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arXiv:0710.0338v1 [hep-lat] 1 Oct 2007

Bicocca-FT-xx-yy June 2007

High-temperature expansions through order 24 for the

two-dimensional classical XY model on the square lattice

P. Butera*1and M. Pernici**2

1Istituto Nazionale di Fisica Nucleare and

Dipartimento di Fisica, Universit` a di Milano-Bicocca

3 Piazza della Scienza, 20126 Milano, Italy

2Istituto Nazionale di Fisica Nucleare and

Dipartimento di Fisica, Universit` a di Milano

16 Via Celoria, 20133 Milano, Italy

(Dated: February 2, 2008)

Abstract

The high-temperature expansion of the spin-spin correlation function of the two-dimensional

classical XY (planar rotator) model on the square lattice is extended by three terms, from order

21 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.Hk

Keywords: XY model, planar rotator model, N-vector model, high-temperature expansions

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Tests of increasing accuracy1of the BKT theory2of the two-dimensional XY model

critical behavior have been made possible by the steady improvements of the computers

performances and the progress in the numerical approximation algorithms. However, the

critical parameters of this model have not yet been determined with a precision comparable

to that reached for the usual power-law critical phenomena, due to the complicated and

peculiar nature of the critical singularities. Therefore any effort at improving the accuracy of

the available numerical methods by stretching them towards their (present) limits should be

welcome. After extending the high-temperature(HT) expansions of the model in successive

steps3from order β10to β21, we present here a further extension by three orders for the

expansions of the spin-spin correlation on the square lattice and perform a first brief analysis

of our data for the susceptibility and the second-moment correlation-length. More results and

further extensions both for the square and the triangular lattice4will be presented elsewhere.

Our study strengthens the support of the main results of the BKT theory already coming

from the analysis of shorter series and suggests a closer agreement with recent high-precision

simulation studies1,5of the model.

The Hamiltonian

H{v} = −2J

?

nn

? v(? r) ·? v(? r′) (1)

with ? v(? r) a two-component unit vector at the site ? r of a square lattice, describes a system

of XY spins with nearest-neighbor interactions.

Computing the spin-spin correlation function,

C(?0,? x;β) =< s(?0) · 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 evaluate

the expansions of the l-th order spherical moments of the correlation function:

m(l)(β) =

?

? x

|? x|l< s(?0) · 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:

ξ2(β) = m(2)(β)/4χ(β). (4)

Our results for the nearest-neighbor correlation function (or energy E per link) are:

E = β +3

2β3+1

26127360β17−1102473407093

1657033646428733

4138573824000

3β5−31

48β7−731

120β9−29239

β19−6986191770643

β23+ O(β25)

1440β11−265427

14370048000β21

5040β13−75180487

645120β15

−

6506950039

2612736000

+

(5)

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For the susceptibility we have:

χ = 1 + 4β + 12β2+ 34β3+ 88β4+658

3

β5+ 529β6+14933

12

β7+5737

2

β8+389393

60

β9

+

2608499

180

66498259799

96768

19830277603399

3110400

811927408684296587

14370048000

83292382577873288741

172440576000

β10+3834323

120

β11+1254799

18

β12+84375807

560

β13+6511729891

20160

β14

+

β15+1054178743699

725760

β18+8656980509809027

653184000

β21+399888050180302157

β16+39863505993331

13063680

β19+2985467351081077

β17

+

108864000

β22+245277792666205990697

1034643456000

β20

+

3448811520

β23

+

β24+ O(β25)

(6)

For the second moment of the correlation function we have:

m2 = 4β + 32β2+ 162β3+ 672β4+7378

3

β5+24772

3

β6+312149

12

β7+ 77996β8

+

13484753

60

3336209179

112

17775777329026559

16329600

206973837048951639371

14370048000

79897272060888843617033

1034643456000

β9+28201211

45

β10+611969977

360

β15+16763079262169

β11+202640986

45

β12+58900571047

5040

β13

+

β14+1721567587879

23040

β18+1697692411053976387

653184000

β21+721617681295019782781

21555072000

β23+2287397511857949924319

90720

β16+5893118865913171

13063680

β19+41816028466101527

6804000

β22

β17

+

β20

+

+

12933043200

β24+ O(β25) (7)

The coefficients of order less than 22 were already tabulated in Refs.3, but for complete-

ness we report all known terms. As implied by eq.(1), the normalization of these series

reduces to that of our earlier papers3by the change β → β/2.

Let us now list briefly the main predictions2of the BKT renormalization-group analysis

to which the HT series should be confronted in order to extract the critical parameters.

