Erratum: Incipient order in the t-J model at high temperatures Leonid P. Pryadko Department of Physics & Astronomy, University of California, Riverside, California 92521 Steven A. Kivelson Department of Physics, Stanford University, Stanford, California 94305 7 Oron Zachar 0 Department of Physics, University of California, Los Angeles, California 90095 0 (Dated: February 2, 2008) 2 n Previoisly, we reported calculations of the high-temperature series for thermodynamical suscep- a tibilities towards a number of possible ordered states in the t-J-V model. Due to an error in the J calculation,theseriesford-wavesupeconductingandextendeds-wavesuperconductingorderswere 3 incorrect. We give the replacement figures. In agreement with our earlier findings, we still find no evidence of any strong enhancement of the superconducting susceptibility with decreasing temper- ] ature. However, because different Pade approximants diverge from each other at somewhat higher l e temperatures than we originally found, it is less clear what this implies concerning the presence or - absence of high-temperature superconductivityin thet-J model. r t s t. Previoisly [1], we reported calculations of the high- Χ 26% ΧHd-SCL: t=J(cid:144)2 V=0,J(cid:144)4 a temperature series for thermodynamical susceptibilities 00..2255 m towards a number of possible ordered states in the t-J- 16% - 00..22 d V model. Due to an error in the calculation, the series n for d-wave supeconducting (d-SC) and extended s-wave x=6% 00..1155 o superconducting (s-SC) orders were incorrect. The first c terms of the correct series are [ 00..11 (1+x)(1−3x) 2 χ(1) =χ(1) =β . v s−SC d−SC 8ln((1−x)/2x) 00..0055 2 4 Here,β =1/T istheinversetemperature,andthedoping 00..55 11 11..55 22 22..55 33 T/J 3 xistheaveragenumberofholespersite. TheSCsuscep- 6 FIG.1: (Coloronline)Superconducting(d-wave)susceptibil- tibilities atsmallx→0aresuppressednotquadratically 0 ity for the “unphysical” case t = J/2 with V = 0 (dashes) 3 but only logarithmically in x. Note also that the value and V = J/4 (dot-dash) from the HTS to 1/T10 (diagrams 0 x = 1/3 is not singular, as the zeros in the numerator with up toNE =11 edges). Shadingrepresentsthestandard / and the denominator cancel. deviation of nondefective Pad´e approximants as discussed in t a We performed the corrected calculations of the series Ref.1. Thecorrespondingcurvesfor s-SCsusceptibility(not m shown)aresimilarbutsomewhatsmalleratlowtemperatures. for thermodynamicalsusceptibilities with an extra order - in powers of β up to the term with β10. The coefficients d n ofthecorrectedseriesareavailableuponrequest. Figs.1 Χ ΧHd-SCL: t=2J V=0 o –3representtheanalyzedresultsforthe correctedseries 00..0055 c and should replace Figs. 2 and 3 in Ref. 1. v: For physically plausible parameters, t > J, the ten- 00..0044 x=26% i dency of the superconducting susceptibility to decrease X with decreasing temperature is not quite as prominent 00..0033 r x=16% a as in our original work. The increased variance between different Pad´es relative to our former results indicate 00..0022 x=6% that the series can only be trusted down to tempera- 00..0011 tures T ∼J, whereas for t=2J, the mean susceptibility reaches a maximum at T/J ranging from T/J ∼ 1.2 at x = 6% to T/J ∼ 1.3 at x = 26%. Fig. 3 shows that 00..55 11 11..55 22 22..55 33 T/J at t = 3J the maximum is shifted up to temperatures FIG. 2: As in Fig. 1 but for t = 2J. Only