Plaquette valence-bond ordering in J_1-J_2 Heisenberg antiferromagnet on the honeycomb lattice

0.74 MB·

by  H. Mosadeq

Plaquette valence-bond ordering inJ1 −J2 Heisenberg antiferromagnet onthe honeycomb lattice
H. Mosadeq∗,1 F. Shahbazi†,1 and S.A. Jafari‡1,2,3
1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
2Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran
3School ofPhysics, InstituteforResearchinFundamental Sciences(IPM),Tehran19395-5531, Iran
(Dated:February1,2011)
WestudyS =1/2Heisenbergmodelonthehoneycomblatticewithfirstandsecondneighborantiferromag-
neticexchange(J1−J2 model),employingexactdiagonalizationinbothSz = 0basisandnearestneighbor
1 singlet valence bond (NNVB) basis. Wefind that for 0.2 < J2/J1 < 0.3, NNVB basis gives a proper de-
1 scription of the ground state in comparison with the exact results. By analysing the dimer-dimer as well as
0 plaquette-plaquettecorrelationsandalsodefiningappropriatestructurefactors,weinvestigatepossiblesymme-
2 trybreakingstatesasthecandidatesforthegroundstateinthefrustratedregion.Weprovidebodyofevidences
n infavorofplaquettevalencebondorderingfor0.2<J2/J1 <0.3. ByfurtherincreasingtheratioJ2/J1,this
a stateundergoesatransitiontothestaggereddimerizedstate.
J
PACSnumbers:75.10.Jm75.10.Kt,75.40.Mg
1
3

I. INTRODUCTION latticegeometryissuchthat,itisnotpossibletominimizethe
]
l interactionenergyofallbondsatthesametime,e.g. intrian-
e
- gularorKagome´latticein2Dandpyrochlorelatticein3D[6].
Quantumspinliquid(QSL)isnonmagneticstateofacor-
r (ii)Whenthereareseveralcompetingexchangeinteractions,
t related matter for which there is no broken symmetry in the
s suchascompetitionbetweenfirstandsecondneighboringAF
. spin part of the ground state wave function. Hence the lo-
t exchange interactions (J − J model). Since the quantum
a cal magnetic moments remain disordered down to absolute 1 2
m fluctuationsare largerin 2D, manyattemptsto find QSL are
zero(T = 0)[1]. The quantumgroundstate forQSL can be
focusedonthequasitwo-dimensionalfrustratedspinsystems
- expressed as the superposition of many different configura-
d withS =1/2[7].
tions, such as linear combinations of the short range singlet
n Two dimensional Heisenberg antiferromagnetsapart from
valence bonds. This state is called resonating valence bond
o their own importance [4], received intensive attention in the
(RVB), originallyproposedby Fazekasand Andersonas the
c
context of layered high-T superconducting (HTSC) mate-
[ ground state of the Heisenberg model on the triangular lat- c
rial[3]. ThegroundstateofS = 1/2Heisenbergmodelwith
tice [2]. The singlet bonds in RVB state can be considered
2 AFnearestneighbor(NN)exchangecouplingon2Dbipartite
aspre-formedCooperpairs,whichundersuitableconditions(
v
latticeshasbeenshowntobeNe´elordered[8–14]. Addition
i.e. hole doping) may coherently propagate throughout the
7
ofnextnearestneighbor(NNN)AFinteractionsfrustratesthe
2 system,hencegiverisetosuperconductivity[3].
systemandgraduallydestroysthe(Nee´l)order. SinceAFex-
1 Many strongly correlated systems are well described by
0 changeinteractionencouragesthesingletformation,thequan-
Hubbard Hamiltonian whose ground state for large on site
. tumgroundstateofAFHeisenbergmodelcanbeexpressedin
7 coulombinteractionis theMottinsulatingstate. Inthisstate
termof overcompletesetof valencebond(VB) basiswhich
0 the electrons are localized on the atoms, nevertheless local
representatotalspinsingletstate[15].
0
charge fluctuations induce an Anti-ferromagnetic (AF) ex-
1 The J − J AF Heisenberg model on square lattice has
changeinteractionbetweenthespinsoftheelectrons. Hence, 1 2
: beenextensivelystudiedandvariousVBstateshavebeenpro-
v AFHeisenbergmodelisaneffectiveHamiltonianfordescrib-
i ingthelowenergyexcitationsoftheMottinsulators[4].Ithas posed to describe its disorder regime [16]. One example of
X
suchquantumstatesisnearestneighborRVB(NNRVB)rep-
been proved that in 1D, the Ground state of the Heisenberg
r resentingaspinliquid,whichbreaksneithertranslationalnor
a model is a gapless (critical) spin liquid for S = 1/2-chain,
rotationalsymmetries. However, in highly frustrated regime
while it is gapped spin liquid for S = 1-chain [5]. Finding
wherethegroundstateisclassicallydisorderedandtheSU(2)
therealizationsofQSLintwoandthreedimensionshasbeen
symmetry of the Hamiltonian is restored, there is no theo-
thesubjectofmanyresearchesinrecentyears[1]. Quantum
rem to prevent breaking of the lattice translational symme-
fluctuationsaswellasfrustrationmaydestroythelongrange
try. Therefore, in spite of earlier proposal of states with no
magneticorderinspinsystems. Whenaspinsystem isfrus-
symmetrybreaking[17],stateswhichbreaktranslationalsym-
trated, itcan notfind a spinconfigurationto fullysatisfy the
metrywereproposed[18–20]. Oneexampleisthestaggered
interactionbetweeneach pairof spins. Thereare two mech-
dimerizedstatewhichbreakstranslationalandrotationalsym-
anismsforfrustration: (i) Geometricalfrustration,wherethe
metries of the lattice. Another candidate, that has been pro-
posed recently for some lattices, is plaquette RVB (PRVB)
wave functions in which the resonance of VBs is limited to
one plaquette [21–25]. PRVB state breaks the translational
∗[email protected]
†[email protected] symmetry,whilepreservestherotationalsymmetryofthelat-
‡[email protected] tice.
