Gauge symmetry and non-abelian topological sectors in a geometrically constrained model on the honeycomb lattice

0.22 MB·English

by  Paul Fendley

Gaugesymmetry andnon-abelian topologicalsectors
inageometrically constrained model onthe honeycomb lattice
Paul Fendley1, Joel E. Moore2,3, and Cenke Xu2
1 Department of Physics, University of Virginia,
7
Charlottesville, VA 22904-4714 USA
0
0 2 Department of Physics, University of California,
2
Berkeley, CA 94720

n
3 Materials Sciences Division, Lawrence Berkeley National Laboratory,
a
J Berkeley, CA 94720
(Dated:January17,2007)
7
1
We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions.

Theyare1)theIsingmodelonthehoneycomblattice,constrainedtohavethreeupspinsandthreedownspins
]
h oneveryhexagon,2)thethree-color/fully-packed-loopmodelonthelinksofthehoneycomblattice,withloops
c around asinglehexagon forbidden, and 3) threeIsingmodelson interleavedtriangular lattices, withdomain
e wallsofthedifferentIsingmodelsnotallowedtocross. Unlikethethree-colormodel,theconfigurationspace
m on the sphere or plane is connected under local moves. On higher-genus surfaces there are infinitely many
dynamical sectors, labeledby anoncontractible set of nonintersecting loops. Wedemonstrate that at infinite
-
t temperaturethetransfermatrixadmitsanunusualstructurerelatedtoagaugesymmetryforthesamemodelon
a
ananisotropiclattice. Thisenablesustodiagonalizetheoriginaltransfermatrixforupto36sites,findingan
t
s entropyperplaquetteS/kB ≈ 0.3661... andsubstantialevidencethatthemodelisnotcritical. Wealsofind
.
t thestrikingpropertythattheeigenvaluesofthetransfermatrixonananisotropiclatticearegivenintermsof
a
Fibonaccinumbers.Wecommentonthepossibilityofatopologicalphase,withinfinitetopologicaldegeneracy,
m
inanassociatedtwo-dimensionalquantummodel.
-
d
n I. INTRODUCTION thesquarelattice2. Obviously,notallconstrainedmodelsare
o
critical: dimersonthe triangularlattice (oranynon-bipartite
c
[ Classical lattice statistical-mechanical models with local lattice)haveexponentiallydecayingcorrelators4.

constraints have been of great interest for decades. By “lo- Thepurposeofthispaperistopresentaconstrainedlattice
2
v cal constraint”, we mean a local rule which restricts the al- modelthathasseveralratherinterestingproperties. Thereare
4 lowedconfigurations. A famousexampleis thatofthehard- threeequivalentwaysofdefiningthemodel.OneisasanIsing
5 core close-packed dimer model1,2. The degrees of freedom modelonthehoneycomblatticewithaconstraintaroundeach
7 aredimersstretchingbetweenadjacentsitesofalattice,while hexagon; one is with a constraint on the three-color model,
0 the hard-core and close-packing constraints mean that each andathirdisasthreecoupledIsingmodels.Thelatterformis
1
siteofthelatticeistouchedbyexactlyonedimer.Anotherfa- mostnaturallygivenintermsofloopsrepresentingIsingdo-
6
mousexampleisBaxter’sthree-colormodel,whereeachlink mainwalls,andisalsotherepresentationwhereitsproperties
0
/ is covered by one of three “colors” of dimers, with the con- aremosttransparent.
t
a straintthateachsiteistouchedbyallthreecolors3. This model is of interest for several reasons. It is defined
m Anotheroft-studiedconstraintistorequirethatthedegrees in terms of simple local degreesof freedom and constraints,
- of freedom be “loops”, i.e. one-dimensionalobjects without yetexhibitsfascinatingconservationlaws. Aswewilldetail,
d
ends. For example, both close-packed hard-coredimers and thetransfermatrixdecomposesintosectorswhicharelabeled
n
the three-colormodel can be viewed as loop models. In the by non-abelian(and non-local)charges. The numberof dis-
o
c lattercase, thelinkscoloredbytwoofthecolors(sayRand tinctsectorsexponentiallyincreasesasthesizeofthesystem
: G) form closed loops of alternating R and G colors. Since increases. Thissymmetryenablesustodoexactdiagonaliza-
v
every vertex has one R and one G touching, the three-color tionof thetransfermatrixfor systemsofsizes upto 36 sites
i
X model is therefore equivalent to a fully-packed loop model across, i.e. a Hilbertspace initially of size 236. We know of
r (everysitehasoneloopgoingthroughit). Eachloopreceives nonon-trivialsystemwithsuchaproperty.
a
a weight 2, since there are two possible ways of ordering R Despite the fact that the conserved charges are non-local,
andGaroundeachloop. theconfigurationspaceofthemodelhasthestrikingproperty
One interesting limit of constrained models is at infinite that it is connected under simple local moves. Even though
temperature, where each allowed configurationhas the same the three-color model is closely related to ours, to relate all
Boltzmann weight. The partition function in this limit is a itsdifferentconfigurationsrequireschangingdegreesoffree-
purelycombinatorialquantity:itsimplycountsthenumberof dom arbitrarily far apart5,6. Since we give a precise relation
configurations.Becauseoftheconstraints,thephysicsofsuch betweenthethree-colormodelandours,wethushavelocated
modelsisstill veryrich. Forexample,the three-colormodel the obstruction to connectivity (the “11th” vertex discussed
is critical, as is the hard-core close-packed dimer model on below)inthethree-colormodel.Notonlydoesthismeanthat
2
ourmodelisamenabletoMonteCarlosituations,butitshould hexagon,i.e.
proveinterestingtostudyitsclassicaldynamics5,7.
Anewreasontobeinterestedintwo-dimensionalclassical model1: M σ =0 (1)
h i
latticemodelswithconstraintscomesfromquantumphysics. ≡iX∈7
Themotivationistofindphaseswithtopologicalorder,where
thereisnonon-vanishinglocalorderparameter,butonlynon- Thisisquiteastrongconstraint,retainingonly20oftheorig-
local ones. The idea for building such a quantum model by inal 64 possibilities for the spins around each hexagon. We
starting with a classical magnet with local constraints came willmostlydiscusstheinfinite temperaturelimitT or
long ago8, and a theoretical triumph in proving they exist K =0,inwhicheachallowedconfigurationhasequa→lwe∞ight.
