-onecorrespondencewithH1(G,Z ),i.e. togetacompleteclassificationoffSPTphasesweneedto
2
group homomorphisms from G to Z . Assume for both understandthealgebraicdataofthedefectswhichsatisfy
2
g,h G the defects are non-Abelian. To be able to a set of consistency conditions [29]. Given a particular
∈
3
fusion rule of defects, there may be more than one dis- Next we consider the non-Abelian fSPT phases. First
tinctsetofalgebraicdata. Ontheotherhand,forcertain we show that given a nontrivial Z homomorphism of G
2
fusion rules it is possible that the consistency conditions there exists at least one non-Abelian fSPT, by explicitly
donotallowanysolutions,inwhichcasethe fusionrules constructing the extension. The fusion rules of the Z -
2
do not correspond to any two-dimensional fSPT phases, odd G-sectors are fixed by the homomorphism, and we
i.e. there are obstructions [29, 33]. chooseatrivial2-cocycleinH2(G ,Z ),i.e. ω 1for
e 2 g,h
Let us start from the cases where all the defects are thefusionrulesoftheZ -evenAbeliansectors. We≡define
2
Abelian, i.e. we choose the trivial homomorphism from a map ϕ from the topological charges of this extension
G to Z . Such fSPTs will be referred to as Abelian to those of the familiar Ising UBTC, which has three
2
fSPTs. The fusion rules of such extensions correspond topological charges I˜,ψ˜,σ˜ :
to 2-cocycles ω H2(G,Z ). We can show that for the { }
∈ 2 ϕ(σ+)=I˜,ϕ(σ−)=ψ˜,ϕ(σ )=σ˜. (8)
obstruction to vanish, the following 4-cocycle must be g g h
trivial (i.e. it is a 3-coboundary): Allthe algebraicdata followfromthis mapandthe data
of the Ising category. We will refer to the corresponding
O(g,h,k,l)=( 1)n(g,h)n(k,l). (5)
− fSPT state as the root non-Abelian fSPT state. We now
argue other non-Abelian fSPT phases with the same Z
We notice that Eq. (5) agrees exactly with the result 2
homomorphismcanallbegeneratedfromtherootphase.
of group super-cohomology [24][50]. Following Ref. [24]
wedenotetheobstruction-freesubgroupofH2(G,Z )by We make use of the fact that a non-Abelian defect must
2
BH2(G,Z ),andthegroupofallAbelianfSPTstatesas localize an odd number of Majorana zero modes. Con-
2 sidertwonon-AbelianfSPTstatesfSPT andfSPT cor-
G . 1 2
+ respondingto the same Z homomorphismofG. Denote
We brieflysketchthe derivationof (5)(Seesupplemen- 2
their sum fSPT =fSPT +fSPT . Suppose we create a
tary material for a detailed derivation). The central 3 1 2
g-defect. If g is a Z -even element, the defect is already
quantityresponsiblefortheobstructionistheF symbols 2
AbelianbothinfSPT andfSPT ,soisinfSPT . ifgisa
for the associativity of defect fusion, defined diagram- 1 2 3
Z -oddelement,becausefSPT andfSPT havethesame
matically as 2 1 2
Z homomorphisms the defect localizes an even number
2
of Majorana zero modes altogether, and can only be an
σλ1 σλ2 σλ3 σλ1 σλ2 σλ3
g h k g h k Abelian one. Therefore all defects in fSPT are Abelian.
