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arXiv:cond-mat/9703065 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Lattice statistics in three dimensions: Solution of layered dimer and layered domain wall models

Vladislav Popkov, Doochul Kim, H. Y. Huang, Fa Yueh Wu

Published Oct 1, 1997• 4 citations
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Abstract

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.

DOI
10.1103/physreve.56.3999

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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.
We present the exact solution of a three-dimensional lattice-statistical model consisting of layers of vertex models coupled with interlayer interactions. For a particular nontrivial interlayer interaction between charge-conserving vertex models and using a transfer matrix approach, we show that the eigenvalues and eigenvectors of the transfer matrix are related to those of the two-dimensional vertex model. As an illustration of the general solution, we analyze the phase transitions in a realistic three-dimensional dimer system and determine its phase diagram and nature of transitions.
The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent α = 12.
We investigate the thermodynamic and critical properties of an interacting-domain-wall model which is derived from the triangular-lattice antiferromagnetic Ising model with the anisotropic nearest- and next-nearest-neighbor interactions. The model is equivalent to the general five-vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovksy-Talapov transitions. The phase diagram exhibits commensurate phases where the domain-wall density q is locked at the values of 0, 1/2, and 1. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite-size-scaling predictions of the conformal-field theory. It takes the value 1/2 in the noninteracting limit and also at the boundaries of q=0 or 1 phase and the value 2 at the boundary of q=1/2 phase, while it varies smoothly throughout the IC region.
An anisotropic statistical model on a cubic lattice consisting of locally interacting six-vertex planes solvable via Bethe ansatz (BA), is studied. Symmetries of BA lead to an infinite hierarchy of possible phases, which is further restricted by numerical simulations. The model is solved for an arbitrary value of the interlayer coupling constant. Resulting is the phase diagram in general three-parameter space. Two new phases of chiral (spiral) character and a new first-order phase transition appear due to the interplane interaction. Exact mapping onto the models with some inhomogeneous sets of interlayer coupling constants is established.
Solvable via Bethe Ansatz (BA) anisotropic statistical model on cubic lattice consisting of locally interacting 6-vertex planes, is studied. Symmetries of BA lead to infinite hierarchy of possible phases, which is further restricted by numerical simulations. The model is solved for arbitrary value of the interlayer coupling constant. Resulting is the phase diagram in general 3-parameter space. Exact mapping onto the models with some inhomogenious sets of interlayer coupling constants is established.
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.