2
tweenthesemi-metallicphasecharacterizedbymasslessdirac
fermions(U/t < 3.4) and the AF-Mott insulating phase for
U/t > 4.3 [32]. The other is the experimentally observed
spin liquid behavior in BMNO which remains magnetically
disordereddowntoT = 0.4K,inspiteofitshighCuri-Weiss
temperatureT ≈−257K[29].
CW
Motivatedbytheaboveconsiderations,inthispaperwein-
vestigate the ground state properties of J −J Heisenberg
1 2
modelwhichisproposedforexplainingthespinliquidbehav-
iorofBMNO.Thepaperisorganizedasfollows.Insection.II
weintroducethespinHamiltonianandusingdiagonalization
innearestneighborVBbasis,wefindevidenceforspinliquid
phaseforarangeofcouplingconstants.Insection.IIIweem-
ploytheexactdiagonalizationinfullHilbertspaceofS =0.
z
FIG.1:Thebipartitehoneycomblattice.Twosublatticesaremarked With exact wave-functions obtained in this manner, we cal-
byblackandwhitecircles. Nearestneighborlatticepointsarecon- culatethedimer-dimercorrelationfunction. Wefindtwodif-
nectedwithsolidlinesandnexttonearestneighborlatticepointsare ferentquantumphasesin frustrate regime. In section. IV by
connectedwithdashedlines.Redarrowsshowthetwoprimitivelat- introducing suitable structure factors, quantum phase transi-
ticevectors. tionpointisdetermined. Attheend,insection.V,wecalcu-
lateplaquette-plaquettecorrelationswhichpointstoapossible
PRVBstate. Section.VIisdevotedtodiscussionandconclu-
Recent fabrication of graphene monolayer and also mag- sion.
netic compounds with quasi 2D honeycomb structure, has
brought the honeycomb lattice to attention of the physicists
frombothexperimentalandtheoreticalpointsof view. Hon- II. MODELHAMILTONIANANDITSGROUNDSTATE
eycomblatticedoeshavecoordinationnumberequaltothree, CANDIDATES
which is minimum among two-dimensional lattices. In the
case of Heisenberg model on honeycomb lattice, the small
J −J AFHeisenbergHamiltonianisdefinedby,
1 2
number of neighboring interactions enhances the quantum
fluctuationsandthereforeseemstobepromisingsystemtoex-
plorespinliquidstates.Honeycomblatticeisabipartitelattice H=J1XSi.Sj +J2 X Si.Sj, (1)
composed of two interlacing triangular sublattices (Fig. 1). hi,ji hhi,jii
Theunitcellofthisnon-bravaislatticecontainstwositesand
in which J > 0 and J > 0 are AF exchange interactions
lattice is constructed by two lattice vectors of the triangular 1 2
betweenfirstandsecondneighboringspins,respectively.The
bravaislattice. Thenon-bravaischaracteroflattice resultsin
first sum is limited to NN sites, while the second sum runs
more exotic aspects that can not be seen in square lattice or
over the NNN lattice sites. Since the square lattice is con-
theotherbravaislattices[26].
nectedwithhighT superconductingmaterials,thestudiesof
c
As some realizations of Heisenberg magnets on the hon- frustrated phases of spin models has been usually limited to
eycomb lattice, one can name recently discovered com- thislattice. Recentlydiscoveredmagneticmaterialswithun-
pounds such as InCu2/3V1/3O3 [28] and Na3Cu2SbO6 [27] derlyinghoneycombgeometryisourmotivationtostudythe
in which the Cu+2 ions in the copper-oxide layers form a abovemodelonthehoneycomblattice.
two-dimensionalS = 1/2 Heisenberg antiferromagneton a Using effective action approach to the frustrated Heisen-
honeycomblattice, Bi3Mn4O12(NO3)(BMNO)inwhichthe berg antiferromagnet in two dimensional system, Einarsson
Mn+4ionswithS =3/2resideonthelatticepointsofweakly et.al.provedthattheS = 1 disordergroundstateonthehon-
coupledhoneycomblayers[29]. ReplacingMn+4 with V+4 eycomblatticeisthree-fold2degenerate[33]. Fouetet. al.[34]
inthiscompoundmayrealizetheS =1/2Heisenbergmodel provided some evidence for staggered dimerized (SD) state
onhoneycomblattice. Alsotherecentprogressinthefieldof forS = 1/2atJ /J = 0.4. Suchstatebreakstherotational
2 1
ultracold atoms and trapping techniques [30] along with the lattice symmetry(C ), while preservesits translationalsym-
3
abilitytotunetheinteractionparametersviatheFeshbachres- metry (Fig. 2). They also speculated that spin liquid phase
onance[31]canbethoughtofanotherwaytorealizeHeisen- andPRVB phasescanbestabilizedforsomerangeofJ /J
2 1
bergspins(oflocalizedfermions)onahoneycombopticallat- (Fig 3). PRVB phase breaks the translational symmetry, but
tice. preserves the rotational symmetry of lattice. Alternatively,
Two recent achievements has raised the hope of finding ReadandSachdev[35]usedlargeN expansionmethod,pro-
QSLinhoneycombgeometries. One,isthelargescalequan- posedanothergroundstatewavefunctionwhichbreaksboth
tumMonteCarlosimulationofthehalf-filledHubbardmodel the translational and the rotational symmetries of the lattice
on the honeycomb lattice, which results a spin liquid phase (Fig4).
with finite spin gap for moderate values of on-site coulomb In the classical limit (large spins), it has been shown that
interaction (3.5 < U/t < 4.3). This phase is located be- the ground state of the above model is Ne´el ordered for
3
FIG.2: Threedegeneracyofstaggereddimerizedwavefunctionon FIG.4: Threedegeneracy of wavefunction proposed byReadand
honeycomblattice. Sachdevonhoneycomblattice.
thatthegroundstateoftheJ −J Heisenbergmodelinthe
1 2
frustratedregime,wherethereis nolongrangeorder,canbe
verywellapproximatedintermsofstatesinNNVBsubspace.
