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

Le Journal de Physique Colloques

CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET

Donald D. Betts, R.V. Ditzian, C. J. Elliott, M. Howard Lee

Published Feb 1, 1971• 0 citations
Publisher

Abstract

The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian [MATH] where the σ's are Pauli matrices defined at the sites i and j of a regular lattice. This model is appropriate for magnetic insulators for which g⊥ >> g|| temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and s.c. lattices and a susceptibility index γ ≈ 4/3. The fourth order fluctuation series yields the estimate Δ ≈ 5/3 from which by the scaling hypothesis all other indices are obtained. A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.

DOI
10.1051/jphyscol:19711122

Related papers

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

Effect of Anisotropy on the Critical Behaviour of Three- Dimensional Heisenberg Ferromagnets

R. Shanker, R. A. Singh

The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.

Zeitschrift für Naturforschung A•Jan 1, 1976•3 citations

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

David Jasnow, Michael Wortis

Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.

Journal of Applied Physics•Mar 1, 1969•2 citations

Critical Properties of the XY Model

Donald D. Betts, M. Howard Lee

The $\mathrm{XY}$ model of a quantum lattice fluid or a ferromagnet is studied by the method of exact high-temperature series expansion. Nine terms are obtained in the free-energy series and seven in the series for the square of the fluctuation in the long-range order. Analysis of these series yields the critical values $\frac{k{T}_{c}}{J}=4.84\ifmmode\pm\else\textpm\fi{}0.06$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, and $\ensuremath{\alpha}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.20$ for the fcc lattice.

Physical Review Letters•Jun 24, 1968•35 citations

Dynamical Properties of the Three-Dimensional XY Model

R.V. Ditzian, Donald D. Betts

The spin-1/2 XY model in three dimensions is a particularly suitable model for the study of dynamical critical properties, both for comparison with experimental systems such as liquid He near Tλ and certain magnetic systems, and for testing of the predictions of dynamic-scaling theory.Sixteen coefficients have been obtained in the high-frequency high-temperature expansion of the perpendicular (order-parameter) susceptibility, [Formula: see text]. Analysis of [Formula: see text] yields estimates of dynamic critical indices not in conflict with experiment or dynamic scaling, but longer series are needed for a definitive comparison.

Canadian Journal of Physics•Jan 15, 1972•3 citations

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

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.

Canadian Journal of Physics•Oct 1, 1979•56 citations
The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian [MATH] where the σ's are Pauli matrices defined at the sites i and j of a regular lattice. This model is appropriate for magnetic insulators for which g⊥ >> g|| temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and s.c. lattices and a susceptibility index γ ≈ 4/3. The fourth order fluctuation series yields the estimate Δ ≈ 5/3 from which by the scaling hypothesis all other indices are obtained. A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.
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.
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
The XY\mathrm{XY}XY model of a quantum lattice fluid or a ferromagnet is studied by the method of exact high-temperature series expansion. Nine terms are obtained in the free-energy series and seven in the series for the square of the fluctuation in the long-range order. Analysis of these series yields the critical values $\frac{k{T}_{c}}{J}=4.84\ifmmode\pm\else\textpm\fi{}0.06$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, and $\ensuremath{\alpha}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.20$ for the fcc lattice.
The spin-1/2 XY model in three dimensions is a particularly suitable model for the study of dynamical critical properties, both for comparison with experimental systems such as liquid He near Tλ and certain magnetic systems, and for testing of the predictions of dynamic-scaling theory.Sixteen coefficients have been obtained in the high-frequency high-temperature expansion of the perpendicular (order-parameter) susceptibility, [Formula: see text]. Analysis of [Formula: see text] yields estimates of dynamic critical indices not in conflict with experiment or dynamic scaling, but longer series are needed for a definitive comparison.
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.