Table Of ContentSU(6)Heisenbergmodelonthehoneycomblattice: competitionbetweenplaquetteandchiralorder
PierreNataf,1 Miklo´sLajko´,2 PhilippeCorboz,3 AndreasM.La¨uchli,4 KarloPenc,5,6 andFre´de´ricMila1
1InstituteofTheoreticalPhysics,EcolePolytechniqueFe´de´raledeLausanne(EPFL),CH-1015Lausanne,Switzerland
2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
3University of Amsterdam, Faculty of Science ITF, NL-1090 GL Amsterdam
4Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria
5Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,
Hungarian Academy of Sciences, H-1525 Budapest, P.O.B. 49, Hungary
6MTA-BMELendu¨letMagneto-opticalSpectroscopyResearchGroup, 1111Budapest, Hungary
(Dated:January6,2016)
6
1 WerevisittheSU(6)Heisenbergmodelonthehoneycomblattice,whichhasbeenpredictedtobeachiralspin
0 liquidbymean-fieldtheory[G.Szirmaietal.,Phys. Rev. A84,011611(2011)]. Usingexactdiagonalizations
2 offiniteclusters,infiniteprojectedentangledpairstatessimulations,andvariationalMonteCarlosimulations
basedonGutzwillerprojectedwavefunctions,weprovidestrongevidenceinfavourofthecompetingplaquette
n
state, whichwasreportedtobehigherbutclosebyinenergyaccordingtomean-fieldtheory. Thisisfurther
a
confirmedbytheinvestigationofthemodelwitharingexchangeterm,whichshowsthatthereisatransition
J
betweentheplaquettestateandthechiralstateatafinitevalueoftheringexchangeterm.
5
PACSnumbers:67.85.-d,71.10.Fd,75.10.Jm,02.70.-c
]
s
a
g With the recent progress towards achieving SU(N) sym- ED: Withthestandardexactdiagonalizationapproachthat
- metrywithultra-coldfermionicatoms,1–6 theinvestigationof takes into account all spatial symmetries but only an abelian
t
n theeffectiveSU(N)Heisenbergmodelonvarious1Dand2D subgroup of the SU(N) symmetry group (color conservation
a latticeshasbecomeaveryactivefieldofresearch. Severalre- pluscycliccolorpermutations), thecurrentlylargestaccessi-
u
markablegroundstatepropertieshavebeenreported, includ- ble cluster with a number of sites multiple of 6 (a require-
q
ing long-range color order,7 algebraic correlations,8 transla- ment for having a singlet ground state) is an 18-site cluster.
.
at tionalsymmetrybreakingvalence-bondsolidstatesinwhich The spectrum is shown in Fig. 1(a). The plaquette state is
m groups of N atoms form local singlets on plaquettes,9,10 and expected tobe 3-folddegenerate inthe thermodynamiclimit
chiralgroundstates,suggestedbyHermeleetal.11,12forMott (onestateattheΓpointandtwostatesatthetwoK pointsin
-
d insulatorsonsquarelatticewithseveralparticlespersite. In- theBrillouinzone),butinthe18siteclustertheplaquettescan
n terestingly, a mean-field calculation even predicted a chiral alsowraparoundthetorus,10artificiallyenlargingthenumber
o
spinliquidintheSU(6)Heisenbergmodelonthehoneycomb
c
lattice with only one particle per site.13 However, the rather
[
naturalplaquettestateinwhichsixSU(6)spinsformsinglets 1.2
1 on nonadjacent hexagons was found to lie very close in en- ×6 Ae
v
ergy.Sothisresultcallsforfurtherinvestigationwithmethods 1 ×1 ΓGB2
59 thatgobeyondmean-fieldtheory. ××26 AΓAo2,ΓB1 ×4 KE ×6
9 In this paper, we have attacked this problem with state- 0.8 ×6 Ae ×2
×4 ΓE1,KA2
0 of-the-art numerical methods: exact diagonalizations (ED),
0 infinite projected entangled pair states simulations (iPEPS), GS 0.6
1. and variational Monte Carlo (VMC) simulations based on −E
0 Gutzwiller projected wave functions. As could be expected E 0.4 ×2
6 fromthequasi-degeneracyofthemean-fieldresults,itturned
1 outtobeverydifficulttosolvetheproblem,andallmethods
0.2
: hadtobepushedtotheirlimittoreachadefinitiveconclusion,
v
Xi buteachmethodledindependentlytothesameconclusionthat 0 ×1 ΓB2 18 sit(eas) ×1 24 sit(ebs)
thegroundstateisactuallyaplaquettestate. Thechiralstate
r is not far in parameter space however, and it does not take a 0 6 0 6
a
large ring-exchange term to stabilize it, as demonstrated by C2 C2
EDandVMC.
