Further extensions of the hightemperature expansions for the twodimensional classical XY model on the triangular and the square lattices
ABSTRACT The hightemperature expansions for the spinspin correlation function of the twodimensional 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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Article: The two dimensional XY model at the transition temperature: A high precision Monte Carlo study
[show abstract] [hide abstract]
ABSTRACT: We study the classical XY (plane rotator) model at the KosterlitzThouless phase transition. We simulate the model using the single cluster algorithm on square lattices of a linear size up to L=2048.We derive the finite size behaviour of the second moment correlation length over the lattice size xi_{2nd}/L at the transition temperature. This new prediction and the analogous one for the helicity modulus are confronted with our Monte Carlo data. This way beta_{KT}=1.1199 is confirmed as inverse transition temperature. Finally we address the puzzle of logarithmic corrections of the magnetic susceptibility chi at the transition temperature. Comment: Monte Carlo results for xi/L in table 1 and 2 corrected. Due to a programming error,these numbers were wrong by about a factor 1+1/L^2. Correspondingly the fits with L_min=64 and 128 given in table 5 and 6 are changed by little.The central results of the paper are not affected. Wrong sign in eq.(52) corrected. Appendix extendedJournal of Physics A General Physics 02/2005;  SourceAvailable from: arxiv.org[show abstract] [hide abstract]
ABSTRACT: The massive continuum limit of the (1+1)dimensional O(2) nonlinear σmodel (XYmodel) is studied using its equivalence to the sineGordon model at its asymptotically free point. It is shown that leading lattice artifacts are universal but they vanish only as inverse powers of the logarithm of the correlation length. Such leading artifacts are calculated for the case of the scattering phase shifts and the correlation function of the Noether current using the bootstrap Smatrix and perturbation theory respectively.Journal of Physics A General Physics 06/2001; 34(25):5237.
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arXiv:0806.1496v1 [heplat] 9 Jun 2008
BicoccaFTxxyy June 2007
Further extensions of the hightemperature expansions for the
twodimensional classical XY model on the triangular
and the square lattices
P. Butera*1and M. Pernici**2
1Istituto Nazionale di Fisica Nucleare
Sezione di MilanoBicocca
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 hightemperature expansions of the spinspin correlation function of the twodimensional
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, Nvector model, hightemperature expansions
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We have recently extended1, from the 21th to the 24th order, the hightemperature
(HT) expansions for the spinspin correlation function of the classical twodimensional XY
BerezinskiiKosterlitzThouless(BKT) lattice model2with nearestneighbor interactions on
the square(sq) lattice and briefly updated the series analysis for the susceptibility and the
secondmoment correlationlength. 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 nongraphical recursive algorithm3,4based on the SchwingerDyson equations
for the spinspin correlations of the XYmodel. The necessary computations were performed
using a few nodes of a pc cluster for an equivalent singleprocessor CPUtime of seven
weeks in the case of the tr lattice and a CPUtime 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
closelypacked.
Let us recall that the lattice XY spin model with nearestneighbor interactions is described
by the Hamiltonian
H{v} = −2J
?
<nn>
? v(? r) ·? v(? r′)(1)
where ? v(? r) is a twocomponent classical unit vector at the site ? r and the sum extends to the
nearestneighbor sites of the lattice.
We have calculated the spinspin 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 nontrivial within
the maximum order reached. In terms of these quantities, we can form the expansions of
the lth order spherical moments of the correlation function:
m(l)(β) =
?
? x
? xl< ? v(?0) ·? v(? x) >(3)
and, in particular, of the reduced ferromagnetic susceptibility χ(β) = m(0)(β).
The secondmoment 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 nearestneighbor
spinspin 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 highorder 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 renormalizationgroup 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 nonuniversal (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 correlationfunction exponent and is predicted to take the
value η = 1/4.
At the critical inverse temperature β = βc, the asymptotic behavior of the twospin
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 renormalizationgroup analysis9of the twodimensional O(2) nonlinear
σmodel, 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 lowerorder coefficients of the sqlattice 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 nearestneighbor spinspin 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
inhomogeneousdifferentialapproximant13(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 βaxis 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 quasidiagonal 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 24th 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 scatterplot showing the distribution of the nearby complex
singularities for a large class of inhomogeneous firstorder 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 complexconjugate 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 sqlattice
twodimensional O(N) σmodel14in 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 inversetemperature βc= 0.3412(4),
sizably improving the precision of our 14thorder estimate βc= 0.340(1) reported in Ref.[4],
but disagreeing with the DA estimate βc = 0.33986(4) obtained in the 15thorder 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 sqlattice
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 βplane is much simpler than that observed in the case of the sq lattice and
consists of a single pair of complexconjugate 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 trlattice 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 correlationmoments), 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
nonuniversal 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 logderivative, 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 MilanoBicocca INFN Section. We thank the
Physics Depts. of MilanoBicocca 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
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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).
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41, 11494, (1990); P. Butera, and M. Comi, ibid 47, 11969 (1993);
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Butera, and M. Comi, Phys. Rev. B 50, 3052 (1994).
5A. J. Guttmann, J. Phys. A 20, 1855 (1987).
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7R. Kenna, condmat/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).
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13A. J. Guttmann, in “Phase Transitions and Critical Phenomena”, edited by C. Domb and
J. Lebowitz (Academic, New York 1989), vol. 13.
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15P. Butera, and M. Comi, Phys. Rev. B 54, 15828 (1996).
6
Page 7
FIG. 1: XY model on the sq lattice. Distribution of the singularities of a class of secondorder
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.
7
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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 temperaturebiased 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 firstorder
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 secondorder 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 nearestneighbor
spinspin 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