V = 0 curves are T ∼ 2J. We also note that the effect of the n.n. repul- shown; curves for V =J/4 (not shown) are similar but more suppressed at low temperatures. sionV onthesuperconductingsusceptibilityismuchless dramatic than we originally reported. As a result, although the corrected series still suggest 2 Χ ΧHd-SCL: t=3J V=0 puted the high-temperature series for both the uniform and real-space equal-time pairing correlation functions 00..0033 x=26% up to 12thorder. These authors deriveaninstantaneous superconducting correlation length which increases with x=16% decreasing temperature, but which never grows as large 00..0022 as one lattice constant in the accessible range of T/J. x=6% Nevertheless, they interpreted their results as an indica- tion that the t-J model has a superconducting ground 00..0011 state. In our opinion, the failure of the thermodynami- cal susceptibility we have computed to show any strong tendency togrowwith decreasingtemperatureinthe ac- 00..55 11 11..55 22 22..55 33 T/J cessible range, as well as the extremely short correlation FIG. 3: Asin Fig. 2 butfor t=3J. lengths involved, call into question the strength of the conclusions drawn in Refs. 2, 3. theabsenceofhigh-temperaturesuperconductivityinthe t-J-V model at t>∼2J, this conclusion cannot be drawn as strongly as in our original Letter [1]. We should also note the papers by Koretsune and [1] L.P.Pryadko,S.A.Kivelson,and O.Zachar, Phys.Rev. Ogata[2]andbyPutikkaandLuchini[3]whichappeared Lett. 92, 067002 (2004). after Ref. 1. In these papers, the authors use a similar [2] T.KoretsuneandM.Ogata,J.Phys.Soc.Japan74,1390 (2005). technique to study not the thermodynamical suscepti- [3] W. O. Putikka and M. U. Luchini, Phys. Rev. Lett. 96, bilities but the equal-time correlation functions for the 247001 (2006). t-J-V model with the specialvalue V =J/4. They com- Incipient order inthet-J model athightemperatures Leonid P. Pryadko∗ Department of Physics, University of California, Riverside, California 92521 Steven A. Kivelson and Oron Zachar Department of Physics, University of California, Los Angeles, California 90095 (Dated:June12,2003) 7 0 Weanalyzethehigh-temperaturebehaviorofthesusceptibilitiestowardsanumberofpossibleorderedstates 0 inthet-J-V modelusingthehigh-temperatureseriesexpansion.Fromalldiagramswithuptotenedges,reliable 2 resultsareobtaineddowntotemperaturesoforderJ,or(withsomeoptimism)toJ/2.Intheunphysicalregime, t<J,largesuperconductingsusceptibilitiesarefound,whichmoreoverincreasewithdecreasingtemperatures, n butfort>J,thesesusceptibilitiesaresmallanddecreasingwithdecreasingtemperature;thissuggeststhatthe a J t-J modeldoesnotsupporthigh-temperaturesuperconductivity. Wealsofindmodestevidenceofatendency towardnematicandd-densitywaveorders. 3 ] el Thediscoveryofhightemperature(high-Tc)superconduc- probethephysicsofthemodelonlength-scalesrelevanttosu- tivityinthecuprateperovskiteslauncheda renewedeffortto perconductivity.ItisimportanttostressthatHTS[8]isfreeof - r developanunderstandingofthe physicsofhighlycorrelated finitesizeeffectsthatplagueothercomputationaltechniques. t s electronicsystems. Itisclearlysignificantthatsuperconduc- Specifically, we have computed the HTS for antiferro- . t tivityarisesinthesematerialsupondopinganearlyideal,spin magnetic(AF), d-wavesuperconducting(d-SC), extendeds- a m 1/2antiferromagneticinsulating“parent”state. Indeed,there wave superconducting(s-SC), nematic (N), and orbital anti- - is a prominent school of thought[1] that holds that high-Tc ferromagnetic(d-DW) susceptibilities, as defined in Eq. (2). d superconductivityismoreorlessinevitableina dopedquasi To extend the temperature range over which the results n two dimensional (2D) antiferromagnet; for this reason enor- can be trusted, we have made use of standard methods of o mous effort has been focused on studies of the t-J model resummation,[9]whichwedescribeexplicitlybelow. Inout- c [ [Eq. (1)], as the simplest model of a doped antiferromag- line, what we do is to construct a set of Pade´ approximants net. However, while the existence of antiferromagnetic or- ofrelatedEuler-transformedseries[seeEq.(3)]withdifferent 2 der in the spin 1/2 Heisenberg model (the zero doping limit values of the parameter β , eliminate all “defective” mem- v 0 2 ofthet-J model)iswellestablished[2], itremainsuncertain bers of this set, and then average over the remaining series. 4 whetherthe2Dt-J model,byitself, supportshigh-Tc super- Thevariancein thisaveragegivesanestimate forassociated 3 conductivity. In addition to antiferromagnetism and super- errors;wherethevariancegetslargeweconcludethatthere- 6 conductivity, various other types of order[3, 4, 5, 6, 7] have sultscannolongerbetrusted.Itispossiblethatbybiasingthe 0 been or may have been observed in the cuprates, including series,usingadditionalinformationaboutthelowtemperature 3 nematic[7](orspontaneousbreakingofthepoint-groupsym- state obtained from other methods, one mightbe able to ex- 0 / metry),charge-stripe,[4]spin-stripe[4]andd-density[6]wave tendtheresultstolowertemperatures.However,withoutsuch t a (alsocalledstaggeredfluxororbitalantiferromagnetic)order. additionalinformation, it is our experiencethat using differ- m Itisthusinterestingtodeterminewhichifanyoftheseorders ent prescriptionsfor resummation, or even adding a few ad- - aregenericfeaturesofadopedantiferromagnet. ditionaltermsto theseries, doesnotsignificantlychangethe d In this paper we report the results of an extensive high- resultsorincreasetheirrangeofvalidity. n o temperature series (HTS) study of the susceptibilities of the Therehavebeenprevioushighqualityseriesstudies[9,10, c 2D t-J model toward various short-period orders. (Long- 11, 12, 13] of this same model which inspired the present : periodstripeordercannotbereadilystudiedusingthesemeth- work,buttheyprimarilyfocusedonextrapolatingtheresults v i ods.) Naturally,theresultsobtainedinthiswayareonlyreli- toT =0.Whatdistinguishesthepresentstudyfromtheseear- X ableatmoderatelyhightemperatures. However,correspond- lier studies is (a) we have obtainedsusceptibilities that were r ingtoanylow-temperaturebrokensymmetrystatetheremust not previously computed[9, 14] and (b) we have contented a be a susceptibility which divergesat the ordering transition; ourselveswithstudyingthehightemperaturebehaviorofthe unlessthe transitionis stronglyfirst order, this susceptibility model. willbelarge,andanincreasingfunctionofdecreasingtemper- Ourprincipalfindingsareasfollows:1.Theresultsweob- atureevenattemperatureswellaboveanyorderingtransition. tain are reliable, without apology, for temperatures T > J, Put another way, the HTS is sensitive to relatively short- and are probably qualitatively correct down to T ∼ J/2. distance physics (the range is determined by the order to However,wehavenotfoundanymethodofanalyzingthese- which the