More on Slavnov Products of Spin Chains and KP Hierarchy Tau Functions
Thiago Araujo
TL;DR
We analyze Slavnov products in quantum spin chains and show that, under rational/trigonometric model conditions, they are $\tau$-functions of the KP hierarchy. The work provides an alternant determinant representation, a basis expansion in terms of other $\tau$-functions, and a homogeneous-limit/Wronskian form, together with a Baker–Akhiezer function expressed as a determinantal object. A tentative multicomponent KP structure is suggested, linking quantum scalar products to classical integrable hierarchies. The results offer a concrete bridge between Bethe-state overlaps and classical soliton theory, with potential applications to correlation functions and thermodynamic limits. Extensions to elliptic or boundary-condition variants and numerical implementations are identified as important future directions.
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.