3
It immediately follows that any non-Abelian fSPT state
=Fσgλ1σhλ2σkλ3 . (6) is equivalent to the sum of the “root” state of the same
Z homomorphismandanAbelianfSPTstate. Thispro-
2
vides a complete classification of the non-Abelian fSPT
They must satisfy consistency conditions known as the phases. In addition, our argument implies that G+ is a
pentagon equations. We can determine the structure normalsubgroupofG withH1(G,Z2)beingthequotient
of F-symbols by bootstrapping from the pentagonequa- group.
tionsbetweenψandthedefects,andobtainthefollowing Letuscommentonthegroupstructureofnon-Abelian
general parametrization of defect F symbols: fSPT phases. From our discussion we see that the root
non-Abelian fSPT is essentially a Z fSPT. We also no-
2
Fσgλ1σhλ2σkλ3 =ν(g,h,k)λn(h,k). (7) tice that given a homomorphism from G to Z2, we can
1 define a Z 2-cocycle on G by pulling back the nontriv-
2
Here λ = labels the two defects in the same sector, ial 2-cocycle in H2(Z ,Z ). We thus conjecture that the
2 2
±
and ν is a U(1) 3-cochain to be determined. Plugging addition of two root non-Abelian fSPT with the same
(7) into the general pentagon equation we get dν = O, homomorphism yields an Abelian fSPT given by this 2-
which implies that O belongs to the trivial cohomology cocycle.
classinH4(G,U(1)),thustheobstructionvanishingcon- Gaugingthefermion parity — Aninterestingprobeof
dition. Oncetheobstructionvanishes,differentsolutions thesymmetry-protectedstatesistoconsidergaugingthe
of ν are given by 3-cocycles in H3(G,U(1)). In fact, we fermion parity, which results in a (bosonic) topological
can see from the derivation that H3(G,U(1)) is a nor- gauge theory. The gauging procedure is well-defined in
malsubgroupofG withBH2(G,Z )beingthe quotient lattice models of fermions. One can choose the Hamil-
+ 2
group. tonian such that the low-energy spectra of the resulting
We can actually further constrain the group structure gauge theory coincide with the original model.
of G, by noticing that the squareof the F symbols (7) is For fSPT states, it is easy to see that the gauged the-
ν2,anddν2 =0,i.e. ν2 Z3(G,U(1)). Thisimpliesthat oryhasfouranyons I,e,m,ψ ,wheremistheZ gauge
2
∈ { }
“adding up” two Abelian fSPTs given by the same class flux,ψisthefermion,e=m ψcanbeconsideredasthe
in H2(G,Z ) results in a bosonic SPT phase labeled by (bosonic)Z charge. Thetop×ologicalorderisidenticalto
2 2
ν2 [34, 35]. that of a Z toric code lattice model. Importantly, the
2
4
gauged theory preserves the G symmetry and is there- Z . This implies that two fSPT corresponding to the
N
fore aG-symmetry enrichedZ gaugetheory. Therefore, nontrivial 2-cocycle can be stacked to form the gener-
2
a symmetric adiabatic path between two fSPT phases ating bosonic SPT. Therefore, the Abelian fSPT phases
mapsexactlytoasymmetricadiabaticpathbetweenthe with Z symmetry form a Z group.
n 2n
corresponding Z topological phases, and consequently Wecanactuallycompletelydeterminethegroupstruc-
2
if the two symmetry-enriched Z topological phases are ture and the classification is given by:
2
distinct, the original fSPTs must be distinct too [51]. In
this approach we can consider anti-unitary symmetries, Zn n is odd
orthefermionscarryingprojectiverepresentationsofthe G =Z2 Z2n n 0(mod 4). (9)
× ≡
symmetry group. Z n 2(mod 4)
4n
Symmetry enrichment in the toric code model can be  ≡
analyzed using the general theory developed in Ref. 29 G=Z Z . We can easily see that H1(G,Z )=Z2,
• 2× 2 2 2
(See also [36] for related discussions). First of all, one correspondingtonon-AbelianfSPTsprotectedbyanyof
needs to specify the symmetry action on the topological the three Z subgroups. The Abelian fSPTs are classi-
2
charge labels of anyons. It is easy to see that besides a fied by H2(Z Z ,Z ) = Z3. Using the cohomology
trivialaction,thereisaZ actionthatpermutestheeand mapping (5) 2we×fin2d t2hat all2these cocycles are free of
2
m particles. These two possible actions on the label set obstructions. Therefore, all these cocycles represent in-
form a Z topological symmetry group. Therefore, the trinsically distinct fSPT phases. Among the eight, four
2
symmetry action on the label set is specified by a group areessentiallythe fSPT phasescorrespondingto the two
homomorphism from G to Z . It is clear that this is the Z subgroups. TheotherfourarenewfSPTphaseswhere
2 2
same Z homomorphism that classifies the non-Abelian a Z fermion parity flux carries two-dimensional projec-
2 2
fusion rules of symmetry defects in fSPT states. tive representations of the symmetry group.
