Measuring space-group symmetry fractionalization in Z$_2$ spin liquids

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by  Michael Zaletel

Measuring space-group symmetry fractionalizationinZ spinliquids
2
Michael P. Zaletel,1 Yuan-Ming Lu,2 and Ashvin Vishwanath2
1Department of Physics, Stanford University
2Department of Physics, University of California, Berkeley, California 94720, USA
Theinterplayofsymmetryandtopologicalorderleadstoavarietyofdistinctphasesofmatter,theSymmetry
EnrichedTopological(SET)phases.HerewediscussphysicalobservablesthatdistinguishdifferentSETsinthe
contextofZ quantumspinliquidswithSU(2)spinrotationinvariance.Wefocusonthecylindergeometry,and
2
showthatgroundstatequantumnumbersfordifferenttopologicalsectorsarerobustinvariantswhichcanbeused
toidentifytheSETphase. Moregenerallytheseinvariantsarerelatedto1Dsymmetryprotectedtopological
5 phaseswhenviewingthecylinder geometryasa1Dspinchain. Inparticularweshow thattheKagomespin
1 liquidSETcanbedeterminedbymeasurementsononegroundstate,bywrappingtheKagomeinafewdifferent
0 ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to
2 connect bosonic and fermionic mean field theories of spin liquids. When fusing quasiparticles, it correctly
predictsnontrivialphasefactorsforcombiningtheirspacegroupquantumnumbers.
n
a
J
CONTENTS 1. TranslatingbetweentheZ PSGandthe1D
7 2
SPTU(1)PSG 11

] I. Introduction 1 C. Identificationof1DSPTorderandthe
el RTR−1T−1PSGs 11
- II. ReviewofZ spin-liquids 3
r 2
t A. Minimallyentangledstates 3 VI. Intrinsictopologicalorder:detectingthetopological
s
flux 12
.
at III. Space-groupquantumnumbers:robustSET A. 1,v vs b,f 12
m invariants 3 B. 1{ vsv} { } 12
- A. Reflectionquantumnumbers 3 C. bvsf 12
d 1. Reviewofmomentumpolarization 12
B. Translationquantumnumbers 4
n 2. A1DSPTinvariantfordetectingfermionic
o
IV. Symmetryenrichedorder:detectingtheProjective topologicalflux 13
c
[ SymmetryGroup 5 3. Quantizationofmomentumpolarizationby
the1DSPTinvariant 13
A. ThePartonConstructionandtheProjective
1
SymmetryGroup 5
v
VII. DetectingSETorderontheKagomelatticeusing
5 B. Minimallyentangledstateswithintheparton
cylinder-DMRG 13
9 construction 5
A. FiniteDMRG 13
3 C. Computationofglobalquantumnumbersfrom
1 thePSG 6 B. InfiniteDMRG 14
0 C. DeterminingRTR−1T−1 14
D. Ratiosofedge-exchangingquantumnumbersin
.
1 differenttopologicalsectors 6
VIII. Conclusions 14
0
1. Spinoninsertion 6
5
2. Visoninsertion 7
1 IX. Acknowledgements 15
: 3. Fusionandunification:Q(f) =Q(v)Q(b) 7
v
E. Thequantumnumbersofthegroundstate 7 References 15
i
X 1. Eigenvalueofplaquette-centeredinversion
r Ip 8
a 2. Eigenvalueofsite-centeredinversionIs 8 I. INTRODUCTION
3. EigenvalueofmirrorreflectionoperatorR 9
4. UnifyingbosonicandfermionicPSGs 9 Incontrasttoconventionalphasesthataredistinguishedby
Landau order parameters, topologically ordered states with
V. Dimensionalreductionandentanglement
emergentanyonicexcitations remain distinct even in the ab-
signatures 9
sence of symmetry. However, the presence of symmetries,
A. Areviewof1DSPTphases. 9 whichisnaturalinmostphysicalcontexts,leadstofurtherdis-
1. Matrixproductstates 9 tinctions,thesocalledsymmetryenrichedtopologicalphases
2. Onsitesymmetries 9 (SETs). Wellknownmanifestationsincludefractionalcharge
3. Time-reversalsymmetry 10 ofanyonsinthe fractionalQuantumHallstates(andinfrac-
4. Reflectionsymmetry 10 tional Chern insulators) and spin-charge separation in quan-
B. Identificationof1DSPTorderandthe tum spin liquids. Recently, rapid progress in the theoretical
space-groupPSGs 10 understandingof SETs is being made [1–11]. This is partly
2
drivenbyconceptualadvancesintherelatedtheoryofstrongly Areflectionquantumnumberisaprobeofquantumentan-
interacting Symmetry Protected Topological phases (SPTs), glement:ifastateisoddunderreflection,thenthetwohalves
where despite the absence of anyon excitations in the bulk, ofthe system are entangled. Buthowcan quantumnumbers
nontrivial edge excitations emerge [7, 12–18]. What would ofaglobalsymmetry,whicharealwaysintegral,berelatedto
beparticularlywelcomeatthisstage,isaphysicallywellmo- symmetryfractionalization? Asa simple example- consider
tivatedexampleofanSET.Tomakeprogressinthisdirection creatingapairofidenticalanyonsfromthevacuum. Saythat
wewillneedtounderstandhowdifferentSETphasescanbe the pair is related to one another by a symmetry (such as a
distinguished. reflection or rotation). If the excitationscarry charge, a unit
Furthermotivationforstudyingmeasurablecharacteristics chargeforthisstate actuallyimplieshalfchargeforeachex-
ofSETsistherecentprogressinthesearchforquantumspin citationsince they areconstrainedbysymmetry. Thisis one
liquidsinfrustratedmagnets,bothinexperimentsandin nu- of the key ideas that we will exploit. Its implementation is
merics. A numberofS=1/2 quantummagnetsin 2Dand3D more involvedwhen the second symmetryis not charge, but
frustratedlatticehavebeenidentified,whichappeartoevade also a space group (or time reversal) symmetry. Neverthe-
magnetic order [19, 20]. These include the S=1/2 Kagome less such arguments establish the relative quantum numbers
material, Herbertsmithite, which shown no sign of ordering betweendifferenttopologicalsectors.
down to temperaturesthat are a thousandtimes smaller than Torelateourresultstoanestablishedclassificationscheme,
theexchangeconstant[21].Whileclearcutevidenceofanen- weuseaspecificmodelofSETsobtainedbyapartondecom-
ergygapinthesematerialsisyettoemerge,arequirementto positionofthespinoperatorintobosonic(Schwingerbosons)
beconsideredtopologicallyordered,thismaybeanextrinsic or fermionic (Abrikosovfermions) partons. Symmetry frac-
effectduetoimpurities(althoughotherexplanationshavealso tionalization is encoded in the Projective Symmetry Group
been suggested [22–24]). Meanwhile, extensivedensity ma- (PSG) [1, 44, 45], which determines how the partons trans-
trixrenormalizationgroup(DMRG)simulationsofthenearest formundersymmetry. Apartonmeanfieldtheorycombined
neighbor Kagome Heisenberg antiferromagnet (KHA) indi- with projection leads to a spin wave-function whose quan-
cate(i)agappedgroundstatethatrespectallsymmetries[25] tum numbers in each topological sector are completely de-
and (ii) a topologicalentanglement[26, 27] entropy of log2. termined, which reflectthe underlyingSET. In particularwe
Inreference[28]itwasarguedthatthegroundstatemustpos- show that the Kagome spin liquid SET can in principle be
sessZ (toriccode)topologicalordertobecompatiblewith(i) uniquelydeterminedbymeasurementsononegroundstatein
2
and(ii).Animportantopenquestionistheidentificationofthe a few different finite-cylinder geometries. This information
precisephaseofmatter,i.e. theSET,realizedbytheKagome is numerically superior to the relative quantum numbers be-
antiferromagnet.Whileacompletesolutionwouldnecessitate tweentopologicalsectors,sinceDMRGnumericsonKHAdo
extensivenumericalinput,andisbeyondthescopeofthispa- not obtain all topological sectors in the finite sized systems
per,herewewillrelateSETphasestophysicalpropertiesthat studied.
