A quantum phase transition from $Z_2 \times Z_2$ to $Z_2$ topological order

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by  Mohammad Hossein Zarei

Z Z Z
A quantum phase transition from × to topological order
2 2 2
MohammadHosseinZarei1
PhysicsDepartment,CollegeofSciences,ShirazUniversity,Shiraz71454,Iran
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] Abstract
h
p Althoughthetopologicalorderisknownasaquantumorderinquantummany-bodysystems,
- it seems that there is not a one-to-one correspondence between topological phases and quantum
t
n phases.Asawell-knownexample,ithasbeenshownthatallone-dimensional(1D)quantumphases
a aretopologicallytrivial[31].Bysuchfacts,itseemsachallengingtasktounderstandwhenaquan-
u tumphasetransitionbetweendifferenttopologicalmodelsnecessarilyrevealsdifferenttopological
q classesofthem. Inthispaper, wemakeanattempttoconsiderthisproblembystudyingaphase
[
transition between two different quantum phases which have a universal topological phase. We
2 define aHamiltonian as interpolation of the toriccode model withZ2 topological order and the
v colorcodemodelwithZ2×Z2 topologicalorderonahexagonallattice. Weshowsuchamodel
6 isexactlymappedtomanycopiesof1DquantumIsingmodelintransversefieldbyrewritingthe
0 Hamiltonianinanewcompletebasis. Consequently,weshowthattheuniversaltopologicalphase
5 ofthecolorcodemodelandthetoriccodemodelreflectsinthe1Dnatureofthephasetransition.
6 WealsoconsidertheexpectationvalueofWilsonloopsbyaperturbativecalculationandshowthat
0
behavioroftheWilsonloopcapturesthenon-topological natureofthequantumphasetransition.
.
1 Theresultonthepointofphasetransitionalsoshowthatthecolorcodemodelisstronglyrobust
0 againstthetoriccodemodel.
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: 1 Introduction
v
i
X
AfterthesymmetrybreakingtheoryofLandau[46],manythoughtsciencehasreachedacomprehen-
r siveunderstandingofdifferentphasesofmatter.However,anewconceptemergedincondensematter
a
physics by quantum Hall effect [2], quantum spin fluids [3, 4, 5, 6] and superconductors [7, 8, 9]
whichwascalledtopologicalorder.Topologicalorderisadifferentphaseofmatterwhichcannotun-
derstoodbyLandautheorysothatonecanfinddifferenttopologicalphaseswiththesamesymmetries
[10].
Thegroundstateofaquantumsystemwithtopologicalorderexhibitsalong-rangeentanglement[11].
Specifically,thetopologicalentanglemententropyhasbeendefinedasanon-localorderparameterof
topologicalorder [12, 13] so that it can characterize a quantum phase transition from a topological
phase [14]. Another interesting property of a topological phase is the behavior of the Wilson loop
where the expectation value of the Wilson loop is related to perimeter of the loop in a topological
phase,whileitisrelatedtoareaoftheloopinanon-topologicalphase[14].
Long-rangeentanglementof the groundstate of a topologicalmatter also leads to some interesting
1email:[email protected]
1
andexoticpropertieswhichhavenotbeenseeninsymmetrybreakingphases. Robustdegeneracyof
the groundstate and exotic statistics of excitationsof the system, which are called anyons, are two
importantcharacteristicsofthetopologicalphases[15,16,17]. Braidinganyonsleadstoanarbitrary
phasefactoronwavefunctionofthesystemintheabelianmodelsandaunitaryoperatorinthenon-
abelian models [18, 19]. The robustness of degeneracy of the ground state and braiding operators
against local perturbationsis a key propertyof the topologicalphases which has convertedthem to
importantcandidatesforfault-tolerantquantumcomputation[20,21,22].
Inspiteofmanyimprovementsondefiningdifferentcharacteristicsofatopologicalphase,recogniz-
ingdifferenttopologicalclassesindifferentquantumsystems[23,24,25,26,27,28]isstillanopen
problemwhichhasreceivedmuchdealofinterest. Sincetopologicalorderisa kindofquantumor-
der, it is clear that the same quantumphases have also the same topologicalphases. Consequently,
consideringquantumphasetransition betweendifferenttopologicalmodelsis a usefulapproachfor
classificationoftopologicalmodels.Specifically,therearesomerecentpaperswhichshowaquantum
phase transition can well reveal different topologicalproperties of two different topological phases
[29,30].