Once the symmetry action ρ on charge labels is spec- •G = ZT2 = {1,T}. We now discuss time-reversal sym-
ified, we can classify patterns of symmetry fractionaliza- metry. Our method of symmetry extension does not ap-
tion, i.e. anyons carrying projective representations of ply because the time-reversal symmetry is anti-unitary.
the symmetry group. This is captured by the group co- However,wecanstillconsidergaugingthefermionparity
homologyH2(G,Z Z ),wherethesubscriptρindicates and study the symmetry action in the toric code. Since
that G has aρ nontr2i×vial2action on the coefficients. How- after gauging one has a spin model, the time-reversal
ever, we need to make sure that the ψ particle always symmetry operatorsatisfies T2 =1. We can then distin-
transforms trivially under the symmetry. This means guishtwocases,wherethefermionψisaKramerssinglet
thatwe canactuallyrestrictthe coefficientofthe second (T2 =1) or a doublet (T2 = 1).
−
group cohomology to be Z = I,ψ , and we drop the Firstletusspecifythesymmetryactionρonthecharge
2
G action ρ. Therefore we recov{er th}e previous classifi- labels. Therearetwopossibilities: (a)T doesnotchange
cation [37]. We notice that this also provides a physi- charge labels at all. (b) T exchanges e and m. Interest-
cal characterization of H2(G,Z ), through local projec- ingly, in the latter case ψ must have T2 = 1 [23, 40].
2 −
tive symmetry actions on the Z fermion parity flux in Therefore we immediately see that there is a 2D fSPT
2
a fermionic system, which can be measured in numerical with T2 = 1 fermions, in which the local fermion par-
−
simulations [38, 39]. ity of a π vortex changes under the time-reversal op-
eration [20, 41, 42], and there are no other symmetry
Examples — In this section we apply the generalthe-
ory to G=Zn,Z2×Z2 and ZT2. fraLcetitonuasliczoantisoidnecrlatshseessdyumemteotrHyρ2f(rZaT2ct,iZon2a×lizZa2t)io=n0c.lass
G = Z . We label the group elements of Z by
• n n of the trivial action on the charge labels, which is clas-
aten=a0d,d1,it.i.v.e,lyn,−i.e1. aand+tbhe=gr[oau+p bm]uwlthiperlieca[tai]onisisawmroitd- sified by H2(ZT2,Z2 × Z2) = Z2 ×Z2. Physically, the
fourclassescorrespondtofourpossiblewaysofassigning
n. First we have H1(Z ,Z ) = Z , i.e. non-Abelian
n 2 (n,2) T2 = 1 to the four charges. For the two classes with ψ
fSPT phases exist for even n. ±
being a Kramers doublet, one of the e or m charges has
We then consider the Abelian fSPTs, classified by to be a Kramers singlet, which means that the π vortex
H2(Zn,Z2) = Z(n,2). Let us focus on the case of even is trivial in the fSPT. Therefore they do not correspond
n. Since H4(Zn,U(1)) is trivial, there are no obstruc- toanynontrivialfSPTphases. Wearethenleftwithone
tions. A representative cocycle in the nontrivial coho- nontrivialfractionalizationclasswithψ beingaKramers
mology class is given by ω(a,b) = einπ([a]+[b]−[a+b]). In- singlet,andbotheandmbeingKramersdoublets. How-
deed, ω([n],[n]) = 1 which is a gauge-invariant quan- ever,we knowthat this classhas anobstructionofbeing
tity since2we2are c−onsidering Z2 coefficients. We can realized in two dimensions [43, 44] and thus does not
also explicitly find the fermionic 3-cocycle: ν(a,b,c) = correspondto a fSPT in 2D. Together with the fact that
eniπ2a([b]+[c]−[b+c]). We notice that ν(a,b,c) is the square H3(ZT2,U(1)) = 0, we conclude that there is only one
rootofthe generatingU(1) 3-cocyclein H3(Z ,U(1))= nontrivial 2D fSPT phase with ZT symmetry, the class
N 2
5
DIII topological superconductor. arXiv:1410:4540.
