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Journal of Physics A Mathematical and General

Is the Zamolodchikov model critical?

R. J. Baxter, Peter J. Forrester

Published Jun 21, 1985• 14 citations
Publisher

Abstract

Evidence is presented in favour of the hypothesis that the Zamolodchikov model (an exactly solvable three-dimensional lattice model without any temperature-like parameters) is critical. The evidence is obtained by generalising the Zamolodchikov model to include a temperature-like variable. The magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier. Also the authors show the two-layer Zamolodchikov model corresponds to a critical free-fermion model.

DOI
10.1088/0305-4470/18/9/028

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Journal of Physical Studies•Jan 1, 2001•0 citations
Evidence is presented in favour of the hypothesis that the Zamolodchikov model (an exactly solvable three-dimensional lattice model without any temperature-like parameters) is critical. The evidence is obtained by generalising the Zamolodchikov model to include a temperature-like variable. The magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier. Also the authors show the two-layer Zamolodchikov model corresponds to a critical free-fermion model.
It is shown that the Zamolodchikov model on the cubic lattice of NNN sites is equivalent to a spin model on a lattice of 2N2N2N sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function ZZZ (for NNN large) factors into a product of single-variable functions. These functions, and hence ZZZ, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
Abstract It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘Z-invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.
The spin-3/2 Ising model described by the most general Hamiltonian with up-down symmetry, −βH=Σ〈ij〉{JS i S j +KS 2 i S 2 j +LS 3 i S 3 j +M/2(S i S 3 j +S j S 3 i )}−ΔΣi S 2 i , is investigated on a square lattice. It is shown that this model is reducible to an eight-vertex model on a surface in the parameter space spanned by the coupling constants J, K, L, and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. Consequently, the critical behavior and, in particular, the critical temperature for the second-order phase transitions of the model is found exactly.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.