came from a quantum eight-vertex model9 and a quantum b. Model 2: The degrees of freedom in the three-color
dimermodelonthe triangularlattice4. Thetwo-dimensional model are three colors, say R, G, and B, which are placed
quantummodelsaredefinedbyusingeachconfigurationina onthelinksonthehoneycomblattice. Theusualconstraintin
two-dimensionalclassicallatticemodelasabasiselementof thethree-colormodelistorequirethatateachsiteofthelat-
the Hilbert space. One characteristic of topological order is tice,allthreecolorsappear. Inotherwords,linksofthesame
thatthenumberofgroundstatesdependsonthegenusoftwo- colorcannevertouch. Whenthepartitionfunctionissimply
dimensional space. Constrained lattice models give natural the sum over all allowed configurations (i.e. in the infinite-
waysofdefiningthedifferentsectorswhich,withappropriate temperaturelimit), themodeliscriticalandintegrable3. Our
choiceofHamiltonian10,correspondtodifferentgroundstates model2isthethree-colormodelwithanadditionalconstraint
inthequantumtheory.Whenwritingtheeight-vertexordimer forbiddingconfigurationswhichhavethesametwocolorsal-
modelsasloopmodels,thedifferentgroundstatesarelabeled ternatingaroundanygivenhexagon.Inapicture,
bythenumber(mod2)ofloopswhichwraparoundcyclesof
thetorus. c′
InsectionIIweintroducethesemodelsandshowthatthey
c c
are equivalent. We relate our model to several others in ap-
model2: forbid (2)
pendixA, enablingus to putupperand lowerboundson the
c′ c′
entropy. In section III, we discuss the dynamicsunder local
moves,showingthattheconfigurationsontheplaneorsphere c
are all connected by simple local moves. On surfaces with
non-contractiblecycles, we classify the infinitely many sep-
arate dynamical sectors. In section IV, we give arguments wherec = c′ canbeanyofR,G,orB. Thisforbids6ofthe
which suggest our model is not critical. We also develop in 66allowe6 dconfigurationsaroundahexagoninthethree-color
section V and solve in section VI a closely related model model.
which has a gauge symmetry. We exploit this symmetry in
Imposingtheconstraint(2)inthefully-packedloopformu-
sectionVIItoshowhowtoreducedramaticallythesizeofthe
lation of the three-colormodel forbidsthe shortest loops, of
originaltransfermatrix. Thisenablesustoexactlydiagonal-
length6. The constraintissymmetricunderpermutationsof
ize the transfermatrixforquitelargelattices, andthe results
R,G,andB,soitforbidsall“short”loops,nomatterwhich
again suggestthatthe modelis notcritical. In section 9, we
twocolorsarechosentoformtheloops.
present our conclusions, and discuss applying our results to
c. Model 3: Consider now Ising spins s = 1 on the
buildaquantummodelwithtopologicalorder. i ±
triangular lattice. Instead of studying a single Ising model
onthislattice,weinsteadconsiderthreeidenticalIsingmod-
els,oneachofthethreeidenticaltriangularsublatticesofthe
II. THEMODEL,ANDITSTHREEDESCRIPTIONS triangular lattice. This has an (Ising)3 critical point when
K = arcsinh(1/√3)/211. The domain walls for an Ising
The model we are introducing can be described in three modelseparateunlikespins;theyliveonthelinksofthedual
equivalentways. Herewepresentthem,andthendemonstrate lattice.ForeachofourthreeIsingmodelsontriangularlattice,
theirequivalence. itsduallatticeisthehoneycomblatticemadeupofthesitesof
a. Model1: ThedegreesoffreedomoftheIsingmodel theothertwoIsing models. Itis notpossibleforthedomain
are“spins”σ takingvaluesof 1ateachsiteiofsomelat- walls of a given Ising model to cross or even touch, but the
i
±
tice.Theenergyingeneralfromnearest-neighborinteractions wallsofthedifferentdecoupledmodelscancrossandtouch.
isgivenby Our constraint couples the three Ising models by not allow-
ing the walls of differentmodelsto cross (althoughthey can
E =J σ σ . touch). A configuration in this model is displayed in figure
i j
X 1. Intermsofthespins,considerahexagononthetriangular
hiji
lattice,comprisedofsixsitessurroundingagivensite. Label
TheIsingmodelonthehoneycomblatticehasacriticalpoint thesixspinsaroundthishexagonbys ,sothats ,s ands
i 2 4 6
whenK = J/k T = arcsinh(√3)/211. Ourmodel1is the areinoneofthethreeIsingmodels,whiles ,s ands arein
B 1 3 5
Isingmodelonthehoneycomblattice,withtheconstraintthat another. Denoted = s s , withthe subscriptsinterpreted
i i i+2
theremustbethreeupspinsandthreedownspinsaroundeach mod6. Adomainwalloccurswhend = 1. Theconstraint
i
−
3
thatdomainwallsnotcrossintermsofthesespinsisthen To show the equivalence of model 1 with model 3, we
reexpress the degrees of freedom in model 1 in terms of
6
model3: 3 (d d d +d d /2)=0 antidomainwalls. Everytimeadjacentspinsaredifferent,we
j i i i+1 i i+3
C ≡ − − draw an antidomain wall on the link of the dual lattice sep-
Xi=1
(3) arating them. These antidomain walls therefore form loops
wherej isthesiteonthetriangularlatticeatthecenterofthe on the dual triangular lattice. Each of the configurationson
hexagon. The allowed domainwalls inside this hexagonare eachhexagonobeyingtheconstraint(1)correspondtooneof
thetypesofantidomain-wallconfigurationsillustratedinfig-
+ + + + + + + + +
ure2. Thereare10differentconfigurationsofthreedifferent
types:theemptyone,six(relatedby60degreerotations)with
+ + + + + + + + + + two antidomain walls, and three (related by 60 degree rota-
tions)withfourantidomainwalls. Isingdomainwallshavea
+ + + − + + + + + weighte−2K perlink. Sinceantidomainwallsaresimplythe
complementof the domain walls, they can be taken to have
+ + − − + − + − + + weighte2K perlink.
+ + + − + + + + + u u u
d d d d d u
+ + + + + + + + + +
u u d u d u
+ + + + + + + + +
d u d
FIG.1: ThreeIsingmodelsonthreetriangularsublattices. Thecon-
straint(3)requiresthatthedomainwallsdonotcross. FIG. 2: The three types of antidomain-wall configurations on a
hexagon in model 1. TheIsing spins here are denoted by uand d
ofthetypesillustratedinfigure3below.Model3canequiva- todistinguishthemfromtheIsingspinsinmodel 3, whicharede-
lentlybedescribedintermsofclosedmutually-avoidingloops notedby±.