series is computed). However, since the super- riesthatwetrusttoanylowertemperatures.2.Inthe“unphys- conductingcoherencelengthin the cupratesis thoughtto be ical” rangeof parameters,t ≤ J, with V = 0, the strongest aroundtwolatticeconstants,itisreasonabletoexpectthatthe incipientorderisAF—theAFsusceptibility,χ ,islargeand AF 10-12termswehavecomputedinthisseriesaresufficientto showsa strongtendencyto increasewith decreasingtemper- 2 χχ χχ χ(AF): J=1 V=0 x=26% χ(d−SC): t=1/2 J=1 11..44 00..77 11..22 t=1/2 00..66 1.0 x=16% x=1% 00..55 00..88 00..44 00..66 x=31% 00..33 x=16% V=0 00..44 00..22 x=26% x=6% 00..22 t=2 00..11 x=16V%=1/4 00..55 11 11..55 22 22..55 T 00..55 11 11..55 22 22..55 T FIG. 1: Temperature dependence of the AF susceptibility for t = FIG.2:Superconducting(d-wave)susceptibilityχd−SCfort=J/2 J/2 (dashes, x = 1%, 16%, 31%) and t = 2J (dot-dashed lines, withV =0(dashes)andV =J/4(dot-dash)fromtheHTSto1/T9 x = 1%,6%)fromtheHTSto1/T11 (diagramswithNE ≤ 10). (diagramswithNE ≤10).ShadingasinFig.1.Pairingfluctuations Shadingrepresentsthestandarddeviationofnon-defectivePade´ ap- arestrongatV =0,butarestronglysuppressedalreadyatV =J/4. proximantsasdiscussedintext. χχ ature, although this tendency gets gradually weaker with in- χ(d−SC): t=2 J=1 creasingdopingx(Fig.1). TheSCsusceptibilitiesarelargest 00..0022 V=0 at values of doping x >∼ 16% (both d-SC shown in Fig. 2 ands-SC,notshown[15];typicallyχ > χ ). How- d−SC s−SC ever, the inclusion of an additional nearest-neighbor (n.n.) V=1/4 repulsion, V = J/4, is sufficient to strongly suppress the pairing fluctuations. 3. In the physical range of parameters, 00..0011 x=26% t>J,wherethebareinteractionsbetweenelectronsaretruly V=0 repulsive, both χ (Fig. 3) and χ (not shown[15]) d−SC s−SC aresmallanddecreasingwithdecreasingtemperaturealready V=1/4 for T <∼ 2J; the pairing fluctuations are further suppressed x=16% by the addition of small n.n. repulsion. Based on these ob- 00..55 11 11..55 22 22..55 T servations we conclude that the 2D t-J model with t > J probablydoesnotsupporthightemperature superconductiv- FIG.3: SameasFig. 2for t = 2J. Pairingfluctuationsareweak ity. 4. For t > J, commensurate AF fluctuations are mod- alreadyatV =0(dashes),anddecreasefurtherwithintroductionof erate for x = 1% butare dramaticallysuppressedalreadyat weakn.n.repulsion(dot-dash). x ≥ 6%(Fig.1). Atlargerx,wefindthatthed-DW(Fig.4) and the nematic (Fig. 5) susceptibilities are both moderate, χχ show a weak tendency to increase with decreasing tempera- χ(d−DW): t=2 J=1 V=0,1/4 ture,andremainvirtuallyunaffectedbytheadditionofaweak n.n.repulsion,V =J/4. x=26% 00..0088 We compute the high-temperature series for the 2D t-J modeldefinedonthesquarelattice, 00..0066 x=16% 1 H =− t c† c +J S ·S − n n +V n n , (1) X ij iσ jσ (cid:16) i j 4 i j(cid:17) i j 00..0044 hiji x=6% where tij = t is the hopping matrix element, ciσ is the 00..0022 electron annihilation operator, Si ≡ (1/2)c†iστσσ′ciσ′ and n ≡ c† c are on-site spin and charge operators, and the i iσ iσ 00..55 11 11..55 22 22..55 T doubly-occupied sites are projected out. The canonical t-J model[1]canbeobtainedfromEq.