Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Canadian Journal of Physics

The spin model. III. Analysis of high temperature series expansions of some thermodynamic quantities in two dimensions

Jos Rogiers, Ernst W. Grundke, Donald D. Betts

Published Oct 1, 1979• 56 citations
Publisher

Abstract

In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.

DOI
10.1139/p79-237

Related papers

The spin model. II. Analysis of high temperature series expansions of some thermodynamic quantities in three dimensions

Jos Rogiers, Donald D. Betts, T. Lookman

In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.

Canadian Journal of Physics•Apr 1, 1978•15 citations

The spin XY model. IV. Analysis of high temperature series expansions of some thermodynamic quantities in four dimensions

Jos Rogiers

The series for the fluctuation of the order parameter and the fourth-order fluctuation of the order parameter of the S = [Formula: see text] XY model on four-dimensional lattices are analysed. Assuming a simple power law we find γ = 1.125 ± 0.010 for the simple hypercubic and γ = 1.113 ± 0.005 for the face-centred hypercubic. An alternative method of analysis which includes a logarithmic correction factor and assumes γ = 1 gives as power for the logarithmic correction p = 0.36 ± 0.05 for the simple hypercubic lattice.

Canadian Journal of Physics•Jan 1, 1980•0 citations

The spin XY model. V. Moments of the spin–spin correlations in three dimensions

Donald D. Betts, Ernst W. Grundke

Ninth degree high temperature expansions have been obtained for all transverse spin–spin correlations of the [Formula: see text] model on the simple cubic and body centred cubic lattices. Analysis of the series for the second and fourth moments of the correlations yields the estimate of v = 0.689 ± 0.010 and estimates of critical amplitudes. No evidence for violation of hyperscaling is found.

Canadian Journal of Physics•Nov 1, 1980•2 citations

On the critical behaviour of the two dimensional spin XY model

Jos Rogiers

Transformation methods are used to analyse the series for the second order fluctuation of the transverse magnetization for the triangular and square lattices. For the triangular lattice some evidence is found for an exponential behaviour of this quantity near the critical point with a tentative estimate for the exponent [Formula: see text].

Canadian Journal of Physics•Mar 1, 1982•2 citations

Phase Transitions in Two-Dimensional Spin Systems

Donald D. Betts, C. J. Elliott, R.V. Ditzian

The first nine coefficients have been derived in the exact high temperature expansion of the fluctuation in the long range order in the XY model on the triangular and square lattices. For the triangular lattice seven coefficients have also been obtained in the expansion of the fourth-order fluctuation. The method of conformal transformations has been developed in a form suitable for the analysis of series of this type. Clear evidence is obtained in favor of a phase transition for which the critical index of the fluctuations is γ ≈ 3/2. The critical temperatures kTC/J for the triangular and square lattices are 1.500 ± 0.006 and 0.900 ± 0.006 respectively.

Canadian Journal of Physics•May 15, 1971•54 citations

Exact high temperature series expansions for the XY model

Donald D. Betts, C. J. Elliott, M. Howard Lee

A method is developed for the exact high temperature series expansion of thermodynamic properties of the spin 1/2 XY model of ferromagnetism or of a quantum lattice gas. Eleven coefficients in the specific heat series for the f.c.c. and b.c.c. lattices and nine coefficients in the series for the fluctuation in the long range order for the f.c.c., b.c.c., and s.c. lattices are calculated. From the fluctuation series the critical temperatures Kc−1 ≡ kTc/J are estimated to be 4.520, 2.902, and 2.02 respectively, while the critical index for the square of the fluctuation in the long range order is γ = 1.35.

Canadian Journal of Physics•Jul 1, 1970•50 citations
In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.
The series for the fluctuation of the order parameter and the fourth-order fluctuation of the order parameter of the S = [Formula: see text] XY model on four-dimensional lattices are analysed. Assuming a simple power law we find γ = 1.125 ± 0.010 for the simple hypercubic and γ = 1.113 ± 0.005 for the face-centred hypercubic. An alternative method of analysis which includes a logarithmic correction factor and assumes γ = 1 gives as power for the logarithmic correction p = 0.36 ± 0.05 for the simple hypercubic lattice.
The first nine coefficients have been derived in the exact high temperature expansion of the fluctuation in the long range order in the XY model on the triangular and square lattices. For the triangular lattice seven coefficients have also been obtained in the expansion of the fourth-order fluctuation. The method of conformal transformations has been developed in a form suitable for the analysis of series of this type. Clear evidence is obtained in favor of a phase transition for which the critical index of the fluctuations is γ ≈ 3/2. The critical temperatures kTC/J for the triangular and square lattices are 1.500 ± 0.006 and 0.900 ± 0.006 respectively.
A method is developed for the exact high temperature series expansion of thermodynamic properties of the spin 1/2 XY model of ferromagnetism or of a quantum lattice gas. Eleven coefficients in the specific heat series for the f.c.c. and b.c.c. lattices and nine coefficients in the series for the fluctuation in the long range order for the f.c.c., b.c.c., and s.c. lattices are calculated. From the fluctuation series the critical temperatures Kc−1 ≡ kTc/J are estimated to be 4.520, 2.902, and 2.02 respectively, while the critical index for the square of the fluctuation in the long range order is γ = 1.35.