onthetriangularlattice, withtheaddedrestrictionthatloops
mustturnby 120degreesateverysite. These antidomain walls in model 1 correspondto domain
±
walls in model 3. The triangular lattice for model 3 is sim-
Thesethreemodelsareequivalenttoeachotherunderlocal plytheduallatticeofthehoneycomblatticeformodel1. The
reformulationsof the degrees of freedom. First let us recall domainwallsinatriangular-latticeIsingmodelmakea 120
the mapping of the three-color model without constraint (2) degreeturnateverysite,justliketheantidomainwallsi±nfig-
toanIsingmodelonthehoneycomblattice12. TheIsingvari- ure 2. If the three Ising modelsin model3 were decoupled,
ables represent chiralities in the three-color model; this chi- therewouldbe16differentdomain-wallconfigurationsgoing
ralityrepresentationoccursinthesuperconducting-arrayreal- through each site of the triangular lattice, because there are
ization of the three-colormodel13,14. Consider given config- fourpossibilitiesforeachofthetwoIsingmodelswhosedo-
urationin thethree-colormodel. Therearesix possiblecon- mainwallsgothroughthispoint. Thereareonly10possibili-
figurations of the three-color model around each site of the tiesinfigure2. Model1andModel3arethereforeequivalent
honeycomblattice: PutanIsingspin+onthesiteifthecol-
orsonthethreelinkstouchingitareRGB clockwise,and
−
ifthethreeareRGB counter-clockwise. Goingaroundeach
hexagon,it iseasy to checkthatthereareeither 0, 3 or 6up
spins. Itisalsoeasytocheckforanyconfigurationwith0,3
or6upspins,onecanreversethemapandfindaconfiguration
inthethree-colormodel.Ignoringboundaryconditions,there
are three configurationsin the three-colormodel for each in
theIsingmodel,sothemapcanbemadeone-to-onebyspec-
ifyingthecolorononelink. Thethree-colormodelatinfinite FIG.3: The10possibledomain-wallconfigurationsinmodel3,i.e.
temperaturethereforemapsontotheIsingmodelonthehon- the10vertices.
eycomb lattice with K = 0 and the requirementthe sum of
theσ aroundhexagonobeysM = 0, 6. Onecangeneral- ifwerestricttothese10,whichareredrawninfigure3. Asis
i h
±
izethethree-colormodeltoincludeinteractionsequivalentto obviousfromthefigures,theonesdisallowedarethosewhere
anon-zeroKifdesired;thisiseasilydoneinthedomain-wall thedomainwallscross. Disallowingcrossingsleavesexactly
formulationgivenbelow. the10,sothenon-crossingconstraintistheonlyone.Infigure
The equivalence of model 1 to model 2 is now obvious. 1,wedrewthedomainwallsforthethreedifferentIsingmod-
Hexagonsinthethree-colormodelwithalternatingcolorsas elswithsolid,dashed,anddot-dashedlines,toemphasizethe
in (2) correspond to having M = 6 in the Ising model. fact that they do not cross, with each forming closed loops.
h
±
Theseareforbiddeninmodel1by(1),andinmodel2by(2). In figure 3, we drew these with the dotted and dashed lines,
4
butthe same 10 verticesoccurat any pointon the triangular Themost-localdynamicsoftheIsingvariablesofmodel1
latticewiththeappropriatetypesoflines. We havetherefore thatconservestheconstraint(1) isto actonhexagonswhere
shown that model 3 is the same as model 1, up to unimpor- the spins alternate up and down around the hexagon. Flip-
tantconstantsinfrontofthepartitionfunctions.Ourmodelis pingeachupspintodownandeachdowntouparoundsuch
thereforea“10-vertexmodel”onthetriangularlattice. These a hexagon preserves the constraint not only on the original
verticesareasubsetofthoseinthe32-vertexmodeldiscussed hexagon, but also on each of its six neighbors as well. We
in15. displaythisflipinfigure5. Inmodel3,thiscorrespondssim-
Theseproofsofcoursemeanthatmodel2isequivalentto
model3aswell,sothethree-colormodelwithandwithout(2) u d
canalsobewrittenintermsofavertexmodelonthetriangular d d u u
lattice. Theusualthree-colormodelalsoallowsthe11thver-
u u d d
texpicturedinfigure4. Thisvertexisasource/sinkofdomain
d u
FIG.5:TheflipintermsofIsingspinsinmodel1
plytoflippingtheIsingspinatthecenterofthishexagon,i.e.
sendings s .
i i
→−
Thisistheonlylocalmovenecessarytoconnectconfigura-
FIG.4: The11thvertexinthethree-colormodelwithoutconstraint
tions. Thisiseasiestto see inthe looprepresentation. Since
(2)
in the model3, the flip changesthe spinat the centerof this
hexagon, it simply flips the model-3 loop variables on the
walls,andsothethree-colormodelwithoutconstraint(2)can-
hexagon surroundingthis hexagon of model 1. An example
notbemappedontothreeIsingmodels. Inmodel1,this11th
isillustratedinfigure6. Forexample,ifallsixofthelinkson
vertexcorrespondstoahexagonwithalluporalldownspins,
i.e. M = 6.
h
±
In the appendix, we relate our model to two others: hard
hexagonsonthetriangularlattice,andageneralizedIsingan-
tiferromagnet. Our model is found from these by relaxing
constraints,sothesemodelsgivelowerboundsontheentropy
ofours.
III. CONNECTINGCONFIGURATIONSBYLOCAL
MOVES
FIG.6: Anexampleoftheeffectofafliponaloop. Thealternating
Isingspinsinmodel1aroundtheshadedhexagonarethoseflipped,
An important question in many physical applications of asillustratedinfigure5.
two-dimensionalgeometricallyconstrainedmodelsiswhether
thespaceofstatesisconnectedunderlocalmoves.Itisessen- thesurroundinghexagonareempty,theflipcreatesaloopof
tial if one is to study either classical or quantum dynamics, minimal length. If they are all full, this is a minimal-length
and is also useful for doing Monte Carlo simulations16. For loopsurroundingthehexagon,andtheflipremovestheloop.
example,tobuildthequantummodelsdiscussedintheintro- Inothercases,itshrinksorexpandstheloopwithoutcreating
duction,withoutconnectivityunderlocalmoves,theHamilto- anylooseends.
nianisnon-local.Thethree-colormodelisnotconnected:any Itis noweasy to see howthe flip connectsconfigurations.
closed loop of bonds of the honeycomb lattice that contains Aloopofminimallengthhasaflippablehexagoninsideit,so
onlytwocolorswillgiveadifferentconfigurationifthosetwo thesecanberemovedbyoneflip.Longerloopscanbeshrunk
colorsare permuted. Eventhoughthere exist shortloopson and then removed by repeatedly flipping. If there are loops
thelattice(theshortestloopisasinglehexagon),thespaceof inside other loops, then the ones inside need to be removed
statesisnotconnectedunlessthedynamicsisabletopermute first. Whenspaceistopologicallyasphere,allconfigurations
arbitrarilylargeloops5,6,7. arethereforeconnectedtotheemptyone. Sinceallprocesses
Inthissectionwediscussthepropertiesofourmodelunder can be reversed, this means all configurations on the sphere
localdynamics. We showthat, unlikethe three-colormodel, areconnected.
