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Journal of Physics A Mathematical and General

A soluble seven-vertex model for clusters with interfacial bending energy

Thomas Blum, Yonathan Shapir

Published May 21, 1990• 30 citations
Publisher

Abstract

A model describing an ensemble of disjoint clusters on a square lattice is introduced. In addition to the usual surface energy terms, in x and in y , the model includes a bending energy in c per corner in the interface. It is formulated as a seven-vertex model which is in general not solvable. For beta in c =ln(2)/2, however, this seven-vertex model becomes free fermionic one and an exact expression for its free energy is derived. In the extreme anisotropic ('Hamiltonian') limit the operator content coincides with that of the 1D Ising model in a transverse field. Three critical lines with Ising-like behaviour are found.

DOI
10.1088/0305-4470/23/10/007

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A model describing an ensemble of disjoint clusters on a square lattice is introduced. In addition to the usual surface energy terms, in x and in y , the model includes a bending energy in c per corner in the interface. It is formulated as a seven-vertex model which is in general not solvable. For beta in c =ln(2)/2, however, this seven-vertex model becomes free fermionic one and an exact expression for its free energy is derived. In the extreme anisotropic ('Hamiltonian') limit the operator content coincides with that of the 1D Ising model in a transverse field. Three critical lines with Ising-like behaviour are found.
A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.
The spin-3/2 Ising model described by the most general Hamiltonian with up-down symmetry, −βH=Σ〈ij〉{JS i S j +KS 2 i S 2 j +LS 3 i S 3 j +M/2(S i S 3 j +S j S 3 i )}−ΔΣi S 2 i , is investigated on a square lattice. It is shown that this model is reducible to an eight-vertex model on a surface in the parameter space spanned by the coupling constants J, K, L, and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. Consequently, the critical behavior and, in particular, the critical temperature for the second-order phase transitions of the model is found exactly.
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.