# Further extensions of the high-temperature expansions for the two-dimensional classical XY model on the triangular and the square lattices

**ABSTRACT** The high-temperature expansions for the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model are extended by two terms, from order 24 through order 26, in the case of the square lattice, and by five terms, from order 15 through order 20, in the case of the triangular lattice. The data are analyzed to improve the current estimates of the critical parameters of the models.

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**ABSTRACT:**High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions d=1, 2, and 3, are expressed in terms of the two variables K1=J1/kT and K2=J2/kT, where J1 and J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the d=3 uniaxial XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter R=J2/J1. Universality with respect to R on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination γ=1.3177(5) and ν=0.6726(8) of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional XY universality class, can be achieved. For the exponents at the multicritical (m,d,N)=(1,3,2) Lifshitz point the estimates γl=1.535(25), ν⊥=0.805(15), and ν∥=0.40(3) are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.Physical review. B, Condensed matter 08/2008; 78(5). · 3.77 Impact Factor

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arXiv:0806.1496v1 [hep-lat] 9 Jun 2008

Bicocca-FT-xx-yy June 2007

Further extensions of the high-temperature expansions for the

two-dimensional classical XY model on the triangular

and the square lattices

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

1Istituto Nazionale di Fisica Nucleare

Sezione di Milano-Bicocca

3 Piazza della Scienza, 20126 Milano, Italy

2Istituto Nazionale di Fisica Nucleare

Sezione di Milano

16 Via Celoria, 20133 Milano, Italy

(Dated: June 9, 2008)

Abstract

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

classical XY (planar rotator) model are extended by two terms, from order 24 through order 26,

in the case of the square lattice, and by five terms, from order 15 through order 20, in the case

of the triangular lattice. The data are analyzed to improve the current estimates of the critical

parameters of the models. We determine βc= 0.5599(7) and σ = 0.499(5) in the case of the square

lattice. For the triangular lattice case, we estimate βc= 0.3412(4) and σ = 0.500(5).

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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We have recently extended1, from the 21th to the 24th order, the high-temperature

(HT) expansions for the spin-spin correlation function of the classical two-dimensional XY

Berezinskii-Kosterlitz-Thouless(BKT) lattice model2with nearest-neighbor interactions on

the square(sq) lattice and briefly updated the series analysis for the susceptibility and the

second-moment correlation-length. Here we present a further extension of the HT expansions

from the 24th order through the 26th order for the sq lattice and from the 15th order to the

20th order in the case of the triangular(tr) lattice. In our derivation of the series coefficients,

we have used a non-graphical recursive algorithm3,4based on the Schwinger-Dyson equations

for the spin-spin correlations of the XY-model. The necessary computations were performed

using a few nodes of a pc cluster for an equivalent single-processor CPU-time of seven

weeks in the case of the tr lattice and a CPU-time ten times as long in the case of the

sq lattice. Arguing as in Ref.[5] that the “effective length” (namely the ability to provide

accurate numerical information) of the HT expansions of a given system on different lattices,

is proportional to the computer time used in the series generation, we expect that the

“effective length” of our expansions for the tr lattice, is only slightly smaller than that of the

sq lattice series and moreover that their convergence will be faster, because the tr lattice is

closely-packed.

Let us recall that the lattice XY spin model with nearest-neighbor interactions is described

by the Hamiltonian

H{v} = −2J

?

<nn>

? v(? r) ·? v(? r′)(1)

where ? v(? r) is a two-component classical unit vector at the site ? r and the sum extends to the

nearest-neighbor sites of the lattice.

We have calculated the spin-spin correlation function

C(?0,? x;β) =< ? v(?0) ·? v(? x) >, (2)

as series expansion in the variable β = 2J/kT, with T the temperature and k the Boltzmann

constant, for all values of ? x for which the HT expansion coefficients are non-trivial within

the maximum order reached. In terms of these quantities, we can form the expansions of

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

m(l)(β) =

?

? x

|? x|l< ? v(?0) ·? v(? x) >(3)

and, in particular, of the reduced ferromagnetic susceptibility χ(β) = m(0)(β).