TheSU(6)HeisenbergmodelisdefinedbytheHamiltonian FIG. 1. Spectrum of the 18- (left) and 24-site (right) clusters as
a function of the quadratic Casimir C . The degeneracies of some
(cid:88) 2
H= P , (1) statesareindicated,aswellasthespatialquantumnumbersforthe
ij
18-site cluster. For the 24-site cluster, the presence of 3 low-lying
(cid:104)i,j(cid:105)
statesisastrongindicationofaplaquettephase(seetextfordetails).
where the operator P = (cid:80) |α β (cid:105)(cid:104)β α | exchanges the Inset: broken-symmetry plaquette state reconstructed from ED. It
ij α,β i j i j
N = 6colorsαandβ oftheatomsonneighboringsitesi,j breaks translations, but the D6 symmetry is preserved. The bond
energyis-0.81(-0.56)forthethick(thin)lines.
ofahoneycomblattice.
2
(a) (b) (c) (d)
FIG.2.(cid:104)P (cid:105)−1/6spin-spin(a)and(cid:104)P P (cid:105)−(cid:104)P (cid:105)(cid:104)P (cid:105)dimer-dimercorrelations(b)intheexactground-stateofthe24-sitecluster.As
0i 12 kl 12 kl
areferencewepresentthedimer-dimercorrelationsofthetranslationalinvariantlinearcombinationofvariational0ππ-fluxprojectedstates
with|t /t |=0.8(c),andofthevariational2π/3-fluxprojectedstate(d). Thepatternofthedimer-dimercorrelationsoftheED(b)andthe
d h
0ππvariationalstates(c)isanindicationoflong-rangeplaquetteordering.
of plaquette coverings to 6. By contrast , the chiral state is −0.9
iPEPS color−ordered state a)
2 × N = 2 × 6 = 12-fold degenerate in the spontaneous iPEPS plaquette state
time-reversal symmetry (TRS) breaking scenario. While the −0.95 VMC chiral state
first three levels ΓB , KA (2×) (plus the symmetry related e VMC plaquette state
level ΓE1 particular2to Ns2= 18) are in agreement with the er sit −1 ED
expectationsforaplaquettestate,10thesestatesareveryclose y p
g
tomanyotherexcitedstates(includingnonsinglets). Soitis ner−1.05
difficulttoprovidestrongevidenceforeitherstateonthebasis E
ofthe18-sitecluster.
