Effective spin model for the spin-liquid phase of the Hubbard model on the triangular lattice

0.33 MB·English

by  H-Y. Yang

Effective spinmodel forthe spin-liquidphase ofthe Hubbard model onthe triangularlattice
Hong-Yu Yang,1 Andreas M. La¨uchli,2 Fre´de´ric Mila,3 and Kai Phillip Schmidt1
1Lehrstuhlfu¨rTheoretischePhysikI,Otto-Hahn-Straße4, TUDortmund, D-44221Dortmund, Germany
2Max Planck Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
3InstituteofTheoreticalPhysics,EcolePolytechniqueFe´de´raledeLausanne, CH-1004Lausanne, Switzerland
Weshowthatthespinliquidphaseofthehalf-filledHubbardmodelonthetriangularlatticecanbedescribed
byapurespinmodel.Thisisbasedonahigh-orderstrongcouplingexpansion(uptoorder12)usingperturbative
continuousunitarytransformations.Theresultingspinmodelisconsistentwithatransitionfromthree-sublattice
long-rangemagneticordertoaninsulatingspinliquidphase, andwithajumpofthedoubleoccupancyatthe
1 transition. Exactdiagonalizationsofbothmodelsshowthattheeffectivespinmodelisquantitativelyaccurate
1 wellintothespinliquidphase, andacomparison withtheGutzwillerprojectedFermiseasuggestsagapless
0 spectrumandaspinonFermisurface.
2
n PACSnumbers:75.10.Jm,75.10.Kt,05.30.Rt
a
J
Although the Hubbard model has been one of the central Thestartingpointisthesingle-bandHubbardmodelonthe
4
1 paradigmsinthefieldofstronglycorrelatedsystemsforabout triangularlatticedefinedbytheHamiltonian
fivedecades,newaspectsofitsextremelyrichphasediagram

l] are regularlyunveiled. Even at half-filling, the popularwis- H = HU +Ht
e dom according to which the model has only two phases, a = U n n −t (c† c +h.c.) (1)
- X i↑ i↓ X iσ jσ
r metalliconeatweakcouplingandaninsulatingoneatstrong i hi,ji,σ
t couplingseparatedbyafirstordertransition[1],hasbeenre-
s
t. cently challenged. This goes back to the work of Morita et where the sum over i runs over the sites of a triangular lat-
a al. [2]onthetriangularlatticewhichrevealedthepresenceof tice, and the sum over hi,ji over pairs of nearest neigh-
m
a non-magnetic insulating phase close to the metal-insulator bors. Toderiveaneffectivelow-energyHamiltonian,weuse
- transitionusing path integralrenormalizationgroup. Further PCUTs[10–13]aboutthelocalizedlimittreatingthehopping
d
n evidencein favour of a phase transition inside the insulating term as a perturbation. The kinetic part can be written as
o phasehasbeenreportedusingavarietyoftheoreticaltools[3– Ht = t(T0+T1+T−1) where Tm changes the number of
c 6]. Morerecently,anintermediatespinliquid(SL)phasehas doubly-occupiedsitesbym. ThePCUTmethodprovidesor-
[
alsobeenidentifiedonthehoneycomblatticeusingQuantum der by order in t/U an effective Hamiltonian Heff with the

3 MonteCarlosimulations[7]. property [HU,Heff] = 0, i.e. Heff is block-diagonal in the
v The precise natureof the SL phase of the Hubbardmodel numberofdoubly-occupiedsites. Thelow-energyphysicsat
9
onthetriangularlatticeisofdirectexperimentalrelevancefor half filling is expected to be contained in the block with no
4
6 the2Dorganicsaltκ-(BEDT-TTF)2Cu2(CN)3[8].Assuch,it doubly-occupiedsite He0ff, whichcan be expressedas a spin
5 hasalreadyattractedalotofattention,butfundamentalques- 1/2modeloftheform
. tions such as the appropriatelow-energyeffective theory re-
6
0 mainunanswered. Sincethe phaseis insulating, aneffective He0ff =const+XJ~n(cid:16)S~~r·S~~r+~n(cid:17) (2)
0 modelwherechargefluctuationsaretreatedasvirtualexcita- ~r,~n
v:1 thiaosnsbesehnoutaldkebnebpyosMsiobtlreu.nOicnhe[9st]e,pwfhoorwparrodpoinsetdhitsoddierescctriiobne + X J~n~n13,~n2(cid:16)S~~r·S~~r+~n1(cid:17)(cid:16)S~~r+~n2 ·S~~r+~n3(cid:17)+...