The second-moment correlation length is defined, as usual, in terms of m(2)(β) and χ(β):

ξ2(β) = m(2)(β)/4χ(β). (4)

After completing our computations, we have been kindly informed by H. Arisue6that,

in the particular case of the sq lattice, he has recently obtained, using the “finite lattice”

method, remarkably longer HT expansions: through the 38th order for the nearest-neighbor

spin-spin correlation C(0,0;1,0;β) and through the 34th order for the susceptibility and the

second and fourth moment of the correlation function.

When also these further extended data will be published, the agreement, through their

common extent, between both sets of results independently obtained by completely different

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methods, will be a significant check of correctness in consideration of the notoriously intricate

nature of the high-order HT computations.

In order to determine the critical parameters of the models, the HT expansions of the

above defined quantities should be confronted to the following main predictions2,7of the

BKT renormalization-group analysis of the XY system.

As β → βc, the divergence of the correlation length is expected to be dominated by the

characteristic singularity

ξ(β) ∼ ξas(β) = exp(bτ−σ)[1 + O(τ)](5)

where τ = 1 − β/βcand the universal exponent σ is expected to take the value σ = 1/2,

while b is a non-universal (and thus lattice dependent) positive constant.

The critical behavior of the susceptibility is predicted to be

χ(β) ∼ ξ2−η

as(β) = exp

?

(2 − η)bτ−σ?

[1 + O(τ)](6)

The parameter η represents the correlation-function exponent and is predicted to take the

value η = 1/4.

At the critical inverse temperature β = βc, the asymptotic behavior of the two-spin

correlation function as |? x| = r → ∞ is expected8to be

< ? v(?0) ·? v(? x) >∼(lnr)2θ

rη

[1 + O(lnlnr

lnr)](7)

The value θ = 1/16 is predicted for the second universal exponent characterizing the

critical correlation function.

Following a recent renormalization-group analysis9of the two-dimensional O(2) nonlinear

σ-model, which should belong to the same universality class as the XY model, we shall

assume that the critical behavior of χ(β) does not contain singular multiplicative corrections

by powers of ln(ξ) (or equivalently singular multiplicative corrections by powers of τ). The

possible existence of such corrections has long been numerically investigated7,10,11,12with

conflicting and essentially inconclusive results. As we will indicate below, our series analysis

is completely consistent with the conclusions of Ref.[9].

Let us now come to our results for the sq lattice: we have added the two terms

χ(β) = ... +376988970189597090587

384296140800

β25+62337378385915430773643

31384184832000

β26+ ...(8)

to the expansion of the susceptibility.

To the expansion of the second moment of the correlation function, we have added the

terms

m(2)(β) = ...+5412508223507386985733313

13450364928000

β25+7139182711315236460182251

7846046208000

β26+... (9)

The lower-order coefficients of the sq-lattice expansions of χ(β) and of m(2)(β) have been

recently tabulated in Ref.[1] and therefore they will not be reproduced here.

On the contrary, we have listed in Table I also the coefficients already known4,12through

15th order in the case of the tr lattice, in addition to the five coefficients that we have

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recently calculated for the nearest-neighbor spin-spin correlation, the susceptibility and the

second moment of the correlation function.

Let us now update the analysis of Refs.[1,4] for the sq lattice expansions by including

all coefficients that we have so far derived.

singularity suggests that the critical parameters should be conveniently obtained by an

inhomogeneous-differential-approximant13(DA) study of the location and the exponent of

the leading singularity of the quantity ln2(χ). This conclusion is supported also by a simple

comparison of the distribution of the singularities on the real β-axis for the DAs of χ(β) and

of ln2(χ) which, independently of theoretical prejudice, suggests that the analytic structure

of the latter form is more suitable to a study by DAs.

particular function of χ(β) does not imply any biasing of σ, as long as we do not make any

selection of the singularities of the DAs for ln2(χ) on the basis of their exponents. In our

previous note1, we had already observed that a possibly more accurate analysis might be

based on the obvious remark that, near the critical point, by eq. (6), we have to expect

ln(χ) = csq

can determine csq

function ln2(χ), defined by L(a,χ) = (a + ln(χ))2, where a is some constant. The relative

strength of the τ−σand τ−2σsingularities in the function L(a,χ) is determined by the value

of a. Therefore it might be numerically convenient to analyze the function L(a,χ) with

a ≃ −csq

then be τ−2σ, with a small or vanishing τ−σcontribution and thus approximately a simple

pole, if σ ≃ 1/2. We can therefore expect that analyzing the DAs of L(−csq

possible to determine with good accuracy not only the position, (which is not very sensitive

to the choice of a), but also the exponent of the critical singularity.