−1.1
Togofurther,wehaveusedanewlydevelopedmethodthat
allows one to take advantage of the full SU(N) symmetry,
0 0.02 0.04 0.06 0.08 0.1
hence to work directly in the irreducible representations of 1/D or 1/N
s
SU(N). ForthesingletandthesmallestvaluesoftheCasimir
operator,thisleadstoHilbertspacesofmuchsmallerdimen- b) 0.8 c)
sionthanthestandardapproach.14 Thespectrumisshownin 0.3
0.6
Fig.1(b). Interestinglyenough,on24sites,thespectrumcon-
0.2
sistsof3low-lyingstatesreasonablywellseparatedfromthe m ∆ E0.4
restofthespectrum, thefirstindicationthatthegroundstate
0.1
might have plaquette order. The spin-spin and dimer-dimer 0.2
correlations are shown in Fig. 2. The spin-spin correlations
0 0
decayquitefast,consistentwithsomekindofspinliquid,and 0 0.05 0.1 0 0.05 0.1
the dimer-dimer correlations are consistent with a plaquette 1/D 1/D or 1/Ns
phase on the honeycomb lattice (see for instance the discus-
sionoftheSU(3)caseinRef.[10]). FIG.3. iPEPSresultsfortheSU(6)Heisenbergmodelonthehon-
eycomblattice. (a)Comparisonofthegroundstateenergyobtained
Asanadditionaltest,wehavedeterminedthespatialquan-
withiPEPS,VMC,andED,asafunctionofinversebonddimension
tumnumbersofthefirstexciteddoubletbyapplyingoneofthe
Dandinversesystemsize. TheboldsymbolsmarkimprovedVMC
twoelementarytranslationsofthelattice. Thecorresponding resultsforN = 24and72(seeTableIandmaintext). Forlarge
s
eigenstatesbelongtothetwoK pointsintheBrillouinzone. bonddimensionwithiPEPStheplaquettestatehasthelowestvari-
Thecorrelationsinthesestatesareverysimilartothoseinthe ationalenergy, inagreementwithVMC.(b)Colororderparameter
groundstate,whichsuggeststhatthesethreestatescouldcor- asafunctionofinverseD. Itisfiniteforthecolor-orderedstateand
respondtothedegenerategroundstateofthethermodynamic vanishes for the plaquette state. (c) Difference in energy between
limitsplitbyfinitesizeeffects. Todemonstratethatthisisthe the strongest bond and the weakest bond in the unit cell, which is
case,wehaveconstructedthesymmetricsumofthesestates, stronglysuppressedinthecolor-orderstate,andfiniteintheplaque-
ttestate,consistentwithplaquettelong-rangeorder.Thedottedlines
which corresponds to the finite-size approximation of a bro-
areaguidetotheeye.
ken symmetry state (a simple task since the numerical wave
functions are real and not complex). In that state, the strong
bonds correspond to a covering of the lattice with hexagons
(seeinsetofFig.1(b)),withadifferencebetweenstrongand betweenthegroundstateandthefirstpairoflowlyingstates.
weak bond energies of 0.25, in good agreement with the ex- So, in the next sections, we turn to the results obtained with
trapolatediPEPSestimate(seebelowFig. 3(c)). othermethods.
However,oneshouldnotforgetthatwehaveaccesstoonly iPEPS: AniPEPSisavariationaltensornetworkansatzto
one cluster with the appropriate number of low-lying states, representa2Dwavefunctioninthethermodynamiclimit.15–17
and that the gap to the next levels is comparable to the gap The ansatz on the honeycomb lattice consists of a unit cell
3
of rank-4 tensors which is periodically repeated on the infi- and the artificial fluxes given by their total phase around the
nitelattice, foreachtensoronephysicalindexcarriesthelo- elementary hexagons (plaquettes). An importance sampling
cal Hilbert space of lattice site, and three auxiliary indices Monte Carlo method was used to calculate the energies and
connecttothenearest-neighbortensors. Theaccuracyofthe correlationsoftheprojectedstates.8 Ourcalculations(shown
ansatz can be systematically controlled by the bond dimen- in Fig. 4) reveal that the lowest energy states are similar to
sion D of the auxiliary indices. For the experts we note that those of Ref. [13]: (i) a configuration with uniform 2π/3-
the contraction of the tensor network is performed using a fluxbeforeprojection,correspondingtoachiralspin-liquid28,
variant18,19 ofthecorner-transfermatrixmethod,20,21 andthe and (ii) a translation symmetry breaking configuration with
optimization is done by an imaginary time evolution using a 0-flux in a center plaquette surrounded by π-flux plaquettes
combined simple and (fast-) full update.22,23 To increase the with non-uniform hopping integrals, corresponding to a pla-
efficiencyofthesimulationswemakeuseofabeliansymme- quetteorderedphase.Whilethemean-fieldresultsofRef.[13]
tries.24,25Asimilarapproachhasbeenusedinpreviouscalcu- slightlyfavoredthechiralphase,theplaquette-orderedphase
lationsofSU(N)Heisenbergmodels,seee.g. Refs.8and10. turned out, after projection, to have a slightly lower energy
ForanintroductiontoiPEPSwerefertoRefs.22and23. for all studied system sizes (see Table I), in agreement with
We have used a 6-site unit cell which is compatible with theothernumericalapproaches.