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6
Supplementary material
REVIEW OF GROUP COHOMOLOGY
In this section, we provide a brief review of group cohomology for finite groups. Given a finite group G, let M be
anAbeliangroupequippedwithaGactionρ:G M,whichis compatiblewithgroupmultiplication. Inparticular,
→
for any g G and a,b M, we have
∈ ∈
ρ (ab)=ρ (a)ρ (b). (10)
g g g
(Weleavethegroupmultiplicationsymbolsimplicit.) SuchanAbeliangroupM withGactionρiscalledaG-module.
Let ω(g ,...,g ) M be a function of n group elements g G for j = 1,...,n. Such a function is called a
1 n j
∈ ∈
n-cochain and the set of all n-cochains is denoted as Cn(G,M). They naturally form a group under multiplication,
(ω ω′)(g ,...,g )=ω(g ,...,g )ω′(g ,...,g ), (11)
1 n 1 n 1 n
·
and the identity element is the trivial cochain ω(g ,...,g )=1.
1 n
We now define the “coboundary” map d:Cn(G,M) Cn+1(G,M) acting on cochains to be
→
n
dω(g1,...,gn+1)=ρg1[ω(g2,...,gn+1)]×Yω(−1)j(g1,...,gj−1,gjgj+1,gj+1,...,gn+1)×ω(−1)n+1(g1,...,gn).
j=1
(12)
One can directly verify that ddω = 1 for any ω Cn(G,M), where 1 is the trivial cochain in Cn+2(G,M). This is
∈
why d is considered a “boundary operator.”
With the coboundary map, we next define ω Cn(G,M) to be an n-cocycle if it satisfies the condition dω = 1.
∈
We denote the set of all n-cocycles by
Zn(G,M)=ker[d:Cn(G,M) Cn+1(G,M)]= ω Cn(G,M) dω =1 . (13)
ρ → { ∈ | }
We also define ω Cn(G,M) to be an n-coboundary if it satisfies the condition ω = dµ for some (n 1)-cochain
µ Cn−1(G,M). ∈We denote the set of all n-coboundaries by Also we have −
∈
Bn(G,M)=im[d:Cn−1(G,M) Cn(G,M)]= ω Cn(G,M) µ Cn−1(G,M):ω =dµ . (14)
ρ → { ∈ | ∃ ∈ }
Clearly, Bn(G,M) Zn(G,M) Cn(G,M). In fact, Cn, Zn, and Bn are all groups and the co-boundary maps
ρ ⊂ ρ ⊂
are homomorphisms. It is easy to see that Bn(G,M) is a normalsubgroupof Zn(G,M). Since d is a boundary map,
ρ ρ
we think of the n-coboundaries as being trivial n-cocycles, and it is natural to consider the quotient group
Zn(G,M)
Hn(G,M)= ρ , (15)
ρ Bn(G,M)
ρ
which is called the n-th group cohomology. In other words, Hn(G,M) collects the equivalence classes of n-cocycles
ρ
that only differ by n-coboundaries.