the connected sectors can be enumerated simply and corre- When space has non-contractible cycles, however, not all
spond to topologicalclasses of sets of nonintersectingloops loops can be removed. In order to use the formulation of
intheplane.Sincewehaveshownthattheconstraint(2)turns model 3, the periodic boundary conditions around a cycle
the three-color model into our model, this result illuminates mustidentifysitesofthesametriangularsublattice.Whenthis
thereasonwhythethree-colormodelisnotconnectedbylo- isdone,theloopsareofthreedistincttypes,asseeninfigure1.
calmoves. Sinceflipscannotmovetwoloopsofdifferenttypesthrough
5
eachother,loopswhichwraparoundanon-contractiblecycle that productsmust be interpreted cyclically, and S is an el-
t
can onlybe removedif they adjacentto another of the same ementof SL(2,Z), the groupof modulartransformationsof
type. The flip illustrated in figure 7 turns two adjacent non- thetorus. SL(2,Z)isgeneratedbyexchangingm n,and
↔
contractibleloopsofthesametypeintotwocontractibleones. shiftingn n+1. Topologicalsectorsnotofthisformare
→
thetrivialsectorS = 1(i.e.noS ),andsectorwithasingle
c t
loop(i.e. S =a,borc),whereS =Z Z .
c t 2 2
×
IV. FIELD-THEORYAPPROACHES
Abasicquestionaboutourmodelisifitiscritical. Wecan
gain insight into this question by studying the field theories
validneartwocriticalpointswhichoccurwhenbyrelaxingor
increasingtheconstraints.
Three decoupled Ising models are critical when K is ap-
propriately tuned. In the continuum limit, the critical point
canbedescribedbyusingconformalfieldtheory18. Oneim-
FIG.7:Aflipwhichconvertsnon-contractibleloopsintocontractible
portantthingconformalfieldtheoryallowsonetodoisclas-
ones.
sifyalltheoperatorsofthetheory. TheIsingmodelhasonly
tworelevantrotationally-invariantones,thespinfield,andthe
Whenspaceisacylinder,thedifferentsectorscanbeenu-
energy operator ǫ. Perturbing the critical point by the latter
meratedsimply: a sectorisgivenbya sequenceof loopcol-
correspondstochangingthetemperature,i.e.takingK away
ors (those encountered reading from left to right along the
fromK ,sointhelatticemodelwecanidentify
c
cylinder, for example), with an even numberof adjacentoc-
currencesofthesamecolorsbeingequivalenttotheidentity. ǫ s s d
i j hiji
Mathematically, this set is isomorphic to the free group on ∼ ∼
three elementsa,b,c, with the relationsa2 = b2 = c2 = 1. so that d = 1 corresponds to a domain wall be-
hiji
−
Puttingtwocylindersnexttoeachotherdefinesagroupaction tween i and j. A useful symmetry of the Ising model is
onthesetoftopologicalsectors,andthisgroupactionisnon- Kramers-Wannier duality, which shows the equivalence of
abelian:forexample,ababisnotthesameasa2b2 =1.Fig.8 high-andlow-temperaturepartitionfunctions.Intermsofthe
showsanexampleofhowthenon-intersectionconstraintcan spins/fields,ittakesd dandǫ ǫ.
preventannihilationoftwoloopsofthesamecolor.Eachsec- Toreachourmodel→3,o−nemustp→ert−urbthe(Ising)3critical
tor correspondsto a conservedchargein the transfermatrix; pointtoenforcetheconstraint = 0from(3). Byconstruc-
j
C
we will define these in section V. When building a quan- tion, = 0whenthedomainwallsthroughthishexagondo
j
C
tummodelbasedonthisclassicalmodel,eachofthesesectors not cross, and = 6 when they cross. Thus to reduce the
j
C
willcorrespondtoa groundstate ofanappropriatelydefined weight of configurations where domain walls cross, we add
Hamiltonian. We will discuss the quantum model further in totheenergywithpositivecoefficientλ,i.e.
j
C
sectionVIII.
E =E +λ
0 j
C
Xj
whereE istheenergyofthreedecoupledIsingmodels. The
0
≠ constraint(3)isenforcedintheλ limit. Thisperturba-
→ ∞
tionisclearlyrelevant,sinceitincludestheenergyoperators
d in the three individualmodels, and marginaltermswhich
j
couplethe two models. A keyfactto noticeis that is not
j
C
invariantunderanyofthedualitiesofthethreeIsingmodels.
Animportantresultintwo-dimensionalstatisticalmechan-
FIG.8: Inequivalent sectorsontheannulus, whichistopologically ics is the existence of a “c-theorem”19. The c-theorem says
equivalent to a finite cylinder. The two outer loops cannot move thatthere is a functionc of the parametersof the theorysat-
throughtheinnerlooptoannihilate,becauseofthenon-intersection isfying a very important property: it cannot increase under
constraint. renormalizationgroupflows. Moreover,atacriticalpointits
valueisknownfromconformalfieldtheory–itisa quantity
Finally, when the modelis definedon a torus, allnoncon- calledthecentralcharge.Thusifonestartsataknowncritical
tractibleloopsmustgoaroundthesamecycle.Thiscyclecan pointandperturbsbyarelevantoperator,thefactthatcmust
belabeledasm~c +n~c ,wheremandnareintegers,and~c decrease means that either the flow must end up at a critical
1 2 1
and~c definethetorus. Mosttopologicalsectorsonthetorus pointwithasmallervalueofc,oratnocriticalpointatall.
2
canbelabeledbyS S ,whereS isanon-trivialelementof TheIsingcriticalpointhasc = 1/2. Sinceourmodelisa
c t c
×
thefreegroupdefinedabovewiththeadditionalrequirement relevant perturbationof three Ising models, this implies that
6
either our model is not critical, or if it is critical, it should convenienttowriteT intheform
havec 3/2.Thethree-colormodel(atinfinitetemperature)
iscritic≤al,andhasc=220,21,22.Thusimposingconstraint2on T =U U −1 (4)
T T
thethree-colormodelshouldmovethemodelawayfromthe
where U moves you to the next zig-zag line, which has the
three-colorcriticalpoint. Thisisinaccordwiththenumerics
propertythatlinksiandi+1meetatavertexwheniiseven.
discussedbelow.
