Rokhsar-Kivelson Models of Bosonic Symmetry-Protected Topological States

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by  Luiz H. Santos

Rokhsar-KivelsonModelsofBosonicSymmetry-ProtectedTopologicalStates
Luiz H. Santos
Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
A platform for constructing microscopic Hamiltonians describing bosonic symmetry-protected topological
(SPT) states is presented. The Hamiltonians we consider are examples of frustration-free Rokhsar-Kivelson
models, which are known to be in one-to-one correspondence with classical stochastic systems in the same
spatial dimensionality. By exploring this classical-quantum mapping, we are able to construct a large class
of microscopic models which, in a closed manifold, have a non-degenerate gapped symmetric ground state
describing the universal properties of SPT states. Examples of one and two dimensional SPT states which
illustrateourapproacharediscussed.
5
1
I. INTRODUCTION SPT states. As we shall see, some of the exactly solvable
0
models previously studied20,21,26 will be identified as special
2
The SO(3) symmetric spin-1 anti-ferromagnetic Heisen- cases of a large class of models to be constructed here. We
r
p bergchain,whichwasshownbyHaldane1–3tohaveasymme- shallalsobeabletoconstructparentHamiltoniansfortwodi-
A trypreservinggappedgroundstate,providestheoldestknown mensionalZ2×Z2paramagnets,whoseeffectiveedgetheory
was shown in Ref. 16 to be in direct relation to non-trivial
example of a bosonic symmetry-protected topological (SPT)
9 3-cocycles.
stateinonedimension. Thetopologicalcharacterofthisstate
2
The classes of gapped bosonic insulators protected a by
is captured by a topological θ-term present in the non-linear
global symmetry G that we shall be concerned with have,
] sigmamodeleffectiveactiondescribinglong-wavelengthde-
el greesoffreedom.1–3 onad-dimensionalclosedmanifold,anon-degenerateground
- An insightful account of the properties of this anti- state
r
.st fLeirerbo,maangdnTetaiscaksipi(nAKchLaTin), wwhaso cgoivnesntrubcytedAafflHecakm,ilKtoenninaendiyn, |ΨG(cid:105)= (cid:112) 1 (cid:88) e−β2EG(s)+iWG(s)|s(cid:105), (1.1)
t Z(β)
a the same phase as the Heisenberg model, where the S = 1 s
m spins emerge from the composition of underlying S = 1/2
where {|s(cid:105)} denotes an orthonormal many-body basis,
d- dAeKgrLeTesmoofdferelehdaosma.n4,o5nW-diethgepneeriroadteicgbroouunnddasrtyatceotnhdaittdiooness,nthoet EG(s) ∈ R is a non-universal local function related to the
n decay of correlations of local operators in the ground state
break any symmetries and is separated from the first excited
o andthephaseW (s)∈Risauniversalpiecethatendowsthe
state by a finite gap. With open boundary conditions, on the G
c ground state (1.1) with its non-trivial topological properties.
[ otherhand,theAKLTmodelmakesitmanifestthateachedge
supportsa“free”S = 1/2degreeoffreedomcontributingto WG(s)plays,atthemicroscopiclevelconsideredhere,arole
3 analogoustothetopologicalθ-term.6,7,17,18
a2-folddegeneracyperedge. Interestingly,whiletheHamil-
v WhenW (s)=0,oneobtainsfromEq.(1.1)thenodeless
tonian is SO(3) symmetric and the bulk degrees of freedom G
6
groundstate
6 transformlinearlyunderthissymmetry,theeffectiveS =1/2
0 spinsontheedgestransformprojectivelyundertheactionof
0 thespinrotationsymmetry: aninitialS = 1/2spinstatero- |ΦG(cid:105)= (cid:112) 1 (cid:88) e−β2EG(s)|s(cid:105). (1.2)
0 tated by 360 degrees about an arbitrary axis is mapped into Z(β) s
.
2 itself up to a minus sign. The inter-connection among the
TheformofthegroundstateEq.(1.2)isveryappealing;for
0 topological θ-term action, the ground state degeneracy with
5 openboundaryconditionsandtheprojectiverepresentationof equal time correlation functions of operators in the diagonal
1 representation{|s(cid:105)},
the global symmetry on the edge degrees of freedom makes
:
v thissystemanon-trivialgappedphaseofmatter.
Xi theIrnesphiraevde bbyeetnhereecxeanmtppleroopfostahlesatnoti-cflearsrsoimfyaggnaeptpicedchSaPinT, (cid:104)ΦG|Oˆa(sˆ)Oˆb(sˆ)|ΦG(cid:105)=(cid:88) Oa(s)Ob(s)e−Zβ(EβG)(s) ,
r s
a phasesofmatterprotectedbyaglobalsymmetryGusingvar- (1.3)
iousmathematicalframeworkssuchasgroupcohomology,6,7
which generalizes the concept of projective representations, can be interpreted as equal time correlation functions of an
topologicalfieldtheories8–16 andnon-linearsigmamodelsin equilibriumd-dimensionalstatisticalmechanicalsystemwith
the presence of a topological θ-term action compatible with classical configurations {s}, each one occurring with prob-
the global symmetry G.17,18 Recently, a number of micro- ability p(s0) = e−βEG(s)/Z(β), where the real parameter β
scopicmodelsofbosonicSPTstateshavebeenstudied,which acquiresthenaturalinterpretationofaneffectiveinversetem-
helptoshedlightontheroleplayedbyphysicalinteractions peratureandthenormalizationfactorofthegroundstate,
inbringingaboutSPTphases.6,7,9,15,16,19–26
Thepurposeofthispaperistoprovideaframeworkforcon- Z(β)=(cid:88) e−βEG(s), (1.4)
structing microscopic models capable of describing bosonic
s
2
isinterpretedasthepartitionfunctionoftheclassicalsystem hence,
in “thermal” equilibrium. Hence, if the associated classical
model described by the partition function Eq. (1.4) is in the |Ψ (cid:105)=W |Φ (cid:105). (1.7b)
G G G
“disordered” phase, then typical correlation functions of lo-
caloperatorsdistantby|r|behavease−|r|/ξ,forsomefinite
correlationlengthξ,andtherepresentation(1.2)canbeasso-
FortheSPTgroundstateEq.(1.1)tobeinvariantunderthe
ciatedwithaquantummany-bodygroundstateinitsgapped
symmetryG,itisrequiredthat
phase.