OBSTRUCTION TO ABELIAN G-CROSSED EXTENSIONS
In this section we derive the obstruction to a consistent Abelian G-extension. We first briefly review the algebraic
theoryofsymmetrydefects,knownastheG-crossedbraidedextensionofabraidedtensorcategory[29]. Forsimplicity,
we assume all defects (as well as anyons in the original theory) are abelian. The collection of all defects is called the
G-extension:
= . (16)
CG MCg
g∈G
We will use the diagrammatic formulation (for a review, see Refs. [29, 31]). The basic data of the G-extension
includes:
7
G-graded fusion rules, i.e. a b = Ncgh c .
• g× h Pcgh agbh gh
F symbols for associativity of fusion.
•
ag bh ck ag bh ck
αeβ = X hFdagghbkhcki(e,α,β)(f,µ,ν) ν fµ . (17)
f,µ,ν
d d
ghk ghk
The F symbols can be viewed as changes of bases for the states associated with quasiparticles.
G action on labels, defined by maps ρ , which acts on the topological charge labels in the following way:
g
•
ρh(ag) hgh−1. We write ρh(ag) as hag. Diagrammatically, this means that when topological charge lines
∈ C
cross, the labels should change accordingly:
hag bh
(18)
b a
h g
In our derivation we just need the obvious fact that ρ (a )=a .
1 g g
G action on fusion spaces, defined by unitary transformations U (a ,b ;c ):
k g h gh
•
ag bh ag bh
cgh ν
= [U (a,b;c)] . (19)
xk µ ¯kb Xν k µν xk
k¯cgh k¯cgh
Importantly, we have the normalization condition U1 11.
≡
Natural isomorphisms η (g,h) on topological charges,which define the projective G actions:
• xk
ag bh h¯g¯xk ag bh h¯¯gxk
¯gx µ
=η (g,h) (20)
x
xk xk
µ
cgh cgh
Similarly, η are normalized: η (1,h)=η (g,1)=1.
x x
G-crossed R symbols, defined by the following diagrammatic relation:
•
ag bh ag bh
µ = Ragbh ν . (21)
Xh cgh iµν
c ν c
gh gh
These data satisfy a set of coherence conditions. For our purpose, we have to solve the pentagon equations and a
generalization of hexagon equations, called heptagon equations (see Fig. 1), to find consistent F and R symbols.
The pentagon equation leads to the following general parametrizationof F symbols:
Fσgλ1σhλ2σkλ3 =ν(g,h,k)θ1−2λ1(h,k). (22)
8
Notice that because we areconsidering Abelian fusion rules, alllabels in the definition (17) of F symbolare uniquely
determined by the three outgoing lines and therefore suppressed here. ν(g,h,k) is a U(1) 3-cochain, and θ is a Z
2
2-cocycle. And they should satisfy
(dν)(g,h,k,l)=θng,h(k,l). (23)
This implies that the right-hand side must be a 4-coboundary.
a b c a b c a b h¯g¯c a b h¯g¯c
F F
R e g R R−1 e g R−1
a b c d d a b c a b h¯g¯c d d a b h¯g¯c
e g e g
d F a b c a b c a b c F d d F a b h¯g¯c a b h¯g¯c a b h¯g¯c F d
U R η R−1
kf f f f
d d d d d d
FIG. 1: Heptagon equations which ensure consistency between G-crossed braiding and fusion of defects.
We now use G-crossed heptagon equations to find θ.