As β → βc, the correlation length ξ2(β) = m(2)(β)/4χ(β) is expected to diverge with the

characteristic singularity

ξ2(β) ∝ ξ2

as(β) = exp(b/τσ)[1 + O(τ)] (8)

where τ = 1 − β/βc. The exponent σ takes the universal value σ = 1/2, whereas b is a

nonuniversal positive constant. At the critical inverse temperature β = βc, the asymptotic

behavior of the two-spin correlation function as |? x| = r → ∞ is expected6to be

< s(?0) · s(? x) >∝(lnr)2θ

rη

[1 + O(lnlnr

lnr)] (9)

Universal values η = 1/4 and θ = 1/16 are predicted also for the correlation exponents.

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A simple non-rigorous argument based on eqs. (8) and (9) suggests that, for l > η − 2,

the spherical correlation moment m(l)(β) diverges as τ → 0+with the singularity

m(l)(β) ∝ τ−θξ2−η+l

This argument was challenged7by a recent renormalization group analysis implying that

the logarithmic factor in eq.(9) gives rise to a less singular correction in the correlation

moments, taking, for example in the case of the susceptibility, the form

as

(β)[1 + O(τ1/2lnτ)] (10)

m(0)(β) ∝ ξ2−η

as(β)[1 + cQ] (11)

where Q =

By eqs.(8) and (10), the ratios rn(m(l)) = a(l)

coefficients of the correlation moment m(l)(β), for large n should behave3as

π2

2(ln(ξ)+u)2+ O(ln(ξ)−5) and u is a non universal parameter.

n /a(l)

n+1of the successive HT expansion

rn(m(l)) = βc+ Cl/(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 usual

power-law critical singularities.

To begin with, let us assume that σ = 1/2 as expected, so that ζ = 2/3. Fig.1 gives

a suggestive visual test of the asymptotic behavior of some ratio sequences rn(m(l)) by

comparing them with eq.(12). The four lowest continuous curves interpolating the data

points are obtained by separate three-parameter fits of the ratio sequences rn(χ), rn(m(1/2)),

rn(m(1)) and rn(m(2)) to the asymptotic form a+b/(n+1)2/3+c/(n+ 1) of eq.(12). In the

same figure, the two upper sets of points are obtained by extrapolating the alternate-ratio

sequence for the susceptibility, first in terms of 1/(n+ 1)2/3and then in terms of 1/(n+ 1).

The values of βc indicated by the fits of the ratio sequences, range between 0.5592 and

0.5611.

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) = c1/τσ+ c2+ ... Therefore, if σ = 1/2,

the relative strength of the 1/√τ and 1/τ singularities in the function L(a,β) = (a +

ln(χ))2is determined by the value of the constant a. If we choose a ≈ 1.19, the function

L(a,β) is approximately dominated by a simple pole and we can expect that the differential

approximants (DAs)8will be able to determine with higher accuracy not only the position,

but also the exponent of the critical singularity. Using inhomogeneous second-order DAs

of L(a,β), we can locate the critical singularity at βc= 0.5598(10). By analysing in the

same way the series data truncated to order 21 which were previously available , we would

get the estimate βc= 0.5588(15). A consistent estimate βc= 0.558(2) had been obtained

in earlier independent3,9studies of the same series using Pad´ e approximants or first-order

DAs. Older studies3of slightly shorter series also indicated values of βcin the same range,

but with notably larger uncertainty. Thus our new series results indicate a stabilization

and a sizable reduction of the spread for the βcestimates. Our uncertainty estimates are

generally taken as the width of the distribution of the values of βcin the appropriate class of

DAs. Fig.2 shows the singularity distribution (open histogram) of the set of quasi-diagonal

DAs which yield our new estimate. These are chosen as the approximants [k,l,m;n] with

17 < k+l+m+n < 22. Moreover, we have taken |k−l|,|l−m|,|k−m| < 3 with k,l,m > 3

and 1 < n < 7. The class of DAs can be varied with no significant variation of the final

estimates, for example by further restricting the extent of off-diagonality, or by varying the

minimal degree of the polynomial coefficients in the DAs. No limitations have been imposed

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on the exponents of the singular terms or on the background terms in the DAs in order to

avoid biasing the βcestimates. Should we require that the exponent of the most singular

term in the approximants differs from -1, for example, by less than 20%, we would obtain

βc = 0.5602(5), well within the uncertainty of our previous unrestricted estimate. The

vertical dashed line in Fig.2 shows the value βc= 0.55995 suggested by the simulation of

Ref.5. Although no explicit indication of an uncertainty comes with this estimate, an upper

bound to its error might be guessed from the statement5that the simulation can exclude

values larger or equal than βc= 0.56045 for the inverse critical temperature.

