On the Trigonometric Felderhof Model with Domain Wall Boundary Conditions

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. 0. Introduction 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 six vertex domain wall partition function can be evaluated explicitly in product form [3]. In the homogeneous limit, it is proportional to a 1-Toda τ function [4]. In this work, we look for analogous results in the context of the trigonometric limit of Felderhof’s model [5], which is a vertex model of free fermions in an external field. In section 1, we recall the definition of the model in the parametrization of Deguchi and Akutsu [6], and formulate it on an N ×N lattice. There are four sets of complex variables: horizontal and vertical rapidities {ui, vj}, and horizontal and vertical external field variables {αi, βj}, where {i, j} ∈ {1, 2, . . . , N}. The weight wij of the vertex vij at the intersection of the i-th horizontal line and j-th vertical line depends on the difference of the rapidities, ui−vj , but depends on the external fields, αi and βj , separately. In section 2, we impose domain wall boundary conditions and obtain an Izergintype determinant expression for the domain wall partition function, under the restriction that the difference of any two rapidity variables is a multiple of 2π √ −1, but for general {αi, βj}. This expression can be evaluated in product form. In the homogeneous limit, it is proportional to a 2-Toda τ function [7, 8]. In section 3, we use the factorized basis of the algebraic Bethe ansatz, [9, 10], to obtain a product expression for the domain wall partition function for general {ui, vj} and {αi, βj}. 0.1. Abbreviations. In the rest of this paper, DWBC stands for ‘domain wall boundary conditions’ and DWPF stands for ‘domain wall partition function’. Z TF is the DWPF of the trigonometric Felderhof model on an N ×N lattice. Z 6V is the DWPF of the six vertex model on an N ×N lattice. 2000 Mathematics Subject Classification. Primary 82B20, 82B23.