Infacttheforegoingclassical-quantumcorrespondencecan
E (gs)=E (s),
bemademoreprecise.27–31Eq.(1.2)isrecognizedasthezero G G (1.8a)
W (gs)=W (s)mod2π,
energygroundstateofaclassofquantumdimer-likemodels G G
attheso-calledRokhsar-Kivelson(RK)point.27 InRef.27,it
ifGisanunitaryglobalsymmetry,and
wasnotedthatdimer-dimercorrelationfunctionsofthesquare
latticequantumdimermodelattheRKpointcanbecomputed
E (gs)=E (s),
exactlyfromthecorrespondingclassicaldimerproblem.32 In G G (1.8b)
Ref. 29, Ardonne, Fendley and Fradkin have established the WG(gs)=−WG(s)mod2π,
quantum-classicalcorrespondencetomoregeneralclassesof
RK Hamiltonians beyond dimer models. In Ref. 30, Henley ifGisananti-unitaryglobalsymmetry.
has observed that any stochastic classical system described NowletHG andHG be,respectively,thequantumHamil-
by a real transition rate matrix M can be interpreted, via a tonians whose non-degenerate ground states are |ΦG(cid:105) and
similaritytransformation,asanRKHamiltonian. InRef.31, |ΨG(cid:105). ThentheunitarymappingEq.(1.7b)establishes
Castelnovo, Chamon, Mudry and Pujol have shown that the
reverseistrue,namely,thatgivenaquantumRKHamiltonian HG =WGHGW−G1. (1.9)
in a “preferred basis” [the basis in which the ground state is
expressed as a linear combination of same-sign coefficients Starting, thus, from a parent Hamiltonian HG for the trivial
asinEq.(1.2)],thereexistsanassociatedstochasticclassical gapped state Eq. (1.2), one can construct a parent Hamilto-
modelwhosespectrumofrelaxationratesis(uptoanoverall nianHGfortheSPTgroundstateEq.(1.1)usingEq.(1.9),if
minussign)thesameastheenergyspectrumofthequantum the unitary transformation connecting the two ground states,
RKHamiltonianandwhoseequilibriumprobabilitydistribu- Eq.(1.7),isknown.
tion is the square of the coefficients in the expansion of the This paper is organized as follows. In Sec. (II) we review
RKquantumgroundstateEq.(1.2). therelevantpointsaboutthemappingbetweenstochasticclas-
In light of the above arguments, if one considers the con- sical systems and quantum RK Hamiltonians. In Sec. (III)
figurations {s} to be made of spins on a lattice, then the weconstructparentHamiltoniansofonedimensionalbosonic
groundstateEq.(1.2)offersanaturalrepresentationofapara- SPT states with Z × Z symmetry, where we shall use
n n
magneticstate,providedthecorrespondingclassicalsystemis the concept of entanglement spectrum degeneracy to deter-
chosentohaveaspectrumofrelaxationrateswithafinitegap minertheunitarytransformationEq.(1.7)thatmapsthetrivial
andcorrelationfunctionsoflocaloperators,Eq.(1.3),exhibit- groundstatetoothern−1topologicalphases.InSec.(IV)we
ingshort-rangebehavior. discusstheonedimensionalSPTstatewithanti-unitarytime-
Asfortheroleplayedbysymmetries,wenowletthequan- reversalsymmetryZT. InSec.(V)weconstructtwodimen-
2
tum system be invariant under a global symmetry group G, sionalmicroscopicmodelsthataccountforallthe8possible
whoseactiononthebasis|s(cid:105)isrepresentedby classesofZ ×Z SPTparamagnets.WeshowthattheZ ×Z
2 2 2 2
symmetrytransformationprojectedontotheonedimensional
S(cid:98)G|s(cid:105)=|gs(cid:105). (1.5) edgeacquiresanon-onsiteform,whichwasstudiedinRef.16
in connection with non-trivial 3-cocycles in the group coho-
ItisthenclearthatthegroundstateEq.(1.2)isauniqueand
mology.Finallywedrawsomeconclusionsandpointtofuture
Ginvariantstateprovidedthelocal“classicalenergy”E (s)
G directionsinSec.(VI).
issymmetricunderthetransformation{s}→{gs}:
E (gs)=E (s). (1.6)
G G
II. CLASSICAL-QUANTUMMAPPING
The central point of this paper is the observation, which In order to make the discussion self-contained, we review
willbesupportedbyconcreteexamples,thatinaclosedman- theessentialaspectsoftherelationshipbetweenquantumRK
ifold,theSPTgroundstateEq.(1.1)canbeobtainedfromthe Hamiltonians and stochastic classical systems.30,31 The con-
trivialinsulatorgroundstateEq.(1.2)viaaglobalsymmetry- tent of Secs. (IIA) and (IIB) closely follows Ref. 31, which
preserving unitary transformation W , whose action on the the reader may consult for further details. In Sec. (IIC) we
G
many-bodybasis{|s(cid:105)}is givethegeneralformoftheSPT-RKHamiltoniansdescribing
bosonicSPTstatesind-dimensionalspaceobtainedfromthe
WG|s(cid:105):=eiWG(s)|s(cid:105), (1.7a) unitarymappingEq.(1.9).
3
A. Rokhsar-KivelsonHamiltonians B. RelationbetweenquantumRKHamiltoniansand
stochasticclassicalsystems
AquantumRKHamiltoniansatisfiesthreeproperties31:
InordertoshowtherelationbetweenquantumRKHamil-
1. The orthonormal elements of the basis Σ = {|s(cid:105)},
tonianandstochasticclassicalsystems,oneconsidersthereal-
whichspantheHilbertspace, valuedmatrixM definedby30,31
(cid:88)
(cid:104)s|s(cid:48)(cid:105)=δs,s(cid:48), 11= s |s(cid:105)(cid:104)s|, (2.1) Mss(cid:48) ≡−e−β2[E(s)−E(s(cid:48))] (HRK)ss(cid:48) , (2.7)
formacountableset. where(H ) =(cid:104)s|H |s(cid:48)(cid:105)denotesthematrixelements
RK ss(cid:48) RK
ofH . FromthefactthatH isarealandHermitianoper-
2. The quantum Hamiltonian can be decomposed into a RK RK
ator,itfollowsthat
sum of positive-semidefinite projector-like Hermitian
operatorsP [Eq.(2.2)].