Let us first consider the heptagon equation with the three outgoing lines being σλ,ψ and ψ (from left to right).
g
RσgλψFσgλψψRψψ =FψσgλψRσgλ×ψ,ψFσgλψψ. (24)
Since Rψψ = 1, we have
−
Rσg−ψ = Rσg+ψ. (25)
−
Let us consider the heptagon equation with the three outgoing lines being σλ1,σλ2 and ψ (from left to right).
g h
Because ψ , the action on the vertex is trivial. We have
1
∈C
Rσgλ1ψFσgλ1ψσhλ2Rσhλ2ψ =Fψσgλ1σhλ2Rσgλ1×σhλ2,ψFσgλ1σhλ2ψ (26)
Following the general parametrization and the normalization condition of ν, we have Fσgλ1ψσhλ2 = Fσgλ1σhλ2ψ = 1 and
Fψσgλ1σhλ2 =θ(g,h). Therefore
Rσgλ1ψRσhλ2ψ
θ(g,h)= . (27)
Rσgλ1×σhλ2,ψ
Now we set λ =λ =+ and notice σ+ σ+ =ψn(g,h) σ+ , we arrive at
1 2 g × h × gh
θ(g,h)=( 1)ng,hRσg+ψRσh+ψ. (28)
− Rσg+hψ
We can further show that Rσg+ψ = 1 by considering the heptagon equation for inverse braiding with the three
outgoing lines being ψ,ψ,σλ. This is±basically what we need, i.e. θ is Z -cohomologically equivalent to ( 1)n.
g 2 −
Therefore, the necessary condition for the extension to exist is that the 4-cocycle:
O(g,h,k,l)=( 1)ng,hnk,l (29)
−
is in the trivial cohomology class, i.e. [O] = 0. Once the obstruction vanishes, different solutions of ν are related to
each other by a 3-cocycle.
9
g1 g2 g3 g4
1 2
g1 g2 g3 g4 4 g1 g2 g3 g4
1 2
3 5
4 4
g1 g2 g3 g4 g1 g2 g3 g4
6 6
3 5
4 4
FIG. 2: The Pentagon equation enforces theconsistency between different sequences of F moves starting and endingwith the
same fusion trees.
PROJECTIVE PENTAGON EQUATION
In this section we give a different derivation of the obstruction mapping, which is closer in spirit to the group
super-cohomology theory.
Instead of considering all the defects σ±, we take a representative, e.g. σ+ from each g-sector. To account for the
g g
fusion rules, we allow the fusion space to be fermionic, i.e. Z -graded. In our case, the fermion parity of the fusion
2
spaceofσ andσ iscompletelydeterminedbyg andh: itisjust( 1)n(g,h). Moregeneralconstructionsoffermionic
g h
−
TQFTs have been studied in Refs. [46, 47].
With this modification, now the F move becomes an operator possibly connecting states with different fermion
parities.
g h k g h k
1 =ν(g,h,k)fn(gh,k)fn(g,h)f−n(h,k)f−n(g,hk) 3 . (30)
2 1 3 2
2 2
Here when we write f we really mean the fermionic creation operator while f−1 means the annihilation one.
Let us substitute the fermionic F moves into the pentagon equation. We will not explicitly write the ν factors
since they just give the standard4-coboundary,and focus on the fermionic part. First evaluate the upper pathof the
Pentagon equation:
fn(g1g2g3,g4)fn(g1g2,g3)f−n(g3,g4)f−n(g1g2,g3g4)fn(g1g2,g3g4)fn(g1,g2)f−n(g2,g3g4)f−n(g1,g2g3g4)
4 3 2 4 4 1 5 4
(31)
=fn(g1g2g3,g4)fn(g1g2,g3)f−n(g3,g4)fn(g1,g2)f−n(g2,g3g4)f−n(g1,g2g3g4).