This thoughtthat a quantumphase transition revealsnecessarily differenttopologicalclasses of the
quantum phases is challenged specifically by 1D quantum models. It has been shown that all 1D
quantumphasesbelongtoatrivialtopologicalphase[31]sothatonecanfindaquantumphasetransi-
tionbetweentwodifferent1Dquantummodelswhicharetopologicallytrivial[32,33]. Infact,Since
topologicalsignatureoftopologicalphasessuchastopologicalentanglemententropyorexpectation
valueoftheWilsonloopoperatorscannotbedefinedfora1Dmodel,topologicalorderof1Dmodels
onlydefinesasasymmetryprotectedtopologicalphase[31].
Moreover1Dquantummodelswhichareinthesametopologicalphases,therearealsotwo-dimensional
(2D)quantummodelswhichbelongtoauniversaltopologicalphase. Awell-knownexampleofuni-
versaltopologicalphaseshasbeenseeninthetoriccodemodel(TC)withZ topologicalorder[22]
2
and the color code model(CC) with Z ×Z topologicalorder [34]. Since the TC and CC model
2 2
havedifferentgaugesymmetries,degeneraciesofthegroundstateofthemaredifferent.WhiletheTC
modelprovidesafour-folddegenerategroundstatewhichcanbeusedasarobustquantummemory,
degeneracyofthe groundstate oftheCCmodelissixteen-fold. Furthermore,unlikethe TCmodel,
thereisalsopossibilityofapplyingunitaryCliffordgroupinatopologicalway[34]. Recently,there
isalsoanexperimentalrealizationofthecolorcodeontrapped-ionqubits[35].
AlthoughdifferentdegeneraciesoftheCCmodelandtheTCmodelshowthattheyareintwodifferent
quantumphases,othertopologicalcharacteristicsofbothmodelsarecompletelythesame. Especially,
itis possibleto showa CC modelandtwo copiesof theTCmodelare in thesame quantumphases
[36, 37]. In [36], the authors have emphasized that since there is an adiabatic evolution without
quantumphasetransitionbetweenaCCmodelandtwocopiesoftheTCmodel,theyhavethesame
topologicalphases. ThisresultshowsthatdifferentquantumphasesoftheTCandtheCCmodelare
completelyrelatedtotheirdifferentgaugesymmetriesandhasnotatopologicalnature.
By above facts, there is a good chance that one considers how the same topologicalcharacteristics
of the CC and the TC modelreflects in the propertiesof a quantumphase transition betweenthem.
Tothisend,inthispaperweconsideraCCmodelonahexagonallatticebesidesasingleTCmodel
insteadoftwocopiesofitonthesamelattice. WeshowtheuniversaltopologicalphaseoftheCCand
theTCmodelreflectsinpropertiesofthequantumphasetransitionwhereweshowthatthequantum
phasetransitionhasa1Dnature.Ourresultisalsoanotherprooffortheuniversaltopologicalphaseof
theCCmodelandtheTCmodel.Weemphasizethatourproofhasarecentandimportantpointwhere
generallyshowshow the universaltopologicalphase oftwo 2D topologicalmodelscan be revealed
eveninpresenceofaquantumphasetransition.
Toderivingresults,wedefineaspecificversionoftheTCmodelonthehexagonallattice. Weshow
that a model Hamiltonian as interpolation of the CC and the TC model is mapped to many copies
of1DIsingmodelsintransversefieldwhichbelongtothe2DclassicalIsinguniversalityclass. The
2
mappingisbasedonre-writingtheHamiltonianofthe modelina newcompletebasis. Ourmethod
isequivalentwithunitarytransformationontheHamiltonianwhichdoesnotchangespectrumofthe
model. WealsostudythebehaviorofWilsonloopsandexplicitlyshowthattheexpectationvalueof
theWilsonloopsinourmodelcapturesthenon-topologicalnatureofthephasetransition.
Fromanotherpointofview,wefindthepointofthephasetransitionofourmodelwhichisthesame
as 1D Ising model in transverse field at gt = 1 [38] where g and g are the couplings of the TC
gc t c
andtheCCmodels,respectively. ItshowswhenweaddtheTCHamiltonianasasmallperturbation
againsttheCCmodel,suchaperturbationcannoteverchangethequantumphaseoftheCCmodel.