s,s(cid:48) (H ) =(H ) (2.8)
RK ss(cid:48) RK s(cid:48)s
3. The ground state [Eq. (2.4)] is annihilated by every
P andthenormalizationconstantofthegroundstate and,fromEqs.(2.2)and(2.7),thatthematrixM satisfies:
s,s(cid:48)
canbeinterpretedasthepartitionfunctionofaclassical
M >0, ifs(cid:54)=s(cid:48), (2.9a)
system[Eq.(2.5)]. ss(cid:48)
and
TheRKHamiltoniantakestheform
s(cid:48)(cid:54)=s
1 s(cid:88)(cid:54)=s(cid:48) Mss =−(cid:88) Ms(cid:48)s. (2.9b)
H = ω P , (2.2a)
RK 2 s,s(cid:48) s,s(cid:48) s(cid:48)
(s,s(cid:48))
One then considers a classical system with a phase space
where
formedbyconfigurations{s},which,asafunctionoftimeτ,
ωs,s(cid:48) ∈R, ωs,s(cid:48) >0, (2.2b) can be visited stochastically with probability ps(τ) evolving
accordingtothemasterequation
Ps,s(cid:48) =−|s(cid:105)(cid:104)s(cid:48)|−|s(cid:48)(cid:105)(cid:104)s| dps(τ) =(cid:88) M p (τ)
+e−β2[E(s(cid:48))−E(s)]|s(cid:105)(cid:104)s|+e−β2[E(s)−E(s(cid:48))]|s(cid:48)(cid:105)(cid:104)s(cid:48)|, dτ s(cid:48) ss(cid:48) s(cid:48)
(2.10)
(2.2c) s(cid:48)(cid:54)=s
(cid:88) (cid:104) (cid:105)
= M p (τ)−M p (τ) ,
whereβ ∈R,E(s)∈RandP isaprojector-likepositive ss(cid:48) s(cid:48) s(cid:48)s s
s,s(cid:48) s(cid:48)
semi-definiteHermitianoperatorsatisfying
(cid:26) (cid:27) where Eq. (2.9b) has been used to achieve the last equality
β
P2 =2 cosh [E(s(cid:48))−E(s)] P . (2.3) of Eq. (2.10). The first term on the r.h.s. accounts for the
s,s(cid:48) 2 s,s(cid:48) transitions out of the configurations s(cid:48) into the configuration
s,whilethesecondtermonther.h.s. accountsfortransitions
Oneeasilyverifiesthatthenodelessstate outoftheconfigurationsintotheconfigurationss(cid:48).
MoreoverEqs.(2.7)and(2.8)areeasilyseentoimply,for
|ΦRK(cid:105)= (cid:112) 1 (cid:88) e−β2E(s)|s(cid:105), (2.4) everypairofindices(s,s(cid:48)),
Z(β)
s
withnormalizationconstant Mss(cid:48)p(s0(cid:48)) =Ms(cid:48)sp(s0), (2.11a)
(cid:88)
Z(β)= e−βE(s), (2.5) where
s
1
satisfies p(0) ≡ e−βE(s). (2.11b)
s Z(β)
P |Φ (cid:105)=0, ∀(s,s(cid:48)). (2.6)
s,s(cid:48) RK Eq. (2.11) implies the condition of detailed balance on the
Thus Eq. (2.4) is the ground state of the RK Hamilto- matrix M as well as that p(s0) is the equilibrium probability
nian Eq. (2.2) with energy zero. The normalization constant distributionassociatedwiththeclassicaldynamicsEq.(2.10).
Eq.(2.5)canbeinterpretedasthepartitionfunctionofaclas- Denoting by λ and ψ(R;n), respectively, the right-
n s
sicalsystemwithclassicalenergyE(s)attheeffectiveinverse eigenvaluesandright-eigenvectorsofM,
temperatureβ. Inthe“infinitetemperature”limitβ = 0,the
Boltzmann factors in the partition function tend to unity and (cid:88) M ψ(R;n) =λ ψ(R;n), (2.12)
ss(cid:48) s(cid:48) n s
Z(0)countsthenumberofallowedconfigurations.
s(cid:48)
4
then the time dependent solution of Eq. (2.10) can be ex- largestcharacteristictimescaleassociatedwiththedecayinto
pressedas the equilibrium configuration is finite in the thermodynamic
limit, thenitfollowsfromEq.(2.14)thatthemany-bodyen-
(cid:88)
p (τ)= a (0)eλnτψ(R;n), (2.13) ergy spectrum of the quantum Hamiltonian possess a finite
s n s
energygapforexcitationsabovegroundstate.
n
wherea (0)arecoefficientsdeterminedbytheinitialcondi-
n
tions. C. SPT-RKHamiltonians
Since Eq. (2.7) establishes, up to an overall minus sign, a
similaritytransformationbetweenH andM,thespectrum WenowgivethegeneralformoftheSPT-RKHamiltonians
RK
ofrelaxationrates{λ }ofM andtheenergyspectrum{ε } withgroundstategivenbyEq.(1.1).
n n
ofH aresimplyrelated: Letthed-dimensionalquantumsystem,protectedbyglobal
RK
symmetryG,initstrivialphasebedescribedbyanRKHamil-
εn =−λn >0. (2.14) tonian HG of the form Eq. (2.2) with the non-degenerate
groundstate|Φ (cid:105),Eq.(1.2),whereweimposethesymmetry
G
constraint Eq. (1.6) upon E (s). Then the unitary transfor-
G
Whentheclassicalsystemwhosedynamicsisdescribedby mationEq.(1.7)yieldstheSPTgroundstate|Ψ (cid:105),Eq.(1.1),
G
Eq. (2.10) has a spectrum of relaxation rates such that the andEq.(1.9)yieldstheSPT-RKHamiltonianH :
G
s(cid:48)(cid:54)=s
1 (cid:88)
H =W H W−1 = ω P , (2.15a)
G G G G 2 s,s(cid:48) s,s(cid:48)
(s,s(cid:48))
where
ω ∈R, ω >0, (2.15b)
s,s(cid:48) s,s(cid:48)
Ps,s(cid:48) =−ei[WG(s)−WG(s(cid:48))]|s(cid:105)(cid:104)s(cid:48)|−ei[WG(s(cid:48))−WG(s)]|s(cid:48)(cid:105)(cid:104)s|+e−β2[EG(s(cid:48))−EG(s)]|s(cid:105)(cid:104)s|+e−β2[EG(s)−EG(s(cid:48))]|s(cid:48)(cid:105)(cid:104)s(cid:48)|.