~r,~n1,~n2,~n3
i theSLphasewith4-spininteractions.However,whetherade-
X
scriptionintermsofapurespinmodelispossibleisfarfrom where~rand~r+~n denotesitesonthetriangularlattice. All
r α
a obvious, in particular since there seems to be a jump in the remainingterms can be written in a similar way as products
double occupancy at the transition from the three-sublattice ofS~~r·S~~r′ duetotheSU(2)symmetryoftheHubbardmodel.
Ne´elphasetotheSL[2,6]. The PCUT provides the magnetic exchange couplings as
InthisLetter,weshowthatthecorrectlow-energytheoryof series expansions in t/U. Since the spectrum of the Hub-
bothinsulatingphases,andinparticularoftheSLphase,isin- bard model is symmetric at half filling under the exchange
deedapurespinmodel.Thishasbeenachievedbyderivingan t ↔ −t, onlyeven ordercontributionsare present[14]. We
effectivespinmodeltoveryhighorderaboutthestrongcou- havedeterminedall2-spin,4-spinand6-spininteractionsup
plinglimitusingperturbativecontinuousunitarytransforma- to order 12. The obtained series are valid in the thermody-
tions(PCUTs),andbyshowingthatitgivesaqualitativeand namiclimit. To this endwe have fullyexploitedforthe first
quantitativeaccountofthetransitionfromthethree-sublattice timeinaPCUTcalculationthelinkedclustertheorembyus-
magnetic order to the SL state. This description gives deep ing graph theory. At order 12 there are 1336 topologically
insight into the nature of the SL phase and clearly provides differentgraphs. The major complicationfor the determina-
theappropriateframeworkforfurtherstudies. tionofhigherorderscomesfromthelargenumberofdifferent
2
-Jd4isJap1g,0 J6sp Energy per site [t]----0000....6543 8(a) xxxxxxxxxxxxx10xxxxxxxxxxxxx 12 NNNNNN======122323217616,,,,,, SSSSHP1ppppIuRiiiibnnnnG4b mmmma Rroooodeddddfeeee. llll[6] 16 xxxx9.xxxx5xxxxxxxxxx9.xx75xxxxxxxxxxxxxxx10xxxxxxxxx1xxx0.x25 (b1)0x.5--00000..00..0000001221GS Energy Crossing
U/t U/t
J 5
Jp4asrpa J1,1 2,0 2U nd444...468 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxx xxx xxx123πQ=(4/3,0)]
JO4s6p 4.24 8(c) xxxxxxxxxxxxxxx10xxxxxxxxxxxxxxxxxxxxx12 14 16 8 xxxxx10xxxxx 12 14 (d)160S[
U/t U/t
FIG.2:(Coloronline)(a)EnergypersiteasafunctionofU/tforthe
FIG. 1: (Color online) The most relevant exchange couplings dis- Hubbardmodelandtheeffectivespinmodel. (b)Levelcrossingsin
playedasafunctionoft/U. The12thorderbareseriesofallcou- theeffectivespinmodelsignalledbytheenergydifferenceofthetwo
plingsappearinguptoorder4andthelargest4-spininteractionJ4sp lowestenergies. (c)Doubleoccupancy (timesU2)asafunctionof
plusthelargest6-spininteractionJ6spappearinginorder6areplOo6t- U/t. (d)Magneticstructurefactoratthe120◦ AFMorderingwave
ted. Greyshadedareamarkstheregionwhereextrapolationeffects vector. Dashed lines denotes results for a simplified2B-4B model
becomeimportant.Verticaldashedlineindicatesthemagneticphase evaluated up to order ten. Continuous lines with symbols are the
transition.Inset:variousPade´approximantsasafunctionoft/U for resultsforthefull2B-4B-6Bmodelconsideredhere.Theobservable
thenearast-neighborHeisenbergexchangeJ1,0. isuptoordertwoinbothcases.
