Separation of variables for the quantum SL(2,R) spin chain

TL;DR

This paper develops a separated variables framework for the quantum SL(2,R) spin chain, providing explicit eigenfunctions, the scalar product measure, and a diagrammatic approach, advancing the analytical understanding of this integrable model.

Contribution

It introduces a novel construction of the SoV representation for the quantum SL(2,R) spin chain, including explicit eigenfunctions and the Sklyanin measure, using Feynman diagram techniques.

Findings

Explicit integral eigenfunctions derived

Sklyanin measure explicitly calculated

Kernel described by pyramid diagram and Baxter Q-operators

Abstract

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.