arereadilymeasurableinnumericalsimulations. A different perspective on our approach is to view it as a
Aprerequisiteisameasurementofthetopologicalorderit- ‘dimensional reduction’ in which we view a 2D SET phase
self. Entanglement entropy [29, 30] providesone signature, on a cylinder as a 1D SPT. The nature of the 1D SPT de-
although it does not uniquely specify the topological order. pends on the topological sector being studied, the geometry
A complete characterization is obtained, from either the en- ofthecylinder,andtheSET.Inadditiontoitsutilityasadi-
tanglementspectrum [31] ( in certain cases), or the S and T agnosticinnumerics,ourapproachprovidesatheoreticaltool
matrices[32–38]someofwhicharewellsuitedtonumerical tostudyconnectionsbetweendifferentrepresentationsofthe
calculations[39–42]. Relatedtechniquescanbeusedtodiag- sameSET.The1DSPT invariantsforthedegenerateground
nose2DSPTphases[43]. states (whichare labeledbyquasiparticlesof the topological
Here we discuss measurable properties that distinguish order)areshowntofollowthesamemultiplicativelawasthe
SETs. For the reasons above we focus on the S=1/2 KHA, fusionrulesfortheAbelianquasiparticles. Thisallowsusto
assumingZ2 (toriccode)topologicalorder,which hasa pair determinefromthePSGsoftwoanyonstypesintheZ2topo-
of emergentS=1/2 spinons (one bosonic and one fermionic) logicalorderthePSGofthethirdanyontype,whichisfound
and a vortex (vison). The different SETs differ in their re- to obtain nontrivial phases in certain cases. Some of these
alization of space groupsymmetriesand their interplay with were not previously known [4] and serve to correctly relate
timereversal. bosonic and fermionic mean field states on the Kagome lat-
tice[5].Inparticularthisleadsustoequatetwopopularstates,
Weconsidersystemsonbothfiniteandinfinitecylinderge-
ometries, which are well suited to DMRG calculations. We the Q1 = Q2 Schwinger boson state [46] and the Z2[0,π]β
fermionicmeanfieldstate[5,47,48].
show that 1) the many-body quantum numbers of a finite-
cylindergroundstate under space groupsymmetriessuch as In addition, we show how dimensional reduction can be
reflection,translationetc.provideapowerfuldiagnosticofthe used to completely identify the four topologicalsectors of a
underlyingSETand2)whenviewinganinfinitelylongcylin- cylinder; this is highly useful for DMRG studies, and does
derasa1Dspinchain,the1DSPTorderofthespinchainfor notrequiresimultaneousknowledgeofallfourgroundstates.
variousgeometriesandtopologicalsectorscompletelydeter- In particular, we have found a 1D SPT invariant that distin-
mines the SET order, at least within the space of mean-field guishesbetweenthebosonicandfermionicspinon.
partonansatz. Earlierworkemployedasimilardimensionalreductionap-
3
proachinthecaseofinternalsymmetrieswithprojectiverep- TheMESbasisistheuniquebasisinwhich a isrealized
resentations, [49] and here we find it to be much more gen- asapermutationofthebasisstatesforalla 1F,xb,v,f . a
∈{ } Fx
erally applicablein the presenceofspace groupsymmetries. actsasapermutationintheMESbasisbecauseeachMEShas
Relatedworkspecializingtothecaseofjusttranslationsym- definitetopologicalfluxthreadingthecylinder(its‘topologi-
metry has recently appeared [50]. The connection between calsector’).
1D SPT invariants and global quantum numbers was previ- In previous discussions of the MES it is often assumed
ously noted [13, 51], and other works have utilized global thateachofthefourMEScanbeuniquelyidentifiedwithan
many body quantum numbers to identify topological phases anyon type 1,b,v,f, so that each MES can be labeled with
[52,53]. ananyontype a . Thenthreadingtopologicalfluxisrealized
as b a = b| ia . This is the case for Z spin-liquids on
Fx| i | · i 2
even circumferencecylinders, but for an S = 1/2modelon
II. REVIEWOFZ SPIN-LIQUIDS an odd circumference cylinder there is a subtlety that arises
2
because the MES double the unit-cell along the x-direction.
According to the argumentsof Hastings, Oshikawa, Lieb, The unit cell doublesbecause S = 1/2 modelsbehave as if
there is a topological flux piercing each unit cell, so (as for
Shultz and Mattis, [54–56] a quantum magnet with half-
a magnetic field) the net topological flux through the cylin-
integer spin per unit cell is either gapless, breaks a symme-
try, or is a gapped‘spin - liquid’ with emergentanyonic ex- derchangesalongthedirectionx. [28]Strictlyspeakingitis
moreprecisetoviewtheMESasatorsorforthefusiongroup,
citations. Inthelattercase,thesimplestpossibilityconsistent
butthissubtletydoesnotaffectthemeasurementsproposedin
withtime-reversalistheZ ‘toriccode’-typespin-liquid.[28]
2
thiswork,sofornotationalsimplicitywewilllabeltheMES
ThisphasehasemergentS = 1/2excitations,the‘spinons,’
byanyontypes.
eventhoughatrulylocalexcitation(themagnons)mustcarry
S = 1. Theseemergentspinonsareanyonicexcitationswith
non-trivialbraidingandstatistics. TheZ spin-liquidhasfour
2
III. SPACE-GROUPQUANTUMNUMBERS:ROBUSTSET
anyon types: the local excitations (‘1’); the bosonic spinon
INVARIANTS
(‘b’), which carries S = 1/2; the vison (‘v’), which be-
haves like a π-flux for the spinon; and the fermionic spinon
Inthissectionweconsiderthespace-groupquantumnum-
(‘f’), formed from the composite of f = bv, and which
bers of a finite length cylinder: even thoughthe structure of
also carries S = 1/2. Each particle is its own anti-particle,
v2 = f2 = b2 = 1 (hence ‘Z ’). The braiding and statis- thetwoedgesisnon-universal,wearguethatthespace-group
2
quantumnumbersare. In particular, considercreatinga pair
tics of the Z spin-liquid are equivalent to Z gauge theory
2 2
of well-separated excitationsfrom the vacuumwhich are re-
(the‘toriccode’).InthelanguageofZ gaugetheory,bisthe
2
latedbyamirrorplane,andseparatingthemouttotheedges
electricchargee;v isthemagneticfluxm;andf isthedyon
of the cylinder. If the reflection quantum number flips sign
f =emcomposedoffluxandcharge.
after this process, the excitations must be anyons which are
ThesimplestSETaspectoftheZ spin-liquidisitsbehav-
2
connected by an invisible string which is odd under reflec-
ior under SO(3) spin rotations: 1 and v carry integer spin,
tion.SincethepairofanyonstogethertransformsasR= 1,
whilethespinonsb,f carryhalf-integerspin.Thehalf-integer
−
it as if R = √ 1 acting on each anyon individually, so we
spincarriedbythespinonsis‘fractionalized’becauseanylo-
−
sayRis‘fractionalized.’
calexcitationoftheconstituentS = 1/2spinstransformsas
Whileglobalquantumnumbersareonlywelldefinedforfi-
S = 1. We can look for additional SET distinctions based
nitesystems,atalaterpointwewillshowthattheyleavetheir
onthetransformationpropertiesofanyonsunderspace-group
imprint on the bulk entanglement spectrum in a way which
symmetries,whichisthesubjectofthiswork.
canbemeasuredonaninfinitelylongcylinderaswell.