spinoperators,whichrequirestocalculateallpossiblematrix - Convergence of the series: As usual, we use Pade´ ap-
elementsforeachgraph. proximants to extrapolate the series, and we consider that
The mostrelevantspin couplingsare shownin Fig. 1 as a series are well converged as long as different Pade approxi-
function of t/U. As already pointed out in Ref. 9, the most mantsgiveconsistentresults. Herethisisdefinitelytrueupto
important correction to the nearest-neighbor Heisenberg ex- t/U ≃ 0.125(U/t ≃ 8), i.e. wellbeyondthelevelcrossing
change at order 4 are the 4-spin interactions on a diamond (see inset of Fig. 1). For all couplings, the absolute change
plaquette. High-order contributions result in more complex betweenorder10and12isbelow10−4(10−3)fort/U =0.1
multi-spin interactions with larger spatial extension but also (t/U = 0.125). Infact, clear indicationsofdivergenceonly
lead to renormalizations of already existing couplings. The appearabovet/U ≃ 0.15,andthisisprobablyrelatedtothe
latter effectcausesfor examplea sizable splitting ofthe two metal-insulatortransition.
4-spininteractionsshowninFig.1,whichhavethesameab- -ComparisonofGSenergywithHubbardmodel: Inview
solute value at order 4. Processes involving sites far apart oftheanomalyobservedatthetransitionbyotherapproaches
have small amplitudes, and multi-spin interactions between [2, 6], it is legitimate to ask whether the GS remains in the
sitesnotfarfromeachotherareclearlythedominantcorrec- sector with no doubly-occupied site. To address this point,
tionstothespinoperatorsatorder4. we have used ED of finite clusters to calculate the GS en-
At large U/t, nearest-neighbourexchangedominates, and ergyofboththeHubbardmodelandtheeffectivespinmodel
thesystemisexpectedtodevelopthree-sublatticelong-range as a functionof U/t (see Figs. 2a and 2b) up to 21 sites for
order[15–17]. Usingexactdiagonalizations(ED)ofclusters the Hubbardmodel, and up to 36 sites for the effective spin
upto36sites,wehaveinvestigatedhowthepropertiesofthe model. For the spin model, the total number of spin opera-
model evolve upon reducing U/t. The most striking feature torsisenormous,andtruncatingthehierarchyofoperatorsis
isalevelcrossinginthegroundstate (GS)atU/t ≃ 10(see necessary. Wehaveincludedinourcalculationallspinoper-
Fig.2). Thislevelcrossingisobservedforallclustersizesat atorsappearinguptoorder6. Additionally,wehaveomitted
essentiallythe samevalue, whichindicatesthepresenceofa for a given ratio t/U all terms whose coupling was smaller
first order transition in the thermodynamic limit. Since this than10−6. Whenembeddedonourlargest36sitecluster,this