(2.15c)
Before we proceed to discuss specific SPT systems, we III. SPTSTATESINONEDIMENSION
closethissectionwithafewimportantremarks:
(a) InSecs.(IIA)and(IIB)wesawthatthestructureofthe
InthissectionwederiveparentHamiltoniansofonedimen-
RK ground state is determined only by βE(s), while sional bosonic SPT states protected by Z ×Z symmetry.
n n
the energy spectrum is determined solely by the cou-
In a chain with periodic boundary conditions, each of these
plingsω .30,31
s,s(cid:48) phases is described by a non-degenerate gapped symmetric
(b) If the couplings ω are such that the energy spec- groundstate.Inachainwithopenboundaryconditions,onthe
s,s(cid:48)
trumhasanon-degenerategappedgroundstate(which otherhand,thereremainsatrivialphasewithanon-degenerate
istheonlycaseconsideredinthiswork),thenEq.(2.4) groundstate,whilen−1phaseshaven-folddegeneracyper
parametrizes a large class of ground states, whereby edge,accountingforatotaln2-folddegeneracyoftheground
βE(s) controls the correlation length associated with statemanifoldinthethermodynamiclimit.
thedecayofcorrelationfunctionssuchasEq.(1.3). OuraimistoconstructtheunitarytransformationsEq.(1.7)
connecting the trivial ground state and the n−1 non-trivial
(c) Intheparticularcaseβ = 0,thegroundstateEq.(2.4)
SPTgroundstates.Ourstrategyinderivingsuchunitarymap-
acquires the form of a trivial product state. It will
pings is to draw on the notion of entanglement spectrum de-
proveusefultoworkinthis“infinitetemperature” limit,
generacy as follows. As we pointed out in the remark (c)
where the correlation length is zero, in order to deter-
of Sec. (IIC), in the “infinite temperature” limit β = 0 the
minetheformoftheunitarytransformationEq.(1.7).
ground state of the trivial SPT chain reduces to a product
(d) While the unitary mapping Eq. (1.7) leaves the energy state. The entanglement structure of the product state is as
spectrum unchanged, it introduces a non-trivial entan- simpleasitgets,fortheSchmidtdecompositionwithrespect
glementstructureintheSPTgroundstateEq.(1.1)that toanypartitioncontainsonlyoneeigenvalue(equalto1). We
isresponsibleforitsuniversaltopologicalproperties. shall find the unitary transformation Eq. (1.7) by demanding
5
that, in the limit β = 0, the entanglement spectrum of the andtheSPTstate
non-trivial SPT ground state, for any partition of the chain,
acquires an n-fold degeneracy. Remarkably, we shall verify |ΨZ2×Z2(cid:105) ≡WθZ2×Z2|ΦZ2×Z2(cid:105)
that these unitary mappings, via Eq. (1.9), endow the parent 1 (cid:88)
≡ eiWZ2×Z2(s)|s(cid:105)
Hamiltonians of non-trivial SPT chains with the required n- 2Ns/2 (3.5)
s
fold degeneracy of the energy spectrum per edge. Once the
unitarytransformationEq.(1.7)isderived,wecanobtainthe = 1 (cid:88) ei(cid:80)j θj(1−sjsj+1)/2|s(cid:105).
most general form of the SPT ground state Eq. (1.1) by al- 2Ns/2 s
lowing β (cid:54)= 0 without changing either the gapped nature of
The unitary transformation Eq. (3.4) endows the state
themany-bodyenergyspectrumorthetopologicalproperties
Eq. (3.5) with an amplitude eiθj for every domain wall be-
of the ground state. That this is true can be seen perturba-
tweenneighborspinsinthemany-bodyconfiguration|s(cid:105).We
tively: movingawayfromtheβ = 0limitwithβ << 1and
seektofindθ forwhichEq.(3.5)describestheSPTground
E (s)alocalfunctioninEq.(2.15),amountstoaddingsmall j
G state.
localsymmetry-preservingperturbationstothegappedβ =0
DuetotheproductstatenatureofEq.(3.2)andthepairwise
theory,whichtherefore,cannotimmediatelydestroytheSPT
entanglement induced by the unitary mapping Eq. (3.4), one
phase. Based on this we expect that, for 0 < β ≤ β , the
c effortlesslyfindsthat,foranypartitionΣ ,thereduceddensity
ground state Eq. (1.1) describes a class of many-body SPT i
operator obtained by tracing over one of the subsystems is
groundstatesadiabaticallyconnectedtotheβ =0limit.
givenbythe2×2matrix
(cid:18) (cid:19)
A. Z2 × Z2SPTstatesind=1 ρ = 1 1 cos(θi) . (3.6)
i 2 cos(θi) 1
Weconsideraspinchain,withanevennumberN ofsites,
s For θ = 0 the above density matrix has a single non-zero
dividedintoevenandoddsublatticesandZ ×Z symmetry i
2 2 Schmidt eigenvalue. The existence of 2 degenerate Schmidt
generatedby
eigenvalues(equalto1/2)isverifiedfor
S(cid:98)Z(1) = (cid:89) Xj, S(cid:98)Z(2) = (cid:89) Xj. (3.1) θi =±π/2. (3.7)
2 2
j∈even j∈odd
Inthefollowing,{X,Y,Z}denotethethreePaulimatrices. Moreover, imposing that the unitary transformation
Eq. (3.4) commutes with either of the Z symmetries in
We start by examining a trivial spin chain in the special 2
Eq.(3.1)yieldsthefinalform
limitwhereitisrepresentedbyaproductstate,
(cid:89)Ns 1 (cid:88) WZ2×Z2 =(cid:89) eiπ4 (1−Z2j−1Z2j)e−iπ4 (1−Z2jZ2j+1).
|ΦZ2×Z2(cid:105)= |→(cid:105)j = 2Ns/2 |s(cid:105), (3.2) j (3.8)
j=1 s
where Xj|→(cid:105)j = |→(cid:105)j and {|s(cid:105)} represents the 2Ns Onethenfinds
many-body spin states in the eigenbasis of Z operators:
Zj|sj(cid:105) = sj|sj(cid:105), for sj ± 1. We recognize Eq. (3.2) as WZ2×Z2X2jW−Z21×Z2 =X2jZ2j−1Z2j+1, (3.9)
thegroundstateEq.(1.2)intheβ =0limit. Ithastheparent WZ2×Z2X2j+1W−Z21×Z2 =X2j+1Z2jZ2j+2.
