Generalized sℓ(2) Gaudin algebra and corresponding Knizhnik–Zamolodchikov equation

I Salom,N Manojlović,N Cirilo António

doi:10.1016/j.nuclphysb.2018.12.025

I Salom, N Manojlović + Show 1 more

Open Access

https://doi.org/10.1016/j.nuclphysb.2018.12.025

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Abstract

The Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.

Highlights

Gaudin model was first proposed almost half a century ago [1,2,3], and has promptly gained attention primarily due to its long-range interactions feature [4,5]
En route to our treatment of the KZ equations, we present a closed form expression for the offshell Bethe vectors and prove the implementation of the algebraic Bethe ansatz in full generality
In the same section we present the novel formula for the scalar product of off-shell Bethe vectors

Summary

Introduction

Gaudin model was first proposed almost half a century ago [1,2,3], and has promptly gained attention primarily due to its long-range interactions feature [4,5]. Hikami comes close to this goal in his paper [10], but does not tackle the issue in full generality – namely, he constrains his analysis to a special case of equal spins at all nodes, fixing these spins to the value He does not provide the expression for the norms of the eigenvectors of the Gaudin Hamiltonians, which can be obtained from the KZ approach. En route to our treatment of the KZ equations, we present a closed form expression for the offshell Bethe vectors and prove the implementation of the algebraic Bethe ansatz in full generality (for arbitrary reflection matrices and to arbitrary number of excitations) Such a development was a result of a suitable change of generalized Gaudin algebra basis (as compared to the one used in [19]), combined with observation of certain algebraic relations that we came across.

Preliminaries

New generators and the eigenvectors

Solutions to the Knizhnik–Zamolodchikov equations

Conclusion