U thereforeimposesthe weightsat L vertices. Theoperator
Anobviousquestionisifourmodeliscriticalatsomevalue
isthetranslationoperator,whichshiftsallthespinsbyone
of K. While it is conceivable, it does not seem likely. One T
site. Thetransfermatrixwithperiodicboundaryconditionsin
can cancel the relevant piece d of by changing the tem-
j Cj bothdirectionsisthereforeZ =tr(TM)foralatticeof2LM
perature of the three Ising models. This leaves the marginal
hexagons. SinceT andT 2 havethesameeigenvectors,the
terms quadratic in d . These marginalterms can change the T
j eigenvectorsofT arethesameasthoseofU ,andtheeigen-
dimensionsofoperators,soafine-tunedmodelcouldbecrit- T
valuesaresimplyrelated.
ical. However,sincetheconstraint = 0violatesdualities,
this critical point is not likely to bCejthe (Ising)3 one. This Some interesting conservation laws follow immediately
fromthefactthatdomainwallsdonotcross. Thetotalnum-
argument does not preclude a flow to a critical point with a
ber of domain walls must be conserved mod 2, so w is
lessercentralcharge. i
conservedmod2. WhenN isamultipleofthree,thisPconser-
One candidatefor a flow is the hard-hexagonmodel. The
vationlawismuchmorepowerful:thetransfermatrixlocally
critical point in this model has c = 4/5; it is in the same
conservesthenumberofdomainwallsmod2oneachofthe
universalityclass as the three-state Potts model15. However,
threesublattices. Namely,justasthesitescanbedividedinto
it occurs very far from the hard-hexagon model of interest,
threesublattices,thelinkscanaswell;theseresultinthethree
for several reasons. First, to get our model, one must allow
types of the domain walls illustrated in figure 1. The power
configurations not present in the hard-hexagonmodel. Sec-
ofthenon-crossingconstraintisthatadjacentdomainwallsof
ond, the latter’s critical point occurs when the weight z per
differenttypescannotchangeplacesorannihilateasthetrans-
hard hexagon is z = z ((1 +√5)/2)5 = 11.09.... To
c fer matrixevolvesthe system acrossthe lattice. The distinct
≡
get model 3 at infinite temperature, the configurations must
sectorsonthecylinderdescribedattheendofsectionIIIarea
all be of equal weight, i.e. z = 1. Perturbing z away from
consequenceofthissymmetry.
z isrelevant. Itis notclear whetherallowingthe additional
c This symmetryis alreadyquite powerful. By studying U,
configurationsisrelevantornot. Itisconceivablethatthetwo
wefindevenmoreremarkableproperties. U commuteswith
perturbationscouldeffectivelycancel,leavingoneatthehard-
2L local symmetry generators, so the model with transfer
hexagoncriticalpoint,butwehavenoevidenceforthis.
matrixU insteadofT hasagaugesymmetry.
TherearenounitarycriticalpointswithZ3 symmetryand There are two different types of local conservation laws.
c < 4/5, so our model cannot be critical with these central Thefirstoneiseasytosee. Saytwoconsecutivelinksmeeting
charges. Thereareseveralwith4/5 < c < 3/2,soitiscon- atavertexarebothoccupied,i.e.w w =1. Thenexam-
2j−1 2j
ceivablethatitcouldbecriticalwiththesecentralcharges,but inethe10verticesinfigure3,andtakethetransfermatrixto
wehavefoundnoevidenceforthis. actintheverticaldirection. Thereisonlyonepossiblevertex
where both incoming links are covered, the last one drawn.
Thisvertexhasbothoutgoinglinkscoveredaswell. Thusact-
ingwithU keepsw w =1,whileallotherverticeshave
V. THETRANSFERMATRIX 2j−1 2j
w w =0beforeandafterU acts. ThusQ w w
2j−1 2j j 2j−1 2j
≡
isconservedbyU foranyintegerj.
Inthissectionwedefinethetransfermatrix,andshowthatit Thesecondlocalconservationlawisnotasobvious. Itin-
possessessomeremarkableandunusualpropertiesatinfinite volves two adjacent vertices connected by a horizontal link.
temperature.Weexploitthesepropertiesinthenextsectionto This horizontal link is of the same type as the links 2j 1
solveourmodelonadifferentlattice,andtheninsectionVII and 2j +2, so an incoming domain wall on these links−can
todonumericsonverylargesystems. turnby 120degreesontothe horizontallink. Thisconserva-
Consider the model formulated in terms of non-crossing tion law arises from the facts that there are no allowed ver-
domain walls on the links of the triangular lattice. Take the tices which have just one or three walls touching them, and
transfer matrix to act perpendicularto one of the three axes. thatthenumberofwallsofagiventypeisconservedlocally
ThetransfermatrixT actsonthespaceofstatesonazig-zag mod 2. To illustrate this, first consider the case where links
line; each link is labeled by an indexi = 1...2L, where L 2j 1 and 2j + 2 are either both occupied, or both unoc-
isthenumberofhexagonsacrosstheoriginallattice. Wetake cup−ied. Computing U requires summing over the two pos-
theconventionthatlinksiandi+1meetatavertexwheni sibilities for the horizontallink. When the horizontallink is
is odd. The degreesof freedomare the domainwalls on the unoccupied,theonlyallowedcontributiontoU istoleavethe
links. We denote wi = 1 if there is a domain wall on link configurationunchanged. When the horizontal link is occu-
i, and wi = 0 if there is not. The space of states is then of pied,theonlyallowedconfigurationisthatboth-occupiedan-
dimension2L. nihilates into both-unoccupied, or vice versa. Therefore U
The interactions are at the vertices of the lattice: the fact here doesnot conservew and w individually,but it
2j−1 2j+2
that there are only 10 vertices must be enforced. It is most doespreservethenumberofincominglinesmod2. Defining
7
R =(2w 1)(2w 1),herewehaveR =1before Withinanyofthesesectors,
j 2j−1 2j+2 j
− −
andafterU acts. Whenoneofthetwolinks2j 1and2j+2
isoccupiedandtheotherunoccupied,Rj = −1. Inthiscase, U = 1 1 .
whenthe horizontallink isunoccupied,U le−avesthe config- (cid:18)1 1(cid:19)
urationunchanged,andwhenthehorizontallinkisoccupied,
U for each of the 7 states with Q = 1 and/or Q = 1 is
thetwoconfigurationschangeplace.ThusR remains 1be- 1 2
j − diagonal:thereisonlyonestateineachsector.
foreandafterU acts. ThusR isalocalconservedquantity.
j ThecrucialpropertyofU atK =0isthatitisproportional
Note that R = ( 1)W, where W = w is the total
j j − i i toaprojectionoperator. Namely,theproductoftwodifferent
numberofQwalls(whichisindeedconservedPmod2). projectionoperatorsiszero, andeach 2 = n , wheren
The quantities Qj and Rj are not conserved in the full isthenumberofstatesinthissector. EPacPh hasPonlya sin-
model,becausetheydonotcommutewith translationopera- P
gle non-zeroeigenvaluen, and the correspondingeigenstate
tor . However,intheinfinite-temperaturecaseK = 0,they
T istheequal-amplitudesumoverallstatesinthesector. Thus
do allow the non-zeroeigenvaluesof the full transfer matrix
most states in the Hilbert space are annihilated by T. The
T tobefoundfrommuchsmallermatrices. Forexample,we
eigenstatesofT withnon-zeroeigenvalueshaveanimportant
show in section VII how this gives the largest eigenvalue of
property, following from the fact that all states in the same
T forL = 6froma5-by-5matrix,considerablysmallerthan
sector end up with the same coefficient after acting with U.
the212 212 transfermatrixobtainedwithoutexploitingany
× SinceU isthelastpartofofT,thefinalstateafteractingwith
symmetries!
