The critical temperature for the attractive Hubbard model on a square lattice
is determined from the analysis of two independent quantities, the helicity
modulus, $\rho_s$, and the pairing correlation function, $P_s$. These
quantities have been calculated through Quantum Monte Carlo simulations for
lattices up to $18\times 18$, and for several densities, in the
intermediate-coupling regime. Imposing the universal-jump condition for an
accurately calculated $\rho_s$, together with thorough finite-size scaling
analyses (in the spirit of the phenomenological renormalization group) of
$P_s$, suggests that $T_c$ is considerably higher than hitherto assumed.Comment: 5 pages, 6 figures. Accepted for publication in Phys. Rev. 

A. Moreo

B. Berche

D.A. Huse

D.J. Scalapino

D.R. Nelson

F.F. Assaad

G. Litak

H. Shiba

J. Luo

J.A. Wilson

J.M. Kosterlitz

M. Randeria

M.E. Fisher

M.P. Nightingale

M.P.A. Fisher

P. J. H. Denteneer

P.J.H. Denteneer

R. Blankenbecler

R. Gupta

R. Micnas

R. T. Scalettar

R.R. dos Santos

R.T. Scalettar

Raimundo R. dos Santos

S.-Y. Hsu

S.R. White

T. Paiva

Thereza Paiva

English

arXiv

Crossref

Critical temperature for the two-dimensional attractive Hubbard model

Paiva, T.

Santos, R.R. dos

Scalettar, R.T.

Denteneer, P.J.H.