Hamiltonian
HZ2×Z2 =ω0 (cid:88)Ns (1−Xj) , (3.3) munoTidtahirfieyeotdrpaPenarsuaftlooirsrmpWiantiZoo2pn×e)Zrwa2thXoircjh(sWiisn−Zc“2ed1×rieZts2issecodab”ntabibyneethdreefrgdoaomrmdeaXdinjawbsyalaal
j=1 operatorZ Z withsupportontheoppositesublattice.
j−1 j+1
So, under the unitary transformation Eq. (3.8), the SPT
whichisanRKHamiltonianoftheformEq.(2.2)withβ =0
Hamiltonianatzerocorrelationlengthis
andω =ω >0,wherethesuminEq.(2.2a)extendsover
s,s(cid:48) 0
paiTrsheofpsrotadtuesct(sst,aste(cid:48))fworhmicohfdthifefegrrboyunadssintagtleeEsqp.in(3fl.2ip).implies HZ2×Z2 ≡WZ2×Z2HZ2×Z2W−Z21×Z2
that its Schmidt decomposition, with respect to any partition (cid:88)Ns (cid:16) (cid:17) (3.10)
=ω 1−Z X Z .
Σ between sites i and i+1 of the lattice, contains a single 0 j−1 j j+1
i
Schmidteigenvalue(equalto1). j=1
Now,forθj ∈R,weintroducetheunitarymapping We note that the model Eq. (3.10) has been constructed in
Ref.21usingtheconceptofdecorateddomainwalls,whilewe
(cid:89)
WθZ2×Z2 ≡ eiθj(1−ZjZj+1)/2 (3.4) hspaevcetraurmrivveidaothneitubnyitaarpypmeaalpinpgintgoEthqe.n(3o.t8io).nofentanglement
j
6
Groundstatedegeneracycanbeeasilyattestedbystudying whichisanRKHamiltoniansoftheformEq.(2.2)withβ =
thismodelwithopenboundaryconditions,where 0,ω = ω > 0,wherethesuminEq.(2.2a)extendsover
s,s(cid:48) 0
pairsofstates(s,s(cid:48))whichdifferbyasingleZ spinflip.
3
N(cid:88)s−1 (cid:16) (cid:17) Wenowconsider,forθj ∈R,theunitarytransformation
Hopen =ω 1−Z X Z . (3.11)
Z2×Z2 0 j−1 j j+1
j=2
ThefactthattheaboveHamiltoniancommuteswithZ1and WθZ3×Z3 =(cid:89) exp(cid:34)iθj(cid:88)2 (ω¯1a−−(σ1)j†(σωja+1−)a1)(cid:35) (3.16)
Z impliesthatthereare2-folddegeneratestatesassociated j a=1
Ns
withtheleftandtherightedgescorrespondingtostateswith
s = ±1 and s = ±1. Thus the universal properties of
1 Ns andtheSPTstate
theSPTstatestudiedhereareencodedintheunitarymapping
Eq.(3.8).
theWmitohrethgeenuenriatalrfyortmranosffotrhmeaStiPoTngErqo.un(3d.8s)t,atweeEqc.an(1o.1b)tabiyn |ΨZ3×Z3(cid:105)≡WθZ3×Z3|ΦZ3×Z3(cid:105)
allowingβ (cid:54)= 0withoutchangingeitherthegappednatureof ≡ 1 (cid:88) eiWZ3×Z3(s)|s(cid:105)
themany-bodyenergyspectrumorthetopologicalproperties 3Ns/2
s
ofthegroundstate. Inthisregard,theHamiltonianEq.(3.10)  
is a particular example of a larger class of SPT models de- 1 (cid:88) (cid:88) (cid:88)2 1−ωa(sj+1−sj)
scribedinEq.(2.15),withthephasefactorseiWZ2×Z2(s)given = 3Ns/2 exp iθj (ω¯a−1)(ωa−1)|s(cid:105).
s j a=1
by acting with the unitary transformation Eq. (3.8). on the
(3.17)
state|s(cid:105).
B. Z3 × Z3SPTstatesind=1 The unitary transformation Eq. (3.16) endows the state
Eq. (3.17) with an amplitude eiθj for every pair of neighbor
Weconsideraspinchain,withanevennumberNs ofsites spinsj andj+1forwhichsj (cid:54)= sj+1 (mod3)inthemany-
andZ ×Z symmetrygeneratedby bodyconfiguration|s(cid:105).
3 3
S(cid:98)Z(1) = (cid:89) τj, S(cid:98)Z(2) = (cid:89) τj. (3.12) wiDseueenttoantghleempreondtuinctdustcaetdebnyattuhreeuonfitEarqy.m(3a.1p4p)inagnEdqth.(e3p.1a6ir)-,
3 3
j∈even j∈odd oneeffortlesslyverifiesthat,foranypartitionΣ ,thereduced
i
density operator obtained by tracing over one of the subsys-
where,ateachsitej,weconsidertheoperators
temsisgivenbythe3×3matrix
   
0 0 1 1 0 0
τj =1 0 0 , σj =0 ω 0  , (3.13)  1 f(θ ) f(θ )
0 1 0 0 0 ω2 1 i i
ρi = 3 f(θi) 1 f(θi) , (3.18)
f(θ ) f(θ ) 1
satisfyingτ3 =σ3 =1,τ†σ τ =ωσ ,whereω =ei2π/3. i i
j j j j j j
Let |s (cid:105), for s = 0,1,2, be the eigenstates of σ :
i i i
σi|si(cid:105) = ωsi|si(cid:105). As in Sec. (IIIA), we start with a triv- wheref(θ)= 1(1+2cos(θ)). Forθ =0theabovedensity
ialZ ×Z paramagnetdescribedbythegroundstate 3 i
3 3 matrix has a single non-zero Schmidt eigenvalue. The exis-
tence of 3 degenerate Schmidt eigenvalues (equal to 1/3) is
|ΦZ3×Z3(cid:105)≡(cid:89) (cid:16)|0j(cid:105)+|√1j3(cid:105)+|2j(cid:105)(cid:17)= 3N1s/2 (cid:88) |s(cid:105), thenverifiedfor
j {s}
(3.14)
2π
θ(p) = p, p=1,2. (3.19)
where{|s(cid:105)}representsthe3Ns many-bodyspinstatesinthe i 3
eigenbasisofσoperators. WerecognizeEq.(3.14)astheRK
ground state Eq. (1.2) in the β = 0 limit. It has the parent
Hamiltonian
Moreover, imposing that the unitary transformation
(cid:88)Ns (cid:104) (cid:16) (cid:17)(cid:105) Eq. (3.16) commutes with either of the Z3 symmetries in
HZ3×Z3 =ω0 2− τj +τj† , (3.15) Eq.(3.12)yieldsthefinalform
j=1
7
W(p) =(cid:89) exp(cid:104)i2πp (cid:88)3 1−(σ2†j−1σ2j)a (cid:105) exp(cid:104)−i2πp (cid:88)2 1−(σ2†jσ2j+1)a (cid:105), p=1,2. (3.20)
Z3×Z3 3 (ωa−1)(ω¯a−1) 3 (ωa−1)(ω¯a−1)
j a=1 a=1
Eq.(3.20)establishestwounitarymappingsbetweenthetriv- Moreoverwefindthat
ial SPT ground state and the two non-trivial SPT ground
(cid:16) (cid:17)−1
states. W(p) τ W(p) =τ (σ σ† )p,
Z3×Z3 2j Z3×Z3 2j 2j−1 2j+1
(cid:16) (cid:17)−1
W(p) τ W(p) =τ (σ† σ )p,
Z3×Z3 2j+1 Z3×Z3 2j+1 2j 2j+2
(3.21)
so that the operator τ gets “dressed” by a domain wall like
j
operator–carryingZ chargep = 1,2–withsupportonthe
3
oppositesublattice.