Let us expand the ground state wave function in terms of
NNVBstatesas
|ψ0i=Xa(cα)|cαi, (2)
α
where|c idenotesallpossibleconfigurationsαofNNVBs:
α
|cαi= Y (i↑j↓−i↓j↑). (3)
(i,j)∈α
FIG.3: ThreedegeneracyofPlaquetteValenceBondwavefunction
onhoneycomblattice. First, we haveto enumeratethe basis |cαi to constructa nu-
merical representation of the Hamiltonian matrix in this ba-
sis. To determinethe basis, the exactPfaffian representation
J /J < 1/6 while for J /J > 1/6 the groundstate con- of the RVB wave function is employed[39]. In this method
2 1 2 1
sists of an infinitely degenerateset of spiral states character- oneexpressestheRVBwavefunctionasthePfaffianofanan-
izedbyspiralwavevectorsq[36]. Okumuraetal[37]useda tisymmetric matrix whose dimension is equalto the number
combinationof low temperatureexpansionand Monte Carlo of lattice points. The NNVB basis is much smaller than the
simulationshowedthatthermalfluctuationscanliftthehuge whole Sz = 0 basis, so that the Hamiltonian matrix can be
degeneracy of the ground state, leading to a state with bro- fully diagonalized with standard library routines. Note that
ken C3 symmetry of honeycomb lattice. According to their sincetheNNVBstates(|cαi)arenotorthonormal,oneneeds
finding, in the vicinity of Nee´l phase boundary, the energy tosolvethegeneralizedeigen-valueproblem
scale associated with the thermalorder-by-disorderbecomes
det[H−EO]=0,
so small that exotic spin liquid behavior, such as ring-liquid
or pancake-liquid can emerge. Mulder et. al. argued that
whereO = hc |c idenotestheoverlapmatrixoftheNNVB
taking the quantum fluctuations into account, some specific β α
configurations.
wave vectors in this manifoldare picked as the groundstate
IntheupperpanelofFig.5wehavecomparedgroundstate
–amanifestationoforderbydisordermechanism. Theyfind
energiesobtainedinthe NNVB basis, andthose obtainedby
for S = 1/2 quantum fluctuations are strong enough to de-
numerically exact diagonalization in the S = 0 basis ver-
stroythespiralorderandstabilizethevalencebondsolidwith z
susJ /J . Inthemiddlepanelwe showtherelativeerrorin
2 1
staggeredordering[38]. Ouraiminthispaperistostudythe
the groundstate energyand the lower panel showsthe over-
groundstateofmodel(1)usingexactdiagonalizationinboth
lap of the exact ground state wavefunction with the ground
S andnearestneighborvalencebond(NNVB)basis.
z state obtained within NNVB basis set. As can be seen in
this figure, the agreement between the two sets of energies
for J /J ∈ ]0.2,0.3[ is remarkable. Since the NNVB ba-
2 1
III. DIAGONALIZATIONINNNVBBASIS sis is not complete, the large error obtained by NNVB ba-
sis for J /J < 0.2 and J /J > 0.3 can be attributed to
2 1 2 1
The valence bond states are a subset of S =0 basis with the fact that longer range valence bonds start to contribute.
z
totalspinmagnitudeS2equaltozero.Inthissectionweshow For J /J < 0.2, where there is Nee´l order in the ground
2 1
4
1.6
J=0.20
2
-0.3 1.4 J2=0.40
NNVB Basis
-0.35 Sz=0 Basis 1.2
1
E/N0 -0-0.4.45 P(w) 00..68
0.4
-0.5
0.2
-0.55
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1 2 3 4 5 6
J /J w
2 1
FIG.6:ProbabilitydistributionoftherelativeamplitudesofVB’sin
0.14
ED|0 0.12 Nthe=gro5u4n.dstatewave-functionforJ2/J1 =0.2,0.4andclustersize
E
V|/| 0.1
B
N 0.08
N
D-E0 0.06 plitude.Thusitcanberepresentedas:
E
E0 0.04
| |RVBi=AX|cαi, (4)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 α
J /J
2 1
in which A = 1 is the normalization coef-
1 ficient. In order(tPoαg,αe′thpcαre|clαim′i)in1/a2ryinsight into the nature of
> 0.8 the ground state obtained in NNVB basis, we compare the
B
NV distributionoftheamplitudesofVBconfigurationsineq.(2)
N 0.6
ψ0 to the uniform amplitude A of the above RVB liquid state.
