More on Slavnov Products of Spin Chains and KP Hierarchy Tau Functions

TL;DR

This paper explores the relationship between Slavnov products in spin chains and tau functions of the KP hierarchy, revealing new structural insights and explicit representations in integrable systems.

Contribution

It extends the connection between Slavnov products and KP hierarchy tau functions to more general models, including their expansions and determinantal forms.

Findings

Slavnov products can be interpreted as KP tau functions under certain conditions.

Homogeneous limits of Slavnov products are expressed as Wronskians of transfer matrix eigenvalues.

Explicit determinantal formulas for Baker-Akhiezer functions are derived.

Abstract

Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally takes the form of determinants. Building on recent results on the structure of rational and trigonometric models, we show that, provided certain conditions are satisfied, the Slavnov product of a given model can be interpreted as a tau function of the KP hierarchy, thus extending known results in a more general setting. Moreover, we show that Slavnov products can be expanded in terms of other tau functions. We also prove that their homogeneous limit can be systematically expressed as a Wronskian of functions related to the eigenvalues of the transfer matrices. Finally, we compute the Baker-Akhiezer functions associated with these Slavnov products and show that, apart from a universal multiplicative factor, they admit a closed determinantal representation.