High-temperature expansions through order 24 for the two-dimensional classical XY model on the square lattice
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High-temperature expansions through order 24 for thetwo-dimensional classical XY model on the square lattice
P. Butera *1 and M. Pernici **2 Istituto Nazionale di Fisica Nucleare andDipartimento di Fisica, Universit`a di Milano-Bicocca3 Piazza della Scienza, 20126 Milano, Italy Istituto Nazionale di Fisica Nucleare andDipartimento di Fisica, Universit`a di Milano16 Via Celoria, 20133 Milano, Italy (Dated: November 1, 2018)
Abstract
The high-temperature expansion of the spin-spin correlation function of the two-dimensionalclassical XY (planar rotator) model on the square lattice is extended by three terms, from order21 through order 24, and analyzed to improve the estimates of the critical parameters.
PACS numbers: PACS numbers: 05.50+q, 11.15.Ha, 64.60.Cn, 75.10.HkKeywords: XY model, planar rotator model, N-vector model, high-temperature expansions of the BKT theory of the two-dimensional XY modelcritical behavior have been made possible by the steady improvements of the computersperformances and the progress in the numerical approximation algorithms. However, thecritical parameters of this model have not yet been determined with a precision comparableto that reached for the usual power-law critical phenomena, due to the complicated andpeculiar nature of the critical singularities. Therefore any effort at improving the accuracy ofthe available numerical methods by stretching them towards their (present) limits should bewelcome. After extending the high-temperature(HT) expansions of the model in successivesteps from order β to β , we present here a further extension by three orders for theexpansions of the spin-spin correlation on the square lattice and perform a first brief analysisof our data for the susceptibility and the second-moment correlation-length. More results andfurther extensions both for the square and the triangular lattice will be presented elsewhere.Our study strengthens the support of the main results of the BKT theory already comingfrom the analysis of shorter series and suggests a closer agreement with recent high-precisionsimulation studies of the model.The Hamiltonian H { v } = − J X nn ~v ( ~r ) · ~v ( ~r ′ ) (1)with ~v ( ~r ) a two-component unit vector at the site ~r of a square lattice, describes a systemof XY spins with nearest-neighbor interactions.Computing the spin-spin correlation function, C ( ~ , ~x ; β ) = < s ( ~ · s ( ~x ) >, (2)(for all values of ~x for which the HT expansion coefficients are non-trivial within the max-imum order reached), as series expansion in the variable β = J/kT , enables us to evaluatethe expansions of the l -th order spherical moments of the correlation function: m ( l ) ( β ) = X ~x | ~x | l < s ( ~ · s ( ~x ) > (3)and in particular the reduced ferromagnetic susceptibility χ ( β ) = m (0) ( β ). In terms of m (2) ( β ) and χ ( β ) we can form the second-moment correlation length: ξ ( β ) = m (2) ( β ) / χ ( β ) . (4)Our results for the nearest-neighbor correlation function (or energy E per link) are: E = β + 32 β + 13 β − β − β − β − β − β − β − β − β + 16570336464287334138573824000 β + O ( β ) (5)2or the susceptibility we have: χ = 1 + 4 β + 12 β + 34 β + 88 β + 6583 β + 529 β + 1493312 β + 57372 β + 38939360 β + 2608499180 β + 3834323120 β + 125479918 β + 84375807560 β + 651172989120160 β + 6649825979996768 β + 1054178743699725760 β + 3986350599333113063680 β + 198302776033993110400 β + 8656980509809027653184000 β + 2985467351081077108864000 β + 81192740868429658714370048000 β + 3998880501803021573448811520 β + 2452777926662059906971034643456000 β + 83292382577873288741172440576000 β + O ( β ) (6)For the second moment of the correlation function we have: m = 4 β + 32 β + 162 β + 672 β + 73783 β + 247723 β + 31214912 β + 77996 β + 1348475360 β + 2820121145 β + 611969977360 β + 20264098645 β + 589005710475040 β + 3336209179112 β + 172156758787923040 β + 1676307926216990720 β + 589311886591317113063680 β + 1777577732902655916329600 β + 1697692411053976387653184000 β + 418160284661015276804000 β + 20697383704895163937114370048000 β + 72161768129501978278121555072000 β + 798972720608888436170331034643456000 β + 228739751185794992431912933043200 β + O ( β ) (7)The coefficients of order less than 22 were already tabulated in Refs. , but for complete-ness we report all known terms. As implied by eq.(1), the normalization of these seriesreduces to that of our earlier papers by the change β → β/ of the BKT renormalization-group analysisto which the HT series should be confronted in order to extract the critical parameters.As β → β c , the correlation length ξ ( β ) = m (2) ( β ) / χ ( β ) is expected to diverge with thecharacteristic singularity ξ ( β ) ∝ ξ as ( β ) = exp ( b/τ σ )[1 + O ( τ )] (8)where τ = 1 − β/β c . The exponent σ takes the universal value σ = 1 /