ED| 0.4 Forthispurposewedefinetheratiow(c ) = a(cα),andlook
ψ0 α A
< at the distributionof relative weights, w. If the groundstate
0.2
hasthe characteristicsof a RVB spin liquid, this distribution
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 is expected to be sharply peaked around w = 1. In Fig. 6
J2/J1 weplottheprobabilitydistributionfunction(PDF)oftherel-
ative amplitudes w for J /J = 0.2 and J /J = 0.4 in a
2 1 2 1
FIG.5: Up: Thecomparisonbetweentheexactgroundstateenergy
cluster of N = 54 spins. As can be seen in this figure, for
evaluatedusingexactdiaganolazitioninSz=0basis(squares)anddi- J /J = 0.2 the PDF is narrower relative to J /J = 0.4,
agonlizationinNNVBbasis(circles)asafunctionofJ2/J1.middle: w2hich1impliesthatthegroundstateforJ /J =20.21ismore
Therelativeerrorsbetweenthegroundstateenergiesobtainedbythe 2 1
twobasissetsdefinedas(ENNVB−EED)/(EED).Down:Theover- similartospinliquidstate. Thebroaderdistributionofampli-
0 0 0
lapoftheexactGSwavefunction|ψ0EDiwiththeGSwavefunction tudesforJ2/J1 >0.4ontheotherhand,indicatessignificant
obtainedinNNVBbasis|ψNNVBi. deviation from QSL behavior, which can be considered as a
0
signofsymmetrybreaking. Toinvestigatetheeffectoffinite
latticegeometryonthegroundstate,wecomparePDFsofrel-
state, it was shown that long-rangedVB states have remark-
(a) (b)
ablecontributioninthegroundstatewavefunction[40].Start-
1.6 1.8
iinnggffrruosmtraJt2io/nJ1str=en0g,ththuepNtoee´Jl2o/rJd1er≈is0d.e2s.trAoytethdisbypoininctr,etahse- P(w) 0011 ....16824 JJ22==00..3250 P(w) 00111 .....168246 JJ22==00..3250
spin-spin correlations will become short ranged and the na- 0.4 0.4
0.2 0.2
tureofthegroundstatecanbeaccuratelycapturedbyNNVB 0 0 1 2 3 4 5 0 0 1 2 3 4 5
wave functions. For J /J > 0.3, the frustrating second w w
2 1
neighborAFexchangecouplingJ inducessingletformation (c) (d)
2 0.6 1.4
bwsueifttfiwhceNieeNnntVnteBox’ctsa.npTetauhrreeersettfhonereetirgiunhebtghorirsosur(enNgdNiosNtnaVttehB.e’FNsu)NrwtVhheBircmbhoacsroeis,miusppeiontne- width 000 ...00345..55545 NN==5504 skewness 01..182 NN==5504
0.3 0.6
increasing J /J beyond0.35, as will be shown shortly, the 0.2 0.24 0.28 0.32 0.36 0.2 0.24 0.28 0.32 0.36
2 1 J2/J1 J2/J1
nearest neighborsinglets in the VB states (dimers) will start
tobecomecorrelated.
FIG.7: (Color online) PDF of relativeamplitude w for J2/J1 =
The RVB wave function (QSL) is defined as linear super- 0.2,0.35for(a)N = 50and(b)N = 54. (c)Widthand(d)skew-
positionofallpossibleVBconfigurationswiththesameam- nessofPDFvs.J2/J1forN =50,54.
5
Trialstate ψSD ψPL ψRS
hPαiavg 0 -0.106(5) 0
α′ β γ δ β γ δ β γ δ
hPαPα′i1 +1/4 +1/4 +1/4 - - - - - - -1/2 1 -1/2 +1/4 -1/2 1
hPαPα′i2 +1/4 +1/4 -1/2 - - - - - - +1/4 +1/4 -1/2 1 -1/2 +1/4
hPαPα′i3 +1 +1 -1/2 - - - - - - -1/2 +1/4 +1/4 +1/4 +1/4 +1/4
hPαPα′iavg 1/2 1/2 -1/4 - - - - - - -1/4 +1/2 -1/4 +1/2 -1/4 +1/2
C(α,α′) 1/2 1/2 -1/4 -0.090(5) 0.170(8) -0.090(5) 0.170(8) -0.090(5) 0.170(8) -1/4 +1/2 -1/4 +1/2 -1/4 +1/2
TABLEI:hPαPα′i−hPαi2 forαfixedandα′ = β,γ,δ (Fig.8). Threeindices1,2,3refertothreedegeneratestates(cf. Figs.2,3,4)
which become orthogonal toeach other in thethermodynamic limit for pure SD and RS states. Subscript avg denotes average over these
threepossibledegeneracies. SincethethreedegeneratePLstatesarenotorthogonaltoeachotherinthethermodynamiclimit,inthiscasethe
correlationshavebeenextractedfromfinitesizescalingofthenumericdata.Digitsintheparenthesisdenoteerrorsinthelastdigit.