bothaplaquettestateandauniformstate. Asinitialstateswe The energy minimum for the 0ππ-flux states, shown in
startedeitherfromcompletelyrandomtensorsorfromapla- Fig. 4, occurs for t /t ≈ −0.85. Now, for t ≤ −t /2,
d h d h
quettestatemadeofSU(6)singletsonhexagons.Intheformer whichincludestheoptimalenergyvalue,thefermionicwave
case,usingbonddimensionsuptoD =24,anewcompeting functionisgaplessattheFermi-energy: thelowestfilledband
stateappears,inwhicheachsiteintheunitcellexhibitsadif- touchestheemptybandaboveitattheΓpoint.10. So,bycon-
ferentdominantcolor.ForD ≤24thiscolororderedstatehas trast to the plaquette phase of the SU(3) Heisenberg on the
a lower variational energy than the plaquette state, as shown honeycomblattice,whichisdescribedbyagappedfermionic
inFig.3(a). However, theslopein1/D islargerforthepla- wavefunction10,theplaquettephasediscussedhereforSU(6)
quettestate. Sowehavepushedthecalculationtoverylarge corresponds to a gapless spectrum before projection, hence
values of D, up to D = 36. Around D = 30 the energies possibly also to a gapless spectrum after projection. Since
of the two ordered states indeed cross such that the plaque- thisgaplesspointisnotprotected(thespectrumisgappedfor
tte state clearly becomes energetically favored. We have not t > −t /2), we suspect that this is an artefact, and that
d h
found a competing uniform chiral state with iPEPS which is adding additional terms in the fermionic Hamiltonian might
anindicationthatatleastforthebonddimensionsstudiedhere open a gap and further lower the variational energy of that
the plaquette state is the lowest energy state, consistent with state, whichisnotasgoodasthatofthechiralstate(seebe-
theVMCresult. low). However,itmightaswellbethatthespectrumisindeed
InFig.3(b)wepresenttheresultsforthecolor-orderparam- gapless. Thispointdeservesfurtherinvestigation.
eterofthetwocompetingstates,givenbythelocalmoment
N 24 24opt 72 72opt 288 meanfield13
(cid:118) s
m=(cid:117)(cid:117)(cid:116)65(cid:88)(cid:18)(cid:104)Sαβ(cid:105)− δα6β(cid:19)2, (2) p2lπaqcuheirttael --11..0003694 --11.0.015074 --11..00007797 --11..00102837 --11..00008727 --11..001205
α,β 3
averaged over all sites in the unit cell, where Sβ = |α(cid:105)(cid:104)β|
α TABLEI.VMCenergiesofGutzwillerprojectedwavefunctionsfor
aretheSU(6)spinoperatorsandα,β runoveralllocalbasis
thecompeting0ππ(plaquette)andthe2π/3fluxconfigurationsfor
states. For the color-ordered state m is large for low D. It
different system sizes. The statistical error of the calculations is
decreaseswithincreasingD buttendstoafinitevalueinthe smallerthanO(10−4).Theoptimizedenergiesareobtainedbycon-
infinite D limit. The local moment of the plaquette state is sideringtheoverlapbetweenprojectedstateswithdifferentboundary
much more strongly suppressed with increasing D, and van- conditionsbeforeprojection.