Thequantumphasetransitionoccursonlywhenaconsiderableperturbationg >g isapplied. Such
t c
a result show that unlike the small robustness of the CC model against local perturbations such as
magneticfieldorIsinginteraction[39,40,41,42,43],itisstronglyrobustagainstatopologicalper-
turbationliketheTCmodel.
Insection(2),wereviewdifferentpropertiesofthetopologicalphasesoftheTCandtheCCmodel.In
Section(3),wepresentourmodelasainterpolationoftheCCandtheTCmodel.Wedefinebothtopo-
logicalmodelsonahexagonallatticeandqualitativelyshowhowaquantumphasetransitionhappens.
Insection(4),weuseamathematicalmethodtomapourmodelto1DIsingmodelsintransversefield
whichhavea well-knownphasetransitionpoint. Finally,in section (5) we considerthe behaviorof
Wilsonloopoperatorsbyaperturbativeapproachtoshowthequantumphasetransitioninourmodel
hasnotatopologicalnature.
2 Brief review of the toric code and the color code model
In this section we review the topological structure of the TC and the CC models. We specifically
emphasizeondifferenttopologicaldegeneraciesanddifferenttopologicalstructuresofthembyrepre-
sentationofthegroundstatesasaloop-condensatestates. Formoredetails,thereisalsoacomparative
studyofthesemodelswhichhasdonein[44].
2.1 Toriccode model
A TC modelis ordinarilydefined as the groundstate of a model Hamiltonianon an oriented graph
where qubitslive on edgesof the graph. Two commutativeoperatorsB and A are defined corre-
p s
sponding to each plaquette and vertex of the graph in the following form, See figure (1, left) for a
squarelattice:
Bp =YZi , As =YXi (1)
i p i s
∈ ∈
wherei ∈ prefersto qubitsaroundofa plaquetteandi ∈ s refersto qubitsaroundofa vertexand
Z ,X arethePaulioperators.TheHamiltonianofthemodelisassummationoftheseoperatorsonall
plaquettesandverticesofthegraphasthefollowingform:
H =−XBp−XAs. (2)
p s
Bythefactthatthevertexandplaquetteoperatorscommutewitheachother,itissimpletoshowthe
followingstateisagroundstateofHamiltonian(2):
|φti=Y(1+Bp)|+++...+i (3)
s
wherethestate|+iiseigenstateoftheoperatorX correspondingtoeigenvalue+1andweignorethe
normalizationfactorforthisstate.
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L1 L1
x z
Z
L0 Bp As e m m
x m
e
X
e e
L0 m
z
Figure 1: (Color online)Left: the plaquette and vertexoperatorshave been shown by two different
colorscorrespondingtoverticesandplaquettesofthelattice. Therearefournon-trivialloopoperators
which describe topologicaldegeneracy. Right: two charge(flux) anyonsare generated by applying
Z (X)operatoroneachqubit. ByapplyingasequenceofZ (X)operatorsonqubitsitispossibleto
moveacharge(flux)anyononthelattice.
There is also a simple representation for the state (3) which helps for better understanding of the
topologicalorder of such a state. To this end, let us span the productof operators(1+B ) on all
p
plaquettesintherelation(3)asthefollowing:
Y(1+Bp)=1+XBp+XBpBp′ +... (4)
p p p,p′
wheretherighthandofthisrelationisthesummationofallpossibleproductsoftheplaquetteopera-
tors. Sinceeachplaquetteoperatorcanbeinterpretedasaloopoperatoronthelattice,righthandof
therelation(4)isthesummationofallpossibleloopoperators.
Finally, since Z|+i = |−i wherethe state |−iis the eigenstateof theoperatorX correspondingto
eigenvalue−1, the groundstate of theTCmodel(3) canbe interpretedasuniformsuperpositionof
allloopconstructionsofqubits|−iintheseaofqubits|+iwhichiscalledtheloopcondensation.
Such a state has a topological order which leads to a robust degeneracy in the ground state of the
modelwhenwedefinethemodelHamiltonianonatorus. Infact,onatorustopology,therearealso
non-contractibleloopoperatorsintheformof:
LσZ = Y Zi (5)
i Lσ
∈
where i ∈ Lσ refers to the qubits which live on a non-contractible loop around the torus in two
differentdirectionsσ = 0or 1,seefigure(1,left). Therefore,fourthefollowingquantumstatesare
thedegenerategroundstatesofHamiltonian(2):
|ψ i=(L0)i(L1)j|φ i (6)
i,j z z t
wheretheindicesi,j ={0,1}refertofourdifferentquantumstates.