TheSPTHamiltoniansintheβ =0limitthenread
(cid:16) (cid:17)−1
HZ(p3)×Z3 =W(Zp3)×Z3HZ3×Z3 W(Zp3)×Z3
(cid:88) (cid:110)(cid:104) (cid:105) (cid:111) (cid:88) (cid:110)(cid:104) (cid:105) (cid:111) (3.22)
=ω 1−τ (σ σ† )p +H.c. +ω 1−τ (σ† σ )p +H.c. ,
0 2j 2j−1 2j+1 0 2j+1 2j 2j+2
j j
forp=1,2. ofthegroundstate. Inthisregard,theHamiltonianEq.(3.22)
providesaparticularexampleofalargerclassofmodelsde-
As in Sec. (IIIA), degeneracy of the ground state energy
scribedinEq.(2.15),withthephasefactorseiWZ3×Z3(s)given
manifold can be checked by placing this system in an open
by acting with the unitary transformation Eq. (3.20). on the
chain, in which case, one finds that the SPT Hamiltonians
state|s(cid:105).
commuteswithσ andσ ,thusimplyinga3-folddegenerate
1 Ns
state associated to the left and right edges. Thus the univer-
salpropertiesoftheSPTstatestudiedhereareencodedinthe C. Z × Z SPTstatesind=1
n n
unitarymappingEq.(3.20).
With the unitary transformation Eq. (3.20) we can obtain We generalize the findings of Secs. (IIIA) and (IIIB) to
the more general form of the SPT ground state Eq. (1.1) by Z × Z SPT states (see the Appendix for definitions and
n n
allowingβ (cid:54)= 0withoutchangingeitherthegappednatureof useful formulas). We arrive at n − 1 Z × Z symmetric
n n
themany-bodyenergyspectrumorthetopologicalproperties unitarytransformations,labeledbyp=1,...,n−1,
W(p) =(cid:89) exp(cid:104)i2πp n(cid:88)−1 1−(σ2†j−1σ2j)a (cid:105) exp(cid:104)−i2πp n(cid:88)−1 1−(σ2†jσ2j+1)a (cid:105). (3.23)
Zn×Zn n (ωa−1)(ω¯a−1) n (ωa−1)(ω¯a−1)
j a=1 a=1
Itcanbeshown,usingEqs.(A3), (A4)and(A5),that thatthisclassofonedimensionalSPTmodelshasbeenstud-
ied in Ref. 26 without reference to the connection between
(cid:16) (cid:17)−1
W(p) τ W(p) =τ (σ σ† )p, unitary transformations and ground state entanglement spec-
Zn×Zn 2j Zn×Zn 2j 2j−1 2j+1 trumthatweareexploringhere. Moreover,inourformalism,
(cid:16) (cid:17)−1
W(p) τ W(p) =τ (σ† σ )p, we do not need to be restricted to the β = 0 limit for, with
Zn×Zn 2j+1 Zn×Zn 2j+1 2j 2j+2 theunitarytransformationEq.(3.23),wecanconstructalarge
(3.24)
class of SPT-RK Hamiltonians of the form Eq. (2.15) with
forp=1,...,n−1. thephasefactorseiWZn×Zn(s)givenbyactingwiththeunitary
TheZ × Z SPTHamiltoniansintheβ = 0limitareof transformationEq.(3.23)onthebasisstate|s(cid:105).
n n
thesameformasEq.(3.22)withp = 1,...,n−1. Wenotice
8
IV. SPTSTATEINONEDIMENSIONWITH due to the fact that, with periodic boundary conditions,
(cid:80)
TIME-REVERSALSYMMETRY (1−s s )/2isanevennumber. Moreover,comparing
j j j+1
Eqs.(4.5)and(4.7)withtheresultsobtainedinSec(IIIA),we
WenowstudytheonedimensionalbosonicSPTstatespro- concludethat, foranypartitionΣi betweensitesiandi+1,
tectedbytime-reversalsymmetrywhich,beingananti-unitary thereare2degenerate(equalto1/2)Schmidteigenvaluesand
symmetry,requiresconditionsEq.(1.8b)tobesatisfied. thattheSPTparentHamiltonianforthegroundstateEq.(4.4)
Theactionoftime-reversalsymmetryshallberepresented reads
bytheanti-unitaryoperator Θ, (cid:88)Ns
H ≡W H W−1 =ω (1+Z X Z ) .
ΘΘ−1 =1, Θ2 =1, (4.1a) ZT2 ZT2 ZT2 ZT2 0 j−1 j j+1
j=1
(4.8)
whereweworkwiththerepresentation
(cid:16) (cid:89) (cid:17) SincethisisessentiallythesamemodelastheZ ×Z spin
Θ = X K, 2 2
j (4.1b) chaindiscussedinSec(IIIA),weconcludethattheZT sym-
j 2
metricSPTHamiltonianhas2-folddegeneracyoftheground
withK denotingthecomplexconjugationoperator. stateperedge.
