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Journal of Physics A Mathematical and General

A relation between the temperature exponents of the eight-vertex and q-state Potts model

M. P. M. den Nijs

Published Oct 1, 1979• 529 citations
Publisher

Abstract

The mapping of the critical points in the q-state model for q<or=4 onto the Baxter line in the eight-vertex model makes it possible, by a comparison of the exactly known critical exponent y T 8v with the approximately known values for y T P , to conjecture the relation (y T 8v -2)(y T P -3)=3. This relation confirms weak universality.
DOI
10.1088/0305-4470/12/10/030

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Results on the critical temperature of the eight-vertex model in any dimension are combined with the Griffiths inequalities for correlations in Ising ferromagnets to yield relations between critical exponents of eight-vertex models of differing dimensionality. Then Baxter's exact results for the two-dimensional, eight-vertex model are able to yield information on the three-dimensional exponents of these models. As in the planar eight-vertex models, the exponents in all dimensions probably vary continuously with the energy parameters.
The critical dimensions x CH (q) of the chiral and x CB (q) of the cubic operator in the two-dimensional q-state Potts model satisfy the extended scaling relations x CH =(n 2 -y 2 )/(4x)+x and x CB =(m 2 -y 2 )/(4x)+x with x+y=2, 2 cos( pi y/2)= square root q, n an odd and m an even integer; n=1 and m=0 give the leading exponents. At q=3 x CH is relevant, but takes the special value x T +1=9/5. The crossover exponent at the percolation point in the bond-diluted random Ising model, determined by x CB at q=1, is equal to one. Along the Baxter line in the Ashkin-Teller, eight-vertex and ANNNI model x CH =x T /4+1/x T .
The general eight-vertex model on a square lattice is studied numerically by using the Corner Transfer Matrix Renormalization Group method. The method is tested on the symmetric (zero-field) version of the model, the obtained dependence of critical exponents on model's parameters is in agreement with Baxter's exact solution and weak universality is verified with a high accuracy. It was suggested long time ago that the symmetric eight-vertex model is a special exceptional case and in the presence of external fields the eight-vertex model falls into the Ising universality class. We confirm numerically this conjecture in a subspace of vertex weights, except for two specific combinations of vertical and horizontal fields for which the system still exhibits weak universality.