ativeamplitudesfortwo cluster sizes N = 50 andN = 54, dimersfor0.2 . J /J . 0.5. Thedimer-dimercorrelation
2 1
and J /J = 0.2 and J /J = 0.35. These are shown in isdefinedby,
2 1 2 1
the two top panels of Fig. 7. It can be easily seen that for
the lattice size N = 50, the PDFs are more symmetric than C(α,α′)=4(h(Si.Sj)(Sk.Sl)i−h(Si.Sj)i2), (5)
N = 54. Thiscanbeattributedtothefactthat,incontrastto
whereα′ =(k,l),andα=(i,j)isthereferencebondrelative
the cluster with N = 50 lattice points, the plaquette wave
towhichthecorrelationsarecalculated. Definethepermuta-
function can be fitted to the cluster with N = 54 subject
tionoperatorby,
to the periodic boundary condition. Therefore the symme-
try breakingtowardplaquette formationis morepronounced 1
P =2(S .S )+ , (6)
forN = 54. Thissignalsatendencyforplaquetteformation, kl k l 2
provideditiscompatiblewiththelatticegeometry.Moreover
intermsofwhichEq. (5)canbealternativelyexpressedas
forN =54,increasingthevalueofJ leadstoemergenceof
2
asecondpeakattherighttailofPDF.Theaverageofrelative C(α,α′)=hPαPα′i−hPαihPα′i. (7)
amplitudesof VB configurationstaking partin the plaquette
wavefunctionturnsoutto be1.56. Interestingly,positionof In table I quantities C(α,α′) for fixed α and α′ = β,γ,δ
the second peak is quite close to this values. Therefore, the (Fig. 8) are shown for three trial wave functions ψ , ψ
SD PL
secondpeakcanbeattributedtothe amplificationofplaque- and ψ , where RS, SD and PL stands for Read-Sachdev,
RS
tte characterinthe groundstate, asa resultofincreasingthe staggered dimerized, and plaquette states, respectively. The
second neighbor exchangeinteraction. To quantify the vari- expectationvaluesoftheoperatorhPα′iforeachofthethree
ations of PDF (p(w)) versus J2/J1, we compute its width degenerate SD and RS states is −1 if bond α′ is occupied
given by the standard deviation (σ = h(w −hwi)2i1/2) and by a dimer, and +1/2, otherwise. hPα′iavg in table I is ob-
also its skewness defined by h(w−hwi)3i. Panel (c) of Fig. 7 tained by averagingoverthree degeneratestates correspond-
3
σ ingtoSDandRStrialwavefunction. ForPLtrialstate,since
is plot of width versusJ /J for N = 54, showing that the
2 1
thethreedegenerateconfigurationsarenotorthogonalinther-
width ofPDF increasesmonotonicallyfromJ /J = 0.2 to
2 1
modynamiclimit, theaveragingisnotvalidandwecompute
J /J = 0.4. Thepanel(d)inthisfigure,representstherise
2 1 C(α,α′) numerically by finite size scaling method. Fig. 9
of PDF skewness for N = 54 in terms of J /J , indicating
2 1
givesagraphicalrepresentationofthecorrelationsobtainedin
thatbyincreasingJ thedistributionfunctionsforthissizeget
2
thisway. Redandbluelinksdenotepositiveandnegativecor-
moreandmoreasymmetric,whileforN =50skewnessdoes
relations,andtheirthicknessisproportionaltothemagnitude
notchange remarkably. This suggeststhat the skewness can
ofcorrelationswiththereferencebondαofFig.8. Therefer-
be consideredas a heuristic indicationof possible symmetry
encebondαisdenotedbyadoublelineinFig.9. Itcanbe
breaking. In summary, since the widths of PDFs considered
herearefinitefortheinterval0.2.J /J .0.35,theabove
2 1
argumentssuggestthatthegroundstate ofJ −J modelin
1 2
thisintervalisnotaperfectQSLstate. Amorepreciseunder-
standingofthegroundstateproperties,requiresthestudythe
correlationbetweendimers.Thiswillbedoneinthefollowing
section.
IV. EXACTDIMER-DIMERCORRELATIONS
Inthissection,employingexactdiagonalizationweobtain
the ground state in S = 0 basis. Using the exact wave
z
functionofthegroundstate,wecalculatecorrelationbetween FIG.8:Thereferencebondα,andthreeindependentbondsβ,γ,δ.
6
(cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)
(cid:1)(cid:5)(cid:3)
FIG.9: Snapshotsofcorrelationscorrespondingfromlefttorightto
(a)RS,(b)PL,and(c)SDtrialstates.Redandbluelinkscorrespond
topositiveandnegativecorrelationswithrespecttoreferencebond
αwhichhasbeendenotedbydoubleline.
FIG. 11: The dimer-dimer correlation for honeycomb lattice with
periodicboundary condition atJ2 = 0.4. Red(Blue)linesdenote
positive(negative)correlation.Thethicknessoflinesisproportional
tothemagnitudeofcorrelations.ThesystemsizeisN =32.
tions,whiletheblueonesindicatenegativecorrelations. The
thicknessofbondsareproportionaltothemagnitudeofcorre-
lations.AscanbeseeninFig.10,forJ /J =0.3,thecorre-
2 1
lationsare decayingwith distance (measuredwith respectto
centralbond). ComparingthiscorrelationmapwithFig.9(b),
showsremarkablesimilaritytothesnapshotcorrespondingto
PLordering.
Thedimer-dimercorrelationpatternatJ /J =0.4shown
2 1
in Fig. 11 is entirely different. First the dimer-dimer corre-
lationsdonotappreciablydecayoverthemaximumdistance
displayed in the figure. Second, one can easily distinguish
FIG. 10: The dimer-dimer correlation for honeycomb lattice with a staggered ordering pattern by noting that, correlations be-
periodicboundary conditionatJ2 = 0.3. Red(Blue)linesdenote tweenbondsparalleltothereferencedimerarepositive,while
positive(negative)correlation.Thethicknessoflinesisproportional others are negatively correlated with the reference dimer.
tothemagnitudeofcorrelations.ThesystemsizeisN =32.
ComparingwithFig.9(c),thiscorrelationsnapshotobviously
suggestsastaggereddimerizedstate atJ /J = 0.4. Thisis
2 1
inagreementwithpreviousstudyofFouetandcoworkers[34].
However,Fouetet. al.speculatedthatatJ /J =0.3thecor-
seeninthisfigurethatthepatternforthesignofcorrelations 2 1
relation pattern resembles a RS state. Based on qualitative
inPLandRSstatesareidentical,exceptfortheirmagnitudes.
symmetry consideration of short-range dimer-dimer correla-
Specifically, in the PL state, the magnitudes of correlations
tions,Fouetandcoworkersproposedthepossibilityofcrystal
betweenthedimerswhichbelongtothesamehexagonasthe
ofhexagonplaquettesasacandicateforthegroundstateinthe
referencebond,arestrongerthantherestofdimers.Butinthe
0.3 < J /J < 0.35range[34]. Inthefollowingwedemon-
RS state all dimershave identicalcorrelationwith respectto 2 1
stratethatfor0.2 . J /J . 0.3,thedominantcorrelations
thereferencebond. 2 1
areofPLtype,ratherthanRS.