(1)bysettingthen.n.re- FIG.4: Staggered-flux susceptibilityχd−DW for t = 2J, at V = pulsion V = 0, while the version of the model used in pre- 0 (dashes) and V = J/4 (dot-dash) obtained from HTS to 1/T9 vious high-temperature series studies[9, 10, 11, 13] can be (NE ≤10).ShadingasinFig.1. obtained by setting V = J/4. All results presented in this 3 χχ (βNE+1forAFordersincetheoperatorO isdefinedonthe AF χ(N): t=2 J=1 V=0,1/4 vertices). Asthelaststep,weperformtheLegendretransfor- 00..0088 mation to obtain the series for the free energy F(x,β), and x=26% reexpress[10](b)theobtainedseriesforχ intermsoftheav- erageholedensityx=1+(∂Ω/∂µ) . 00..0066 T All obtained series for χ, Ω, and F were carefully com- paredwith series computedanalyticallyto β3, andalso with 00..0044 x=16% series to β5 obtained by a direct differentiation of the free energy expression generated with an independently-written Mathematica[17]program. Amongotherconsistencychecks, 00..0022 we verified the cancellation of low-order terms in the irre- x=6% ducible weights. We have also compared the obtained free energyand relatedspecific heatseries with those forthe t-J 00..55 11 11..55 22 22..55 T modelwith V = 1/4fromRefs. 9, 11, andalso the specific FIG.5:TemperaturedependenceofthenematicsusceptibilityχNfor heat series at x = 0 with the corresponding series for the t=2J andV =0(dashes)andV =J/4(dot-dash)obtainedfrom Heisenbergmodel[18]. HTSto1/T9 (NE ≤ 10). ShadingasinFig.1. Then.n.repulsion A standard way of extrapolating a power series in β is to termV hasvirtuallynoeffectonχN. construct a ratio of polynomials p (β)/q (β) with match- n m ing expansionin powersof β, referred to as an (n,m) Pade´ approximant. However, when only a few first terms of the paperrefertothesetwovaluesofV. series are known, and without a detailed knowledge of the The susceptibilities (per unit site) χ can be expressed via structureofthesingularitiesofthefunction,theprocedureis the second derivatives of the free energy with respect to ap- plaguedby spuriousdivergences. Indeed, the coefficientsof propriatelychosenperturbingparameters,orastheirreducible the power series are only weakly modified if the numerator thermodynamiccorrelation functions of (properly projected) andthedenominatorofthefractionhavecloseroots. Theac- operatorsO, curacyof the extrapolationwould notbe affectedby cancel- 1 β lationofsuchfactors,whichamountstousingsmallernand χ ≡ T dτ eHτOe−HτO† , β ≡1/T. (2) m. Furthermore,itisnotgenerallyclearwhetheraseriesinβ O N τZ β 0 (cid:10)(cid:10) (cid:11)(cid:11) wouldgivebetterextrapolationthanaseriesinarelatedvari- We take the staggered magnetization O = eiQ·rSz, able β˜ = f(β). As a result, it hasbecomestandardpractice AF r r toaverage[9]overalargenumberof“non-defective”Pade´ap- Q ≡ (π,π), for AF ordering, the anisotroPpic part of the kinetic energy O = c† c − c† c + proximantsgeneratedforafamilyoffunctionsf(β,β0)with N r r+xˆσ rσ r+yˆσ rσ differentβ ,andusethecorrespondingdispersiontoestimate h.c. /2 for nematic orderingP, th(cid:0)e staggered orbital currents 0 theerrors. Aspecificdifficultyofextrapolatingtheseriesfor Od−(cid:1)DW = i reiQ·r c†r+yˆσcrσ −c†r+xˆσcrσ −h.c. /2for susceptibilities [comparedto non-singularquantities such as d-DW orderiPng[16], a(cid:0)nd the isotropic (anisotropic) (cid:1)part of n(k)] is that interesting susceptibilities can actually diverge the uniform pairing OSC = r(∆r,r+xˆ ±∆r,r+yˆ)/2 for atacriticaltemperature.Additionally,thepeaksinχdevelop s-SC (d-SC) ordering, wherePthe pair annihilation operator atrelativelylargespatialscale,meaningthattheybecomepro- ∆ij ≡ ci↑cj↓ + cj↑ci↓. Respectively, the first coefficients nouncedonlyatsufficientlyhighorders. of the correspondingseries are