In our analysis of the HT series, we have restricted to a class of quasi-diagonal second-

order DAs quite similar to that used in Ref.[1], namely to the [k,l,m;n] DAs defined by the

conditions: 20 ≤ k+l+m+n ≤ 24 with k ≥ 6;l ≥ 6;m ≥ 5 and such that |k−l|,|l−m| ≤ 2,

and 1 < n < 7. For the shorter tr lattice series, these conditions have to be modified in an

obvious way. We have always made sure that our final estimates, within a small fraction of

their spread, are independent of the precise definition of the DA class examined.

In the case of the sq lattice, from an unbiased DA analysis of ln2(χ) we obtain βc =

0.5599(7), in complete agreement with the results of our previous analysis of the 24-th order

series and also with the MonteCarlo analysis of Ref.[11] which yielded βc = 0.55995(15).

Notice that the uncertainty of this estimate was not given explicitly in Ref.[11], and that

the value that we have nevertheless indicated is only a reasonable guess. We have obtained

the estimate σ = 0.499(5) for the singularity exponent by DAs of L(−csq

same class defined above and biased with the value of βc= 0.55995 given in Ref.[11]. Its

uncertainty accounts also for the spread in the estimate of the critical temperature used to

bias the DA calculation. These results are illustrated in Fig.1.

In Fig.2, we have compared how the estimate of σ depends on the value βbias

bias the DAs, when either the quantity L(−csq

βbias

c

= βc = 0.55995. Clearly the results obtained from the analysis of ln2(χ) are much

more sensitive to the bias value.

Taking advantage of our extension of the HT series, we can also get some hint of the

analytic structure of the susceptibility in a vicinity of the origin of the complex β plane.

In Fig.3, we have reported a scatter-plot showing the distribution of the nearby complex

singularities for a large class of inhomogeneous first-order DAs of L(−csq

The expected form eq.(5) of the critical

Notice that the choice of this

1/τσ+ csq

2+ ... Assuming σ ≃ 1/2, a simple fit to the asymptotic form of ln(χ),

2≃ −1.5. We are then led to study also the simple generalization of the

2, rather than simply ln2(χ), because the dominant singularity of L(−csq

2,χ) should

2,χ) will make it

2,χ), chosen in the

c

used to

2,χ) or ln2(χ) is analyzed in a vicinity of

2,χ). We have

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discarded only a few spurious real singularities with modulus less than 0.9βc.

notice that, in addition to the physical singularities at ±βcand to many randomly scattered

complex singularities, which we believe to represent only numerical noise, there is evidence

of five pairs of complex-conjugate singularities, located just beyond the convergence disc of

the series. They are indicated by clusters of DA singularities, which are likely to coalesce

around tips of cuts with increasing series order. These results, which do not essentially

depend on the value of the constant a in L(a,χ), are reminiscent of the regular pattern of

the nearby unphysical singularities (cuts) which were exactly determined for the sq-lattice

two-dimensional O(N) σ-model14in the large N limit and conjectured15to exist also for

finite N ≥ 3.

Let us now perform a similar update of the analysis for the tr lattice series. As shown in

Fig.4, an unbiased DA study of ln2(χ) yields a critical inverse-temperature βc= 0.3412(4),

sizably improving the precision of our 14th-order estimate βc= 0.340(1) reported in Ref.[4],

but disagreeing with the DA estimate βc = 0.33986(4) obtained in the 15th-order study

of Ref.[12]. From an analysis of DAs of ln2(χ) biased with the critical temperature βc=

0.3412(4), we are led to the estimate σ = 0.53(2). If, however, in analogy with the sq-lattice

analysis, we consider the quantity L(a,χ) with a = −ctr

asymptotic behavior of ln(χ) as indicated above, we arrive at the estimate σ = 0.500(4).