In Fig. 10 and Fig. 11, we have shown the exact diago-
ForquantitativecharacterizationofthenatureofVBcrys-
nalizationresultsforthedimer-dimercorrelationfunctionsat
tallinestate,wedefinethefollowingstructurefactor:
J /J = 0.3andJ /J = 0.4,respectivelyforalatticewith
2 1 2 1
N = 32sites, subjecttoperiodicboundaryconditions. Note Sλ =Xελ(α′)C(α,α′), (8)
thatinordertoimplementsymmetriesofinfinitelatticeonfi-
α′
nite size systems, the size N is limited to specific numbers
N = 24,32,42,50,54,.... Since the dimension of Hilbert whereC(α,α′)isgivenbyEq.(7)andε (α′)isthephasefac-
λ
spacegrowsexponentiallywith N, the exactdiagonalization tor,appropriatelydefinedforeachofthethreestatesλ ≡SD,
in the whole S = 0 subspace is not feasible, and hence for PL,RS [24]. Thephasefactorsε ,ε andε areshown
z SD PL RS
N > 32wecarriedoutthecalculationsinNNVBbasis. The inFig.12. Sincethesignofdimer-dimercorrelationsforPL
correlationsarecomputedwithrespecttothemiddle-bondin- andRSstatesarethesame,theirphasefactorsmustbeequal.
dicated by double lines. Red bonds denote positive correla- ScalingbehaviorofS foralatticewithN sitesandN bonds
λ b
7
(a)
(cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)
N=24
N=42
0.18
N=54
0.14
b
N
/SD 0.1
S
0.06
0.02
FIG.12: Redand blue linkscorrespond to+1, −1phase factors,
0.2 0.25 0.3 0.35 0.4 0.45 0.5
respectively,whilethedashedlinksstandfor0. Referencedimeris J /J
2 1
identifiedbydoublelink.(a)PhasefactorconventionforPLandRS (b)
states.(b)Phasefactorsforstaggereddimerizedstate.
N=24
N=42
0.07 N=54
∞
isgivenby, C PL
Nb 0.05
S A /L
λ =C∞+ . (9) SP
Nb λ N 0.03
Usingthe abovephasefactors, andthecorrelationsC(α,α′)
0.01
givenintableIwehavecalculatedthecorrespondingC∞ in
λ
tableIIforeachofthethreetrialstatesψ . 0.2 0.25 0.3 0.35 0.4 0.45 0.5
λ
In Fig. 13 we have shown S and S versus J /J for J2/J1
SD PL 2 1
N = 24,42,54. Note that for N = 24, we have done ex-
act diagonalizationin S = 0 basis, while for N = 42,54, (c)
z
NNVBbasishasbeenused.NotethattheNNVBcalculations
is valid only in the region0.2 < J2/J1 < 0.35. With these SPL 0.02
three sizes, we have performed finite size scaling according
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
to Eq. (9) to obtain CS∞D and CP∞L. Fig. 13(a) and (b) show N-1
SD and PL structure factors, respectively, for various sizes.
ForvaluesofJ /J & 0.35we donothavereliabledatafor
2 1
N = 42,54. Hence we have not reportedfinite size scaling
forthesevaluesofJ2/J1. For0.2 < J2/J1 < 0.35,theSD FIG.13: ThestructurefactorcomputedforlatticewithN =24(di-
structurefactorin(a)remainsmuchsmallerthanCS∞D = 1/4 amond), N = 42 (triangle) and N = 54 (square). Nb stand for
(tableII)forallthreesizes. Ontheotherhand,thePLstruc- thenumberofdimers. Structurefactorscorrespondto(a)Staggered
turefactorin(b)canbeextrapolatedtoafinitevaluebetween dimerized state, (b) Plaquette state. In (b), empty square with er-
0.07and0.1(emptysquares)whicharecomparablewiththe ror bar indicates extrapolation to infinite lattice size. (c) shows a
finitesizescaling according toEq. (9) for PLstructurefactorsand
exactvalue0.125ofthepurePLstate(tableII).
AsuddenjumpobservedinFig.13(a)and(b)suggeststhe J2/J1 =0.3. AscanbeseenforaspecificvalueofJ2/J1between
0.35and0.4,thereisasuddentincraseforSDstructurefactor,which
existenceofafirstorderphasetransitionfromPLtoSDstate
isaccompaniedbyasuddendecreaseinthePLstructurefactor.
as one increasesJ /J beyonda certain valuebetween 0.35
2 1
and0.4. Inpanels(b)ofFig.13theaverageratioofstructure
factor (averagedoverthe range0.2 < J /J < 0.35)to the
2 1
correspondingC∞ isgivenby(S /N )/C∞ ≈0.66,while
PL PL b PL
thisratioforRSstateinthesameregionis(SRS/Nb)/CR∞S ≈ In addition we calculated the exact value of hPα′i as a
0.08/0.375 ≈ 0.21. Hence for 0.2 < J /J < 0.35 we function of J /J for N = 24 sites. For J /J from 0.2
2 1 2 1 2 1
expectthegroundstatetobedominatedbyplaquettevalence to 0.35,the expectationvaluehPα′iincreasesmonotonically
bondorder. from−0.21to−0.12andshowsasuddenjumpsto0.001for
J /J = 0.4. In view of the ≈ −0.1 value for the expec-
2 1
tationvalueofpermutationoperatorinthe PLstate (tableI),
TABLEII:Intensivestructurefactorinthermodynamiclimitforthe thenegativevaluesintherange0.2 < J2/J1 < 0.35canbe
threetrialstates. considered as an extra support in favor of plaquette valence
bondsolidinthisregime. Guidedbytheaboveevidencesfor
Trialstate ψSD ψPL ψRS
C∞SD 1/4 0 0 plaquetteorderinginregion0.2 < J2/J1 < 0.35,inthenext
C∞PL 0 0.125(5) 3/8 sectionwecalculatetheplaquette-plaquettecorrelationusing
exactdiagonalizationmethod.
8
TABLE III: Relative plaquette-plaquette correlations for hexagons
withdifferentdistancetoreferencehexagon.
