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Physical Review Letters

Remarks on the Eight-Vertex Model and the Ashkin-Teller Model of Lattice Statistics

Chungpeng Fan

Published Jul 17, 1972• 14 citations
Publisher

Abstract

By using a spinor representation, it is shown that the four-component statistical model of Ashkin and Teller is equivalent to a special case of a staggered eight-vertex model, while the eight-vertex model is equivalent to an asymmetrical Ashkin-Teller model.
DOI
10.1103/physrevlett.29.158

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Symmetry Properties of the Ashkin-Teller Model and the Eight-Vertex Model

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A new approach to the statistical model of Ashkin and Teller (AT) is developed with a view to compare it with the recently solved eight-vertex (8-v) model. The four-component nearest-neighbor model of atoms is transcribed into a system of arrow graphs on a square lattice of unit cells. A bidirectional transfer matrix is constructed in terms of spin operators to represent the change in arrow directions at each unit cell. Symmetry properties of the system are considered by means of algebraic analysis on the spin operators. The existence of a dual transform permits determination of the transition temperature in the most general case of the problem. The mathematical condition on the weight factors for locating this transition temperature is found to be identical to the corresponding one for the 8-v model. Furthermore, when expressed as a function of a set of modified weights, the partition function of the AT model also obeys the same symmetry. It is conjectured that the AT model, although possessed with a similar feature of continuously varying critical indices, is topologically distinct from the 8-v model.

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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.

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New model Hamiltonian with four-spin interactions: Phase diagram and formation of domains and walls

Santiago D’Elía, Horacio Ceva

A model Hamiltonian related to the Ashkin-Teller and eight-vertex models is introduced and studied in a two-dimensional square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. It is formulated in terms of a vector variable $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ with four states, and has two-body (${K}_{2}$) and four-body ($\ensuremath{\lambda}$) interactions. If ${K}_{2}=0$, the ground state of this system is highly degenerate and almost all of these states have domains and walls. We also study the region $x\ensuremath{\gg}1$ of the phase diagram ${K}_{2}={K}_{2}(x)$, where $x=\frac{\ensuremath{\lambda}}{{K}_{2}}$, obtaining conditions for the existence of these walls. We find that for $\frac{\ensuremath{\lambda}\ensuremath{\sim}x}{N}\ensuremath{\gg}1$ there is a crossover toward an Ising-type ordering, without walls.

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Eight-Vertex Model in Lattice Statistics and One-Dimensional Anisotropic Heisenberg Chain: II. Equivalence to a Generalized Ice-type Lattice Model

R. J. Baxter

Abstract We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.

Advanced series in mathematical physics•Mar 1, 1990•0 citations
A new approach to the statistical model of Ashkin and Teller (AT) is developed with a view to compare it with the recently solved eight-vertex (8-v) model. The four-component nearest-neighbor model of atoms is transcribed into a system of arrow graphs on a square lattice of unit cells. A bidirectional transfer matrix is constructed in terms of spin operators to represent the change in arrow directions at each unit cell. Symmetry properties of the system are considered by means of algebraic analysis on the spin operators. The existence of a dual transform permits determination of the transition temperature in the most general case of the problem. The mathematical condition on the weight factors for locating this transition temperature is found to be identical to the corresponding one for the 8-v model. Furthermore, when expressed as a function of a set of modified weights, the partition function of the AT model also obeys the same symmetry. It is conjectured that the AT model, although possessed with a similar feature of continuously varying critical indices, is topologically distinct from the 8-v 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 model Hamiltonian related to the Ashkin-Teller and eight-vertex models is introduced and studied in a two-dimensional square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. It is formulated in terms of a vector variable $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ with four states, and has two-body (K2{K}_{2}K2​) and four-body ($\ensuremath{\lambda}$) interactions. If K2=0{K}_{2}=0K2​=0, the ground state of this system is highly degenerate and almost all of these states have domains and walls. We also study the region $x\ensuremath{\gg}1$ of the phase diagram K2=K2(x){K}_{2}={K}_{2}(x)K2​=K2​(x), where $x=\frac{\ensuremath{\lambda}}{{K}_{2}}$, obtaining conditions for the existence of these walls. We find that for $\frac{\ensuremath{\lambda}\ensuremath{\sim}x}{N}\ensuremath{\gg}1$ there is a crossover toward an Ising-type ordering, without walls.