Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Hecke algebraic approach to the reflection equation for spin chains

Anastasia Doikou, Paul Martin

Published Jun 10, 2002• 28 citations
OpenAlex

Abstract

We use the structural similarity of certain Coxeter Artin Systems to the Yang--Baxter and Reflection Equations to convert representations of these systems into new solutions of the Reflection Equation. We construct certain Bethe ansatz states for these solutions, using a parameterisation suggested by abstract representation theory.

Related papers

Hecke algebraic approach to the reflection equation for spin chains

Anastasia Doikou, P P Martin

We use the structural similarity of certain Coxeter Artin systems to the Yang–Baxter and reflection equations to convert representations of these systems into new solutions of the reflection equation. We construct certain Bethe ansatz states for these solutions, using a parametrization suggested by abstract representation theory.

arXiv:hep-th/0206076•Journal of Physics A Mathematical and General•Feb 19, 2003•42 citations

Affine Hecke algebras, reflection equation and boundary symmetries

Anastasia Doikou

Solutions of the reflection equation for the Uq ( sln) case are obtained by implementing certain realisations of the affine Hecke algebra. The boundary non–local charges associated to nondiagonal solutions of the reflection equation are constructed systematically, and the symmetry of the corresponding open spin chain is investigated. We prove that the symmetry of the open spin chain is essentially a remnant of the Uq(gln) algebra.

•Jan 1, 2004•0 citations

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

Anastasia Doikou, Paul Martin

Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

arXiv:hep-th/0503019•Journal of Statistical Mechanics Theory and Experiment•Jun 8, 2006•30 citations

A Q-operator for open spin chains I. Baxter’s TQ relation

Bart Vlaar, Robert Weston

We construct a Q -operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter’s TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of U q ( s l ^ 2 ) . The resulting Bethe equations are the same as those arising from Sklyanin’s algebraic Bethe ansatz.

arXiv:2001.10760•Journal of Physics A Mathematical and Theoretical•Apr 9, 2020•13 citations

Analytical Bethe Ansatz for closed and open gl(n)-spin chains in any representation

D. Arnaudon, Nicolas Crampé, Anastasia Doikou +2 more

We present an "algebraic treatment" of the analytical Bethe Ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe Ansatz. It allows us to deal with a generic gl(n)-spin chain possessing on each site an arbitrary gl(n)-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. As examples, we recover the usual closed and open spin chains, we treat the alternating spin chains and the closed spin chain with impurity.

•Jan 1, 2005•13 citations

A New Approach to Quantum Spin Chains at Finite Temperature

Junji Suzuki, Yasuhiro Akutsu, Miki Wadati

Presented is a new approach to the study of one dimensional quantum spin systems at finite temperature. The approach is based upon a general structure of solvable models; the Yang-Baxter equation and the Bethe ansatz equation, and in addition the evaluation of finite size corrections. Provided that the ground state is constructed by the Bethe ansatz method, exact low temperature expansions of physical quantities are given without further assumption.

Journal of the Physical Society of Japan•Aug 15, 1990•99 citations
Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.
We present an "algebraic treatment" of the analytical Bethe Ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe Ansatz. It allows us to deal with a generic gl(n)-spin chain possessing on each site an arbitrary gl(n)-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. As examples, we recover the usual closed and open spin chains, we treat the alternating spin chains and the closed spin chain with impurity.