C(i,0)/C(0,0) ψPL ψRVB ψED
i=1 -0.12 -0.04 -0.08
i=2 0.28 0.13 0.175
i=3 -0.12 -0.04 -0.07
i=4 -0.12 -0.008 -0.05
i=5 0.28 0.05 0.11
in S = 0 basis for N = 24 sites at J /J = 0.3,0.5, re-
z 2 1
spectively. Comparisonofpanels(b)and(c)of Fig.14 with
panel(a),indicatessubstantialPLorderingatJ /J =0.3.It
2 1
is remarkableto note that even the ratio of strengthsof pos-
itiveand negativecorrelationsin (b)(∼ 0.29 : 0.14)and(a)
(0.35: 0.13)agreewitheachother. Thisratioin(c)becomes
0.04: 0.04whichsignificantlydeviatesfromthecorrespond-
ing value for a pure PL state (a). Fig. 14(d) is the same as
(b),forlargersizeN =54andJ /J =0.3. Ascanbeseen,
2 1
for N = 54 too, a substantial plaquette correlation pattern
can be observed. Moreover, for this size we calculated the
PL-PL correlationsfor all hexagons(denotedby 1,2,3,4,5in
Fig.14(d))withrespecttothereferenceonetowhich,number
0 is assigned. In last column of table III we have listed the
relativecorrelations,definedbyC(i,0)/C(0,0),obtainedby
exactdiagonalizationinNNVBbasisandcomparethemwith
FIG.14: Red(blue)circlesdenotepositive(negative)correlations. correspondingvaluesforPLandRVBstatesgiveninfirstand
The radius of circles are proportional to the value of plaquette- secondcolumns,respectively. Ascanbeseen,forRVBstate
plaquette correlation. Thereference plaquette isdepicted byblack thisquantitydecaysrapidlywithdistancefromthereference
dashed line. (a) The plaquette-plaquette correlation map calcu- hexagon,whiletheexactdatadoesnotshowsucharapidde-
lated for PL trial state. The exact plaquette-plaquette correlation
caying. ThisagainisanevidenceinfavorofthePLnatureof
evaluated using exact diagonalization onhoneycomb latticefor (b)
thegroundstate.
J2/J1 =0.3and(c)J2/J1 =0.5.(d)PL-PLcorrelationpatternfor
N =54andJ2/J1 =0.3.
VI. CONCLUSION
V. PLAQUETTEORDERINFRUSTRATEDREGIME
In summary, diagonalization of J − J antiferromagnet
1 2
Amoredirecttooltodetectplaquetteorderinginfrustrated HeisenbergHamiltonian on honeycomblattice in both S =0
z
regimeistoinvestigatetheplaquette-plaquettecorrelationde- basis and NNVB basis show a striking agreement between
finedby, thesetwoapproachesintheparameterrange0.2 < J /J <
2 1
0.3. Therefore, in this region the ground state can be well
C(p,q)=hQ Q i−hQ i2, described in terms of the singlet bonds between the nearest
p q p
1 neighbor spins. Analysis of the exact dimer-dimer correla-
Qp = (Πp+Π−1p), (10) tions, structure factors, and also plaquette-plaquette correla-
2
tions,suggeststheexistenceofplaquettevalencebondcrystal
where p,q stand for different plaquette and Π (Π−1) is the in this range of couplings. In fact the emergence of such
cyclic exchangeoperatorwhich permutessix spins arounda PLorderingcanbeattributedtothequantumfluctuationsdue
hexagoninclockwise(counter-clockwise)direction.Thiscor- to the tendency of second neighbors to form singlets. This
relationfunctionwasintroducedrecentlyandhasbeenusedto study also reveals a phase transition from the plaquette or-
investigate plaquette ordering in frustrated Heisenberg mag- deredtothestaggereddimerizedstateatapointintheinterval
netsonthecheckerboardandsquarelattice[21–23,25]. J /J ∈]0.35,0.4]. Similar results, regarding plaquette or-
2 1
In Fig. 14-(a) we have depicted the plaquette correlation dering on the square lattice, have been previously obtained
in PL state. Red (blue) circles indicate positive (negative) forJ −J −J Heisenbergantiferromagnetinitsmaximally
1 2 3
correlations, with the radii of circles are proportional to the frustratedregion,J +J ∼J /2,andforJ ≤J [24]. Our
2 3 1 2 3
magnitudeof correlation. The referenceplaquetteis marked resultsareincontrastwiththoseobtainedbyQMCsimulation
with a dashed circle. Fig. 14-(b) and (c) represent the pla- oftheHubbardmodelinintermediateinteractionregime[32],
quettecorrelationfunctionobtainedbyexactdiagonalization in a sense that QMC results in, a RVB liquid phase with no
9
brokensymmetryforthisregion.Meng,et. al. haveonlycal- Vice Chancellor for Research Affairs of the Isfahan Univer-
culated short range dimer-dimer correlation. Therefore this sity of Technology(IUT). S. A. J was supported by the Na-
differencemightbeduetothefactthatthegeometryoffinite tional Elite Foundation(NEF) of Iran. We appreciate useful
clustersusedintheirsimulationisnotcompatiblewithPLor- commentsfromG.Baskaran,A.Paramekanti,B. Kumar,M.
dering.Basedonpresentstudy,webelievethatitisnecessary Vojta,A.V.Chubakov.
toconsiderlatticegeometriescompatiblewithPLstateandto
calculatetheplaquette-plaquettecorrelationforprecisedeter-
minationofthebrokensymmetriesinthegroundstate.