AonedimensionalchainofN sitesinthetrivialZT insu- WiththeunitarytransformationEqs.(4.5)and(4.7)wecan
s 2
latingphasecanbedescribedbytheHamiltonian obtainthemostgeneralformoftheSPTgroundstateEq.(1.1)
by allowing β (cid:54)= 0 without changing either the gapped na-
(cid:88)Ns ture of the many-body energy spectrum or the topological
HZT =ω0 (1−Xj) , (4.2) properties of the ground state. In this regard, the Hamilto-
2
j=1 nianEq.(4.8)providesaparticularexampleofalargerclass
whichhasthetime-reversalsymmetricgroundstate of models described in Eq. (2.15), with the phase factors
1 (cid:88) eiWZT2(s)giveninEq.(4.7).
|ΦZT2 (cid:105)= 2Ns/2 |s(cid:105), (4.3)
s
V. SPTSTATESINTWODIMENSIONS
where{|s(cid:105)}representsthe2Ns many-bodyspinstatesinthe
eigenbasisofZ operators: Zj|sj(cid:105) = sj|sj(cid:105), forsj = ±1. A. Z SPTstatesind=2
2
We recognize Eq. (4.3) as the ground state Eq. (1.2) in the
β = 0limitandEq.(4.2)asanRKHamiltonianoftheform
TheclassificationofbosonicSPTstatesintwodimensions,
Eq.(2.2)withβ = 0andω = ω > 0, wherethesumin
s,s(cid:48) 0 protectedbyZ symmetry,assertstheexistenceofngapped
Eq.(2.2a)extendsoverpairsofstates(s,s(cid:48))whichdifferbya n
phasesofmatter.6–8Fromthepointofviewoftheonedimen-
singlespinflip.
We now obtain the unitary mapping W , which, applied sional edge, these n phases can be distinguished by the way
ZT2 right-andleft-movingdegreesoffreedomtransformunderthe
onthetrivialgroundstateEq.(4.3),yieldsthenon-trivialSPT
symmetry: while in the trivial phase right- and left-moving
state
modescarrythesameZ charges,theirZ chargesarediffer-
n n
|ΨZT2 (cid:105)≡WZT2 |ΦZT2 (cid:105)≡ 2N1s/2 (cid:88) eiWZT2(s)|s(cid:105), (4.4) eanctofnosreeqaucehncoef,thineeoathcheronf−the1senonn−-tri1vinaolnS-PtrTivsiatalteSsP.8T,9,s1t5aAtess
s
theedgestatescannotbegappedandsymmetrypreservingat
whereintheactionofWZT onthemany-bodybasis{|s(cid:105)}is thesametime.
2
definedas LevinandGuhaveconstructedamodelonatriangularlat-
ticethatdescribesaZ SPTparamagnetintwodimensions.20
WZT2 |s(cid:105)=eiWZT2(s)|s(cid:105). (4.5) TheZ2symmetryisg2eneratedby
(cid:89)
S(cid:98)Z2 = Xj, (5.1)
Invariance of the SPT ground state |Ψ (cid:105) under time-
ZT j
2
reversalsymmetryEq.(4.1),accordingtoEq.(1.8b),implies
where {X,Y,Z} denote the three Pauli matrices and
WZT(s)=−WZT(−s)+2π×integer. (4.6) Zj|sj(cid:105)=sj|sj(cid:105),forsj =±1.
2 2 InRef.20thetwokindsofparamagneticgroundstatesare
representedby
In light of the discussion in Sec. (III), if we assume for
1 (cid:88)
WZT2 anansatzthatleadstotheentanglementofnearestneigh- |ΦZ2(cid:105)= 2Ns/2 |s(cid:105), (5.2)
borspins, thantheonlytransformation(uptoatrivialglobal s
phase)consistentwithEq.(4.6)has forthetrivialparamagnet,and
π (cid:88) 1 (cid:88)
WZT2(s)= 4 (1−sjsj+1) (4.7) |ΨZ2(cid:105)= 2Ns/2 eiπL(s)|s(cid:105), (5.3)
j s
9
for the non-trivial paramagnet. N denotes the number of icalpropertiesofthegroundstategiverisetotheedgephysics
s
sitesofthetriangularlatticeandL(s)∈Ncountsthenumber studiedinRef.16.
of loops that separate domain-wall regions for each one of
the 2Ns many-body spin configurations |s(cid:105) (see Figs. 1, 2,
and3). ThuswhilethecoefficientsintheexpansionEq.(5.2)
all have the same sign, in the non-trivial SPT ground state
Eq.(5.3), thesignofthecoefficientsdependsonthenumber
ofloopsL(s). Thisnon-trivial(real)phasefactorwasshown
inRef.20toberesponsiblefortheuniversalpropertiesofthe
Z non-trivialparamagneticstate.
2
FromEqs.(5.2)and(5.3),weidentifythegroundstatesof
Ref. 20 with Eqs. (1.2) and (1.1) in the limit β = 0, and the
unitary mapping WZ between the two classes of paramag-
2
nets,definedbyitsactiononthebasis{|s(cid:105)},tobe
WZ |s(cid:105)=eiWZ2(s)|s(cid:105)=eiπL(s)|s(cid:105). (5.4) FIG.2. (Coloronline)Aparticularmany-bodyconfiguration|s(2)(cid:105)
2
ofspinsZ(2)(blue).
j
Given that the unitary transformation defined in Eq. (5.4)
encodes the universal properties of the two dimensional Z
2
SPT paramagnet, we can move away from the “infinite tem-
perature” limit β = 0 studied in Ref. (20) by allowing a
largerclassofZ SPTparamagnetswithgroundstategivenby
2
Eq.(1.1)andmicroscopicHamiltoniansgivenbyEq.(2.15).
B. Z ×Z SPTstatesind=2
2 2
FIG. 3. (Color online) A particular many-body configuration
|s(12)(cid:105)ofthespinsZ(12) = Z(1)Z(2) (green). Notehowthedi-
j j j
rectionofthearrowsateverysiteisconsistentwithFigs.1and2.
InaddressingSPTstateswithZ ×Z symmetry,weallow,
2 2
at each site j of the lattice, two spin species represented by
Pauli operators {X(1),Y(1),Z(1)} and {X(2),Y(2),Z(2)},
j j j j j j
wheretheZ ×Z symmetryisgeneratedby
2 2
S(cid:98)Z(12) =(cid:89) Xj(1), S(cid:98)Z(22) =(cid:89) Xj(2). (5.5)
j j
FIG.1. (Coloronline)Aparticularmany-bodyconfiguration|s(1)(cid:105)
of spins Z(1) (red), where the loops are defined along the domain Weimmediatelyrealizethatthereshouldexistatleastfour
j
wallsandhavesupportontheduallattice. AsinRef.20, inorder kindsofparamagnets,whichareobtainedbystackingtwode-
todeterminetheprojectionoftheZ symmetryontheedge,wetake coupled Z symmetric systems. These four states of matter
2 2
theouterspinsinareferencestatewheretheyareallpointingup. are parametrized by two integer numbers, p ,p ∈ {0,1},
1 2
which contribute to the expansion of the ground state with
the (real) phase factor exp(cid:8)iπ (cid:2)p L(s(1))+p L(s(2))(cid:3)(cid:9),
In Ref. 16, the classification of two dimensional bosonic 1 2
SPTstateswithZ ×Z symmetrywasstudiedfromthepoint whereL(s(1))andL(s(2))count,respectively,thenumberof
n m
ofviewoftheeffectiveonedimensionaledgedegreesoffree- loopsdefinedbydomainwallconfigurationsformedbyspins
dom. Thenon-trivialSPTedgestateswerethereshowntobe ofspecies1and2,asinFigs.1and2.