TopologicalorderoftheTCmodelareunderstoodbythreeimportantproperties,robustdegeneracy,
non-localorderparameterandanyonicexcitations.
Therobustdegeneracyisduetothisfactthatfourdegenerategroundstates(6)cannotbeconverted
toeachotherbyanylocalparametersothedegeneracyisrobustagainsteachlocalperturbation.
Another property is that any local operator can not distinguish four degenerate ground states. In
fact,therearetwonon-localoperatorswhichhavedifferentexpectationvaluesindifferentdegenerate
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groundstates. Suchoperatorsaredefinedinthefollowingform:
LσX = Y Xi (7)
i Lσ
∈
where i ∈ Lσ refers to the qubits which live on a non-contractible loop around the torus in two
differentdirectionsσ = 0or1,seefigure(1,left). Theexpectationvaluesoftheseoperatorsineach
oneofthefourdegenerategroundstatesareasfollows:
hψ |L0|ψ i=1 , hψ |L1|ψ i=1
00 x 00 00 x 00
hψ |L0|ψ i=−1 , hψ |L1|ψ i=1
01 x 01 01 x 01
hψ |L0|ψ i=1 , hψ |L1|ψ i=−1
10 x 10 10 x 10
hψ |L0|ψ i=−1 , hψ |L1|ψ i=−1 (8)
11 x 11 11 x 11
Thereforethesetwonon-localoperatorscandistinguishthedifferentgroundstates.
Finally,excitationsoftheTCmodelarequasi-particleswithanyonicstatistics. Anexcitationisgener-
atedbyapplyingthePaulioperatorsX orZ onaqubitofthelattice,seefigure(1,right). Anoperator
X on a qubit does not commute with two plaquette operators which are shared in that qubit and it
is interpreted as two flux anyons m in two corresponded plaquettes. Also an operator Z does not
commutewithtwoneighborvertexoperatorsanditisinterpretedastwochargeanyonseintwocorre-
spondedvertices.ByapplyingastringofX (Z)operators,aflux(charge)anyonmovesonplaquettes
(vertices)ofthelattice,seefigure(1,right).Itissimpletoshowthatifachargeanyonwindsarounda
fluxanyon,itleadstoaminussignonwavefunction.Suchafactorshowsthatchargeandfluxanyons
arenotfermionsorbosons.
2.2 Colorcodemodel
TheCCmodelisanotherwell-knownkindofthetopologicallatticemodelswithZ ×Z topological
2 2
order. Although topological properties of this model seems similar to the TC, additional freedom
degreeofcolorinthismodelleadstosomeimportantdifferenceswiththeTC.SpecificallytheCCis
moreefficientthantheTCforcomputationaltasks. ForexamplesintheCConanhexagonallattice,
alltheCliffordoperatorscanbeappliedinatopologicalwaywhileitisnotpossibleintheTCmodel
[34].
TheCCmodelcanbedefinedonthree-colorablelatticeswhichtechnicallycalledcolexesandcanbe
generalizedtoarbitrarydimensions[45]. Asanexampleweconsiderahexagonallatticewherequbits
liveonverticesofthelattice,seefigure(2). Correspondingtoeachhexagonalplaquetteofthelattice
whichisdenotedbysymbol”h”,wedefinetwooperatorsinthefollowingform:
hx =YXi , hz =YZi (9)
i h i h
∈ ∈
wherei∈hreferstoallqubitsbelongingtotheplaquetteh. TheHamiltonianofthemodelisdefined
as
H =−Xhx−Xhz. (10)
h h
Sinceallplaquetteoperatorscommutewitheachother,thegroundstateofthisHamiltonianisinthe
followingsimpleform:
|φci=Y(1+hz)|++...+i (11)
h
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L1,r L1,g L1,b
z z z
L0,b L0,b
z x
L0,r L0,r
z x
L0,g L0,g
z x
L1,g L1,r L1,b
x x x
Figure2: (Coloronline)Left: thequbitsliveontheverticesofthe latticeandtwooperatorsh and
x
h areattachedtoeachhexagonalplaquette. Edgesofthelatticearecoloredbythreedifferentcolors
z
correspondingtothreedifferentcolorsofthehexagonalplaquettes. Therearetwelvenon-trivialloop
operatorscorrespondingtotwodirectionsontorusandthreedifferentcolorsoftheplaquettes.
