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Journal of Physics A Mathematical and General

Pure and dilute Z(N) spin and generalised gauge lattice systems: duality and criticality

Francisco Castilho Alcaraz, Constantino Tsallis

Published Feb 1, 1982• 22 citations
Publisher

Abstract

Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.

DOI
10.1088/0305-4470/15/2/025

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Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.
The exact solution of a general Z(4) gauge Potts model with a single and double plaquette representation of the action is found on a subspace of gauge-coupling parameters on the square and triangular lattices. The two Ising-type critical lines of a second-order phase transition for the model on a square lattice are found. For the model on a triangular lattice the two critical surfaces of an Ising-type and two nontrivial lines of a second-order phase transition with different critical behavior than on the critical surfaces are found. It is shown that a two-dimensional (2D) general Z(4) gauge Potts model with single and double plaquette representation of the action and a 2D spin-3 / 2 Ising model belong to the same universality class.
A scaling transformation is developed for the pure square-lattice-spin- 1 / 2 Ising model with different exchange couplings J 1 , J 2 in the two lattice directions. The transformation preserves Onsager's exact duality relationship and so can treat cases of strong lattice anisotropy. It is generalised to the dilute case and used to obtain critical exponents and critical curves (transition temperature T c as a function of concentration p of magnetic bonds for various exchange ratios gamma =J 2 /J 1 ). For gamma small, corresponding to weakly coupled chains, the transition temperature falls rapidly with dilution, then levels off, finally vanishing at the two-dimensional percolation concentration. Extensions are made to three-dimensional dilute magnets, including weakly coupled layer and chain magnets as special cases. Critical curves and exponents are calculated, and a quantitative description is obtained of various crosscovers between low- and high-dimensional behaviour which occur in the quasi-low-dimensional magnets as the transition temperature falls with dilution and the weak exchanges become significant.
Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.