Phase Model Expectation Values and the 2-toda Hierarchy

We show that the scalar product of the phase model on a finite rectangular lattice is a (restricted) τ -function of the 2-Toda hierarchy. Using this equivalence we then show that the wave-functions of the hierarchy correspond to certain classes of boundary correlation functions of the model. 0. Introduction In [1], it was observed that the N × N domain wall partition function, ZN , of the six vertex model is, up to a multiplicative factor, a τ -function of the KP hierarchy. In [2], the length M XXZ spin12 chain and its associated scalar product, 〈{λ}|{μ}〉, were considered. Restricting either the initial or the final state to a Bethe eigenstate, the resulting expression is again a KP τ -function. In this work we further extend the known correspondences between integrable quantum lattice models and classical hierarchies of non linear partial differential equations in the following way. We show that the scalar product of the phase model [4,5], up to a multiplicative factor, is a (restricted) τ -function of the 2-Toda hierarchy [6,7], where the Toda time variables are power sums of the rapidities. We then consider the two types of wave-functions from the Toda theory, ŵ(∞) and ŵ, and show that they correspond to a certain class of boundary correlation function from the phase model perspective. We additionally give a single determinant form for each of these correlation functions. In section 1, we recall known results about the finite 2-Toda hierarchy, its construction from the wave-matrix initial value problem and the τ -function as a finite bilinear sum of character/Schur polynomials. In 2, we introduce the phase model and the aforementioned lattice model/hierarchy correspondence. Additionally we use known combinatorial bijections to express the state vectors of the model as weighted sums of various objects. In 3, we use the weighted sum expressions of the state vectors to show that the Toda wave-functions correspond to specific classes of boundary correlation functions and give their single determinant form. In 4, we offer some remarks. 1. The finite 2-Toda hierarchy 1.1. Definition of the hierarchy. The details of this section can mostly be found in [6,7]. We begin by giving the definition of the 2-Toda hierarchy, with two sets of (n−m− 1) time variables, in terms of four distinct Lax type systems of first order differential equations. Defining the following shift matrices, Λ [m,n) ≡ (δk±j,l)k,l∈{m,...,n−1} 2000 Mathematics Subject Classification. Primary 82B20, 82B23.