Right: correspondingtoeachhexagonalplaquetteofthelattice,thereisatriangularplaquettewhere
twoqubitsliveoneachedge.Inthisway,therearethreetriangularlatticeswiththreedifferentcolors.
Since thereare two triangularplaquetteswith differentcolorscorrespondingto each hexagonalpla-
quette,onlytwocolorsareenoughforre-presentationofplaquetteoperatorsonthetriangularlattices.
where we ignore the normalization factor. Similar to the TC model, there is a loop re-presentation
for the state (11). To this end, let us span productof operators(1+h ) in the relation (11) as the
z
followingform:
Y(1+hz)=1+Xhz+Xhzh′z +.... (12)
h h h,h′
We colorall plaquettesofa hexagonallattice by threedifferentcolors, red, blue, greenso thatany
twoneighborplaquettesofthelatticearenotinthesamecolor,seefigure(2,left). wealsocolorall
the edgesof the hexagonallattice with three colorsso that eachtwo plaquetteswith the same color
connecttogetherwithanedgewiththesamecolor. Eachhexagonalplaquetteofthehexagonallattice
canalsobeconsideredasatriangleplaquetteofatriangularlatticewheretwoqubitsofeachhexago-
nalplaquetteliveoneachedgeofthistriangularplaquette,seefigure(2,right).Inthisway,theedges
ofthelatticeinsertinthreecategoriescorrespondingtoeachcolorandwecandrawatriangularlattice
correspondingtoeachcategoryofcolorededges,seefigure(2,right).
Finally,weinterpreteachplaquetteoperatoroftheCCmodelbyatriangularcoloredloopononeof
thethreetriangularlattices. By suchainterpretation,the summationintherelation(12) isregarded
assuperpositionofallpossibleloopoperatorswiththreedifferentcolorsonthetriangularlattices. A
closerlooktothehexagonallatticeshowsthataplaquetteoftheinitialhexagonallatticecorresponds
totwodifferenttrianglesoftwodifferentcoloredtriangularlattices, seefigure(2, right). Therefore,
wehavethisfreedomtoselectoneofthecolorsforeachhexagonalplaquette.Bysuchafreedom,itis
simpletoshowthattherelation(12)canbeinterpretedassuperpositionofallloopoperatorswithonly
two differentcolors. Therefore,it is well understoodthatthe topologicalorderin the CC is similar
totheTCwithadditionalfreedomdegreeofcolor. SuchaorderiscalledZ ×Z topologicalorder
2 2
versustheZ topologicalorderfortheTC.
2
The topological order in the CC also leads to some important properties similar to the TC such as
robustdegeneracy,non-localorder parameterand anyonicexcitations. as a sample, if we insert the
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hexagonallatticeonatorus,therewillbemanynon-contractibleloopoperatorswhichleadtodegen-
eracyofthegroundstates. TheseloopoperatorsaredefinedsimilartotheTCmodelwithadifference
that there are three kinds of loop operators corresponding to each color, see figure (2, left), in the
followingform:
Lσz,r = Y Zi
i Lσ,r
∈
Lσz,g = Y Zi
i Lσ,g
∈
Lσz,b = Y Zi (13)
i Lσ,b
∈
where i ∈ Lσ,r(g,b) refersto the qubitswhich live ona non-contractiblelooparoundthe toruson a
red(green,blue)triangularlatticeintwodifferentdirectionsσ =0or1. Sincethreeaboveoperators
are not independentso that the product of them as Lσ,rLσ,gLσ,b is equal to a productof plaquette
z z z
operators, we can generatethe sixteen degenerategroundstates of the model by applyingonly two
coloredloopoperatorsasthefollowingform:
|φ i=(L0,r)i(L1,r)j(L0,b)k(L1,b)l|φ i (14)
i,j,k,l z z z z c
wheretheindicesi,j,k,l = {0,1}refertosixteengroundstatesofthemodel. SimilartoTCmodel,
therearealsofournon-localorderparametersasthefollowingformwhichcancharacterizedifferent
groundstatesoftheCCmodel,seefigure(2,left).
Lσx,r = Y Xi
i Lσ,r
∈
Lσx,b = Y Xi (15)
i Lσ,b
∈
whereweapplyoperatorsX onnon-contractibleloopswithtwodifferentcolors.
3 Interpolation of the CC and the TC
In this section, we consider both the TC and the CC model on the same lattice and define a new
Hamiltonianasthefollowingform:
H =−g H −g H (16)
t t c c
whereH andH areHamiltoniansoftheTCandtheCCmodel,respectively.SuchaHamiltonianis
t c
definedonthequbitswhichliveontheverticesofahexagonallattice.