T musthavethesameproperty: allstateswiththesameval-
The key simplification in the K = 0 limit is that U be-
ues of Q and R have the same coefficientin the end.
comes a sum of projection operators. Precisely, for each set { j} { j}
ThismeansthatatK = 0,alleigenstatesofT withnon-zero
of valuesof the Q and R , define a matrix ( Q , R )
j j P { j} { j} eigenvaluemusthavethesamepropertyaswell!
acting on the 22L states on the zig-zag line. The matrix el-
Wecanthereforeworkinaspaceofstatesvastlyreducedin
ements ( Q , R ) are definedto be1 ifbothstatesa
P { j} { j} ab size,bykeepingjustonestateineachsector.Howtoworkout
andbhavethecharges Q , R ,and0ifeitherorbothdo
{ j} { j} the explicittransfermatrix in this reducedbasisis explained
not. Thentheresultisthat
in section VII. We emphasize that the Q and R are not
j j
conservedchargesforthefulltransfermatrixT,liketheyare
U = ( Q , R ) whenK =0, (5)
j j
P { } { } forU. TheeigenstatesofT donothavedefinitevaluesofthe
X
Q and R , but are a sum over states with different values.
where the sum is over all possible values of Q = 0,1 and j j
j
Our result here says that for eigenstates of T with non-zero
R = 1. Notethatnotallvaluesarepossible: forexample,
j
± eigenvalues, all states in a given sector must have the same
ifQ =1andQ =1,thenR mustbe1aswell. Thede-
j j+1 j
coefficient. This is not a symmetry, because the coefficients
composition (5) follows from an extension of the arguments
arenotthesameforeigenstateswithzeroeigenvalue.
whichledto[U,Q ] = [U,R ] = 0. Therewesawthateach
j j
initial state leads to an outgoingstate with the same charges
atmostonce. Since K = 0, all allowedconfigurationshave
VI. THEGAUGE-SYMMETRICMODEL
thesameweight1,soeveryentryofU mustbe0or1. More-
over,byexplicitlyexaminingallthepossibilitiesforeachset
offoursuccessivesites2j 1,2j,2j+1,2j+2andthehor- Sincethe matrixU commuteswithallthe localsymmetry
izontal links touching the−two vertices, it is easy to see that generators,usingitasatransfermatrixresultsinamodelwith
U takesanyinitialstatewithagivenvaluesofQ ,Q ,R agaugesymmetry.Becauseofthegaugesymmetry,theresult-
j j+1 j
andR ,toanyfinalstatewiththesamevalues. ThusU in- ing “model U” can be reduced to a one-dimensional model
j+1
deedisblockdiagonal,witheachblockgivenbytheoperator and solved exactly. In this respect it is quite similar to the
two-dimensional Ising gauge theory. However, the solution
( Q , R ).
j j
P L{et u}s{give}an explicit example with L = 2 and periodic ofmodelU hassomeverystrikingpropertiesofitsown: the
eigenvalues of the transfer matrix are given in terms of Fi-
boundaryconditions. Wedenoteastatewithdomainwallsat
i,j,kby(i,j,k),andtheemptystateas(). Considerthesec- bonaccinumbers.Wederivethishere.
torwhichhashavethesameconservedchargesastheempty ModelU istheIsingmodelwithconstraint(1)aroundeach
state, which are Q = Q = 0, and R = R = 1. The plaquette of the lattice pictured in figure 9. It is the square
1 2 1 2
lattice, with an extrasite addedto allthe horizontallinks. It
other states which have these charges are (1,4),(2,3). It is
theneasytocheckthatonthesethreestates isthereforenotrotationallyinvariant.Wefindexplicitexpres-
sionsfortheeigenvaluesofU inthelimitK = 0,wherewe
1 1 1 canexploitthefactthatitstransfermatrixU canbewrittenas
U = ( 0,0 , 1,1 )=1 1 1 thesum(5). Thismeansthattheeigenstatesarethesumover
P { } { } 1 1 1 allstatesin a givensector, andthe correspondingeigenvalue
  isthenumberofstatesin thatsector. Thisturnsoutto bean
There are two states in each of the other sectors with Q = amusingcombinatorialproblem.
1
Q = 0. When R = R = 1, the sector is comprisedof Letusconsiderthesectorincludingtheemptystate,which
2 1 2
−
(2) and (3), when R = R = 1 it consists of (1) and hasallQ =0andallR = 1. HavingQ = w w = 0
1 2 j j j 2j−1 2j
− −
(4),andwhenR =R = 1,itconsistsof(1,3)and(2,4). means that the links 2j 1 and 2j are not both occupied
1 2
− −
8
Ingeneral,whenagivenR = 1,onesimplyinsertsAatthe
k
−
kthsite. ThusifR =R = 1withallothersremaining
k k+a
−
1,wehaveeigenvalue
Λ =tr(AVaAVL−a).
a,L−a
ByusingvariousidentitiesforFibonaccinumbers,onefinds
Λ =Λ F F .
a,L−a 0 L−a a
−
Letting some of the Q be 1 can be handled in a similar
j
fashion.Asnotedabove,havingQ =1meansthatthewalls
FIG.9:ThelatticeformodelU. k
onlinksk 2andk+3automaticallyfollowfromknowing
−
R andR .Thisishandledinthetransfer-matrixformalism
j−1 j
by walls. Having R = 1 means that either both of the byinsertingthematrixVBatthesiteofeveryQ =1,where
j k
links 2j 1 and 2j + 2 are occupied, are neither one is.
−
Theeigenvaluefor ( 0,0,... , 1,1,... )isthenthenum- 0 0
P { } { } B = .
ber of states Λ0 satisfying these constraints. To countthese, (cid:18)0 1(cid:19)
note that if both links 2j 1 and 2j +2 are occupied, then
links2j and2j+1mustb−eunoccupied,inordertopreserve ThuswhenQk =1forsomekwhileallotherQj =0andall
Rj =Rj+1 =0. Butiftheselattertwolinksareunoccupied, Rj =1,wehaveeigenvalue
thenlinks2j 3and2j+4mustbeunoccupiedaswell, to
keepQj−1 =−Qj+1 = 1. Thisrulegivesa wayof counting tr(BVL)=FL−1 =Λ0−FL+1
theconfigurationsinthissectorusingone-dimensionaltrans-
fermatrixV,whichpropagatesthesystembytwosites. Start ThiseigenvalueissmallerthanΛ1andΛa,L−a;theeigenvalue
atoneend. Ifw1 =w4 =0,thenw3 =w6 canbeeither0or Λ0isthelargest,withΛ1thenexthighest.
