A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and mutually commuting quantities -which ensure the integrability of the system -are written in terms of the fundamental generators of the new algebra. Relation with the deformed Dolan-Grady integrable structure recently discovered by one of the authors and Terwilliger's tridiagonal algebras is described. Remarkably, this (in)finite dimensional algebra is a "q−deformed" analogue of the original Onsager's algebra arising in the planar Ising model. Consequently, it provides a new and alternative algebraic framework for studying massive, as well as conformal, quantum integrable models.andwith fixed scalar ρ and k, l ∈ N. Finite dimensional representations are obtained, and examples of quantum integrable systems (XXZ spin chain, Sine-Gordon and Liouville field theories) which enjoy this symmetry are given. More generally, our framework opens the possibility of analyzing a large class of quantum integrable models from a new point of view, in the spirit of Onsager's approach [1]. The paper is organized as follows. In Section 2, the fundamental relations (4) are derived using its relation with a class of quadratic algebra, namely the reflection equation. Indeed, finite dimensional representations of its fundamental generators are shown to be generated from general solutions of the reflection equation. In particular, the closure of the algebra is ensured by the existence of a set of linear relations among the generators. Explicit expressions of mutually commuting quantities in terms of W −k , W k+1 , G k+1 ,G k+1 , generalizing (3) are obtained. Also, we argue that similarly to the undeformed case, the integrable structure generated from our "q−deformed" Onsager's algebra coincides with the deformed Dolan-Grady integrable structure recently discovered in [13,14]. In particular, we exhibit the correspondence in the simplest cases. In the last Section, we give some examples of quantum integrable systems which enjoy this (in)finite dimensional symmetry. Structure of the algebraIn the last thirty years, one of the most important progress in the approach of quantum integrable systems has been based on the star-triangle relations which originated in [1,15] and led to the Yang-Baxter equations, the theory of quantum groups as well as the quantum inverse scattering method. Although the star-triangle relations
The transfer matrix of the XXZ open spin-1 2 chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q−Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al. and Cao et al.. In the latter case the complete family of eigenvalues and eigenstates splits in two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.
The XXZ open spin chain with general integrable boundary conditions is shown to possess a q−deformed analogue of the Onsager's algebra as fundamental non-abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.