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Canadian Journal of Physics

Exact high temperature series expansions for the XY model

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

Published Jul 1, 1970• 50 citations
Publisher

Abstract

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.

DOI
10.1139/p70-198

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