1. However,ifw = w = 1, thenw = w = 0. Iterating Continuinginthisfashion,oneobtainsthegeneralformula
1 4 3 6
thisprocedurealongthewholelinegives
Λ( Q , R ) = tr(VX VX ...VX VX ), (6)
j j L L−1 2 1
{ } { }
Λ0 = tr(VL), X = A(1+Rj)/2B1−Qj (7)
j
1 1
V =
(cid:18)1 0(cid:19) The conservation laws still hold when K = 0, so U re-
6
mainsblockdiagonal.However,theblocksarenolongerpro-
The first row and column of V correspond to unoccupied jection operators, so there is genericallymore than one non-
links, while the second correspond to occupied ones. It is zeroeigenvalueperblock.Wesuspecthoweverthatthegauge
simpletoshowbyinductionthat symmetrymakesitpossibletheeigenvalueshereintermsofa
one-dimensionaltransfermatrixlike(6,7).
F F
Va = a+1 a
(cid:18) Fa Fa−1(cid:19)
VII. NUMERICALRESULTS
whereF istheathFibonaccinumber(F = 0,F = 1,and
a 0 1
F =F +F fortherest). Thus
a a−1 a−2
To use exact diagonalization on the transfer matrix of the
Λ0 =FL+1+FL−1 full model at K = 0, we utilize the trick described in sec-
tion V to reduce its size. This enables us to find its largest
whichforlargeLgrowsasτL+1, whereτ = (1+√5)/2is
eigenvalueforcylindersofuptoL = 18hexagons(36Ising
thegoldenmean.
sites).
Usingatransfermatrixinonedimensionmakesitpossible
Eachstateinthisnewspaceislabeledbythevalues Q
j
to write an expression for all the eigenvalues. First consider { }
and R ,whichforshortwe callB. AfterU acts, giving
j
thecasewithallR =1exceptR = 1,andallQ = 0. If { } T
j 1 j everyelementinthe same blockthe samecoefficient, we la-
−
w1 =1 w4 =0,thenw3 =1,butifw1 =1 w4 =1,then beltheblocksB′. Toworkoutthetransfermatrixinthisnew
− −
w canbeeither0or1. Thusthe1dtransfermatrixforj =1
3 basis, firstoneneedsto listall thestates ina givenblockB.
isVA,where
Pick one and act with , i.e. just shift the whole thing over
T
by one site. Computethe new valuesof Q and R af-
0 1 j j
A= . { } { }
(cid:18)1 0(cid:19) ter the shift, or equivalently, compute Qj = w2jw2j+1 and
R = (2w 1)(2w 1), whichwe collectivelylabel
TheeigenvalueΛ forthecasewhereoneoftheR isflipped j 2j − 2j+3 − e
1 j B. The block B′ reached from acting with U on this ele-
to 1istherefore e T
− mentB is thenlabeledby Q and R , whereQ = Q
e { j} { j} j j
Λ =tr(AVL)=2F =Λ F . andR = R forallj. Onedoesthisforeachelementinthe
1 L 0− L−3 j j e
e
9
Width(7) S/7 cest f0,est
Model3 Three-color Model3 Three-color Model3 Three-color
3 0.4621 0.4621
6 0.3911 0.4028 1.880 1.569 0.3674 0.3830
9 0.3771 0.3900 2.000 1.829 0.3659 0.3798
12 0.3722 0.3853 1.990 1.914 0.3660 0.3793
15 0.3700 1.965 0.3660
18 0.3688 1.942 0.3661
. . .
. . .
. . .
∞ 0.3661 0.3791 1.99 0.3661 0.3791
Theory 0.379114 2 0.379114
TABLEI:Resultsofnumericaltransfer-matrixcalculationsonmodel3withperiodicboundaryconditions, comparedtothree-colormodel.
Theestimatedcentralchargecestandbulkfreeenergyf0,estareobtainedfornhexagonsbyfittingtheentropyvaluesfornandn−3hexagons
toequation(8).Extrapolationsofentropypersitetotheinfinitesystemfitthelastthreepointstoc0+c2L−2+c4L−4.
blockB: workoutB andthenB′ foreach,andthenincrease L=6isthesameasthatofthe5-by-5matrix
theelement B′B byone. Goingthroughalltheblocksgives
R e
thereducedtransfermatrix . 4 6 6 0 2
R
Since the eigenvectorsof U are the same as those of T, 2 6 4 2 0
T
and the eigenvaluesare simply related, we focuson this. To 1 2 2 0 0
giveanexample,forL=2,wehave  
0 1 0 0 0
 
1 0 0 0 0
1 0 0 1 1  
1 0 0 1 1 Inthisandalltheexampleswehaveexamined, isupperleft
R
U =1 0 0 1 1 , triangular.
T   Wehaveusedthisreducedtransfermatrixforthedomain-
0 0 1 0 0
  wall looprepresentationin numericalsimulationsof transfer
0 1 0 0 0
  matrices with width up to 18 hexagonsin model1 (36 Ising
variables). The resulting largest eigenvalues for widths that
whichhaseigenvalues2, 1,0,0,0. Note thatitis notsym-
are multiples of 3 are shown in table 1. The entropy per
−
metric. There are three blocks here. The block B = 1 has
hexagonconvergestoanumberrightinthemiddleoftheup-
threestates(),(14),(23),theblockB = 2hasjust(12),and
per (from the three-color model) and lower (from the hard-
theblockB = 3 hasjust(34). Uponactingwith , ()goes
hexagonmodel)boundsgivenin(A2).
T
totheblock(), whichmeansU takesittoallthemembers
Expandingthe largesteigenvalueina powerseriesin 1/L
T
ofthisblock. Thusweincrease by1. Actingwith on
11 givesadditionalvaluableinformation. Whenthesystemisat
R T
(14)takesitto(12),soweincrease by1. Actingwith
21 aconformallyinvariantcriticalpoint,thesubleadingpieceis
R T
on(23)gives(34),so 31 = 1. Doingthisfortheothertwo universalandproportionaltothecentralcharge23,whichmust
R
blocksgives
obeyc 1/2inanysystemwithpositiveBoltzmannweights.