It is also particularly interesting to notice that the pattern of the singularities of L(a,χ) in

the complex β-plane is much simpler than that observed in the case of the sq lattice and

consists of a single pair of complex-conjugate clusters, located farther from the border of the

convergence disc than in the case of the sq lattice. This is an indication that the convergence

of the tr-lattice HT series will be faster than in the sq lattice case, and so the argument that

the effective lengths of the HT series for the two lattices are comparable, receives further

support.

In spite of the sizable extension of the series, both in the sq and in the tr lattice cases,

the study of the indicator function H(β) = ln(1+m(2)/χ2)/ln(χ) =

ogous functions of different correlation-moments), does not show very sharp improvements

in the accuracy of the determination of the exponent η, for which we consistently obtain the

estimate η = 0.25(2).

Assuming σ = 1/2, essentially the same estimate of η is obtained from a DA analysis

of the series expansions of τσln(χ) = (2 − η)b + O(τσ) and of τσln(ξ2/β)) = 2b + O(τσ)

both in the case of the sq and of the tr lattices. Assuming η = 1/4, we can estimate the

non-universal parameters bsq= 1.77(1) and btr= 1.70(1).

In order to get alternative estimates of η, we may also try to analyze directly the large-

order behavior of the expansion coefficients of χ(β) and of ξ2(β). Assuming σ = 1/2 and

using the known value of βc, we can fit the series coefficients of these quantities to their

expected3asymptotic behavior cn ∼ β−n−1exp[B(n + 1)1/3+ O(n−1/3)] with B = Bχ =

3/2((2−η)b)2/3

(βc/2)1/3

in the case of the susceptibility, while B = Bξ2 =

ξ2(β). Both in the case of the sq and of the tr lattices, using the sets of coefficients of

the susceptibility and of the correlation length, we can estimate η = 0.25(1) from the ratio

Bχ/Bξ2. This estimate is however suspect because, assuming η = 0.25, we can consistently

estimate b = 1.46(10) in the case of the sq lattice and b = 1.36(10) in the case of the tr

lattice. Thus we must suppose that the series coefficients are not yet near enough to their

asymptotic values that the value of b can be reliably estimated by this direct analysis, in

spite of the fact that the ratio Bχ/Bξ2 already takes the expected value.

Finally, it is also interesting to notice that, as indicated in our previous analysis1, a study

We can

2≃ 1.1, obtained from a fit of the

η

2−η+O(τσ) (or of anal-

3/2(2b)2/3

(βc/2)1/3 in the case of

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of the function R(β) = χ(β)/ξ2−η(β), of its logarithm and of its log-derivative, supports

the arguments of Ref.[9] concerning the absence of singular multiplicative corrections by

powers of ln(ξ) to the critical behavior of χ and gives results completely consistent with the

assumption that η = 1/4. Indeed if η ?= 1/4 or if η = 1/4, but there were in χ singular

multiplicative corrections, either Pad´ e approximants or DAs of the above quantities should

detect some singular behavior as β → βc, which, at the present orders of expansion, is not

seen at all.

In conclusion, our recent HT series data confirm the BKT predictions for the critical

behavior of the XY system and confirm or improve the existing estimates of the critical

parameters for both the sq and the tr lattices.

I. ACKNOWLEDGEMENTS

We thank H. Arisue for informing us about his recent results. Our computations have

been performed on the pc cluster Turing of the Milano-Bicocca INFN Section. We thank the

Physics Depts. of Milano-Bicocca University and of Milano University for their hospitality

and support. Our work was partially supported by the MIUR.

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

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

1P. Butera and M. Pernici, Phys. Rev. B 76, 092406 (2007).

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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).

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

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

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).

5A. J. Guttmann, J. Phys. A 20, 1855 (1987).

6H. Arisue, Progr. Theor. Phys. 118, 855 (2007) and unpublished work.

7R. Kenna, cond-mat/0512356 (unpublished); Condensed Matter Phys. 9, 283 (2006).

8D. J. Amit, Y. Goldschmidt and G. Grinstein, J. Phys. A 13, 585 (1980).