associatedwithnon-trivial3-cocyclesinthegroupcohomol- In order to obtain Z2 × Z2 symmetric states that go be-
ogy.However,nomicroscopictheoryforthetwo-dimensional yondthetensorproductoftwoZ2 symmetricstates,wereal-
bulkstateshasbeeninvestigated. Inthissection,weconstruct izethat, ateachsitej, wecanconsideranindependentIsing
amicroscopictheoryfortheZ ×Z bosonicSPTstatesusing spinZ(12) ≡Z(1)Z(2)witheigenvaluess(12) =s(1)s(2).
2 2 j j j j j j
theformalismofRKHamiltoniansandshowhowthetopolog- Wethenpropose
10
|Ψ2Zd2×Z2(cid:105)= 2N1s (cid:88) (cid:88) eiπ[p1L(s(1))+p2L(s(2))+p12L(s(12))]|s(1);s(2)(cid:105) (5.6)
s(1) s(2)
asadescriptionofthe8classesofZ ×Z symmetricground whichisthetensorproductoftwotrivialZ paramagnets.
2 2 2
states parametrized by 3 binary indices p ,p ,p ∈ {0,1}.
1 2 12
Weadoptthenotation|s(1)(cid:105)⊗|s(2)(cid:105) ≡ |s(1);s(2)(cid:105). When
Eq. (5.6) corresponds to the choice of ground state in the
p = p = p = 0, Eq. (5.6) describes the trivial ground
1 2 12 “high temperature” limit β = 0 and the SPT ground states
state
can be obtained from the trivial one, Eq. (5.7), via the uni-
|Φ2d (cid:105)= 1 (cid:88) (cid:88) |s(1);s(2)(cid:105), tarytransformationW2Zd2×Z2,whoseactiononthemany-body
Z2×Z2 2Ns (5.7) basis{|s(1);s(2)(cid:105)}reads
s(1) s(2)
W2Zd2×Z2|s(1);s(2)(cid:105)≡eiWZ22d×Z2(s(1);s(2))|s(1);s(2)(cid:105)=eiπ[p1L(s(1))+p2L(s(2))+p12L(s(12))]|s(1);s(2)(cid:105). (5.8)
In Eqs. (5.6) and (5.8), in order to simplify the notation, we describes a trivial paramagnetic systemand has Eq. (5.7) as
omittheindicesp ,p andp onwhichtheSPTgroundstates itsgroundstate. ItisthesumoftwoRKHamiltoniansofthe
1 2 12
andtheunitarytransformationdependupon. formEq.(2.2)withβ =0,ω =ω >0,wherethesumin
s,s(cid:48) 0
Eq.(2.2a)extendsoverpairsofstates(s,s(cid:48))whichdifferbya
singlespinflip.
1. ParentHamiltoniansoftheZ ×Z stateind=2
2 2
TheHamiltonian Then applying the unitary transformation W2d in
H2d =ω (cid:88) (cid:104)(cid:16)1−X(1)(cid:17)+(cid:16)1−X(2)(cid:17)(cid:105) Eq.(5.8)totheHamiltonianEq.(5.9)yieldsthemicZr2o×sZc2opic
Z2×Z2 0 j j (5.9) modelrealizingthe8classesofZ2×Z2SPTparamagnets:
j
H2d =W2d H2d (cid:0)W2d (cid:1)−1
Z2×Z2 Z2×Z2 Z2×Z2 Z2×Z2
=ω0 (cid:88) (cid:110)1−Xj(1)eiπ(p1+p12)+iπ4 (cid:80)<(cid:96)(cid:96)(cid:48);j>(cid:104)p1(cid:16)1−Z(cid:96)(1)Z(cid:96)((cid:48)1)(cid:17)+p12(cid:16)1−Z(cid:96)(12)Z(cid:96)((cid:48)12)(cid:17)(cid:105)(cid:111)
(5.10)
j
+ω0 (cid:88) (cid:110)1− Xj(2)eiπ(p2+p12)+iπ4 (cid:80)<(cid:96)(cid:96)(cid:48);j>(cid:104)p2(cid:16)1−Z(cid:96)(2)Z(cid:96)((cid:48)2)(cid:17)+p12(cid:16)1−Z(cid:96)(12)Z(cid:96)((cid:48)12)(cid:17)(cid:105)(cid:111),
j
where the notation < (cid:96)(cid:96)(cid:48); j > denotes that the summation onamicroscopicscale,originatesfromanon-onsitesymme-
in the exponent runs over pairs of nearest neighbors (cid:96) and tryrealizationontheedgedegreesoffreedom.9,15,16,20Forthe
(cid:96)(cid:48), which themselves belong to the 6 nearest sites around a Z ×Z SPT states considered here, the non-onsite symme-
2 2
given site j of the triangular lattice. When p = 0 and trytransformationsontheedgewereshowninRef.16tobea
12
p = p = 1, Eq. (5.10) describes, up to an overall energy manifestationofthenon-trivialtype-IIcocyclesofgroupco-
1 2
shift, two decoupled Levin-Gu Hamiltonians.20 On the other homology.
hand,thecasep =1encodesstrongentanglementbetween
12 ThegroundstateEq.(5.7)isasimpleproductstateand,as
thetwoIsingsystems.
such, the edge states are disentangled from the bulk. On the
other hand, in the other SPT ground states whose expansion
coefficientsdependuponthedomainwallconfigurations,the
2. EdgesymmetrytransformationsoftheZ2×Z2stateind=2 effectofflippinganedgespincanchangethenumberofloops
L,implyinganentanglementbetweenedgeandbulkdegrees
The physical properties of the edge of two dimensional offreedom. Thereforebystudyingthepropertiesoftheedge
bosonic SPT states can be understood from the chiral ac- states, we can gain information about the nature of the bulk
tion of the symmetry on the edge modes8,9,15,16,20,33 which, statesandviceversa.