A. Minimallyentangledstates
A. Reflectionquantumnumbers
TheZ spinliquidhasa4-foldtopologicalgroundstatede-
2 Consideralargebutfinitecylinderwithareflectionsymme-
generacyon the torus or cylinder. Throughoutthis work we
tryRˆ thatexchangesitstwoedges. Wearguethattheglobal
rely on a special basis for the ground-state manifold called x
Rˆ quantumnumberofanysymmetricstate Λ,a withnoex-
the‘minimallyentangledstates’(MES).[33]Toconstructthe x
| i
citationsintheinteriorofthecylinderdependsonlyona)the
MES basis, let x run along the infinite length of the cylin-
der, and let a denote the adiabatic process in which a pair SETorderofthebulkphaseofmatterb)thedetailsofthege-
Fx ometry,suchasitsdimensions,whichwedenoteby‘Λ’,and
of anyons a/a¯ are created from the vacuum and dragged in
opposite directions xˆ out to infinity. The process a re- c) the topological sector a of the cylinder. Throughout this
± Fx paperwewilldenotetheseglobalquantumnumbersbyQ,
turnsthesystemtothegroundstate,soisaunitaryoperation
oinnitcheflugxroau’ntdh-rsotuatgehmthaencifyollidn.deWr.eSisnacyetthhaetmFoxad‘etlhirseaAdbsealinayn-, Rˆx|Λ,ai=QRx(Λ,a)|Λ,ai. (1)
a b a·b, where a b denotes the fusion of Abelian Toelaborateonc),notethataninfinitelylongcylinderhasthe
FxFx ∝ Fx ·
anyons. sametopologicalgroundstatedegeneracyasthetorus. While
4
the edges may reduce the ground state degeneracy, we only B. Translationquantumnumbers
requirethattherearenoexcitationsinthebulkofthecylinder,
so are left with the same bulk degeneracy as the torus. We
InamagneticfieldthetranslationsT ,T forma‘magnetic
x y
labelthesetopologicalsectorsofthefinitecylinderbya,and algebra’ T T T−1T−1 = eiΦ which is a projective repre-
asdiscussedearlierweassumeaindexesaspecial‘minimally x y x y
sentation of the translation group. Even in the absence of a
entangled’basiswhichhasdefinitetopologicalfluxathread-
physicalmagnetic field, in a topologicallyorderedphase the
ingthecylinder. ThisworkfocusesmainlyonZ topological
2 anyonsmayexperienceaneffectivemagneticfield. Themag-
order because the states Λ,a will break R if a is not it’s
| i x neticfieldexperiencedbyanyonaisencodedintheprojective
ownanti-particle.
relation(T T T−1T−1)(a) =η(a).
x y x y xy
TheglobalquantumnumberQ isinsensitiveto anyde-
Rx For an Abelian theory the projective relations must obey
tails of the edge state or bulk Hamiltonian. To show insen-
thefusionruleη(a)η(b) =η(ab),sinceη(a) istheBerryphase
sitivity to the edge, note that perturbing the edges amounts xy xy xy xy
acquired when a circles a unit cell. For a Z spin-liquid in
to acting with unitariesU ,U localized at the edges of the 2
L R anS = 1/2model, we alwayshave therelationη(v) = 1.
cylinder.WhenU andU arespatiallywellseparated,using xy
L R −
R symmetrywecanrequirethatU =Rˆ U Rˆ−1.Sincethe ThiscanbearguedinthelanguageofZ2gaugetheory,where
x R x L x all S = 1/2 objects are the source of Z electric flux (for
perturbationU Rˆ U Rˆ−1 commuteswith Rˆ , the quantum 2
L x L x x example the spinons b,f, which map on to the electrically
number is unchanged. Q is insensitive to the bulk Hamilto-
chargede,f particlesinthegaugetheory). Thisincludesthe
nian because Q = 1, so is quantizedand can only change
± microscopicS =1/2ineachunitcell,whichimpliesthatthe
duringabulkphasetransition.
system behaves as if there is electric flux piercing each unit
However,QRx(Λ,a)candependonthetopologicalsector cell. Consequentlyundere-mduality,thefluxm(thevison)
a, since changing the topologicalsector from a b a re- experiences a background flux of π per unit cell, so η(v) =
wquhiircehssisepaasratrtiinngg-alinkeanoypoenraptiaoirn.b/W¯beouwtitlolfithnedetdhgaet→sthuesidn·egpFenxb-, −1. Sinceηx(1y) = 1,ηx(vy) = −1,andηx(by)ηx(vy) = ηx(fy),thxeyreis
asinglesignleftundetermined,whichisthemostbasicSET
dence on Λ cancels if look at the relative quantum number
distinctionbetweenZ spin-liquids.
betweentopologicalsectors,[57] 2
Toprobeη ,consideracylinderoflengthL andcircum-
xy x
ferenceL in topologicalsectora, andmeasurethemomen-
y
Q(Rbx) ≡ QQRRxx((ΛΛ,,baa)). (2) tLuymiqsuoadndtutmhenuMmEbSerdQouTbyl(eΛt,hae)u=niTˆtyc|eΛll,aini.tNheotxetdhiartecwtihoenn,
sowe mustrestricttoL even. T symmetryalonedoesnot
y y
protectQ ,sincetheedgeexcitationscancarryanarbitrary
TheseratioshaveaparticularlysimplerelationshiptotheSET Ty
momentum, but the combination of T and T allows us to
order:ifQ(Rbx) =−1,itimpliesthatapairofanyonsrelatedby define a robust ‘momentum per unit lxength.’ yRecall that in
R eachcarryhalftheR = 1quantumnumber,whichwe
x x theLandaugauge,the momentumT ofa particleina mag-
− y
consider to be ‘fractional.’ In contrast, our earlier argument
netic field is proportionalto its position x. Since Λ,a has
impliesthatapairoftrulylocalexcitationsmustalwayshave | i
ana particlelocalizednearthe edge, aswe growL itsmo-
x
Q =1.
Rx mentumgrowslinearlywithx, ie, itisamomentumperunit
NowsupposethereisanadditionalreflectionsymmetryR length of cylinder. To define the momentum per unit length
y
whichdoesnotexchangethetwoedges.Intheabsenceofthe operationally,we needtogrowthe lengthofthe cylinderLx
edge-exchangingR symmetry,theR quantumnumbersare while keeping the topological flux and edge state the same,
x y
notrobust,becausenothingthenpreventstheperturbationUL asotherwiseQTy couldcontainaspuriouscontributioncom-
from beingodd under R while U is even. But if both R ingfromthechangingedgestate. Concretely,werequirethe
y R x
andRy arepresent,wecaninsteadmeasurethecombination reduced density matrix for the edge be kept constant as Lx
I = RxRy; since I exchanges the edges, according to our grows.Themomentumperunitlengthηx(ay) isthen
earlierreasoningQ isalsoaprotectedinvariant.
