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International Journal of Modern Physics B

CRITICAL SPIN-1 VERTEX MODELS AND O(n) MODELS

Bernard Nienhuis

Published Apr 1, 1990• 75 citations
Publisher

Abstract

A method is presented by which critical and multicritical points of spin-1 (three-state) vertex models and classical O(n) models on two-dimensional lattices are determined. It is a straightforward generalization of the ideas that earlier led to the determination of critical points and critical exponents of a honeycomb O(n) model. On the square lattice the methods leads to tricritical as well as critical loci. For n=2 a larger critical manifold is found than for other values of n. At the critical and multicritical points thus produced the models turn out to be soluble. The method is applicable to O(n) models and spin-1 vertex models on any planar lattice.

DOI
10.1142/s0217979290000449

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A method is presented by which critical and multicritical points of spin-1 (three-state) vertex models and classical O(n) models on two-dimensional lattices are determined. It is a straightforward generalization of the ideas that earlier led to the determination of critical points and critical exponents of a honeycomb O(n) model. On the square lattice the methods leads to tricritical as well as critical loci. For n=2 a larger critical manifold is found than for other values of n. At the critical and multicritical points thus produced the models turn out to be soluble. The method is applicable to O(n) models and spin-1 vertex models on any planar lattice.
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Critical properties of two-dimensional classical models and (1 + 1)-dimensional quantum models are obtained by solving a general dimerized spin chain. The staggered-electric exponent of the FFF model is derived. The central role of umklapp scattering in controlling renormalization and determining Potts-model critical exponents is pointed out. The thermal exponent of the Potts model is evaluated, and the validity of den Nijs conjecture is established. Logarithmic singularities at the multicritical point are obtained.
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A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.