Scalar product for the XXZ spin chain with general integrable boundaries

TL;DR

This paper derives a determinant formula for scalar products of Bethe states in the XXZ spin-1/2 chain with general integrable boundaries, advancing the understanding of boundary effects in quantum integrable models.

Contribution

It provides a new compact determinant expression for scalar products with general boundary conditions, extending previous results to more general boundary integrable systems.

Findings

Derived a linear system for scalar products using off-shell equations.

Expressed scalar products as a determinant involving Jacobian and q-Pochhammer polynomials.

Applicable to general integrable boundary conditions in the XXZ spin chain.

Abstract

We calculate the scalar product of Bethe states of the XXZ spin-$\frac{1}{2}$ chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.