I
Finally, if lattice doesn’t have a C4 symmetry (as for the Q (Λ,a)=q(∂Λ,a) η(a) Lx. (3)
Kagomemodel)therearedistinctwaystocompactifythege- Ty xy
ometry into a cylinder: for one choice, R exchanges the (cid:16) (cid:17)
x
edges, while for the second, R does. We can then measure q(∂Λ,a)dependsontheedge,whilethebulkcontributionre-
y
quantum numbers Q in the first cylinder, Q under the vealstheSETinvariantη(a).
Rx Ry xy
secondcylinder,andQI ineither.Thisgivethreeindependent Themomentumperunitlengthistrivialtomeasureinany
quantumnumbersforeachanyontype,Q(b/f/v) ,whichwe tensor network ansatz. For MPS, it is the invariant η(a) =
Rx/Ry/I xy
will find almost fully characterizes Z SETs (at least within r definedinEq.(48))whenviewingg = T asan‘onsite’
2 Ty y
the PSG framework). The remaining information relates to symmetry in the 1D representation of the cylinder. r is a
Ty
the commutationrelations of time-reversal and the reflec- byproductofthealgorithmusedtocalculatethemomentum-
T
tions R, which will lead to protected edge degeneracies we resolvedentanglementspectrumofinfinite-DMRGstudies,so
discussinSec.VC. presumablyhasalreadybeencomputedinexistingstudies.
5
IV. SYMMETRYENRICHEDORDER:DETECTINGTHE we enforcethe constraintviaGutzwiller projectionto obtain
PROJECTIVESYMMETRYGROUP anS =1/2wavefunction:
σ Ψ =
1 2 i
Theprecedingdiscussionisindependentof anyclassifica- h↑ ↓ ··· ···| i
0f f f MF , (6)
tion of space-group SETs, since we have argued on general h | r1,↑ r2,↓··· ri,σi···| i
grounds that these quantum numbers are robust SET invari- and similarly in the bosonic construction. Note that in the
ants. Nevertheless, we would like to identify these invari- fermionic case, we must choose and fix an ordering of sites
ants within a general classification scheme. Currently there r ,r , in order to maintain the correct relative sign be-
1 2
is not a complete classification of space-group SETs. How- tween d··if·ferentspin configurations. For the purposesof cal-
ever, the parton constructionprovidesa rich zoo of Z2 spin- culation MF isusuallytakentobeafreewavefunction,such
liquids whose symmetry properties can be analyzed using asamean|-fieildBCSsuperconductororpair-superfluidforthe
Wen’s‘projectivesymmetrygroup’(PSG).[1]ThePSGpro- fermionic/bosonicconstructionsconsideredhere.Gutzwiller
videsatleastapartialclassificationofspace-groupSETs. In projecting the creation of a single parton f†/b† results in a
thissection, we showhowto computethe quantumnumbers highlynon-trivialexcitation: anS = 1/2anyonicexcitation,
QU(Λ,a)withinthepartonconstruction,therebyidentifying thespinon.
themwithinvariantsofthePSG. A crucial question in the parton construction is how the
symmetries U SG of the S = 1/2 wavefunction(such
{ ∈ }
as global SO(3) spin rotations and space-group symmetries)
A. ThePartonConstructionandtheProjectiveSymmetry are realized in the partonsand their mean-fieldansatz MF .
Group | i
Thesimplestpossibilityisthatthepartonsformalinearrep-
resentation of the symmetry group SG. But is this the only
The resonating valence bond (RVB) picture proposed by option?Theanswerisno,becauseaccordingtoEq.(4)apply-
Phil Anderson provided the first intuition for a spin liquid ingagaugetransformationb eiφrb leavesthephysi-
r,σ r,σ
ground state as a quantum superposition of different dimer calspinoperatorsunchanged,an→daccordinglytheGutzwiller
configurationscoveringa lattice of spin-1/2 particles. Each projectioninEq.(6)isamany-to-onemappinginsensitiveto
dimer(denotedby )isasingletpairformedbytwospin- U(1)gaugetransformations. Consequentlyundera seriesof
•−•
1/2particles: symmetryoperations U SGwhichyieldtheidentityop-
i
eratione { }∈
= .
1 2 1 2
|•−•i |↑i |↓i −|↓i |↑i U U U =e (7)
The state is a ‘liquid’ because the quantum superposition of 1 2··· n
dimerpatternsrestoresthe translationalsymmetry. Onetype a single-parton operator b (or f) may acquire a nontriv-
r,σ
ofelementaryexcitationinthesesystemsiscreatedbybreak- ial phase factor eiφ = 1 instead of remaining invariant. In
6
ing a dimer into a pair of particles carrying spin-1/2 each, this case the partons transform projectively, rather than lin-
whichwerecoinedspinons. early, under the symmetry group SG. The symmetry oper-
Thepartonconstructionisasystematicformalismforwrit- ations U SG are accompanied by certain gauge trans-
{ ∈ }
ingdownansatzRVBwavefunctionsinwhicheachspinonis formations G U SG on partons, forming a “projec-
U
{ | ∈ }
realizedeitherasafermionicpartonf orbosonicpartonb , tive symmetry group” PSG G U U SG which is a
σ σ U
≡ { | ∈ }
whereσ = , .Inthefermionicdescription,eachdimerisre- central extension of the symmetry group SG. [1] The cen-
↑ ↓
alizedasans-waveCooperpairofpartons;breakingaCooper terofsuchanextensioniscalledthe“invariantgaugegroup”
pairgeneratesapairofspinons.ThemicroscopicspinsS~ are IGG PSG/SG. The IGG are those gaugetransformations
r ≡
relatedtothepartonsthroughthebilinears whichleavethemean-fieldansatz MF invariant.
| i
FortheZ spinliquidswhicharethefocusofthiswork,the
2
1 1
S~r = 2 fr†,α~σα,βfr,β = 2 b†r,α~σα,βbr,β.(4) ionfvsayrimanmtegtaruygoepgerroautipoinssIGUG=inZ2(.7)T,heiascmhepaanrstounndaecrqausireersieas
α,Xβ=↑,↓ α,Xβ=↑,↓ Z phasefactorof 1. T{hisii}sconsistentwiththemean-field
2
where~σarethethreePaulimatrices.Inorderforthismapping ansatz being a sing±let BCS superconductor of spinons. For
togenerateasensibleS =1/2wavefunction,thepartonscan- example, for symmetry a symmetry UU = e we have the
notbe free particles: they obeythe “single-occupancy”con- relation
straintofonepartonperlatticesite:
(Gˆ Uˆ)2f (Gˆ Uˆ)−2 =ηff , ηf = 1. (8)
U ri,σi U U ri,σi U ±
f† f = b† b =1, latticesiter. (5)
r,σ r,σ r,σ r,σ ∀ Theprimarygoalofthisworkistoshowhowtodetectthese
σX=↑,↓ σX=↑,↓ 1phasesassociatedwithsymmetrygroupSG.
