Incipient order in the t-J model at high temperatures

0.22 MB·English

by  Leonid P. Pryadko

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,
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B 57, 1422 (1998); S.-C. Zhang, Science 275, 1089 (1997);
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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
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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.
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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
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amodestenhancement. [9] R. R. P. Singh and R. L. Glenister, Phys. Rev. B 46, 11871
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(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-
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