ratio is comparableto the critical ratio where other methods correspondsto∼16’000distincttermsintotal. Theresulting
havedetectedatransitionintoaSLphase,itseemsplausible totalenergiespersitearedisplayedinFig.2. Theenergyper
thatthislevelcrossingcorrespondstothistransition. Further siteoftheeffectivespinmodeldependsverylittleonthesize,
investigations below U/t ≃ 10 confirm this guess (see be- andforthe21-sitecluster,theenergiesoftheHubbardmodel
low), but let us first investigate to which extent the effective andoftheeffectivespinmodelareinverygoodagreementon
spinmodelprovidesanaccuratedescriptioninthisparameter both sides of the transition. This last observationclearly es-
range.Theproofthatthisissoreliesonfourobservations: tablishesthatthe effectivemodelremainsaccuratein the SL
3
-10.8 xxxxxxxx x xx xxxx -10.8 0.25xxxxxxxx xxxxxx xxxxxx 0.25
x xxxx x x xxxx xxxx xxx xxx
Total Energy [t]---11110-11...2911 xxxxHxxxxubxxxbxxxxxxardxxxxxx mxxoxxdxxxxxxxxxxxxel, Uxxxxxxxxxx/t=9xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx K a c b K Spin model2-4-6 body ----11111110...219Total Energy [t] GS shifted energy [t]00..0001..5521xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 120o AFM xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 120o AFM xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx o120 AFM 0000....120155GS shifted energy [t]
-11.3 Ns=21 Ns=21 -11.3 0xxxxxxxxxxxx xxxxxxxxx xxxxxxxxx 0
K,A K,E1K,E2 K,B a b c K,A K,E K,A K,E1K,E2 K,B a b c K,A K,E xxxxNMIxxxx a) Ns=21 xxxxNMIxx b) Ns=27 xxNMIxxxxc) Ns=36
Symmetry sector Symmetry sector xx8xxxxxx12 16 20 xx8xxxx 12 16 20 xx8xxxx 12
U/t U/t U/t
FIG. 3: (Color online) Comparison of the low-energy spectrum of
a Ns = 21 Hubbard model at U/t = 9(left panel) toresults ob- FIG.4:(Coloronline)EDspectraofthetruncatedspinmodelfor(a)
tainedwiththeeffectivespinmodel(rightpanel). Thex-axislabels Ns =21,(b)Ns =27,and(c)Ns =36asafunctionofU/t.Open
thedifferentsymmetrysectors,whileempty(filled)symbolsdenote (black) symbols denote non-magnetic levels. Filled (red) symbols
Sz = 1/2(Sz = 3/2)levels. a,bandclabelmomentaintheinte- areenergylevelscorrespondingtoexcitationswithfinitespinabove
rioroftheBrillouinzone. theGS.ThegreyshadedareasignalstheextensionoftheSLphase.
phase. TheGSenergiesforboththeHubbardmodelandthe effective model since they are not present in the thermody-
effectivemodelareactuallysignificantlylowerthanrecentes- namiclimit. Thesystematicerrorinthethermodynamiclimit
timatesbasedonPIRG[6],seeFig.2a. ismuchsmallersinceitonlycomesfromtheprecisionofthe
- Double occupancy: The comparison of the GS energies coefficientsandtheneglectoftruncatedterms.
