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Journal of Physics A Mathematical and General

q colourings of the triangular lattice

R. J. Baxter

Published Oct 1, 1986• 142 citations
Publisher

Abstract

Nienhuis (1982,4) has shown that the critical O(n) model, on the honeycomb lattice is equivalent to a zero-temperature antiferromagnetic Potts model on the triangular lattice, i.e. to the chromatic polynomial of the triangular lattice. Here the critical O(n) model is solved by the Bethe ansatz method, thereby giving the large-lattice limit of the chromatic polynomial.

DOI
10.1088/0305-4470/19/14/019

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Nienhuis (1982,4) has shown that the critical O(n) model, on the honeycomb lattice is equivalent to a zero-temperature antiferromagnetic Potts model on the triangular lattice, i.e. to the chromatic polynomial of the triangular lattice. Here the critical O(n) model is solved by the Bethe ansatz method, thereby giving the large-lattice limit of the chromatic polynomial.
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A seven-vertex model on the honeycomb lattice is solved exactly by the Bethe ansatz method. The vertex model is equivalent to the critical O(n) model on the honeycomb lattice. The equivalence is made exact for lattices wrapped on a cylinder with one finite and one infinite direction, by the introduction of a seam into the vertex model. Thus asymptotic finite-size amplitudes of the critical O(n) model are obtained exactly. Applications of the theory of conformal invariance, which relate these amplitudes to the central charge and critical exponents, confirm the existing results and conjectures for these quantities. The finite-size amplitude corresponding to the temperature exponent was not obtained from the exact solution. However, this amplitude was accurately determined from numerical finite-size results obtained by the transfer-matrix method. This result also agrees with recent predictions.
We study the O(N) loop model on the Honeycomb lattice with real value N≥1N \geq 1N≥1 by means of a cluster algorithm. The formulation of the algorithm is based on the equivalence of the O(N) loop model and the low-temperature graphical representation of a NNN-color Ashkin-Teller model on the triangular lattice. The latter model with integer NNN can be simulated by means of an embedding Swendsen-Wang-type cluster method. By taking into account the symmetry among loops of different colors, we develop another version of the Swendsen-Wang-type method. This version allows the number of colors NNN to take any real value N≥1N \geq 1N≥1. As an application, we investigate the N=1.25,1.50,1.75N=1.25, 1.50, 1.75N=1.25,1.50,1.75, and 2 loop model at criticality. The determined values of various critical exponents are in excellent agreement with theoretical predictions. In particular, from quantities associated with half of the loops, we determine some critical exponents that corresponds to those for the tricritical q=N2q=N^2q=N2 Potts model but have not been observed yet. Dynamic scaling behavior of the algorithm is also analyzed. The numerical data strongly suggest that our cluster algorithm {\it hardly} suffers from critical slowing down.
Abstract We study the phase diagram of the triangular-lattice Q -state Potts model in the real ( Q , v ) -plane, where v = e J − 1 is the temperature variable. Our first goal is to provide an obviously missing feature of this diagram: the position of the antiferromagnetic critical curve. This curve turns out to possess a bifurcation point with two branches emerging from it, entailing important consequences for the global phase diagram. We have obtained accurate numerical estimates for the position of this curve by combining the transfer-matrix approach for strip graphs with toroidal boundary conditions and the recent method of critical polynomials. The second goal of this work is to study the corresponding A p − 1 RSOS model on the torus, for integer p = 4 , 5 , … , 8 . We clarify its relation to the corresponding Potts model, in particular concerning the role of boundary conditions. For certain values of p , we identify several new critical points and regimes for the RSOS model and we initiate the study of the flows between the corresponding field theories.