±
This constraint can be implemented by a gauge field which
couplestothepartons.
Inpractice,weusethepartonconstructiontocreateansatz B. Minimallyentangledstateswithinthepartonconstruction
wave-functions. If MF isa‘mean-fieldansatz’stateforthe
| i
partonswhichneednotobeythesingleoccupancyconstraint We first must review how to generate the MES within the
(forexample,aBCSsuperconductoroffermionicpartonsf ), partonconstruction.
σ
6
There is a two-fold degeneracy associated with threading U(i),thewavefunctiontransformsas
a vison v for both finite and infinite cylinders. In the parton
ansatzthisarisesbecausetheboundaryconditionsofacylin- h{σi}|UΨi=h{σU−1(i)}|Ψi=h0| fri,σU−1(i)|MFi,
der of circumference L can be either periodic (P) or anti-
y Yi
periodic(AP).Atthe levelofthe partonHamiltonian,thisis (13)
accomplishedbyassigninganadditionalsignof 1toallma-
trix elements which cross a line at some fixed−y = y ; the andsimilarlyforthebosonicansatz.Wecansplittheresulting
0
choiceof y is a gaugechoice. Whenacting with T or R , quantumnumberintotwocontributions. First, thereisapart
0 y y
thelocationofthetwisty0 mustberestoredbyanadditional QU(Λ,bc)comingfromtheGutzwillerprojection,
contributiontothegaugetransformationsG ,leadingto
newPSGrelations: Ty/Ry h0| fri,σU−1(i) =QU(Λ,bc)h0| fri,σiGˆUUˆ, (14)
i i
Y Y
(Gˆ Tˆ)Ly =( 1)bc
Ty y − which dependsonlyon the geometryΛ and the PSG (which
(9)
ismodifiedbybc); we willshowhowto computethisin the
(Gˆ Tˆ)Ly/2(Gˆ Rˆ )(Gˆ Tˆ)−Ly/2(Gˆ Rˆ )−1 =( 1)bc subsequentsection. Thesecondcontributioncomesfromthe
Ty y Ry y Ty y Ry y −
(10) quantumnumberof MF . InsertingEq.(14)intoEq.(13)we
| i
findthatthetotalquantumnumberfactorizesas
where bc = 0/1 for P/AP. Hence in all that follows the
PSGimplicitlydependsontheboundarycondition(bc)ofthe GˆUUˆ MF QU(MF) MF (15)
| i≡ | i
mean-fieldansatz|MFi Uˆ Ψ =QU(Λ,bc)QU(MF) Ψ (16)
The two-fold degeneracy associated with threading a | i | i
spinonisabitmoresubtle,asinthefinitecasethedegeneracy
is split by the edges. If we make a bipartition of the cylin-
D. Ratiosofedge-exchangingquantumnumbersindifferent
deratsomex0, thepartonparity( 1)Nb/f inthelefthalfof topologicalsectors
−
themean-fieldansatz MF fluctuatesacrossthecut(notethe
| i
partonnumberitselfisnotconserved). AfterGutzwillerpro-
Inthefirstscenario,wesupposewehaveaccesstoallsev-
jection, Eq. (6), the parton parity to the left is fixed by the
eral topologicalsectors on the same geometry. We find that
numberofsitestotheleft. Butintheinfinitecasethereisan
foredge-exchangingsymmetries,theratiobetweenthequan-
ambiguity, since the number of sites is infinite. This means
tumnumberbeforeandafterthreadinganyonicfluxareveals
that when Gutzwiller projecting we can freely choose either
thePSGofanyona.
thesectorwitheven(E)orodd(O)partonparitytotheleftof
thecutatx ,whichgeneratesanadditional2-folddegeneracy
0
ontheinfinitecylinder.
1. Spinoninsertion
ThechoiceofP/APboundaryconditionscombinedwithE
/O partonparitygeneratesthe4-folddegeneracyofthe infi-
We fix thegeometryΛandcomputetherelativereflection
nitecylinder.Thesesectorsareidentifiedwiththeanyontypes
quantumnumberQ betweentwo stateswhichdifferbythe
inamannerthatdependsonthepartonconstruction: R
insertion of a pair of spinons at the edges. To generate the
appropriatepairofmean-fieldansatz MF and c MF which
1,v,b,f (P,E),(AP,E),(P,O),(AP,O) (bosonic) (11)
| i | · i
↔ differbyspinonfluxc=b/f (dependingontheconstruction),
1,v,b,f (AP,E),(P,E),(AP,O),(P,O) (fermionic)
↔ letc† createanarbitrarybosonic/fermionicpartonnearthe
(12) L
left edge. To ensure that c MF is symmetric under R we
Note the role of P/AP is flipped between the two construc- must create a corresponding·spinon on the right using c†
R ≡
tions. (Gˆ Rˆ)c†(Gˆ Rˆ)−1,so
R L R
For an evencircumferencecylinderthe E / O parityis the
sameforallcutsx , sinceanevennumberofsites intervene c MF c† (Gˆ Rˆ)c†(Gˆ Rˆ)−1 MF . (17)
0 | · i≡ L R L R | i
betweencuts. Butforanoddcircumferencecylinder,theE/
Oassignmentalternateswithx0. Thisalternationdoublesthe QU(Λ,bc)isunchanged,anditisstraightforwardtoverify
physicalunitcell.
Gˆ Rˆ b/f MF =( 1)F(Gˆ Rˆ)2Q (MF) b/f MF
R R R
| · i − | · i
(18)
C. ComputationofglobalquantumnumbersfromthePSG
wherethesign( 1)F occursforfermionicpartonsaswemust
−
exchangethecreationoperatorsandweuse(Gˆ Rˆ)2todenote
SincethePSGdetermineshowthepartonstransformunder R
symmetry operations, we can compute the (crystal) symme- thepartonPSGassociatedwithRˆ2 =e:
tryquantumnumbersofanyprojectedwavefunction(Eq.(6))
(Gˆ Rˆ)2c (Gˆ Rˆ)−2 =ηc c , c ,
constructedfromapartonmean-fieldansatz. IfU isaspace- R i R R i ∀ i
group operation which permutes the sites according to i (Gˆ Rˆ)2 ηc. (19)
→ R ≡ R
7
Consequently for any geometry the change in the quantum a b f v Q(f) =Q(v)Q(b)
numberQR afterinsertingabosonic/fermionicspinonis Q(Rax) ηRbx −ηRfx 1 ηRbx =−ηRfx
Q(a) ηb −ηf 1 ηb =−ηf
QR(b/f) ≡ QRQ(ΛR,(Λb/,fM·FM)F) =(−1)FηRb/f. (20) QR(Iapy/)b ηRbxηRbyηRbx,RRyy −ηRfxηRfyηRfx,RRyy 1 ηRRbyx,Ry =R−yηRfx,Ry
Q(a) ηb ηb ηb ηb −ηf ηf ηf ηf -1 ηb =−ηf
HencethespinonPSGcanberecoveredbymeasuringtherel- Is xy Rx Ry Rx,Ry xy Rx Ry Rx,Ry xy xy
ativequantumnumberbetweendifferenttopologicalsectors.