DefinitionoftheCConsuchalatticeisthesameasweexplainedintheprevioussection.Sincequbits
liveontheverticesofthislatticewecannotdefineanordinaryTCmodelonsuchalattice. However,
thereisa simpleway topresenta TCmodelonsuchlattice. Tothisend, we divideeachhexagonal
plaquette to two Trapezoid-shaped parts as it is shown in figure(3). Then we paint all trapezoids
as chess-patternwith dark and light colors. In this way, we can use a rotated version of the Kitaev
model [44] where we relate an operator B = Z Z Z Z to each light plaquette and an operator
p 1 2 3 4
A = X X X X toeachdarkplaquette,seefigure(3). Itisverysimpletoshowthatsuchamodel
s 1 2 3 4
isexactlythesameTCmodel.Specifically,onecancheckthatthegroundstateofthismodelis:
|ψi=Y(1+Bp)|++...+i (17)
p
7
B
p
L0,b L0,b L0 A L0
z x x s z
L0,r
L0,r x
z
Figure3: (Coloronline)Eachhexagonalplaquetteofthelattice hasbeendividedtotwo Trapezoid-
shaped parts. By a chess-pattern painting of such a lattice, a TC model can be defined where the
operatorsB arerelatedtolightplaquettesandtheoperatorsA arerelatedtodarkplaquettes. Two
p s
non-contractibleloopswhicharetopologicallydifferentintheCCmodelconverttogetherwithproduct
ofoperatorsA orB whichareinvolvedbythenon-contractibleloops.Itshowthatnon-contractible
s p
loopswithdifferentcolorsaretopologicallythesameintheTCmodel.
Suchastateisthesameasequation(3)inthedefinitionoftheTCmodel.
AfterdefinitionoftheTCandtheCCmodelonthesamehexagonallattice,wearereadytostudythe
propertiesoftheHamiltonian(16). Ontheonehand,inlimitofg ≫g ,wehaveaTCmodelwhich
t c
haveafour-folddegeneracy.Ontheotherhand,inlimitofg ≪g ,wehaveaCCmodelwhichhave
t c
asixteen-folddegeneracy. Hence,bytuningofthecouplingg fromzerotoinfinityweexpecttosee
t
aquantumphasetransition.
Sincenon-contractibleloopoperatorsgeneratedegeneracyintheCCandtheTCmodels,itisuseful
toconsiderhowthesubspaceofthegroundstatesintheCCmodelchangesbyaddingtheTCmodel.
To this end, consider non-trivialloop operatorsin the TC and the CC on the hexagonallattice. As
itisshowninfigure(3), weconsiderfournon-trivialloopoperatorsL0,r,L0,r, L0,b andL0,b forthe
x z x z
CCmodelandtwonon-trivialloopoperatorsL0 andL0 fortheTCmodel. Itispossibletodescribe
x z
sixteen-folddegeneratesubspaceoftheCCbytheprojectorswhichareconstructedbynon-trivialloop
operatorsasthefollowingform:
(1+(−1)iL0,r)(1+(−1)jL0,b)(1+(−1)kL0,r)(1+(−1)lL0,b) (18)
x x z z
wherei,j,k,l = {0,1}. LetuscompareabovesubspacewithdegeneratesubspaceoftheTCmodel.
Asitisshowninfigure(3), fournon-trivialoperatorsoftheCC modelonthehexagonallattice can
alsobeconsiderednon-trivialoperatorsoftheTCmodel. Butthereisadifferencethattwooperators
L0,r andL0,bareinthesamehomologyclassintheTCsothattheyconverttoeachotherbyapplying
x x
productofplaquetteoperatorsA whichareinvolvedbythetwonon-trivialloops.Thesamesituation
s
isforoperatorsL0,r andL0,bwheretheyconverttoeachotherbyapplyingproductofplaquetteoper-
z z
atorsB whichareinvolvedbytwonon-trivialloopsL0,r andL0,b. Inthisway,byperturbingtheCC
p x x
modelbytheTCperturbation,homologyclassesofthenon-trivialloopschangesothattwonon-trivial
loopswithdifferentcolorsbelongtothesamehomologyclassandaquantumphasetransitionoccurs.
