Direct proof for the Scalar Product with Bethe eigenstate in Spin chains

TL;DR

This paper provides a straightforward proof of the determinantal formula for scalar products involving Bethe eigenstates in spin chains, enhancing understanding of their mathematical structure.

Contribution

It introduces a simple, direct proof of the scalar product formula using the factorizing operator, clarifying previous complex derivations.

Findings

Derivation of a determinantal formula for scalar products

Use of the factorizing operator in the proof

Comparison with previous methods

Abstract

We present the simple and direct proof of the determinantal formula for the scalar product of Bethe eigenstate with an arbitrary dual state. We briefly review the direct calculation of the general scalar product with the help of the factorizing operator and the construction of the factorizing operator itself. We also comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state.