TABLEI.RelativequantumnumbersQ(a)betweentopologicalsec-
The relative quantum numbers can be computed for any U
tors a of a square lattice spin-liquid. I is a plaquette or bond
space-group symmetry which exchanges the edges; by us- p/b
centeredπ-rotation,whileI isasite-centeredπ-rotation.
s
ing different cylinder compactifications we can measure the
R ,R , and I = R R quantum numbers. There may be
dixstincytπ-rotationsdxepeyndingonwhetherthe rotationis site a b f v Q(f) =Q(v)Q(b)
orbond/plaquettecentered;onasquarelattice,forexample, Q(Rax) (−1)p2+p3 −ησ 1 (−1)p2+p3 =−ησ
I′ = TxRxTyRy, reveals an independent PSG relation. In Q(Ray) (−1)p2 −ησησC6 1 (−1)p3 =ησC6
Fthiegss.quIaarendanIdI wKeagtoambuelaltaettitchees.relative quantum numbers for QQ(I(Iaahs)) (−1()−p11+)pp33 −η1−2ηηCC66 -11 ((−−11))pp11 ==−−ηηC126ησC6
TABLEII.RelativequantumnumbersQ(a)betweentopologicalsec-
2. Visoninsertion U
torsainaKagomelatticespin-liquid. I isahexagon/sitecen-
h/s
teredπ-rotation.ThebosonicPSGsareexpressedthroughtheinvari-
WeagainfixΛ,andcomputetherelativereflectionquantum ants(p ,p ,p )ofRef.[44],whilethefermionicPSGsareexpressed
1 2 3
numberQ betweentwostates whichdifferbythe insertion throughtheinvariantsηofRef.[47].Thefermionicinvariantssatisfy
R
ofavison. Asdiscussed,threadingavisonswitchesbetween η12ηC6ησC6 ≡1tautologically.
P/APboundaryconditions,soQ (Λ,bc)maychangedueto
U
the PSG relations of Eq. (9). The PSG relations associated
with U2 = e are only modified if U takes an odd number 3. Fusionandunification: Q(f)=Q(v)Q(b)
of sites across the twist boundarycondition modified by the
vison. For both the square and Kagome lattice, for the ge-
Our derivation of the vison quantum numbers shows that
ometriesinwhichU exchangestheedgesofthe cylinderwe
therelativereflectionquantumnumbersobeyfusion,Q(f) =
have U
Q(v)Q(b),becausevisoninsertionchangesthequantumnum-
U U
QRy(Λ,AP)=QRy(Λ,P) (21) ber by Q(v) regardless of the parton parity at the edge. As
Q (Λ,AP)=Q (Λ,P) (22) tabulatedin Table I andTable II, we can use this fusionrule
Rx Rx
to equate the bosonic and fermionic PSGs, unifying the two
Q (Λ,AP)=Q (Λ,P) (23)
Ib/p Ib/p approaches.
Q (Λ,AP)= Q (Λ,P) (24)
Is − Is
whereI isaisbondorplaquettecenteredπ-rotation,while
b/p
I itsite-centered. E. Thequantumnumbersofthegroundstate
s
NextwearguethatQ (MF)isunchangedwhenthreading
U
a vison. To start, suppose MF is in the same phase as the A given geometryΛ generically has a lowest energystate
| i
ground state of a BCS superconductor / pair super-fluid. In whichisSO(3)symmetriconbothedges,andtwo-folddegen-
thesubsequentsection,weshowthatQU(MF)=1regardless eracyforvisoninsertion. Wecancomputethequantumnum-
of boundary condition, so is unchanged by vison insertion. bersforsuchanSO(3)symmetricstate byassuming MF is
Nowsupposewemodifythegroundstate MF witharbitrary inthesameuniversalityclassasaBCSsuperconducto|r/piair
| i
U-symmetric edge perturbationsof fixed parton parity. The superfluid in the fermionic / bosonic constructions. MF is
resultingQU(MF)candependonlyonthepartonparityofthe then invariant under any symmetry: QU(MF) = 1.|Thisi is
edgeperturbation,butnotontheboundarycondition,because becauseinbothconstructionswehave
thevison-modifiedPSGsofEq.(9)willnecessarilyactonU-
relatedpartonsandthesignswillcancel.ThisshowsQU(MF) MF =exp g(i j)c† c† 0 , (26)
isunchangedbyvisoninsertion,andweconcludethat | i i,j − i,↑ j,↓ | i
g(i hj)P=( 1)Fg(j i) i
Q(v) =1, Q(v) =1, Q(v) =1, Q(v) = 1. − − −
Ry Rx Ib/p Is − wherec = f/band( 1)F = 1/1forAbrikosov-fermions
(25)
− −
and Schwinger-bosons respectively. Because MF always
| i
Apossibleloopholeinourargumentisthatweassumedthat contains the parton Fock vacuum 0 as a component in the
| i
for the ground state, MF could be taken to be in the same Taylorexpansionoftheexponential,and 0 isneutralunder
| i | i
phase as a BCS / pair superfluid, but our result agrees with any symmetry operation, MF must also be neutral. There-
| i
earlierdiscussions.[5] forethequantumnumberofthegroundstatedependsonlyon
8
thegeometryΛandthePSG: 2. Eigenvalueofsite-centeredinversionI
s
Q (Λ,bc,MF)=Q (Λ,bc) (27)
U U
Nowlet’stakeonemoresteptoconsideraninversionsym-
where bc will dependon the two-folddegeneracyassociated metry I whose inversion center lies on one or more lattice
s
visoninsertionandQU(Λ,bc)isdefinedinEq.14.Thisresult sites. Let’sassumeinversioncenterscontainNI sitesandNs
is particularly useful numerically, since by modifying Λ we isthetotalnumberoflatticesites. Forthose(N N )sites
s I
canprobethePSGusingonlyasingletopologicalsector. other than the inversion centers, their contributio−n to the I
s
Inthefollowingwespecificallyillustratehowtoobtainthe eigenvalue follows exactly the same form as (31) and (32),
eigenvaluesQU(Λ,bc)oftwocrystalsymmetryoperators,in- exceptthatweneedtoreplaceNsbyNs NI.
versionI and mirrorreflection R, for a projectedwavefunc- WhataboutthecontributionfromtheN−inversioncenters?
I
tion(6)onafinite-sizelatticefrompartonPSGs. First of all, if there is an odd number of inversion centers
(N =odd), the inversion eigenvalue is not a gauge invari-
I
antquantitysincethesymmetryoperationsonasinglespinon
1. Eigenvalueofplaquette-centeredinversionI
p canalwaysbefollowedbyanarbitrarygaugetransformation.
In the case when N =even, if the inversion centers are not
I
First we consider an inversion symmetry Ip whose inver- related by any other symmetry, again they can acquire extra
sion center lies on a plaquette. For a finite-size lattice with gaugetransformationsindependentlyundersymmetryopera-
plaquette-centeredinversionIp,thenumberoflatticesitesNs tions, and againthe inversioneigenvalueis nota topological
must be even. All lattice sites must be exchanged in pairs invariant.
underinversionoperation,sincenolatticesiteremainsinvari-
Ifanevennumberofinversioncentersarerelatedbysym-
antunderIp operation. Morespecifically,spinonfri,σi must metry,ontheotherhand,onecancomputetheircontribution
appear altogether with its inversion counterpart fIˆpri,σi = to Is eigenvalue from parton PSGs in a universal manner.
