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Journal of Physics A Mathematical and General

The critical behaviour of two-dimensional isotropic spin systems

Anthony John Guttmann

Published Mar 1, 1978• 22 citations
Publisher

Abstract

The high-temperature susceptibility and internal energy series have been reexamined for four two-dimensional isotropic spin models on a triangular lattice. They are the planar classical Heisenberg (PCH), infinite spin X-Y, classical Heisenberg and step models. For all four models evidence of a phase transition is found. This evidence is good for the first two models, and weak for the last two. A method of series analysis is developed which permits us to rule out, with some degree of confidence, an algebraic singularity for either the susceptibility or the specific heat for all four models. For the PCH model the susceptibility fits the form suggested by Kosterlitz (1974) while for the X-Y model susceptibility a similar conclusion may be drawn, with a lesser degree of confidence.

DOI
10.1088/0305-4470/11/3/014

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The high-temperature susceptibility and internal energy series have been reexamined for four two-dimensional isotropic spin models on a triangular lattice. They are the planar classical Heisenberg (PCH), infinite spin X-Y, classical Heisenberg and step models. For all four models evidence of a phase transition is found. This evidence is good for the first two models, and weak for the last two. A method of series analysis is developed which permits us to rule out, with some degree of confidence, an algebraic singularity for either the susceptibility or the specific heat for all four models. For the PCH model the susceptibility fits the form suggested by Kosterlitz (1974) while for the X-Y model susceptibility a similar conclusion may be drawn, with a lesser degree of confidence.
For pt.I see J. Phys. C: Solid St. Phys., vol.6, p.2691 (1973). The critical behaviour of the step model of phase transitions on the square, simple cubic and body-centred cubic lattices is investigated by the method of exact series expansions. The model is found to be in agreement with two aspects of universality, namely the dependence of critical exponents on spin-space dimensionality and the lattice independence of critical exponents for fixed lattice and spin-space dimensionality. Series expansions for the three-dimensional susceptibility and specific heats are generated and analysed, and the critical exponents are found to be gamma =1.335+or-0.01 and alpha =-0.04+or-0.10 respectively. In two dimensions the existence and nature of a phase transition remains an open question. No evidence of either a conventional algebraic singularity or a vortex induced essential singularity is found.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
Critical properties of a lattice model of isotropically interacting 'continuous spins' are studied using high-temperature series expansions. Both the spin-spin and spin-field interactions are different from the planar classical Heisenberg model, yet the ground-state symmetry is the same. The model is found to behave in most respects like the planar classical Heisenberg model, and in three dimensions the following values of critical indices are found: gamma =1.32+or-0.04 and alpha =-0.04+or-0.06. In two dimensions there is flimsy evidence for a divergent susceptibility with an exponent gamma approximately=3'. The results support the belief that for finite-range interaction and fixed dimensionality, unrenormalized critical indices are determined by the underlying symmetry.
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 high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.