A note on the eigenvectors of long-range spin chains and their scalar products

TL;DR

This paper proposes a new method to construct eigenvectors and scalar products for long-range spin chains with su(2) symmetry, including models relevant to gauge theory and integrability, especially at large chain lengths.

Contribution

It introduces an explicit expression for eigenvectors and scalar products of long-range spin chains using Dunkl operators, applicable to models like Inozemtsev and BDS, and relates to gauge theory operators.

Findings

Provides eigenvectors for long-range spin chains up to three-loop order.

Derives scalar products for these eigenvectors.

Suggests implications for three-point functions in N=4 gauge theory.

Abstract

In this note, we propose an expression for the eigenvectors and scalar products for a class of spin chains with long-range interaction and su(2) symmetry. This class includes the Inozemtsev spin chain as well as the BDS spin chain, which is a reduction of the one-dimensional Hubbard model at half-filling to the spin sector. The proposal is valid for large spin chains and is based on the construction of the monodromy matrix using the Dunkl operators. For the Inozemtsev model these operators are known explicitly. This construction gives in particular the eigenvectors of (an operator closely related to) the dilatation operator of the N=4 gauge theory in the su(2) sector up to three-loop order, as well as their scalar products. We suggest how this will affect the expression for the quasi classical limit of the three-point functions obtained by I. Kostov and how to include the all-loop interaction.