≥
Ifthesystemisnotcritical,thispieceshouldfallofftozeroas
1 1 1 L . Thepreciseformulaforourcaseis
→∞
=1 0 0
R logΛ πc√3 1
1 0 0 f = =f + +.... (8)
  L 0 6 2 L2
Thishaseigenvalues2, 1,0aswewant,andissymmetric– Here L is the width in hexagons, Λ is the largesteigenvalue
−
itjustlostsomezeroeigenvalues. ofU , andthegeometricalfactor√3/2resultsfromthera-
T
The ground-state sector for L = 3 is even easier. There tio between the width and length of the transfer matrix step.
are 4 states in the block 1: (),(14),(25),(36). These all go Theresultingestimatesofcentralchargeforourmodeldonot
tothesameblockunderU ,sothereducedtransfermatrixis convergeevenatthelargestsystemsizes,whileforthethree-
T
simplyanumber:4.Thisisindeedthelargesteigenvaluehere. colormodel, extrapolationfromsmaller sizes givesa central
For higherL, the size of still increases exponentially,but chargeconsistentwiththeexpectedvaluec=220,21,22.
R
notasquickly. Togiveanexampleofhowmuchthisreduces The conclusion of this transfer-matrix study is that our
the size of the matrix, exploiting translation invariance and modelismostlikelynotdescribedbythec=2criticaltheory
parity as well means that the largest eigenvalue of U for of the three-colormodel, even thoughlargesystem sizes are
T
10
requiredto see the difference. Since the centralchargedoes ttes. ItissimpletowriteH explicitlyintermsofIsingspins,
1
notseem to be convergingto anything,the numericalresults but the expression is quite unwieldy. The lowest eigenvalue
areinharmonywiththefield-theoryargumentsofsectionIV ofH iszero,sinceitisthesumofprojectorsanda positive
1
in suggesting that our model is non-critical(i.e., has a finite diagonalterm. Thegroundstateonthesphereisunique,and
correlationlength). Wecannotcategoricallyruleoutthatitis consists of the equal-amplitude sum over all configurations
criticalwithc<2,buthavenoevidenceforthisscenario. satisfyingconstraint(1).Onsurfaceswithcontractiblecycles,
thelocalflipcannotchangethesectorsdescribedabove.There
willbeagroundstate foreachofthesesectors,consistingof
VIII. FURTHERDIRECTIONS:THEQUANTUMTHEORY theequal-amplitudesumofalltheconfigurationsinthesector
satisfyingtheconstraint.
Inthispaperwediscussedconstrainedclassicallatticemod- The reason for the interest in quantum models of this
els. ByimposingsomesimpleconstraintsonIsingspins, we type is that they often have topological order. Topologi-
found a variety of intriguing properties. In particular, we cal order means that there is no local order parameter with
showed that the space of states on the sphere is connected a non-vanishing expectation value, but only non-local ones.
underlocalmoves,andthatonsurfaceswithnon-contractible By using the Rokhsar-Kivelson trick, it was demonstrated
cycles,differentsectorscanbelabeledbyloopconfigurations. that a quantum eight-vertex model9 and a quantum dimer
Wealsopresentedsubstantial(ifnotconclusive)evidencethat model on the triangular lattice4 indeed have topological or-
themodelisnotcritical. der. ThequantummodelwithHamiltonianH1 indeedshould
Intheintroduction,wementionedaquantummotivationfor have topological order, since the number of ground states
studyingclassicallatticemodelswithconstraints. Theresults depends on the genus of the surface, a telltale sign. One
of this paper imply that our model has the right characteris- reason why models with topological order are interesting is
ticstoyielda quantummodelwithatopologicalphase, with that they can lead to excitations with fractional statistics.
theaddedintriguingpossibilitythattheexcitationshavenon- Non-topologicalsolidandsuperfluidquantumphasesnearthe
abelianstatistics. We thereforewillconcludethispaperwith Rokhsar-Kivelsonpointcorrespondingtoourmodel,andpos-
adiscussionofthequantummodelinmoredetail. sible unconventionalphase transitions, are discussed using a
The connection between quantum and classical models two-componentquantumheightmodelinRef. 17.
comes from a trick due to Rokhsar and Kivelson10. Let the The excitations in the quantum model with Hamiltonian
basis elements for the Hilbert space for the quantum model H1 correspondtohexagonswithdifferentnumbersofupand
consistofconfigurationsinthetwo-dimensionalclassicallat- down spins. The specific Mh we chose means the lowest-
tice model. Then one can construct a quantum Hamiltonian energydefectshaveMh = 2. Theseareillustratedinfigure
±
actingonthesestateswithagroundstateconsistingofasuper- 10. Inthelooplanguage,theycorrespondtojoiningloopsof
position of these states, with each term having an amplitude
correspondingtoitsweightintheclassicalmodel.Correlators u u u
in the groundstate of the quantummodelare then related to d d d u d u
thecorrelatorsintheclassicalmodel.
Inthequantummodel,oneneednotimposetheconstraints u u d u u d
directlyontheHilbertspace,butratheronecanaddanenergy u u u
penalty for configurationswhich violate the constraint. The
groundstate then contains only configurationssatisfying the FIG.10:Defectsoccurringinmodel1whenMh=2.
constraint. Violating the constraint locally then corresponds
toaquasiparticleexcitation.Thusforagivenclassicalmodel,
differenttypes. WiththeHamiltonianH ,thesedefectshave
one can obtain verydifferentquantummodelsdependingon 1
nodynamics,butonecanofcourseaddtermsallowingthem
whichdefectsareallowedandwhicharenot.
tomove.
Letusmakethisexplicitintermsofmodel1. Herewecan
Changingthepotentialtofavorotherkindsofdefectsgives
taketheHilbertspacetobecomprisedoftwo-stateIsingvari-
different theories. Allowing just M = 4 defects gives a
ables on the sites of the honeycomblattice. One simply im- h ±
variationonthe “odd”Isinggaugetheorydescribedin depth
posesanenergypenaltyonconfigurationsviolatingconstraint
in24; these defectscorrespondto allowingloops(the domain
(1),i.e.foreachhexagonhoneincludesM2 intheHamilto-
h walls ofmodel3)to cross. AllowingjustM = 6defects
nian. Theoff-diagonaltermsinHamiltonianaregivenbyflip h ±
givesaquantumversionofthethree-colormodel,i.e.thede-
we defined in section III, as displayedin figure 5. The trick
fects are the 11th vertex shown in figure 4. Note however
of Rokhsar and Kivelson is to add a potential which, com-
thateventhoughthegroundstatesofallthreemodelswehave
binedwiththeflip,isaprojector. Namely,weaddapotential
introduced are identical, their local defects can be quite dif-
term which counts the number of flippable hexagons. Then
ferent. The defects just described are nonlocal in the color
theHamiltonianis
representation of model 2, because the three colors become
H = +(M )2 rotateduponcircling,i.e.,eachbondnolongerhasauniquely
1 Nh−Fh h definedcolor13. Likewise,anotherkindofdefectwecouldin-
Xh (cid:0) (cid:1)
troducewouldbetotreattheloopsinmodel3asthedegrees
where istheflipand isthenumberofflippableplaque- offreedomforthequantummodel.Thenwecanallowdefects
h h
F N