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FACTORIZED DOMAIN WALL PARTITION FUNCTIONS IN TRIGONOMETRIC VERTEX MODELS

Omar E. Foda, M Wheeler, Mathew L. Zuparic

Published Nov 21, 2012• 11 citations
OpenAlex

Abstract

Abstract. We obtain factorized domain wall partition functions for two sets of trigonometric vertex models: 1. The N-state Deguchi-Akutsu models, for N ∈ {2, 3, 4} (and conjecture the result for all N ≥ 5), and 2. The sl(r+1|s+1) Perk-Schultz models, for {r, s ∈ N}, where (given the symmetries of these models) the result is independent of {r, s}. Domain wall partition functions (DWPF’s) were first proposed and evaluated in determinant form, for the spin-1 2 vertex model1 on a finite square lattice, in [1, 2]. At the free fermion point of the spin-1 2 model, this determinant is in

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Abstract. We obtain factorized domain wall partition functions for two sets of trigonometric vertex models: 1. The N-state Deguchi-Akutsu models, for N ∈ {2, 3, 4} (and conjecture the result for all N ≥ 5), and 2. The sl(r+1|s+1) Perk-Schultz models, for {r, s ∈ N}, where (given the symmetries of these models) the result is independent of {r, s}. Domain wall partition functions (DWPF’s) were first proposed and evaluated in determinant form, for the spin-1 2 vertex model1 on a finite square lattice, in [1, 2]. At the free fermion point of the spin-1 2 model, this determinant is in
Abstract. We consider the trigonometric Felderhof model, of free fermions in an external field, on a finite lattice with domain wall boundary conditions. The vertex weights are functions of rapidities and external fields. We obtain a determinant expression for the partition function in the special case where the dependence on the rapidities is eliminated, but for general external field variables. This determinant can be evaluated in product form. In the homogeneous limit, it is proportional to a 2-Toda τ function. Next, we use the algebraic Bethe ansatz factorized basis to obtain a product expression for the partition function in the general case with dependence on all variables. In [1], Korepin introduced domain wall boundary conditions for the six vertex model on a finite square lattice, and proposed recursion relations that determine the corresponding domain wall partition function. In [2], Izergin obtained a determinant solution of Korepin’s recursion relations. At the free fermion point, the
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Abstract We study the domain wall partition function Z N for the (Izergin–Korepin) integrable nineteen-vertex model on a square lattice of size N . Z N is a symmetric function of two sets of parameters: horizontal and vertical rapidities. For generic values of the parameter q we derive the recurrence relation for the domain wall partition function relating Z N +1 to , where P N is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables and . After setting the recurrence relation simplifies and we solve it in terms of a Jacobi–Trudi-like determinant of polynomials generated by P N .