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

Dynamical Properties of the Three-Dimensional XY Model

R.V. Ditzian, Donald D. Betts

Published Jan 15, 1972• 3 citations
Publisher

Abstract

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.

DOI
10.1139/p72-024

Related papers

Dynamical Properties of the Three-Dimensional XY Model. II. Singularity Structure of the Frequency Dependent Susceptibility

S. R. Mattingly, Donald D. Betts

Twenty coefficients are presented in the exact high frequency, high temperature expansion of the perpendicular susceptibility of the spin-1/2 XY model on the f.c.c. lattice. Padé approximant analysis reveals a weak high frequency, temperature independent singularity at ω ≈ (14 ± 2)J/h which we identify with a "quasi-atomic" resonance at ω = 12J/h. We also find a strong singularity at a lower frequency which decreases with decreasing temperature, and for which we have not yet found a satisfying explanation.

Canadian Journal of Physics•Oct 15, 1972•3 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

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 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

CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET

Donald D. Betts, R.V. Ditzian, C. J. Elliott +1 more

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.

Le Journal de Physique Colloques•Feb 1, 1971•0 citations

Critical exponents of the three-dimensional classical plane-rotator model on the sc lattice from a high-temperature-series analysis

P. Butera, M. Comi, Anthony John Guttmann

High-temperature-series expansions of the spin-spin correlation function for the plane-rotator (or XY) model on the sc lattice are extended by three terms through order ${\mathrm{\ensuremath{\beta}}}^{17}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order l=0,1,2. Our analysis of the series leads to fairly accurate estimates of the critical parameters.

arXiv:hep-lat/9305002•Physical review. B, Condensed matter•Nov 1, 1993•15 citations
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.
Twenty coefficients are presented in the exact high frequency, high temperature expansion of the perpendicular susceptibility of the spin-1/2 XY model on the f.c.c. lattice. Padé approximant analysis reveals a weak high frequency, temperature independent singularity at ω ≈ (14 ± 2)J/h which we identify with a "quasi-atomic" resonance at ω = 12J/h. We also find a strong singularity at a lower frequency which decreases with decreasing temperature, and for which we have not yet found a satisfying explanation.
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.
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.
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.