Gˆ Iˆf (Gˆ Iˆ)−1inthemany-spinonoperator f . Without loss of generality, let’s consider a pair of inversion
Ip p ri,σi Ip p i ri,σi
Notethatinprojectedwavefunction(6)thereisalwaysapar- centers related by certain crystal symmetry Pˆ (e.g. it could
ticularorderingforthereal-spacepositions r oftQhemany- beamirrorreflectionoratranslationonafinitecylinder),this
i
{ }
spinonoperator ifri,σi. Herewesimplychooseaordering spinonpairoperatortransformsunderinversionIˆs as
in which a pair of spinons related by inversion show up to-
Q
getheri.e. Gˆ Iˆ f Gˆ Pˆf (Gˆ Pˆ)−1 (Gˆ Iˆ)−1
Is s r,σ· P r,σ P Is s
ifri,σi ≡ ′ifri,σi ·GˆIpIˆpfri,σi(GˆIpIˆp)−1. (28) =GˆIsIˆsfr,σ(Gˆ(cid:2)IsIˆs)−1·GˆIsIˆsGˆPPˆfr,σ((cid:3)GˆPPˆ)−1(GˆIsIˆs)−1
wrehlaetreed′Qbdyeninovteersstihoen.QpCroldeaurcltyouvnedrehrailnfvlearttsiicoensoitpeesrathtiaotnaIrˆeuthne- =ηIfs,PGˆIsIˆsfr,σ(GˆIsIˆs)−1·GˆPPˆGˆIsIˆsfr,σ(GˆIsIˆs)−1(GˆPPˆ)−1
p =ηf ηf f Gˆ Pˆf (Gˆ Pˆ)−1 . (33)
abovemany-spinonoperatortransformas Is,P Is r,σ· P r,σ P
Gˆ Iˆ f (Gˆ Iˆ)−1 = wherewedefinedspinonP(cid:2)SGs (cid:3)
Ip p i ri,σi Ip p
′iGˆIpIˆpfri,σi(GˆIQpIˆp)−1·(GˆIpIˆp)2fri,σi(GˆIpIˆp)−2. Gˆ IˆGˆ Pˆf (Gˆ Pˆ)−1(Gˆ Iˆ)−1 = (34)
Is s P r,σ P Is s
BydQefinitionofPSGswehave ηf Gˆ PˆGˆ Iˆf (Gˆ Iˆ)−1(Gˆ Pˆ)−1,
(Gˆ Iˆ)2f (Gˆ Iˆ)−2 =ηf f , ηf = 1.(29) Is,P P Is s r,σ Is s P
Ip p ri,σi Ip p Ip ri,σi Ip ± (GˆIsIˆs)2fr,σ(GˆIsIˆs)−2 =ηIfs fr,σ. (35)
since I2 = e yields the identity operation. We used ηf to
p
denotethePSGsforfermionicspinons(Abrikosovfermions) Consequently,fora N -site lattice with N inversioncen-
s I
and ηb for bosonic spinons (Schwinger bosons). Notice ters (Ns,NI =even) which are pairwise related by crystal
that for Abrikosov-fermion representation, exchange of two symmetryPˆ,theinversioneigenvalueofprojectedwavefunc-
spinons f and f gives rise to an extra 1 sign due to tion(6)isgivenintermsofspinonPSGsby
ri Iˆpri −
Fermistatistics. Asaresultweobtain
GˆIpIˆp i fri,σi(GˆIpIˆp)−1 =(−ηIfp)Ns/2 i fri,σi (30) QIs(Ns,N=I)(=η(f−)ηNIfss)/(2N(s−ηNfI)/2)(NηIIf/s2η.Ifs,P)NI/2 (36)
Y Y − Is − Is,P
Hence the eigenvalue of plaquette-centered inversion I for
p
projectedwavefunction(6)onaN -sitelattice(N =even)is forAbrikosovfermionsand
s s
QIp(Ns)=(−ηIfp)Ns/2 (31) QIs(Ns,NI)=(ηIbs)Ns/2(ηIbs,P)NI/2. (37)
forAbrikosov-fermionrepresentation. On the otherhand, in
forSchwingerbosons.
theSchwinger-bosonrepresentation,exchangeoftwospinons
won’tyielda 1signandwehave A crucial point is that the PSG ηIs,P can depend on the
− boundary condition of the cylinder (for example, if P =
QIp(Ns)=(ηIbp)Ns/2 (32) TyL/2).
9
3. EigenvalueofmirrorreflectionoperatorR V. DIMENSIONALREDUCTIONANDENTANGLEMENT
SIGNATURES
TheeigenvaluesofmirrorreflectionoperatorRcanbecom-
putedcompletelyinparalleltothecaseofinversionsymmetry A 2D modeldefined on a cylindercan be viewed as a 1D
as discussed previously. Again let’s assume N lattice sites system by grouping one ring of the cylinder into a single
R
lieonthemirrorreflectionaxisonaN -sitelattice.Asargued super-site. Thispointofviewis usefulbecausetheinterplay
s
earlier,onlywhenN isevenandtheseN sitesarerelatedto ofsymmetry,topologyandentanglementhasbeencompletely
R R
eachotherbyothercrystalsymmetries,willtheReigenvalue understoodin1Dthroughtherecentclassificationof1DSPT
be a topological invariant that is fully determined by parton phases. [12–15] In this section we explain how 2D SET or-
PSGs. Let’sassumetheseN sitesareexchangedinpairsby der manifests itself as 1D SPT order under this dimensional
R
crystalsymmetryPˆ. Similartothecaseofinversionsymme- reduction. In particular, we find the Z2 PSG relations have
try I we can computethe reflectioneigenvalueof projected a one-to-onecorrespondencewith the U(1) projective repre-
s
wavefunction(6)as sentationsthatclassify1DSPTphases. WhilegenerallyU(1)
projectiverepresentationsare a coarse-grainedversion of Z
2
Q (N ,N )=( ηf)Ns/2( ηf )NR/2. (38) projective representations, space-group symmetries actually
R s R − R − R,P
have an anti-unitary character under the dimensional reduc-
forAbrikosovfermionsand tion, and for this special case the correspondence becomes
one-to-one.
Q (N ,N )=(ηb)Ns/2(ηb )NR/2. (39)
R s R R R,P
forSchwingerbosons. ThepartonPSGsaredefinedas
A. Areviewof1DSPTphases.
Gˆ RˆGˆ Pˆf (Gˆ Pˆ)−1(Gˆ Rˆ)−1 = (40)
R P r,σ P R
While the classification of 1D SPTs can be discussed in
ηRf,PGˆPPˆGˆRRˆfr,σ(GˆRRˆ)−1(GˆPPˆ)−1, termsofSchmidtdecomposition,themostcompacttreatment
(Gˆ Rˆ)2f (Gˆ Rˆ)−2 =ηff . (41) usestheformalismofmatrixproductstates. We refertopre-
R r,σ R R r,σ viousworksforamoredetailedreview.[12–15]
forAbrikosovfermionsandsimilarlyforSchwingerbosons.