χ(1) = β(1−x)/4, χ(1) = TogeneratethecurvesinFigs.1–5,foreachχ(β)atagiven AF N χ(1) =βx(1−x),andχ(1) =χ(1) =βx2. setofparameterswe constructeda largenumberofPade´ ap- d−DW d−SC s−SC Thedefinitionofsusceptibilitiesintermsofthederivatives proximants (all n, m with n + m ≥ (n + m)max − 4) for ofthefreeenergyoffersaconvenientwayforconstructingthe boththeoriginalseriesandanumberofseriesintermsofthe clusterexpansion[8]. Foreachinequivalentlatticeplacement Euler-transformedvariable[19] C of a given diagram (connected graph) with nE edges, the β˜=β/(β +β), (3) relevanttraces(groupedbythenumberofparticles)arecom- 0 puted using block-diagonal matrices of the cluster Hamilto- using 0.3 ≤ β ≤ 10. We then eliminated as “defective” 0 nianandtheoperatorO. Thetracesarecombinedtoproduce theapproximantswithpositiverealrootsindenominatorsand the coefficients of the inverse-temperature expansion of the numerators[susceptibilitiesarenonnegative,seeEq.(2),and clustersusceptibilityχ (β,y;C)andthethermodynamicpo- wedonotexpectanactualphasetransitionwithadivergentχ O tentialΩ(β,y;C)dependentonthevariabley =z/(1+2z)re- atsuchhightemperatures],aswellastheapproximantswith latedtofugacityz =eβµ. Afterthesubclustersubtractionwe close numerator-denominatorcomplexrootpairs, and calcu- obtaintheirreducibleweightsofthecluster[whoseexpansion lated the averages and the corresponding rms deviations for startswithO(βnE−1)orhigherpowerofβ].Then,combining differenttemperaturesT =1/β. theclusterweightsfordiagramswithuptoN edges,wegen- Asanindependentmethodofanalysis,wehavealsolooked E eratetheseriesexactinthethermodynamiclimitto∼βNE−1 at the behavior of the untransformed and most closely bal- 4 anced(mcloseton)Pade´ approximantswithevenm. When C: Superconductivity 158, 192 (1989); U. Lo¨w, V. J. Emery, they are non-defective, these “best” approximants produce K. Fabricius, and S. A. Kivelson, Phys. Rev. Lett. 72, 1918 curves that look qualitatively similar to the average, and lie (1994);O.Zachar,S.A.Kivelson,andV.J.Emery,Phys.Rev. B 57, 1422 (1998); S.-C. Zhang, Science 275, 1089 (1997); wellwithintheshadedregionsoftheplots.Wehavealsotried S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, anumberofvariationsontheaveragingprocedure,testingon 550 (1998); S. Chakravarty, R. B. Laughlin, D. K. Morr, and the AF susceptibility χ at x = 1%, where we expect a AF C.Nayak,Phys.Rev.B63,094503(2001);S.Chakravarty,H.- prominentpeakatlowtemperatures. In particular,we found Y.Kee,andC.Nayak,Int.J.Mod.Phys.B21,2901(2001). that the hyperbolictangenttransformationβ˜ = tanh(β/β0) [4] J.M.Tranquada, B.J.Sternlieb,J.D.Axe,Y.Nakamura,and which was popular in previous HTS studies[9, 11], strongly S.Uchida,Nature375,561(1995). suppresses all peaks, effectively flattening the extrapolated [5] For review of thestatus of experimental search for stripeand functionsforallsusceptibilitieswecomputed. nematic order in the cuprates see S. A. Kivelson, E. Fradkin, V. Oganesyan, I. P. Bindloss, J. M. Tranquada, A. Kapitul- To summarize, our results indicate that, although in the “unphysical” region, t <∼ J, the t-J model displays a sharp nik, and C. Howald, How to detect fluctuating order in the high-temperaturesuperconductors,cond-mat/0210683[unpub- increase in the pairing fluctuations as the temperature goes lished]. down, this is no longer the case for t > J. Also, super- [6] H. A. Mook, P. Dai, and F. Dogan, Phys. Rev. B 64, 012502 conductingfluctuationsare strongly suppressed by introduc- (2001); H. A. Mook, P. Dai, S. M. Hayden, A. Hiess, J. W. tion of a small n.n. repulsion V. Thus we conclude that Lynn,S.