4 3 2 1 5 4
Then the lower path gives:
fn(g1g2,g3)fn(g1,g2)f−n(g2,g3)f−n(g1,g2g3) fn(g1g2g3,g4)fn(g1,g2g3)f−n(g2g3,g4)f−n(g1,g2g3g4)
3 1 6 3 × 4 3 5 4 (32)
fn(g2g3,g4)fn(g2,g3)f−n(g3,g4)f−n(g2,g3g4)
× 5 6 2 5
We move the first fs(g1g2g3,g4) to the left most and then f−s(g1,g2g3g4) to the right most and the resulting fermionic
4 4
sign is the following:
( 1)n(g1g2g3,g4)[n(g1g2,g3)+n(g1,g2)−n(g2,g3)−n(g1,g2g3)]( 1)−n(g1,g2g3g4)[n(g2g3,g4)+n(g2,g3)−n(g3,g4)−n(g2,g3g4)] (33)
− −
From the 2-cocyle condition
ω(g ,g g )ω(g ,g )=ω(g g ,g )ω(g ,g ), (34)
1 2 3 2 3 1 2 3 1 2
10
together with ω =( 1)n, we have
−
n(g ,g g )+n(g ,g )=n(g g ,g )+n(g ,g )(mod 2). (35)
1 2 3 2 3 1 2 3 1 2
Therefore the term (33) is identically 1.
After this manipulation the expression (32) can be greatly simplified:
fn(g1g2g3,g4)fn(g1g2,g3)fn(g1,g2)f−n(g3,g4)f−n(g2,g3g4)f−n(g1,g2g3g4). (36)
4 3 1 2 5 4
So the upper and the lower paths (36) just differs by exchanging fn(g1,g2) and f−n(g3,g4) which results in a sign:
1 2
O(g ,g ,g ,g )=( 1)n(g1,g2)n(g3,g4). (37)
1 2 3 4
−
In order to satisfy the pentagon equation, this sign has to be compensated by dν:
(dν)(g,h,k,l)=( 1)n(g,h)n(k,l). (38)
−
This is the obstruction-free condition.
SYMMETRY FRACTIONALIZATION IN Z2 TORIC CODE
WederivetheH2(G,Z )classificationthroughthesymmetryfractionalizationinZ toriccode. Ourargumenthere
2 2
is somewhat heuristic and we refer the readers to Ref. [29] for more rigorous discussions.
To be more specific, let us consider the symmetry action on a generalquasiparticle state on a sphere/disk without
loss of generality, and assume that the permutation ρ is trivial for simplicity. The global symmetry operator R can
g
be decomposed into operators localized on each quasiparticle U (a). They only form projective representations of G,
g
i.e. U (a)U (a)=η (g,h)U (a), however η (g,h) must be consistent with fusion rules:
g h a gh a
η (g,h)η (g,h)=η (g,h),if Nc >0. (39)
a b c ab
This stems from the fact that R R = R must hold on vaccume state. In particular, we have η2 = η2 = 1,η =
g h gh e m ψ
η η .
e m
On the other hand, from the associativity of operator products U U U , we have
g h k
η (h,k)η (g,hk)=η (g,h)η (gh,k). (40)
a a a a
So naively, one may conclude that gauge-inequivalent classes of η are classified by H2(G,Z Z ). However, since
2 2
×
we areconsideringsymmetry-enrichedtoriccode fromgauginga fSPT phase,the ψ quasiparticlehas to forma linear
representation,implyingη canbechosento1andthereforeη =η . SotheactuallyclassificationisjustH2(G,Z ).
ψ e m 2
We also notice that any Abelian phase on anyon a that satisfies a relation like (39) must be the braiding phase of
a (fixed) Abelian anyon with a, which actually holds for any topological phase following from modularity. The fact
that we have η = η means η (η ) is the braiding phase of either I or ψ with e(m). Physically, we can think of
e m e m
U (a) as taking the σ+ defect around a. If η = η = 1, the only possiblity is that when σ+ fuses with σ+, we
g g e m − g h
obtainσ+ andalsoa fermionψ whichwhentaking a fullbraidarounde(m) generatesthe phasesη (η ). Therefore,
gh e m
if we write η (g,h)=η (g,h)=( 1)n(g,h) with n(g,h)=0,1 being the linearized Z 2-cocycle, we have
e m 2
−
σ+ σ+ =ψn(g,h)σ+ . (41)
g × h gh