1. Matrixproductstates
4. UnifyingbosonicandfermionicPSGs
Let j spanthe localHilbertspacesofa spin chainwith
n
| i
sitesatn. AMPS Ψ ischaracterizedbyasequenceofrank-
desIcfrainbeAsbthrieksoasmove-fZe2rmspioinnlsitqauteidanstdatae,Stchhewirinsygmerm-beotsroynqsutaante- a3ntseantzsors {Γjαnnαn+|1}iand rank-1 vectors {sαn} through the
tum numbers on any finite lattice must be the same for ar-
bitrary crystal symmetries. Therefore from the eigenvalues χn
of inversion and reflection symmetries summarized previ- j Ψ = s Γjn . (47)
h{ n}| i αn αnαn+1
ously,wecanachieveaunificationofAbrikosov-fermionand
αXn=1Yn
Schwinger-bosonrepresentation: i.e. theirPSGsmustsatisfy
thefollowingcorrespondence: Theindicesαn whichare summedoverare called theauxil-
laryindices,withdimensionχ . WehaveassumedtheMPS
n
−ηIfp =ηIbp; (42) sisetinofthSech‘mcaindotnwiceaiglhfotsrmfo,r’awbhiipcahrtmitoeannosftthhaetseyascthemsαbnetiwsethene
−ηIfs =ηIbs; (43) sitesn−1,n.
ηf =ηb , (44) The MPS ansatz includes both finite and infinite spin
− Is,P Is,P chains. InthefinitecasewithLsites,χ =χ =1. Inthe
crystalsymmetryP satisfyingPI =I P; 1 L+1
∀ s s infinitecasewithaunitcelloflengthL,wecanalwayschoose
−ηRf =ηRb; (45) the tensorsto share this unitcell: Γjαnn++LLαn+1+L = Γjαnnαn+1
ηf =ηb , (46) andlikewiseforsandχ.
− R,P R,P
crystalsymmetryP satisfyingPR=RP.
∀
These relationsare in agreementwith ourconclusionsbased 2. Onsitesymmetries
on the relative quantum numbers. In the next section we’ll
establish the correspondence between Schwinger-boson and Ifaspinchainisinvariantunderanonsitesymmetryg G
∈
Abrikosov-fermion representations for those PSGs concern- (e.g. a spin rotation), it is naturalto ask how the symmetry
ing time reversalsymmetry. Thisis achievedby relating the is encodedin thetensorsΓ. The representationof the onsite
2D parton PSGs to 1D SPT invariants by considering pro- symmetrydecomposesintoitsactiononeachsite,gˆ= gˆ .
n n
⊗
jectedwavefunctionsonathinbutlongcylinder. For notationalsimplicity, we will drop the site index n. An
10
MPSissymmetricundergifandonlyiftheΓandstransform Thereisalsoaninterplaybetweentheothersymmetriesand
as[58] . If G contains H ZT G as a subgroup, where H is
T × 2 ⊂
onsite,forh H thereisaprojectiverelation
gjkΓkαβ =rg Ug;αα′Γjα′β′Ug†;β′β (48) ∈
ω( ,h)
Xk αX′,β′ UTUh∗UT−1 = ω(hT, )Uh (56)
Ug;αβsβ =sαUg;αβ (49) T
Forourpurposesitwillbe sufficientto understandA = Z ;
wheretheU areunitarymatricesandr U(1). 2
Thephasegsrg formaU(1)representagtio∈nofthegroup,and sgianucgee-aiTnvairsiaintsterleflaatiuonnitary Z2 symmetry we can form the
encodetheg-chargeperunitlength. ButfortheunitariesU
h
thereisanotherpossibility: Ug maybeaprojectiverepresen- U U∗ =γ = 1, (when h2 =1), (57)
tation, meaning that the requirementsof a group representa- hT hT hT ±
tionaresatisfiedonlyuptoU(1)phases. Thissubtletyarises anadditionalZ invariant.Alternatively,wehavetherelation
2
becausethephaseofU isn’tfixedbythetransformationlaw
g
of Eq. 48. Arbitrarily fixing the phase of each Ug, the U(1) (Uh2)(UTUh∗UT−1Uh−1)=γhTγT (58)
phasesωareencodedintherelations
UgUh =ω(g,h)Ugh, ω(g,h) U(1). (50) 4. Reflectionsymmetry
∈
Thephasesωarecalledcocycles,orthefactorset. Ifwecon-
Finally, considera reflectionR which spatially invertsthe
siderthephaseambiguityU θ(g)U ,θ(g) U(1)tobea
‘gaugetransformation,’we sgee→that ω isgnotga∈ugeinvariant. 1Dchain,R|jni = uR;jn,k−n|k−ni. TheunitarymatrixuR
encodesanyinternalrotationinthedefinitionofthereflection.
Theclassificationofgauge-inequivalentωisgivenbythe2nd
Thetransformationlawis[13]
groupcohomology[ω] 2[G,U(1)],resultinginaclassifi-
∈H
cmaetitoryngorfo1uDpGph=asZess,ytmhemreetarriecnuondperrojGec.tiNveotreepthreastefnotratsiyomns-, uR;jk ΓT kαβ =rR UR;αα′Γjα′β′UR†;β′β (59)
N k α′,β′
andhenceno1DSPTphases. X (cid:0) (cid:1) X
AnimportantphysicalsignatureofanSPTphaseunderon- UR;αβsβ =sαUR;αβ (60)
site G is degenerate edge states. It can be shown there are
wherer = 1providesthefirstZ invariant,the‘parityper
edgestateswhichtransformunderGwiththesameprojective R ± 2
unitlength.’ Somewhatsurprisingly,combiningthistransfor-
representation[ω]astheU ;if[ω]isnon-trivial,theprojective
g
mation law with those of an onsite h we find the projective
representationmustbemulti-dimensional,implyingadegen-
relationsarethesameasthoseofaanti-unitarysymmetry:
eracy.
U U∗ =ω(R,h)U , (61)
R h Rh
U U =ω(h,R)U ω U(1). (62)
3. Time-reversalsymmetry h R hR
∈
TheoriginofthissimilarityisthattranspositionT andcom-
The transformation laws are modified for time-reversal
plexconjugationbehaveanalogouslywhenactingontheuni-
= uˆ KbecauseofthecomplexconjugationK.Similar
n T tary U . This point is important, as it implies inversion has
T ⊗ h
tobefore,theMPScanbetakentotransformas thesameZ invariantsastime-reversal :
2
T
uT;jkΓ¯kαβ = UT;αα′Γjα′β′UT†;β′β (51) URUR∗ =γR =±1 (63)
Xk αX′,β′ U U∗ =γ = 1 (when h2 =1) (64)
U s =s U (52) hR hR hR ±
T;αβ β α T;αβ (U2)(U U∗U−1U−1)=γ γ = 1 (65)
h R h R h hR R ±
(notewe canremover by redefiningthe U(1) phaseof the
T
state). Butforananti-unitarysymmetryliketime-reversal ,
theprojectiverelationsaremodifiedto T B. Identificationof1DSPTorderandthespace-groupPSGs
U U∗ =ω( ,h)U , (53) Earlier we argued that space-group quantum numbers are
T h T Th
U U =ω(h, )U ω U(1) (54) topological invariants in the presence of reflection symme-
h T hT
T ∈ tries, and calculated these quantum numbersusing the PSG.
In contrast to an onsite G = Z , for the anti-unitary time- We now show that under the dimensional reduction these
2
reversal(G=ZT)wecanformthegauge-invariantrelation quantumnumberscanbecalculatedfromthe1DSPTinvari-
2
ants r ,γ ,γ . This clarifies the origin of their stability,
R R hR
γ U U∗ = 1. (55) since 1D SPT phases are robust in the presence of symme-
T ≡ T T ±
tries, andprovidesa dictionarybetweenthe2D SET and1D
Thisγ givesaZ SPTclassification. SPTorder.
T 2