9J. Balog, J. Phys. A 34, 5237 (2001); J. Balog, M. Niedermaier, F. Niedermaier, A. Pa-

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

10R. Kenna and A. C. Irving, Phys. Lett. B 351, 273 (1995).

11M. Hasenbusch, J. Phys. A 38, 5869 (2005).

12M. Campostrini, A. Pelissetto, P. Rossi and E. Vicari, Phys. Rev. B 54, 7301 (1996).

13A. J. Guttmann, in “Phase Transitions and Critical Phenomena”, edited by C. Domb and

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

14P. Butera, M. Comi, G. Marchesini and E. Onofri, Nucl.Phys.B 326, 758 (1989).

15P. Butera, and M. Comi, Phys. Rev. B 54, 15828 (1996).

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FIG. 1: XY model on the sq lattice. Distribution of the singularities of a class of second-order

DAs of ln2(χ) vs their position on the β axis (open histogram). The central value of the open

histogram is βc= 0.5599(7). The bin width is 0.0007. The vertical dashed line shows the critical

value βc= 0.55995 indicated by the simulation of Ref.[11], for which one can guess an uncertainty

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

σ obtained from DAs of L(−csq

2,χ), biased with βc= 0.55995, vs their position on the σ axis. The

central value of the hatched histogram is σ = 0.499(5) and the bin width is 0.003.

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FIG. 2: A comparison of the estimates of the exponent σ obtained from a class of DAs of ln2(χ)

and of L(−c2,χ), biased with the inverse critical temperature. Results are shown for both the sq

and the tr lattices. We have varied the value of βbias

of the values βsq

lattice. The temperature-biased exponent estimates are plotted vs x = βbias

the sq lattice and vs x = βbias

c

/βtr

triangles) show the results obtained from the analysis of ln2(χ) in the case of the sq lattice (resp.

tr lattice) and the open squares (resp. open triangles) those obtained from the study of L(−c2,χ)

in the case of the sq lattice (resp. tr lattice).

c

, used to bias the DAs, in a small vicinity

c = 0.3412, in the case of the tr

c = 0.55995, in the case of the sq lattice, and βtr

c

/βsq

c

in the case of

c in the case of the tr lattice. The black squares (resp. black

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FIG. 3: XY model on the sq lattice. A scatter plot of the singularities of a class of first-order

DAs for L(−csq

2,χ) in the complex β plane. Here x = Re(β) and y = Im(β). The central circle

has radius βc. The small circles are drawn to enclose clusters of singularities which are likely to

coalesce around tips of cuts.

FIG. 4: XY model on the tr lattice. Distribution of the singularities of a class of second-order inho-

mogeneous DAs of ln2(χ) versus their position on the β axis. The central value of the distribution

is βc= 0.3412(4). The bin width is 0.0003.

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TABLE I: XY model on the tr lattice. The series expansion coefficients for the nearest-neighbor

spin-spin correlation C(0,0;1,0;β), the reduced susceptibility χ(β) and the second moment of the

correlation function m(2)(β).

order

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

C(0,0;1,0;β)

0

1

2

7

2

5

35

6

14

3

−81

16

−3769

72

−165161

720

−7821

10

−20160371

8640

−27984359

4320

−87289819

5040

−10256893919

226800

−3357272555039

29030400

−1400375733941

4838400

−26431737035251

37324800

−55206137402197

32659200

−6827447251427903

1741824000

−480824970393676609

54867456000

χ(β)

1

6

30

135

570

2306

18083

2

276657

8

777805

6

14339641

30

208590287

120

8995595389

1440

3199713875

144

65793037351

840

165647319078571

604800

4600845479023849

4838400

1983863997387623

604800

24492996075345043

2177280

1671043059049640293

43545600

37817672635562705657

290304000

4025832361031298767249

9144576000

m(2)(β)

0

6

72

579

3834

22520

121754

4952033

8

36001013

12

839474407

60

1264157753

20

400323755461

1440

860379412817

720

12688548065393

2520

3152231835174739

151200

411232110524237321

4838400

826713976281365323

2419200

4220638202244829129

3110400

662836750489256935

124416

2575741048252255298333

124416000

728769389306358221619671

9144576000

10