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Journal of Physics A Mathematical and General

Critical behaviour of an isotropic spin system. II

Anthony John Guttmann, A Nymeyer

Published Jun 1, 1978• 11 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/11/6/014

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Journal of Physics C Solid State Physics•Apr 21, 1974•19 citations
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.
Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.
Low temperature Ising model series in the usual variable z = exp(−4 J/kT ) are analysed. The positions of the singularities in the disk|z| <or=|z c | and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.
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 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 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. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.