suggests that the GS is in the sector of the effective model Thetruncatedspinmodelisthusanexcellentstartingpoint
with no doubly-occupied site in the U/t range considered. toinvestigatethequantummagnetismoftheHubbardmodel
This result seems to be incompatible with the jump of the onthe triangularlattice. Consequently,we turnto a detailed
doubleoccupancyn reportedbyothermethodsfortheHub- discussionofitsphysicalpropertiesasafunctionofU/t. The
d
bard model [2, 6], but in fact it is not because, when calcu- low-energy ED spectra for clusters of N = 21,27, and 36
s
lating the expectation value of observables from the effec- sitesareshowninFig. 4. Forclaritythe lowestlevelateach
tive model, one must apply the same unitary transformation U/tvalueistakenasthereferenceenergyandsettozero,and
[12,18]which,inthepresentcase,leadstoanon-zerovalue the spatial symmetry quantum numbers of the levels are not
ofn . Inpractice,itissimplertousetheFeynman-Hellmann specified.Forthe36-sitecluster,thecalculationoftheexcita-
d
theoremwhichallowstodeduceitdirectlyfromtheenergyper tionspectrumis numericallyveryexpensive,andithasbeen
site as n = ∂ hHi/N. The resultsare depictedin Fig. 2c. performedonlyforthreevaluesofU/tclosetothetransition,
d U
A jump is clearly present at the transition for all finite-size and for a simplified two- and four-spin coupling model (in
samplesasaconsequenceofthelevelcrossingoftheeffective contrast to the ground state energy calculations in Fig. 2(a)
model,anditsmagnitudeisconsistentwiththatcalculatedfor which have been performed based on the more demanding
the Hubbardmodel on the 21-site cluster (see Fig. 2c). The model with the aforementioned cutoff 10−6). At large U/t
significantdifferencebetween21and27sitesdoesnotallow (down to U/t ∼ 15), the level structure revealed by the 21
to performa reliable finite-size scaling, butthis does notaf- and27-siteclustersistypicalofthetowerofstatesofa120◦
fecttheconclusionthattheeffectivespinmodelcapturesthe antiferromagnet (AFM) [15]. For smaller U/t ratios, some
doubleoccupancyjumpatthetransition. levelsstarttobenddownandtodeviatefromtheleading1/U
- Low-energyspectrumandGS momentum: Itis also im- behavioroftheHeisenbergregime. Drasticchangesoccurin
portanttocheckhowwelltheeffectivemodelreproducesthe the spectrum at U/t ≈ 10. First of all, on all clusters, ex-
low-energy spectrum of the original model. To address this cited states with different quantum numbers cross the large
issue, we compare the low-energy spectrum at U/t = 9 of U/tGSlevel.Besides,alargenumberofsingletsaccumulate
a N = 21 Hubbard model with the low-energy excitations at low energyin both the effective spin modeland the Hub-
s
of the spin modelon the same cluster [25] in Fig. 3. Quali- bardmodelforU/t≤10.
tatively, the agreementis good. In particular, the non-trivial To better characterize this change of behaviour, we have
symmetrysectorsoftheGSandofthefirstexcitedstatesare calculated the structure factor corresponding to the three-
correctly reproduced. Quantitatively, the difference between sublattice 120◦ order as a function of t/U by implementing
theenergies(afractionofapercent)isdominatedbythefact the unitarytransformationon the observableat ordert2/U2.
that the effective model on a finite cluster should include, Abovethe levelcrossing, this structure factor increases with
starting from order 6, a few processes which wrap around thesize,asexpectedforalong-rangeorderedphase. Bycon-
thecluster. Theseprocessesarenottakenintoaccountinour trast,belowthelevelcrossing-i.e. intheputativeSLregion
4
- the structure factor is finite size saturated, implying either theory of the non-magnetic phase of the half-filled Hubbard
short range correlations, or an algebraic decay with a size- model on the triangular lattice. It is a pure spin model with
abledecayexponent.AtlargeU/t,theslightreductionofthe multi-spinexchange.Furthermore,wehaveperformedanED
structure factor is mainly a consequenceof the admixtureof investigationofthismodel. Severalaspectsoftheresultsare
emptyanddoublyoccupiedsites,whichreducestheeffective compatible with the SBM picture predicting a gapless spec-
localspinlengthS2 below3/4. However,asweapproachthe trum with a spinon Fermi surface. It will be interesting to
levelcrossing,theextratermsinthespinHamiltonianfurther extend our investigation to the anisotropic triangular lattice
reduce the structure factor, as can be seen by comparingthe [23, 24], andto comparethe resultsdirectly to the SL mate-
resultsforthe’full’effectivemodelwiththoseforatruncated rial κ-(BEDT-TTF)2Cu2(CN)3 and to the other members of
modelwhere onlythe 2-spinand 4-spintermsthatappearat the family. Finally, this approachis also applicableto other,
fourthorder(butevaluatedusingthefullseries)arekept(see possiblyfrustratedlatticetopologiesforwhichsimilarlyinter-
Fig.2d). Soonecannotexcludethatthelong-rangemagnetic estingMottphasescanbeexpected.
orderdisappearsslightlybeforethelevelcrossing.