Leiden University Scholary Publications

razilPHYSICAL REVIEW B 69, 184501 ~2004!Critical temperature for the two-dimensional attractive Hubbard modelThereza Paiva,1 Raimundo R. dos Santos,1 R. T. Scalettar,2 and P. J. H. Denteneer31Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, 21945-970 Rio de Janeiro, Rio de Janeiro, B2Department of Physics, University of California, Davis, California 95616-8677, USA3Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands~Received 26 August 2003; revised manuscript received 19 February 2004; published 3 May 2004!The critical temperature for the attractive Hubbard model on a square lattice is determined from the analysisof two independent quantities, the helicity modulusrs and the pairing correlation functionPs . These quantitieshave been calculated through quantum Monte Carlo simulations for lattices up to 18318, and for severaldensities, in the intermediate-coupling regime. Imposing the universal-jump condition for an accurately calcu-latedrs , together with thorough finite-size scaling analyses~in the spirit of the phenomenological renormal-ization group! of Ps , suggests thatTc is considerably higher than hitherto assumed.DOI: 10.1103/PhysRevB.69.184501 PACS number~s!: 74.20.2z, 71.10.Pm, 74.25.Dw, 74.78.2wyntCc-iramishioum-thoreularsu,osefththperths,n,esleityhalfte-e-ergeatan-The attractive Hubbard model1,2 has been successfullused to elucidate a number of important and fundameissues in both conventional and high-temperature~cuprate!superconductivity. The nature of the crossover between Bsuperconductivity~at weak coupling, or small on-site attration! and Bose-Einstein condensation of tightly bound pa~strong coupling! has been shown to be smooth.3,4 The ap-pearance of preformed pairs within a certain range of pareters in the normal phase, especially below a charactertemperature, has been related to pseudogap behavior oftemperature superconductors.5,6 Further, this model allowsone to introduce disorder on the fermionic degreesfreedom7,8 and investigate the behavior near the quantcritical point of the two-dimensional insulatorsuperconductor transition; this provides an alternative todirty-boson picture9 to discuss the universal conductivity.10The attractive Hubbard model with a periodic modulationU has been used to interpret superconductivity in layestructures.11A basic concern has run through many of these calctions, in particular, those based on quantum Monte Ca~QMC! simulations. In two dimensions, there is a consenthat the early QMC phase diagram12,13—in the space of criti-cal temperatureTc , electronic densitŷn&, and magnitude ofthe on-site attractionuUu—is qualitatively correct. Howeversome serious quantitative discrepancies have emergedthe years, pointing towards higher critical temperatures;e.g., the Bogoliubov-Hartree-Fock~BHF! approach of Ref.14. Our purpose here is to examine the dependence oTcwith ^n&, for fixed U, by resorting to a much wider~namely,larger system sizes and several electronic densities! s t ofQMC data, together with alternative procedures to locatecritical temperature. Establishing an accurate value formost fundamental property of the model is important, escially as the physics of variants of the attractive HubbaHamiltonian is explored, and comparisons are made tooriginal system.The model is defined by the HamiltonianH52t (^ i,j &,s~cis† cjs1H.c.!2m(i~ni↑1ni↓!2uUu3(iS ni↑212D S ni↓212D , ~1!0163-1829/2004/69~18!/184501~5!/$22.50 69 1845alSs-ticgh-fefd-losvere,eis-dewherecis (cis† ) destroys~creates! an electron with spins onsite i of a square lattice,̂i,j & denotes nearest-neighbor sitenis[cis† cis , uUu is the strength of the attractive interactioand m is the chemical potential. From now on, all energiare expressed in units of the hopping amplitudet and we alsosetkB51.At half filling ~which corresponds to the particle-hosymmetric point,m50), the degeneracy of charge-denswave ~CDW! and singlet superconducting~SS! correlationsleads to a three-component order parameter;15 the transitiontemperature is therefore suppressed to zero. Away fromfilling, CDW correlations are suppressed and a finitemperature Kosterlitz-Thouless~KT! transition16 into a SSphase takes place;13,17 this phase has only algebraically dcaying correlations for 0,T<Tc . Further, close to half fill-ing an exact mapping onto the two-dimensional Heisenbantiferromagnetic model in a magnetic field leads to12,13 Tc.22pJ/ lnu12^n&u, so thatTc rises sharply from zero as ondopes away from̂n&51.We start by employing the analysis of Ref. 13 to new dfor the SS pairing correlation function,Ps5^D†D1DD†& ~2!withD†51AN (i ci↑†ci↓†. ~3!For 0,T<Tc , one expectsG~r ![^ci↑† ci↓† cj↓cj↑1H.c.