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Theoretical and Mathematical Physics

On the application of field-theory methods in the Ising model II. Calculation of the critical index η

Igor Vasil'evich Volovich, E. A. Dynin, Valentin Anatol'evich Zagrebnov, Valeri P. Frolov

Published Jun 1, 1973• 0 citations
Publisher
DOI
10.1007/bf01094570

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The critical temperatures Tc and the mean-field critical coefficients of the susceptibility χ are calculated for the S=1/2 Heisenberg-Ising model on the simple cubic lattice by the Kikuchi approximation and its simplified versions. χ̄ shows a coherent anomaly. It means that the coherent-anomaly method (CAM) is applicable to evaluate the true critical temperature T*c and the non-classical critical exponent γ for the susceptibility. As a result, the dependence of T*c on an anisotropy parameter η is given. It is shown that the behavior of γ versus η approaches a fiat line step by step as the corrections up to the higher order approximation are taken into account for 0≦η<1. It implies that the anisotropy parameter η of the Heisenberg-Ising model is irrelevant for critical exponents, if η is less than 1, as expected by the renormalization-group argument.
The critical phase surface of a system, in general, can depend on one or more parameters. We show that by calculating the Gini index (ggg) of any suitably defined response function of a system, the critical phase surface can always be reduced to that of a single parameter, starting from g=0g=0g=0 and terminating at g=gfg=g_fg=gf​, where gfg_fgf​ is a universal number for a chosen response function in a given universality class. We demonstrate the construction with analytical and numerical calculations of mean field transverse field Ising model and site diluted Ising model on the Bethe lattice, respectively. Both models have two parameter critical phase surfaces -- transverse field and temperature for the first case and site dilution and temperature in the second case. Both can be reduced to single parameter transition points in terms of the Gini index. We have additionally demonstrated the validity of the method for a mean field two parameter opinion dynamics model that includes a tri-critical point. The method is generally applicable for any multi-parameter critical transition.
The replica Monte Carlo method has been used to investigate the critical behavior of a threedimensional antiferromagnetic Ising model on a body-centered cubic lattice, taking into account interactions of the adjacent behind neighbors. Investigations are carried out for the ratios of the values of exchange interactions behind the nearest and next nearest neighbors k = J 2/J 1 in the range of k ∈ [0.0, 1.0] with the step Δk = 0.1. In the framework of the theory of finite-dimensional scaling the static critical indices of heat capacity α, susceptibility γ, of the order parameter β, correlation radius ν, and also the Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is kept in the interval of k ∈ [0.0, 0.6]. It is established that a nonuniversal critical behavior is observed in the range k ∈ [0.8, 1.0].