WethankR.McKenzieforfruitfuldiscussionsandT.Yosh-
TheissueofthespingapintheSLphaserequiresacareful
iokaforprovidingPIRGdataforcomparison.K.P.S.acknowl-
discussion. Indeed,theenergyofthefirstmagneticexcitation
isnearlyconstantbetween21and27sitesanddropsdramat- edges ESF and EuroHorcs for funding through his EURYI.
TheEDhavebeenenabledthroughcomputingtimeallocated
ically for 36sites. So dependinguponwhetherone includes
at ZIH TU Dresden and the MPG RZ Garching. F.M. ac-
the 36 cluster or not, finite-size effects suggest a vanishing
gap or a finite gap. Now, the significant difference between knowledgesthefinancialsupportoftheSwissNationalFund
andofMaNEP.
the36-siteclusterandtheothertwomightbeduetoaneven-
oddN effect,ortotheuseofasimplifiedmodel,anditmight
s
bebetternottoincludeitinthefinite-sizeanalysis.Now,con-
centrating on the 21 and 27 site clusters, the energy of the
firstmagneticexcitationisindeedalmostconstant,butifone
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(2008).
todecideonthepresenceofaspingapbasedontheavailable
[5] L.F.Tocchioetal.,Phys.Rev.B.78,041101(2008).
ED data alone. In the nextparagraphwe show that the data
[6] T.Yoshioka,A.Koga,andN.Kawakami,Phys.Rev.Lett.103,
canbeinterpretedfromabroaderperspective.
036401(2009).
In his seminal paper Motrunich [9] proposed a plain [7] Z.Y.Mengetal.,Nature464,847(2010).
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phases [20] (see Ref. [21] for a related scenario), which
[12] C.Knetter,K.P.Schmidt,andG.S.Uhrig,J.Phys.A36,7889
has been successfully applied to a one-dimensionaltriangu-
(2003).
lar strip modelwith two-andfour-spininteractions[20, 22].
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Interestingly,thenumericallow-energyspectrumofthetrian- 70,245124(2004).
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spectrumweobserveonthetriangularlattice,withrespectto B37,9753(1988).
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(1992).
gletsandtheirregularfinitesizebehaviorofthespingap[22].
[16] L.Capriotti,A.E.Trumper,andS.Sorella,Phys.Rev.Lett.82,
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3899(1999).
MotrunichandourEDanalysiscomesfromthequantumnum-
[17] S.R.WhiteandA.L.Chernyshev,Phys.Rev.Lett.99,127004
bersof the groundstate wave functionin the SL region. On (2007).
theNs = 27site sampleforexamplethenoninteractinghalf [18] J.-Y.P.Delannoyetal.,Phys.Rev.B72,115114(2005).
filledFermiseahasasixfolddegenerategroundstatemomen- [19] W.LiMingetal.,Phys.Rev.B62,6272(2000).
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79,205112(2009).
tumnumberoftheeffectivespinmodelintheSLregionadja-
[21] S.-S.LeeandP.A.Lee,Phys.Rev.Lett.95,036403(2005).
centtothemagneticallyorderedphase. Ontheothersamples
[22] A.D.Klironomosetal.,Phys.Rev.B76,075302(2007).
(N =21,36)theFermiseagroundstatedegeneracyislarger
s [23] H.C.Kandaletal.,Phys.Rev.Lett.103,067004(2009).
andthuslessconstraining,butthegroundstatequantumnum- [24] K.Nakamuraetal.,J.Phys.Soc.Jap.78,083710(2009)
bersoftheSLregimearestillcompatiblewiththispicture. [25] The largest Hilbert space sector of the Hubbard model (spin
Inconclusion,wehavederivedtheappropriatelow-energy model)hasdim∼6×109(dim∼105).