&;r 2h(T), ~4!wherer[u i2 j u, andh(T) increases monotonically betweeh(0)50 andh(Tc)51/4.16,18The finite-size scaling behavior ofPs is therefore ob-tained upon integration ofG(r ) over a two-dimensional system of linear dimensionL. One then has13Ps5L22h(Tc) f ~L/j!, L@1,T→Tc1 ~5!with©2004 The American Physical Society01-1thmCn0imransitiesliktizew-lesult-edpseEq..llestl-ef-deel-Inav-here--maydi-PAIVA, DOS SANTOS, SCALETTAR, AND DENTENEER PHYSICAL REVIEW B69, 184501 ~2004!j;expF A~T2Tc!1/2G ; ~6!in the thermodynamic limit, one recoversPs;j7/4. For com-pleteness, one should mention that sinceh→0 asT→0, thesystem displays long-range order in the ground state, soa ‘‘spin-wave scaling’’ is expected to hold,19PsL25uD0u21CL, ~7!whereD0 is the superconducting gap function at zero teperature, andC is a uUu-dependent constant.Similarly to Ref. 13, here we use the determinant QMalgorithm20 to calculatePs . Typically our data have beeobtained after 500 warming-up steps followed by 50 0sweeps through the lattice. The discretized imaginary-tinterval20 was set toDt50.125, which is small enough fothe results not to depend on this choice in any significway.In Fig. 1 we show raw data forPs as a function of (1/T),for fixed density,^n&50.5 andU524, and for differentlattice sizes. The crossover between temperature- andlimited regimes is described by finite-size scaling~FSS!theory21 and appears as a leveling off ofPs below a certaintemperature for each system size. Before a more quantitascaling analysis, we can already see a suggestion thatTc isaround 1/6 from the rawPs data. In general, at temperaturfor which correlations are short ranged, a structure factorPs is independent of lattice size. AsT is decreased, the poinat which the structure factor begins growing with lattice ssignals the temperature at which the correlation lengthj isbecoming large~comparable to the lattice sizeL), thus pro-viding a crude estimate ofTc . The subsequent plateau at lotemperatures occurs whenj@L. This crossover is contemplated by the FSS form, Eq.~5!, which can be invoked todetermineTc by plotting L27/4Ps as a function of ofwFIG. 1. ~Color online! Ps as a function ofb[1/T for ^n&50.5 and different lattice sizesL.18450at-0etze-vee[L exp@2A/(T2Tc)1/2#, at a givenU, for different systemsizes, withTc andA being adjusted to give the best possibdata collapse, as done in Ref. 13. Figure 2 shows the reing scaling plot, in which the valuesA50.4 andTc50.045were determined in Ref. 13. With our substantially increasamount of data points it becomes clear that the data collaonto a single curve with the parametersA andTc of Ref. 13becomes rather unsatisfactory. We furthermore note that~6! is expected to hold only fort&1022 @ t[(T2Tc)/Tc#,see, e.g., Ref. 22. For the value ofTc used in Fig. 2, only fewdata obtained forL54,6,8,10 in Ref. 13 satisfy this criterionWe therefore obtain new values ofA and Tc from ourexpanded data set. We disregard the data from the smasystem sizes,L54 ~which, additionally, has a special topoogy, being equivalent to a 2323232 four-dimensional lat-tice!, L56, L58, and alsoL510; as we will see below, theSS pairing correlation function presents large finite sizefects for these lattice sizes. Furthermore, we only includata points for temperaturesT for which Eq.~6! is expectedto hold ~see above!. Figure 3 clearly shows that, for thlarger latices, our newly determined parametersA50.1 andcritical temperatureTc50.13 render a much better data colapse than the old parameters.The present analysis shows that the estimates ofTc ob-tained in Ref. 13 can be quantitatively quite unreliable.our opinion, this is due to the fact that the finite-size behior of Ps , Eqs.~5! and~6!, follows from an analysis which isvalid only for large enough lattice sizes, since it involves tbinding-unbinding of rather large structures~vortices! in theKT transition.16 Moreover, the parametersA andTc that haveto be found via data fitting both reside in an exponent,sulting in large uncertainties for the individual fitted parameters. Although the lattice sizes used in the present studynot be large enough to determineTc with high accuracy, ourresult is bound to be an improvement and in any case incates that the actualTc may be much larger~by even a factorof 3! than believed so far.FIG. 2. ~Color online! Rescaled Ps as a function of w[Lexp@2A/(T2Tc)1/2# for ^n&50.5 and different lattice sizesL.The values forA andTc are the ones determined in Ref. 13.1-2apdsinaree-ela-,l-tooallatecanl-intemsultsti-edthton-asa-it-CRITICAL TEMPERATURE FOR THE TWO- . . . PHYSICAL REVIEW B 69, 184501 ~2004!This tendency towards higher critical temperaturespears as well in a completely independent analysis, basethe behavior of the helicity modulus~HM!. The latter is ameasure of the response of the system in the ordered phaa ‘‘twist’’ of the order parameter,23 and can be expressedterms of the current-current correlation functionsfollows.24rs5Ds4pe2514@LL2LT#, ~8!whereDs is the superfluid weight, andLL[ limqx→0Lxx~qx ,qy50,vn50! ~9!andLT[ limqy→0Lxx~qx50,qy ,vn50! ~10!are, respectively, the limiting longitudinal and transversesponses, withLxx~qW ,vn!5(,WE0bdteiqW•,WeivntLxx~,W ,t!, ~11!wherevn52npT,Lxx~,W ,t!5^ j x~,W ,t! j x~0,0!