Inner products of Bethe states as partial domain wall partition functions

TL;DR

This paper reveals a novel connection between Bethe state inner products in the XXX spin chain and partition functions of the six-vertex model with partial domain-wall boundary conditions, providing new insights into integrable models.

Contribution

It demonstrates that the inner product of on-shell and generic Bethe states can be expressed as a partition function of the six-vertex model, linking algebraic Bethe ansatz to statistical mechanics.

Findings

Inner product of Bethe states equals a six-vertex model partition function.

The inner product of an on-shell and a generic state relates to a 2M-magnon state partition.

Provides a new combinatorial interpretation of Bethe state inner products.

Abstract

We study the inner product of Bethe states in the inhomogeneous periodic XXX spin-1/2 chain of length L, which is given by the Slavnov determinant formula. We show that the inner product of an on-shell M-magnon state with a generic M-magnon state is given by the same expression as the inner product of a 2M-magnon state with a vacuum descendent. The second inner product is proportional to the partition function of the six-vertex model on a rectangular Lx2M grid, with partial domain-wall boundary conditions.