Biasing with βc= 0.5598(10) the set previously specified of second-order DAs of L(a,β),

leads to the exponent estimate σ = 0.50(1). Fig.2 also shows the distribution of the exponent

estimates (hatched histogram) from this biased set. The uncertainty we have reported for σ

accounts not only for the width of its distribution shown in Fig.2, but also for the variation

of its central value as the bias value of βcis varied in the uncertainty interval of the critical

inverse temperature. Essentially the same value of σ would be obtained from the analysis

of a series truncated to order 21.

While, as one should expect, the DA estimate of βcis rather insensitive to the choice of

a, the estimate of the exponent σ and the width of its distribution are fairly improved by

our choice of a. Taking for example a = 0, we would find σ = 0.53(4), which shows how the

convergence of the exponent estimates is slowed down by the more complicated singularity

structure of 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 the

usual indicators are still slightly larger than expected. For example, by studying the function

H(β) = ln(1 + m(2)/χ2)/ln(χ) (or analogous functions of different moments), we can infer

η = 0.260(10). The function7D(β) = ln(χ) − (2 − η)ln(ξ) and its first derivative are also

interesting indicators of the value of η. Taking η = 1/4, Pad´ e approximants and DAs do

not detect any singular behavior of D(β) or of its derivative as β → βc, thus confirming the

complete cancellation of the leading singularity in D(β). Moreover, this behavior seems to

exclude the form eq.(10) of the corrections which implies the presence of weak subleading

singularities, while it is compatible with eq.(11).

In conclusion, our analysis suggests that, in spite of their diversity, the HT extended series

approach and the latest most extensive simulation are competitive and lead to consistent

numerical estimates of the highest accuracy so far possible.

I. ACKNOWLEDGEMENTS

We thank Prof. Ralph Kenna for a useful correspondence. This work was partially

supported by the italian Ministry of University and Research.

*Electronic address: paolo.butera@mib.infn.it

**Electronic address: mario.pernici@mi.infn.it

1R. Kenna, cond-mat/0512356; Condens. Matter Phys. 9, 283 (2006).

2V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 59, 907 (1970); [Sov. Phys. JETP 32, 493 (1971)]; Zh.

Eksp. Teor. Fiz. 61, 1144 (1971); [Sov. Phys. JETP 34, 610 (1973)]; J. M. Kosterlitz and D.

J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz, C 7, 1046 (1974).

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FIG. 1: Ratios of the successive HT-expansion coefficients vs. 1/(n + 1)2/3: for the susceptibility

χ (open circles), for m(1/2)(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 its

leading asymptotic behavior eq.(12). The data points represented by crossed circles are obtained

by extrapolating the sequence of the susceptibility alternate ratios with respect to 1/n2/3, and the

continuous line interpolating them is the result of a two-parameter fit of the last few points to

the 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 black

circles is drawn only as a guide to the eye. The horizontal broken line indicates the critical value

βc= 0.55995 suggested by the simulation of Ref.5

3P. Butera, M. Comi and G. Marchesini, Phys. Rev. B 33, 4725 (1986); ibid. B 40, 534 (1989);

ibid. B 41, 11494, (1990); P. Butera, and M. Comi, Phys. Rev. B 47, 11969 (1993); ibid. B 54,

15828 (1996).

4P. Butera, R. Cabassi, M. Comi and G. Marchesini, Comp. Phys. Comm. 44, 143 (1987); P.

Butera, and M. Comi, Phys. Rev. B 50, 3052 (1994).

5M. Hasenbusch, J. Phys. A 38, 5869 (2005).

6D. J. Amit, Y. Goldschmidt and G. Grinstein, J. Phys. A 13, 585 (1980).

7J. Balog, J. Phys. A 34, 5237 (2001); J. Balog, M. Niedermaier, F. Niedermaier, A. Patrascioiu,

E. Seiler, and P. Weisz, Nucl. Phys. B 618, 315 (2001).

8A. J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and

J. Lebowitz (Academic, New York 1989) , Vol. 13.

9M. Campostrini, A. Pelissetto, P.Rossi and E. Vicari, Phys. Rev. B 54, 7301 (1996).

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FIG. 2: Distribution of singularities for a class of second-order inhomogeneous DAs of L(1.19,β) =

(1.19 + lnχ)2versus their position on the β axis(open histogram). The central value of the open

histogram is βc= 0.5598(10). The bin width is 0.0007. The vertical dashed line shows the critical

value βc= 0.55995 indicated by the simulation of Ref.5for which one can guess an uncertainty at

least twice smaller than ours. The hatched histogram represents the distribution of the exponent

σ obtained from DAs of L(1.19,β) biased with βc= 0.5598, vs. their position on the σ axis. The

central value of the hatched histogram is σ = 0.500(1) and the bin width is 0.0015. The variation

of the central value of σ as βcvaries in its uncertainty interval is 0.01. This value can be taken as

a more reliable estimate of the uncertainty of σ.

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