ishes in the large D limit, consistent with a singlet without
colororder. Ring exchange term: Since the energy difference between
Figure 3(c) shows the difference between the highest and theplaquetteandchiralphasesfoundbyVMCisverysmall,
lowestbondenergyintheunitcellwhichmeasuresthemag- itistemptingtospeculatethatthechiralphasemightbestabi-
nitude of the plaquette order. For the color ordered state it lizedbyaringexchangetermaroundthehexagons. Wehave
is strongly suppressed with increasing D and vanishes for thusconsidered
large D, in contrast to the plaquette state which exhibits a H=cosθ(cid:88)P +sinθ (cid:88) i(cid:0)P −P−1(cid:1) (3)
largedifferenceinbondenergy,wherethestrongbondsform ij
hexagonalplaquettes. (cid:104)i,j(cid:105) plaquttes (cid:55) (cid:55)
VMC: Gutzwiller projected wave functions26,27 offer a wherethesuminthesecondtermrunsoverallhexagonalpla-
qualitative and potentially quantitative description for both quettes,andtheoperatorsP andP−1permutetheconfigu-
typesofcompetingscenariosfoundbymean-fieldstudy.13 In rationonahexagonclockwiseandanticlockwise(alsocalled
(cid:55) (cid:55)
thismethodweprojectouttheconfigurationshavingmultiple ring exchange terms). The new term directly couples to the
occupancyfromtheFermi-seaconstructedfromamean-field scalar chirality on the hexagons, breaks time-reversal invari-
model.Thevariationalparametersarethehoppingamplitudes ance, and is a bona-fide SU(6) generalization of an SU(2)
4
-0.8 -0.98 (c) ofbehaviorbetweenthesmallθrange,withatwofoldexcited
72 288 0 0
-0.85 -1 0ππ0π0 0(π) 0(π) 0(π) state well separated from the rest of the spectrum, and the
πππ 0 0 0 range above θ (cid:39) 0.2, where a manifold of 6 singlet states
E/Ns -0.9 -1.02-1.2 -1 -0.8 φ0φ0φ0 00(π) 0(π) 00(π) breesctoomfethseaslmpeocsttrudmeg.eTnweroaoteftahnedsevestraytewsealrlesaetptahreatΓedpofrionmt,atnhde
-0.95
theremainingfourareattheK points,inagreementwiththe
(d)
-1 π π momentaofthesixchiralVMCstates(discussedbelow). So,
0(π)
(a) 0(π) 0(π) the ED results are clearly consistent with a phase transition
-1.05 π π π
0.5 (b) 〈〈PPhd〉〉 π00 0ππ πππ 000 0π(π) 0(π) 0π(π) btheetwrienegneaxcphlaaqnugeettteerpmh.asNeoatendthaatcthhieradlepgheanseeraucpyonofinthcerecahsiirnagl
stateisonlyequalto6andnot12becausetheHamiltonianof
P〉d 0 (e) Eq.(3)explicitlybreaksthetimereversalsymmetry.