2, whereas b is anonuniversal positive constant. At the critical inverse temperature β = β c , the asymptoticbehavior of the two-spin correlation function as | ~x | = r → ∞ is expected to be < s ( ~ · s ( ~x ) > ∝ (ln r ) θ r η [1 + O ( lnln r ln r )] (9)Universal values η = 1 / θ = 1 /
16 are predicted also for the correlation exponents.3 simple non-rigorous argument based on eqs. (8) and (9) suggests that, for l > η − m ( l ) ( β ) diverges as τ → + with the singularity m ( l ) ( β ) ∝ τ − θ ξ − η + las ( β )[1 + O ( τ / ln τ )] (10)This argument was challenged by a recent renormalization group analysis implying thatthe logarithmic factor in eq.(9) gives rise to a less singular correction in the correlationmoments, taking, for example in the case of the susceptibility, the form m (0) ( β ) ∝ ξ − ηas ( β )[1 + cQ ] (11)where Q = π ξ )+ u ) + O (ln( ξ ) − ) and u is a non universal parameter.By eqs.(8) and (10), the ratios r n ( m ( l ) ) = a ( l ) n /a ( l ) n +1 of the successive HT expansioncoefficients of the correlation moment m ( l ) ( β ), for large n should behave as r n ( m ( l ) ) = β c + C l / ( n + 1) ζ + O (1 /n ) (12)with ζ = 1 / (1 + σ ), to be contrasted with the value ζ = 1 which is found for the usualpower-law critical singularities.To begin with, let us assume that σ = 1 / ζ = 2 /
3. Fig.1 givesa suggestive visual test of the asymptotic behavior of some ratio sequences r n ( m ( l ) ) bycomparing them with eq.(12). The four lowest continuous curves interpolating the datapoints are obtained by separate three-parameter fits of the ratio sequences r n ( χ ), r n ( m (1 / ), r n ( m (1) ) and r n ( m (2) ) to the asymptotic form a + b/ ( n + 1) / + c/ ( n + 1) of eq.(12). In thesame figure, the two upper sets of points are obtained by extrapolating the alternate-ratiosequence for the susceptibility, first in terms of 1 / ( n + 1) / and then in terms of 1 / ( n + 1).The values of β c indicated by the fits of the ratio sequences, range between 0.5592 and0.5611.A more accurate analysis can be based on the simple remark that, near the critical point,by eq.(8) and (10) (or eq.(11)), one has ln( χ ) = c /τ σ + c + .. . Therefore, if σ = 1 / / √ τ and 1 /τ singularities in the function L ( a, β ) = ( a +ln( χ )) is determined by the value of the constant a . If we choose a ≈ .
19, the function L ( a, β ) is approximately dominated by a simple pole and we can expect that the differentialapproximants (DAs) will be able to determine with higher accuracy not only the position,but also the exponent of the critical singularity. Using inhomogeneous second-order DAsof L ( a, β ), we can locate the critical singularity at β c = 0 . β c = 0 . β c = 0 . studies of the same series using Pad´e approximants or first-orderDAs. Older studies of slightly shorter series also indicated values of β c in the same range,but with notably larger uncertainty. Thus our new series results indicate a stabilizationand a sizable reduction of the spread for the β c estimates. Our uncertainty estimates aregenerally taken as the width of the distribution of the values of β c in the appropriate class ofDAs. Fig.2 shows the singularity distribution (open histogram) of the set of quasi-diagonalDAs which yield our new estimate. These are chosen as the approximants [ k, l, m ; n ] with17 < k + l + m + n <
22. Moreover, we have taken | k − l | , | l − m | , | k − m | < k, l, m > < n <
7. The class of DAs can be varied with no significant variation of the finalestimates, for example by further restricting the extent of off-diagonality, or by varying theminimal degree of the polynomial coefficients in the DAs. No limitations have been imposed4n the exponents of the singular terms or on the background terms in the DAs in order toavoid biasing the β c estimates. Should we require that the exponent of the most singularterm in the approximants differs from -1, for example, by less than 20%, we would obtain β c = 0 . β c = 0 . . Although no explicit indication of an uncertainty comes with this estimate, an upperbound to its error might be guessed from the statement that the simulation can excludevalues larger or equal than β c = 0 . β c = 0 . L ( a, β ),leads to the exponent estimate σ = 0 . σ accounts not only for the width of its distribution shown in Fig.2, but also for the variationof its central value as the bias value of β c is varied in the uncertainty interval of the criticalinverse temperature. Essentially the same value of σ would be obtained from the analysisof a series truncated to order 21.While, as one should expect, the DA estimate of β c is rather insensitive to the choice of a , the estimate of the exponent σ and the width of its distribution are fairly improved byour choice of a . Taking for example a = 0, we would find σ = 0 . L (0 , β ). Similar values of σ were found in previous studies of shorter series.Probably for the same reason, also the central values of the η estimates obtained from theusual indicators are still slightly larger than expected. For example, by studying the function H ( β ) = ln(1 + m (2) /χ ) / ln( χ ) (or analogous functions of different moments), we can infer η = 0 . D ( β ) = ln( χ ) − (2 − η )ln( ξ ) and its first derivative are alsointeresting indicators of the value of η . Taking η = 1 /
4, Pad´e approximants and DAs donot detect any singular behavior of D ( β ) or of its derivative as β → β c , thus confirming thecomplete cancellation of the leading singularity in D ( β ). Moreover, this behavior seems toexclude the form eq.(10) of the corrections which implies the presence of weak subleadingsingularities, while it is compatible with eq.(11).In conclusion, our analysis suggests that, in spite of their diversity, the HT extended seriesapproach and the latest most extensive simulation are competitive and lead to consistentnumerical estimates of the highest accuracy so far possible. I. ACKNOWLEDGEMENTS
We thank Prof. Ralph Kenna for a useful correspondence. This work was partiallysupported by the italian Ministry of University and Research. * Electronic address: [email protected] ** Electronic address: [email protected] R. Kenna, cond-mat/0512356; Condens. Matter Phys. , 283 (2006). V. L. Berezinskii, Zh. Eksp. Teor. Fiz. , 907 (1970); [Sov. Phys. JETP , 493 (1971)]; Zh.Eksp. Teor. Fiz. , 1144 (1971); [Sov. Phys. JETP , 610 (1973)]; J. M. Kosterlitz and D.J. Thouless, J. Phys. C 6 , 1181 (1973); J. M. Kosterlitz,
C 7 , 1046 (1974). IG. 1: Ratios of the successive HT-expansion coefficients vs. 1 / ( n + 1) / : for the susceptibility χ (open circles), for m (1 / (rhombs), for m (1) (squares) and for m (2) (triangles). The four low-est continuous curves are obtained by separate three-parameter fits of each ratio sequence to itsleading asymptotic behavior eq.(12). The data points represented by crossed circles are obtainedby extrapolating the sequence of the susceptibility alternate ratios with respect to 1 /n / , and thecontinuous line interpolating them is the result of a two-parameter fit of the last few points tothe expected asymptotic form a + b/n . The small black circles are obtained by a further extrap-olation of the latter quantities with respect to 1 /n . The continuous line interpolating the blackcircles is drawn only as a guide to the eye. The horizontal broken line indicates the critical value β c = 0 . P. Butera, M. Comi and G. Marchesini, Phys. Rev. B , 4725 (1986); ibid. B , 534 (1989); ibid. B 41 , 11494, (1990); P. Butera, and M. Comi, Phys. Rev. B , 11969 (1993); ibid. B ,15828 (1996). P. Butera, R. Cabassi, M. Comi and G. Marchesini, Comp. Phys. Comm. , 143 (1987); P.Butera, and M. Comi, Phys. Rev. B , 3052 (1994). M. Hasenbusch, J. Phys. A , 5869 (2005). D. J. Amit, Y. Goldschmidt and G. Grinstein, J. Phys. A , 585 (1980). J. Balog, J. Phys. A , 5237 (2001); J. Balog, M. Niedermaier, F. Niedermaier, A. Patrascioiu,E. Seiler, and P. Weisz, Nucl. Phys. B , 315 (2001). A. J. Guttmann, in
Phase Transitions and Critical Phenomena , edited by C. Domb andJ. Lebowitz (Academic, New York 1989) , Vol. 13. M. Campostrini, A. Pelissetto, P.Rossi and E. Vicari, Phys. Rev. B , 7301 (1996). IG. 2: Distribution of singularities for a class of second-order inhomogeneous DAs of L (1 . , β ) =(1 .
19 + ln χ ) versus their position on the β axis(open histogram). The central value of the openhistogram is β c = 0 . β c = 0 . for which one can guess an uncertainty atleast twice smaller than ours. The hatched histogram represents the distribution of the exponent σ obtained from DAs of L (1 . , β ) biased with β c = 0 . σ axis. Thecentral value of the hatched histogram is σ = 0 . σ as β c varies in its uncertainty interval is 0 .
01. This value can be taken asa more reliable estimate of the uncertainty of σ ..