Itisalsointerestingifweexplainequivalencyofnon-trivialoperatorsintheTCmodelinaanyonic
picture. InfactintheCCmodeltwooperatorsL0,r andL0,b canbeinterpretedasgeneratingoftwo
z z
chargeanyonse and e which turnaroundtorusand annihilateagain. In the CC model, these two
r b
chargeanyonsarenotequivalentandtheycannotfusetogether. Consequentlytheygeneratenewde-
generatestateswhileintheTCmodel,thereisonlyonechargeanyonsothate ande areequivalent
r b
andtheycanfusetogether.
8
Such an interpretation of the quantum phase transition emphasizes on the role of color in the CC
modelsothatwecanclaimthattheCCmodelistopologicallyequivalentwithtwoTCmodelswith
twodifferentcolors.Thisresultisexplicitlyderivedin[36]whereauthorsshowedthattheCCmodel
islocalequivalentwithtwocopiesoftheTCmodel. Inthenextsection,we showthisresultbyex-
plicitanalysisofthequantumphasetransitionwherewe showthequantumphasetransitionhasnot
a topologicalnature. As another point, it is also usefulto emphasize on the role of differentgauge
symmetriesoftheTCandtheCCmodelinthequantumphasetransition. Infact,theTCisrelatedto
theuniversalityclassofthe2DIsingmodel(classical)withZ symmetry,whiletheCCisrelatedto
2
theuniversalityclassofthe2Dthree-bodyIsingmodelwithZ ×Z symmetry[46]. Therefore,itis
2 2
expectedthatsuchaquantumphasetransitioncanbecharacterizedasasymmetrybreakingprocess
andhasnotatopologicalnature.
4 Explicit analysis of the quantum phase transition
Asitwasexplainedintheprevioussection,accordingto[36],becauseofequivalencyofaCCmodel
withtwocopiesofTCmodel,bothmodelshaveauniversaltopologicalphase.Inthissection,wewant
to considerpropertiesof the phase transition betweenthe CC andthe TC modelto understandhow
theuniversaltopologicalphaseofthemrevealsinthepointofquantumphasetransition. Werewrite
Hamiltonian Eq.(16) in a new basis and show that it convertsto many copiesof 1D Ising modelin
transversefieldwhichbelongsto2DIsinguniversalityclass. Finally,weconcludethattheuniversal
topologicalphaseofthosemodelsreflectsin1Dnatureofthephasetransition.
Considerasetofquantumstatesasthefollowingform:
|ψ{i,rjp,ws}i=(1+(−1)iL0Z)(1+(−1)jL0x)Yp (1+(−1)rpBp)Ys (1+(−1)wsAs)|φci (19)
where|φ i = (1+h )|Ωi is thegroundstate ofthe CC model. Using |φ i inabovedefenition
c Qh z c
isnecessarytofind theeffectofoperatorsoftheCC modelonthe state (19). Thevaluesof0,1for
indicesrefertodifferenteigenstatesofthisbasis. Suchabasishasbeenconstructedbytheprojector
operatorsrelatedtoloopoperatorsoftheTCmodel. Itissimpletocheckthatsuchstatesgeneratea
completebasis. Infact,theindicesi,jcharacterizetopologicaldegeneratesubspaceoftheTCmodel
andr s(w s)refertobinaryvariableswhichcorrespondtoeachlight(dark)plaquetteofthelattice,
p s
wecallthemthevirtualspinswhichareequivalentwithanyonicexcitationsoftheTCmodel.Inother
words, the values 0 and 1 for a virtual spin correspond to presence or absence of an anyon in the
relatedplaquettes.
ItisclearthattheoperatorsBpandAsinthisnewbasishaveasimpleform.SinceBp(1+(−1)rpBp)=
t(h−e1s)trapte(1|ψ+i,j(−1)ripBasp|)r,w,rec,o..n.ic|lwud,ewtha,t..B.ipw|ψh{ie,rrjpe,w|rs}iia=nd(−|w1)irpa|rψe{ie,rijpg,ewnss}tia.teWsoefcoapnearlastoorreZproefstehnet
virtualspin{srwp,whisc}hlivei1nt2hecente1rof2eachlightandpdarkplasquetteofthelattice, respectively. By
suchadefinition,SinceBp|r1,r2,...i|w1,w2,...i = (−1)rp|r1,r2,...i|w1,w2,...i,itisclearthatthe
operatorB playsroleofthePaulioperatorZ onavirtualspincorrespondingtolightplaquettep.