P, 〉〈h 2π/3 2π/3 2π/3 This interpretation is further supported by the comparison
〈 -0.5 2π/3 withVMCon24sites.Toaccessthelowenergyspectrumand
2π/3 2π/3
not just the ground state, we have constructed a large family
2π/3
-1 ofGutzwillerprojectedstatesbychangingtheboundarycon-
-1.5 -1 -0.5 0 0.5 1 1.5 -tthh ttddei2π/3 ditions (BC) of the fermionic wave-functions,32 considering
td/th up to 30 differentBCs for the 2π/3 flux states, and upto 90
forthe0ππ-fluxstates(30foreachtranslationbreakingstate),
FIG. 4. Energies of Gutzwiller projected wave functions (a) and andwehavediagonalizedtheoverlapmatrixandtheHamilto-
thebondenergiesontd andth bondsafterprojectionforthediffer- nianinthisvariationalsubspace.33,34 Theresultsaresumma-
entfluxconfigurationsasafunctionoft /t forN = 72. (c)-(e)
d h s rized in Fig. 5. For the chiral state, this parton construction
showstheconsideredfluxconfigurations,blackbondsrepresenthop-
leads to 6 (and only 6) significant eigenvalues of the overlap
pingamplitudet ,whiledarkandlightpurplebondsdenotehopping
d matrix,whichthemselvesleadto6low-lyingstatesveryclose
amplitudes t and −t , respectively. In case of the uniform 2π/3
h h in energy35, while for the plaquette states, there is not such
fluxconfiguration,redarrowsrepresentcomplexhoppingamplitude
∝ei2π/3,forwhicht =t∗. a clear cutoff, and the three low-lying states are not so well
ji ij
splitfromtheotherstates. Althoughthevariationalplaquette
andchiralstatesarehigherinenergy,theiroverallbehavioris
qualitatively consistent with ED. In particular, the energy of
-1 0.1 theplaquettestateisminimalatθ =0,whilethatofthechiral
λ states is minimal around θ = 0.36, and their energies cross
OV10-3
aroundθ =0.16.
10-5 SimilaroverlapcalculationswerecarriedoutforNs = 72
θE()/Ns-1.05 2x 5 10 15 20 sfoitrese,acwhitthra3n0sladtiifofenrebnretaBkCinsgfsotrattehe(326πi/n3tofltuaxl)cfoasret,hean0dπ1π2-
fluxcase. Theenergycorrectionsforthe0ππcaseturnoutto
belarger(seeTableI),againpromotingtheplaquetteordered
2x
-1.1 NNs==2244 02πππ/3 phaseoverthechiralliquidphaseattheHeisenbergpoint.36
Ns=24, ED Discussion: Tosummarize,thenumericalevidenceclearly
s
0 π/8 π/4 pointstoaplaquettegroundstatefortheSU(6)modelonthe
θ honeycomb lattice, but with a chiral phase close by in pa-
rameter space. Even if it led to the wrong conclusion, the
FIG.5. ComparisonoftheEDspectrum(blackpoints)ofthemodel mean-fieldapproachshouldbegivencreditforidentifyingthe
ofEq.(3)withthevariationalenergies(continuouslines)basedon right candidates with very similar energies.13 This lends fur-
Gutzwillerprojectedwave-functionsforthe0ππplaquettephaseand thersupporttothemean-fieldpredictionbyHermeleetal.11,12
the2π/3chiralphase.Theinsetshowstheeigenvalues(λ )ofthe ofachiralphaseforseveralparticlespersitesincetheredoes
OV
overlapmatricesoftheprojectedstateswithdifferenttwistedbound- not seem to be competing VBS states too close in energy in
aryconditionsbeforeprojection. thatcase. Numericalworkalongthelinesofthepresentpaper
totestthispredictionisinprogress.
Acknowledgements: This work has been supported by the
Hamiltonian on the kagome lattice which has been shown to Swiss National Science Foundation, the JSPS KAKENHI
give rise to an extended SU(2) chiral spin liquid phase29,30. GrantNumber2503802, bytheHungarianOTKAGrantNo.
Alternatively it can be viewed as a drastically truncated ver- K106047, by the National Science Foundation under Grant
sionofaparentHamiltonianforaSU(N)chiralspinliquid31. No. NSFPHY11-25915,bytheAustrianScienceFundFWF
In the following, we will discuss the properties of that (F-4018-N23 and I-1310-N27) and by the Delta-ITP consor-
model as a function of θ, noting that θ = 0 corresponds to tium(aprogramoftheNetherlandsOrganisationforScientific
thepureHeisenbergmodel(1). Research(NWO)thatisfundedbytheDutchMinistryofEd-
TheEDspectrumon24sites(Fig.5)showsaclearchange ucation,CultureandScience(OCW)).
5
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