The quantum group structure of long-range integrable deformations

TL;DR

This paper reveals the quantum group structure underlying long-range integrable deformations of spin chains, showing they are obtained via a twist leading to a non-associative algebra with a Drinfeld associator.

Contribution

It provides a quantum group-theoretical framework for understanding long-range deformations of Yang-Baxter integrable spin chains, including explicit Lax and R-matrices.

Findings

Deformations are realized through a twist of the quantum group algebra.

The resulting algebra is generally non-associative with a non-trivial Drinfeld associator.

Perturbatively, the deformed quantum group contains an associative substructure.

Abstract

Quantum integrable spin chains are known to possess a large family of long-range deformations generated by the local, boost and bilocal operators. Although these deformations are well-understood on the level of the pairwise commuting charges, the underlying quantum group structures had not yet been recognised. In this paper, we provide a quantum group-theoretical description for the family of long-range deformations of arbitrary homogeneous Yang-Baxter integrable spin chains up to first order in the deformation parameter. In particular, we show that the deformations are obtained via a twist of the algebraic structure of the underlying quantum group. This twisting results in a generally non-associative algebra that has a non-trivial Drinfeld associator. The Drinfeld associator is then shown to encode the information about the long-range interaction terms for the integrable spin chain. Moreover, the deformed quantum group is shown to contain a large perturbatively associative substructure, thus ensuring the perturbative integrability of the long-range model. The deformed quantum group provides explicit expressions for the Lax operators and R-matrices of the long-range deformed models, which manifestly satisfy the RLL relation and the Yang-Baxter equation up to first order in the deformation parameter. In order to derive the results, we introduce algebra elements that we call the algebraic charge densities. As a side result, we provide a conjecture for the explicit expressions of the undeformed charge densities in terms of these algebra elements.