p p
ThereissimilarsituationforoperatorsAs. SinceAs(1+(−1)wsAs)=(−1)ws(1+(−1)wsAs),we
wPaiulllihoavpeerAatso|rψZ{i,rjpo,wnsa}ivi=rtu(a−ls1p)iwnsc|ψor{ir,rjeps,pwos}nid.inCgotnosdeaqrukepnltalyq,utehteteosp.eTrahteorreAfosrea,ltshoepTlaCyms roodleeloinf tthhee
s
newbasisiswrittenasthefollowingform:
Hkitaev =−XZp−XZs (20)
p s
ItisclearthattheoperatorsoftheTCmodeldonoteverchangeindexesi,jinthebasis(19).Inother
9
B B B
1 2 3
h h'
A A A
a b c
Figure4: (Coloronline)Left: anoperatorh (h )doesnotcommutewithtwolight(dark)plaquette
x z
operatorB (A ) and B (A ) and it is equivalentwith operatorX X (X X ) on virtualspins in
1 a 2 b 1 2 a b
the center of the light (dark) plaquettes. Similarly, the operatorh (h ) is equivalentwith operator
′x ′z
X X (X X ) on virtual spins living in the center of light (dark) plaquettes which are denoted by
2 3 b c
greencircles(redstatrs). Right: Byapplyingallhexagonaloperatorsinthenewbasis, wewillhave
2N copiesof1DIsingmodelcorrespondingtoeachrowofthelattice.
words,wehavefourdegeneratesubspaceswheretherearethesameformsfortheHamiltonianofthe
TCmodel. InterestingpointisthatthereisthesamesituationintheHamiltonianoftheCCmodel. It
issimpletoshowthatthenon-trivialoperatorsL0 andL0 commutewith alloperatorsh andh in
z x x z
theHamiltonianoftheCCmodel. Consequently,theHamiltonianoftheCCmodelhasalsothesame
forminthefourdegeneratestatesoftheTCmodelsothattheHamiltonianoftheCCcannotgenerate
anytransitionbetweenthosestates.
Byattentiontoaboveargument,weconsiderjustoneofthedegeneratesubspacesoftheTCmodelfor
rewritingHamiltonian(16). Finally,weconsiderfollowingstatesasnewbasis:
|ψ{rp},{ws}i=(1+L0Z)(1+L0x)Y(1+(−1)rpBp)Y(1+(−1)wsAs)|φci. (21)
p s
We are ready to find the new form of the CC modelin above basis. In the CC modelwe have two
operatorsh andh correspondingtoeachhexagonofthehexagonallattice. weshouldfindtheeffect
x z
ofsuchoperatorsonthestates(21).
To this end, consider a hexagon h of the hexagonal lattice as it has been shown in figure(4). The
correspondedhexagonaloperatorh involvessixqubitswhicharecommonwith fewdarkandlight
x
plaquettesoftheTCmodel. Itisclearthattheoperatorh commuteswithalloperatorsA ondark
x s
plaquettes and all operators B on light plaquettes except of two light-plaquette operators B and
p 1
B which have only one joint qubit with the hexagon h, see figure (4, left). Therefore, we have
2
h (1+(−1)r1B )(1+(−1)r2B )=(1+(−1)r1+1B )(1+(−1)r2+1B )h . Sinceh |φ i=|φ i,
x 1 2 1 2 x x c c
weconcludethattheeffectofoperatorh onthestate(21)leadstorisingbinaryvariablesr andr .
x 1 2
SuchaoperationisthesameasoperatorX X onthebasisofvirtualspinsin(21).
1 2
Thesituationisthesameforoperatorsh . Thisoperatordoesnotcommutewithtwodarkplaquettes
z
A andA ofthe TCmodel, see figure(4, left). Therefore,the effectofoperatorh isthe same as
a b z
operatorX X onthebasis(21).
a b
Inthenextstep, consideranotherhexagonalplaquetteinthe neighboroftheplaquetteh, we denote
it by h, see figure(4, left). similar to plaquette h , operator h dose not commute with two light-
′ ′x
plaquette operatorB and B so that it is equalto a operatorX X on virtualspins in (21). If we
2 3 2 3
repeatthisworkforotherplaquettesofthelatticewhichareinthesamerowwithhandh’,wewillhave
anIsingmodelas X X onvirtualspinswhichliveinlight-plaquettesofthecorrespondedrow
P i,j i j
ofthelattice,seefiguhrei(4,right). Thesamesituationisforoperatorh whereitisequaltooperator
′Z
10