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The $q$-Racah polynomials from scalar products of Bethe states II

Pascal Baseilhac, Rodrigo A. Pimenta

TL;DR

The paper builds a bridge between the representation theory of the Askey–Wilson (AW) algebra and the algebraic Bethe ansatz by using Leonard pairs and a Leonard triple of $q$-Racah type. It derives explicit, normalized scalar products of on-shell versus off-shell Bethe states in terms of $q$-Racah polynomials, and shows these overlaps satisfy Belliard–Slavnov linear systems, leading to determinant representations. It also establishes relations between inhomogeneous and homogeneous Bethe roots via decompositions of $q$-Racah polynomials and provides concrete $s=\tfrac{1}{2}$ examples validating the framework. The work offers a functional-analytic view on discrete orthogonal polynomials within the Bethe framework and points toward generalizations to tridiagonal pairs and open boundary integrable models such as the XXZ chain.

Abstract

The theory of Leonard triples is applied to the derivation of normalized scalar products of on-shell and off-shell Bethe states generated from a Leonard pair. The scalar products take the form of linear combinations of $q$-Racah polynomials with coefficients depending on the off-shell parameters. Upon specializations, explicit solutions for the corresponding Belliard-Slavnov linear systems are obtained. It implies the existence of a determinant formula in terms of inhomogeneous Bethe roots for the $q$-Racah polynomials. Also, a set of relations that determines solutions (Bethe roots) of the corresponding Bethe equations of inhomogeneous type in terms of solutions of Bethe equations of homogenous type is obtained.

The $q$-Racah polynomials from scalar products of Bethe states II

TL;DR

The paper builds a bridge between the representation theory of the Askey–Wilson (AW) algebra and the algebraic Bethe ansatz by using Leonard pairs and a Leonard triple of -Racah type. It derives explicit, normalized scalar products of on-shell versus off-shell Bethe states in terms of -Racah polynomials, and shows these overlaps satisfy Belliard–Slavnov linear systems, leading to determinant representations. It also establishes relations between inhomogeneous and homogeneous Bethe roots via decompositions of -Racah polynomials and provides concrete examples validating the framework. The work offers a functional-analytic view on discrete orthogonal polynomials within the Bethe framework and points toward generalizations to tridiagonal pairs and open boundary integrable models such as the XXZ chain.

Abstract

The theory of Leonard triples is applied to the derivation of normalized scalar products of on-shell and off-shell Bethe states generated from a Leonard pair. The scalar products take the form of linear combinations of -Racah polynomials with coefficients depending on the off-shell parameters. Upon specializations, explicit solutions for the corresponding Belliard-Slavnov linear systems are obtained. It implies the existence of a determinant formula in terms of inhomogeneous Bethe roots for the -Racah polynomials. Also, a set of relations that determines solutions (Bethe roots) of the corresponding Bethe equations of inhomogeneous type in terms of solutions of Bethe equations of homogenous type is obtained.
Paper Structure (22 sections, 21 theorems, 172 equations, 1 table)

This paper contains 22 sections, 21 theorems, 172 equations, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic Bethe ansatz and Leonard pairs of $q$-Racah type
  3. Dynamical operators and the Askey-Wilson algebra
  4. Leonard pairs and off-shell/on-shell Bethe states
  5. The Leonard pair $(A,A^*)$ and related eigenbases
  6. Off-shell and on-shell Bethe states
  7. Scalar products of Bethe states from the Leonard triple
  8. The Leonard triple $(A,A^*,A^{\diamond})$
  9. Eigenbases and respective actions of the Leonard triple
  10. Transition matrices
  11. Computation of scalar products
  12. Relating inhomogeneous and homogeneous Bethe roots
  13. Scalar products for $M"=2s$ and Belliard-Slavnov linear system
  14. Belliard-Slavnov linear system associated with a Leonard pair
  15. Determinant formula and $q$-Racah polynomials
  16. ...and 7 more sections

Key Result

Lemma 3.1

Assume that the structure constants in (aw1)-(aw2) take the form: The triple $(A,A^*,A^{\diamond})$ satisfy the Askey-Wilson relations:

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Remark 2
  • Remark 3
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 41 more