&, ~12!wherej x~,W ,t!5eHtF i t(s~c,W 1 x̂,s†c,W ,s2c,W ,s†c,W 1 x̂,s!Ge2Ht~13!is thex component of the current density operator; see R24 for details.At the KT transition, the following universal-jump relation involving the helicity modulus holds:25FIG. 3. ~Color online! Same as Fig. 2, but withA andTc deter-mined from the present data. Inset shows same system sizes, wAandTc from Ref. 13.18450-one tos-f.Tc5p2rs2 , ~14!wherers2 is the value of the helicity modulus just below thcritical temperature. Thus we can obtainTc by plottingrs(T), and looking for the intercept with 2T/p. This proce-dure has been used before, withrs calculated within a BHFapproximation;14,26 since transverse current-current corretions were neglected,rs is likely to have been overestimatedand the ensuingTc’s may have been too high. Here we caculate bothLL and LT by QMC simulations to obtainrsthrough Eq.~8!; a typical example ofrs(T), for ^n&50.5, isshown in Fig. 4. We see that finite-size effects are notdrastic, since all curves cross the straight line within a smrange of temperatures; that is, from Fig. 4 we can estimTc50.1460.02.In order to check the robustness of this method, weextractTc from Ps through a ‘‘phenomenological renormaization group’’ ~PRG! ~Refs. 27 and 28! analysis, providedsome subtleties peculiar to the KT transition are keptmind. Sincej→` for all T,Tc , Eq. ~5! implies that curvesfor L27/4Ps(L,b), when plotted as functions ofb, and fordifferentL, should allmergefor b.bc . Figure 5 shows thatthis characteristic feature only sets in for the largest syssizes, namely,L>12, from which we can inferbc57.560.25; these error bars are somewhat arbitrary, and refrom visual inspection. It should be stressed that this emate forbc agrees remarkably well with the one obtainfrom the helicity modulus, indicating the robustness of boprocedures to extractTc . Interestingly, we should notice thathe curves forL56 and 8crosseach other~as in an ordinarysecond-order transition! at b.7, which is very close tobcestimated from the larger systems. Therefore, within the ctext of PRG, for the smallest sizes a KT transition appearsan ordinary transition, only crossing over to the merging feture for the largest sizes.hFIG. 4. ~Color online! Helicity modulus as a function of temperature for̂ n&50.5 and different lattice sizesL. The straight linecorresponds to 2T/p.1-3leonFtthritaslyteive-istedatathanap-reoan-toeF-y af tdtsPAIVA, DOS SANTOS, SCALETTAR, AND DENTENEER PHYSICAL REVIEW B69, 184501 ~2004!The critical temperature has been estimated for other etronic densities,̂ n&50.1 ~HM and PRG!, 0.3 ~PRG!, 0.7~PRG!, and 0.875~HM and PRG!; all PRG plots display thecrossing and merging tendency observed for^n&50.5. Theresulting phase diagram is shown in Fig. 6; for compariswe also plot the early QMC results,13 the parquet data fromLuo and Bickers,29 and the estimates from the BHapproximation.14 While close to half filling all results~butBHF! are in fair agreement, for larger dopings agreemenonly found between the results from PRG and those fromhelicity modulus. The inescapable conclusion is that the ccal temperature for the superconducting transition in thetractive Hubbard model is actually higher than previouassumed.In summary, we have established very reliable estimafor the critical temperature of the square-lattice attractFIG. 5. ~Color online! Logarithm-linear plot for the rescaledPsas a function ofb, for ^n&50.5 and for different lattice sizesL,symbols are the same as in Fig. 2. The inset shows a blowup oregion centered aboutb57. No parameters are adjusted.hyats-devz,18450c-,isei-t-seHubbard model. This is done by finding quantitative agrement between entirely different procedures to extractTcfrom two independent correlation functions~both computedby determinant QMC!. As a result, the critical temperaturefound to be substantially higher than the currently accepvalues, also obtained using QMC, but with a different danalysis; as expected, they are also substantially lowerestimates obtained within a Hartree-Fock/mean-fieldproximation.The authors are grateful to S. de Queiroz and A. Mofor discussions. T.P. and R.R.dS. acknowledge partial fincial support by Brazilian Agencies FAPERJ, CNPq, Institudo Milênio para Nanocieˆncias/MCT, and Rede Nacional dNanociências/CNPq; R.T.S. acknowledges support by NSDMR-0312261. This research was further supported bjoint CNPq-690006/02-0/NSF-INT-0203837 grant.heFIG. 6. ~Color online! Critical temperature as a function of banfilling, obtained by different methods. All lines through data poinare guides to the eye.ett.on-.R.ev.Lett.Lett.1R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. P62, 113 ~1990!.2J.A. Wilson, J. Phys.: Condens. Matter13, R945~2001!.3A.J. 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Critical temperature for the two-dimensional attractive Hubbard Model

Critical temperature for the two-dimensional attractive Hubbard Model

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