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Journal of Physics A Mathematical and General

On the variation of the critical exponent γ with spin

J. L. Gammel, D C Power

Published Dec 1, 1983• 1 citation
Publisher

Abstract

The rational approximation method is used to examine estimates for the first and second derivatives of the critical exponent of the Ising model with respect to a variable related to the spin on the three-dimensional body centred cubic lattice. A variation of the method of critical point renormalisation is used to eliminate bias due to uncertainty in the location of the critical point. The results support the concepts of universality of gamma with respect to the spin.

DOI
10.1088/0305-4470/16/17/004

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The rational approximation method is used to examine estimates for the first and second derivatives of the critical exponent of the Ising model with respect to a variable related to the spin on the three-dimensional body centred cubic lattice. A variation of the method of critical point renormalisation is used to eliminate bias due to uncertainty in the location of the critical point. The results support the concepts of universality of gamma with respect to the spin.
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.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.