VII. ACKNOWLEDGMENT
Numerical analysis in this work have been carried out by
Spinpack package[41]. Thisresearchwassupportedbythe
[1] L.Balents,Nature464,199(2010). [19] V.N.Kotov,etal,Phil.Mag.B.,80,1483(2000)
[2] P.FazekasandP.W.Anderson,Phil.Mag.30,423(1974) [20] R.R.P.Singhetal,Phys.Rev.B.,60,7278(1999)
[3] P.W.Anderson,Science,235,1196(1987) [21] L.CapriottiandS.Sorella,Phys.Rev.Lett.,84,3173(2000)
[4] P. Fazekas, Magnetism and electron correlations in strongly [22] A.Gelle´,A.M.La¨uchli,B.Kumar,andF.Mila,Phys.Rev.B,
correlatedsystems,Woldscientific(2001). 77,014419(2008)
[5] F.D.M.Haldane,Phys.Rev.Lett.,50,1153(1983) [23] M.E.Zhitomirsky,Phys.Rev.B,54,9007(1996)
[6] H.T. Diepand H.Giacomini inFrustratedSpin Systems, Ed. [24] M.Mambrini,A.La¨uchli,D.Poilbanc,andF.Mila,Phys.Rev.
H.T.Diep,WorldScientific(2004). B,74,144422(2006)
[7] A. Olariu. et al, Phys. Rev. Lett. 100, 087202 (2008); [25] J. B. Fouet, M. Mambrini, P. Sindzingere, and C. Lhuillier,
H.Yoshida. et al, J. Phys. Soc. Jpn 78, 043704 (2009); Y. Phys.Rev.B,67,054411(2003)
Kurosaki.etal,Phys.rev.Lett.95,177001(2005);Y.Okamoto, [26] I.Affleck,Phys.Rev.B.,37,5186(1988)
H.Yoshida,andZ.Hiroi,J.Phys.Soc.Jpn.78,033701(2009); [27] Y. Miura, R. Hiari, Y. Kobayashi, and M. Sato, J. Phys. Soc.
Z.Hiroi.etal,J.Phys.Soc.Jpn.70,3377(2001);T.Itou.etal, Jpn.,75,084707(2006)
Phys.Rev.B77,104413(2008). [28] V.Kataev,A.Mo¨ller,U.lo¨w,W.Jung,N.Schittner,M.Kriener,
[8] D.A.Huse, Phys.Rev. B,37, 2380 (1988), D.A.HuseandV. andA.Freimuth,J.Magn.Magn.Mater.,290-291,310(2005)
Elser,Phys.Rev.Lett.,60,2531(1988) [29] O. Smirnova, et al, J. Am. Chem. Soc. 131 8313 (2009); S.
[9] J.D.RegerandA.P.Young,Phys.Rev.B,37,5978(1988) Okubo,etal,J.Phys.Conf.Ser.200022042(2010)
[10] S.TangandH.Q.Lin,Phys.Rev.B,38,6863(1988) [30] S. Aubin, S. Myrskog, M. H. T. Extavour, L. J. LeBlanc, D.
[11] J.OitmaaandD.D.Betts,Can.J.Phys.,56,897(1978) McKay,A.Stummer,andJ.H.Thywissen,NaturePhys.2,384
[12] J.D.Reger,J.A.RieraandA.P.Young,J.Phys.:Condens.Mat- (2006).
ter,1,1855(1989) [31] S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M.
[13] J.Oitmaa,C.J.HamerandZ.Weihong,Phys.Rev.B,45,9834 Stamper-Kurn,andW.Ketterle,Nature392,151(1998).
(1992) [32] Z.Y.Meng,T.C.Lang,S.Wessel,F.F.Assaad,A.Muramatsu,
[14] Z.Weihong,J.OitmaaandC.J.Hamer,Phys.Rev.B,44,11869 Nature464847(2010).
(1991) [33] T.EinarssonandH.Johannesson,Phys.Rev.B,43,5867(1991)
[15] A.Auerbach, InteractingElectronsand Quantum Magnetism, [34] J. B. Fouet, P. Sindzingre, C. Lhuillier, Eur. Phys. J. B., 20,
Springer(1994). 241-254(2001)
[16] P.Chandraand B.Doucot, Phys.Rev.B,38, 9335 (1988); E. [35] N.Read,S.Sachdev,Phys.Rev.B.,42,4568(1990)
Dagotto and A. Moreo, Phys. Rev. Lett. 63,2148 (1989); D. [36] S.Katsura,T.IdeandT.Morita,J.Stat.Phys.42,381(1986)
Poilblanc,E.Gagliano,S.Bacci,andE.Dagotto,Phys.Rev.B [37] S.Okumura,H.Kawamura,T.OkuboandY.Motome,J.Phys.
43,10970(1991);H.J.SchulzandT.A.L.Ziman,EuroPhys. Soc.Jpn.79,114705(2010).
Lett.,18,355(1992);P.Chandra,P.Coleman,andA.I.Larkin, [38] A.Mulder,R.Ganesh,L.Capriotti,andA.Paramekanti,Phys.
Phys.Rev. Lett64, 88(1990); M. P.Gelfand, R.R. P.Singh, Rev.B,81,214419(2010).
andD.A.Huse,Phys.Rev.B40,10801(1989);O.P.Sushkov, [39] S. M. Bhattacharjee, Z. Phys. B: Condensed Matter, 82 323
J. Oitmaa, and Z.Weihong, Phys. Rev. B 66, 054401 (2002); (1991).
L.Capriotti,F.Becca,A.Parola,andS.Sorella,Phys.Rev.B [40] Z.Noorbakhsh,F.Shahbazi,S.A.Jafari,G.Baskaran,J.Phys.
67,212402(2003); Soc.Jpn.78,054701(2009).
[17] F.Figueiridoetal.,Phys.Rev.B.,41,4619(1989) [41] www-e.uni-magdeburg.de/jschulen/spin/index.html
[18] N.ReadandS.Sachdev,Phys.Rev.Lett.,66,1773(1991)