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International Journal of Modern Physics B

FERMIONIC STRINGS AND THE EXACT SOLUTION OF EDGE MODELS IN THREE DIMENSIONS

Peter Orland A. B. C

Published Aug 20, 1991• 7 citations
Publisher

Abstract

A three-dimensional statistical system, called here the six-edge model, is shown, for a particular domain of parameters, to be equivalent to a theory of Fermionic strings. This model has a local U(1) gauge invariance, similiar to the global invariance of the two-dimensional six-vertex model. Further restricting the parameters gives a system named here the five-edge model, which is exactly solvable. The conformal anomaly of the surfaces is calculated. The transfer matrix is diagonalized, and the free energy, phase diagram and correlation length are determined exactly. The surfaces of the model display a roughening transition similiar to that analyzed by Blöte, Hilhorst and Nienhuis.

DOI
10.1142/s0217979291000948

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A three-dimensional statistical system, called here the six-edge model, is shown, for a particular domain of parameters, to be equivalent to a theory of Fermionic strings. This model has a local U(1) gauge invariance, similiar to the global invariance of the two-dimensional six-vertex model. Further restricting the parameters gives a system named here the five-edge model, which is exactly solvable. The conformal anomaly of the surfaces is calculated. The transfer matrix is diagonalized, and the free energy, phase diagram and correlation length are determined exactly. The surfaces of the model display a roughening transition similiar to that analyzed by Blöte, Hilhorst and Nienhuis.
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.
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.
Some features of integrable lattice models are reviewed for the case of the six-vertex model.By the Bethe ansatz method we derive the free energy of the six-vertex model.Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears.We discuss the connection of the six-vertex model to the conformal field theory with c = 1.We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models.We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group.The graphical approach is emphasized in this review.We explain the meaning of the Yang-Baxter equation by its diagram.Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation.We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models.As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Some features of integrable lattice models are reviewed for the case of the six-vertex model. By the Bethe ansatz method we derive the free energy of the six-vertex model. Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears. We discuss the connection of the six-vertex model to the conformal field theory with c=1. We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models. We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group. The graphical approach is emphasized in this review. We explain the meaning of the Yang-Baxter equation by its diagram. Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation. We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models. As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
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.