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Physical Review Letters

Eight-Vertex Model of Two-Dimensional Domain Walls

Franz S. Rys

Published Sep 5, 1983• 14 citations
Publisher

Abstract

A statistical model of interacting linear domain walls (occurring, e.g., in monolayer adsorbates) is solved on the square lattice with use of exact and numerical results of an equivalent eight-vertex model. For attractive walls a commensurate and an incommensurate phase are separated by a first-order line for stiff walls and by a fluid phase for flexible walls. The phase boundaries with the fluid phase are Ising-like. For repulsive stiff walls an intermediate striped phase with a nonuniversal boundary occurs which vanishes for higher flexibilities. Moreover, disorder lines are located.

DOI
10.1103/physrevlett.51.849

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Analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions. We show that both models are equivalent to a five-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions determined as a function of the strength of the interlayer interaction.
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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.