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Physical Review Letters

Connections between the Critical Behavior of the Planar Model and That of the Eight-Vertex Model

Leo P. Kadanoff

Published Oct 10, 1977• 109 citations
Publisher

Abstract

Lines of critical points which emerge from the two-dimensional planar model's multicritical point are analyzed. These lines seem to be in the same universality class as both the Ashkin-Teller and the eight-vertex (8V) models' critical lines, with the multicritical point being isomorphic with the 8V model at both the points $tanh2\ensuremath{\lambda}=\frac{\sqrt{2}}{2} \mathrm{and} \ensuremath{-}1$. Expansions of critical indices about these values of $\ensuremath{\lambda}$ are performed. If this analysis is right, then this work permits the evaluation of a whole class of interesting correlation functions at the critical point of the $F$ model and that of the four-state Potts model.

DOI
10.1103/physrevlett.39.903

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Lines of critical points which emerge from the two-dimensional planar model's multicritical point are analyzed. These lines seem to be in the same universality class as both the Ashkin-Teller and the eight-vertex (8V) models' critical lines, with the multicritical point being isomorphic with the 8V model at both the points tanh2\ensuremathλ=22and\ensuremath−1tanh2\ensuremath{\lambda}=\frac{\sqrt{2}}{2} \mathrm{and} \ensuremath{-}1tanh2\ensuremathλ=22​​and\ensuremath−1. Expansions of critical indices about these values of \ensuremathλ\ensuremath{\lambda}\ensuremathλ are performed. If this analysis is right, then this work permits the evaluation of a whole class of interesting correlation functions at the critical point of the FFF model and that of the four-state Potts model.
It is widely believed that the critical properties of several planar lattice models, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective Quantum Field Theory obtained as formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones by Kadanoff, [13], and by Luther and Peschel, [16]. 1 Introduction and main results Integrable models in statistical mechanics, like the Ising or the Eight vertex (8V) models in two dimensions, provide conceptual laboratories for the understanding of phase transitions. Integrability is however a rather delicate property requiring very special features, and it is usually lost in more realistic models.
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.
We use Monte Carlo simulations and tensor network methods to study a classical monomer-dimer model on the square lattice with a hole (monomer) fugacity zzz, an aligning dimer-dimer interaction uuu that favors columnar order, and an attractive dimer-dimer interaction vvv between two adjacent dimers that lie on the same principal axis of the lattice. The Monte Carlo simulations of finite size systems rely on our grand-canonical generalization of the dimer worm algorithm, while the tensor network computations are based on a uniform matrix product ansatz for the eigenvector of the row-to-row transfer matrix that work directly in the thermodynamic limit. The phase diagram has nematic, columnar order and fluid phases, and a nonzero temperature multicritical point at which all three meet. For any fixed $v/u<\ensuremath{\infty}$, we argue that this multicritical point continues to be located at a nonzero hole fugacity zmc(v/u)>0{z}_{\mathrm{mc}}(v/u)>0zmc​(v/u)>0; our numerical results confirm this theoretical expectation but find that ${z}_{\mathrm{mc}}(v/u)\ensuremath{\rightarrow}0$ very rapidly as $v/u\ensuremath{\rightarrow}\ensuremath{\infty}$. Our numerical results also confirm the theoretical expectation that the corresponding multicritical behavior is in the universality class of the four-state Potts multicritical point on critical line of the two-dimensional Ashkin-Teller model.