-H.Lee,andF.Dogan,ibid.,66,144513(2002). high-temperaturesuperconductivityisprobablynotageneric [7] Y.Ando,K.Segawa,S.Komiya,andA.N.Lavrov,Phys.Rev. Lett.88,137005(2002). feature[20]ofdopedAFs. Apartfromχ atverysmalldop- AF [8] C.DombandM.S.Green,eds.,Phasetransitionsandcritical ing, none of the studied susceptibilities display remarkably phenomena,vol.3(Academic,London,1974);J.G.A.Baker, strongfluctuationsinthe“physical”ranget > J asthetem- Quantitative theory of critical phenomena (Academic, San peraturegoesdowntoT >∼J/2,butχN andχd−DW doshow Diego,1990). amodestenhancement. [9] R. R. P. Singh and R. L. Glenister, Phys. Rev. B 46, 11871 In the future, we intend to continue studying the t-J- (1992);46,14313(1992). V model within the high-temperature series approach, for a [10] K. Kubo and M. Tada, Progr. Theor. Phys. (Japan) 69, 1345 (1983);71,479(1984). widerrangeofordersinhopeofidentifyingthemostrelevant [11] W.O.Putikka,M.U.Luchini,andT.M.Rice,Phys.Rev.Lett. fluctuationsinthepseudogapregion. Wealsointendtostudy 68, 538 (1992); R. Hlubina, W. O. Putikka, T. M. Rice, and theinteractionoforderparameters,bylookingatvarioussus- D.V.Khveshchenko,Phys.Rev.B46,11224(1992). ceptibilitiesinmodifiedmodelswhereanordering(e.g.,hop- [12] D.F.B.tenHaaf,P.W.Brouwer,P.J.H.Denteneer,andJ.M.J. pinganisotropy)isimposedattheHamiltonianlevel. Within vanLeeuwen,Phys.Rev.B51,353(1995). thisgeneralapproach,wealsoplantostudyarangeofrelated [13] W.O.Putikka, M.U.Luchini, andR.R.P.Singh, Phys.Rev. models,inparticularanarrayofcoupledt-J ladders. Lett.81,2966(1998);W.O.PutikkaandM.U.Luchini,Phys. Rev.B62,1684(2000). Acknowledgments:WeareindebtedtoSudipChakravarty, [14] Note that χAF was previously computed tothesameorder in Mattias Ko¨rner, Rajiv Singh, and Matthias Troyer for en- Ref.9(a)andinA.Sokol,R.L.Glenister,andR.R.P.Singh, lighteningdiscussions. ThisworkwassupportedbytheNSF Phys.Rev.Lett.72,1549(1994). grantDMR01-10329(S.A.K.)andtheDOEgrantDE-FG03- [15] SeeadditionalplotsatURLhttp://faculty.ucr.edu/ 00ER45798(O.Z.). The calculationswereperformedin part ∼leonid/hts-all.pdf. The calculated series are avail- on the clustered computerat the Institute of Geophysicsand ablefromtheauthorsuponrequest. PlanetaryPhysics,UC,Riverside. [16] Thegauge-invariantd-DWsusceptibilitydefinedasthesecond derivative of the free energy with respect to staggered fluxes hasaqualitativelysimilarbehavior for x >∼ 11% (whereitis positive). [17] S.Wolfram,Mathematica,WolframResearch,Inc,Champaign, IL(2001). ∗ [email protected] [18] J.OitmaaandE.Bornilla,Phys.Rev.B53,14228(1996). [1] P.W.Anderson, Science235, 1996 (1987); Thetheory of su- [19] M.TroyerandM.Ko¨rner(2003),privatecommunication. perconductivityinthehigh-Tccuprates(PrincetonUniv.Press, [20] Webelievethattheincreaseofχd−SCwithdecreasingT found Princeton,NJ,1997). byS.R.White,D.J.Scalapino,R.L.Sugar,N.E.Bickers,and [2] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. R. T. Scalettar, Phys. Rev. B 39, 839 (1989) for the Hubbard Lett.60,1057(1988);Phys.Rev.B39,2344(1989). modelwithU = 4tisnotinconflictwithourstrong-coupling [3] I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988); results. J. Zaanen and O. Gunnarsson, ibid., 40, 7391 (1989); H. J. Schulz,Phys.Rev.Lett.64,1445(1990);K.Machida,Physica