Construction of Monodromy Matrix in the F- basis and Scalar products in Spin Chains

TL;DR

This paper simplifies the theory of the factorizing operator in spin-1/2 chains, providing explicit matrix element expressions and deriving a general scalar product formula, bridging different computational bases.

Contribution

It introduces explicit expressions for the factorizing operator's matrix elements and derives a general scalar product formula in the F-basis for quantum spin chains.

Findings

Explicit matrix elements of the factorizing operator are obtained.

A general scalar product formula in the F-basis is derived.

Connections between calculations in F-basis and the usual basis are established.

Abstract

We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin - 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis.