An Izergin-Korepin procedure for calculating scalar products in six-vertex models

TL;DR

This paper develops a determinant formula for scalar products in inhomogeneous XXZ and XX spin-1/2 chains using an algebraic Bethe Ansatz approach inspired by the Izergin-Korepin procedure, extending previous results to inhomogeneous and magnetic field cases.

Contribution

It introduces a novel set of conditions for scalar products in inhomogeneous spin chains and derives explicit determinant and factorized formulas under Bethe equations.

Findings

Derived determinant expression for scalar products in inhomogeneous XXZ chain.

Extended the Izergin-Korepin procedure to models with external magnetic fields.

Obtained factorized scalar product formulas in specific inhomogeneous cases.

Abstract

Using the framework of the algebraic Bethe Ansatz, we study the scalar product of the inhomogeneous XXZ spin-1/2 chain. Inspired by the Izergin-Korepin procedure for evaluating the domain wall partition function, we obtain a set of conditions which uniquely determine the scalar product. Assuming the Bethe equations for one set of variables within the scalar product, these conditions may be solved to produce a determinant expression originally found by Slavnov. We also consider the inhomogeneous XX spin-1/2 chain in an external magnetic field. Repeating our earlier procedure, we find a set of conditions on the scalar product of this model and solve them in the presence of the Bethe equations. The expression obtained is in factorized form.