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

arXiv:1412.7511 · Nuclear Physics B

Modified algebraic Bethe ansatz for XXZ chain on the segment – II – general cases

S. Belliard, Rodrigo A. Pimenta

Published Mar 27, 2015• 85 citations
arXivHTMLPDF

Abstract

The spectral problem of the Heisenberg XXZ spin-12 chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to N, the length of the chain, and which satisfies a set of Bethe equations with an additional term.

DOI
10.1016/j.nuclphysb.2015.03.016

Related papers

Modified algebraic Bethe ansatz for XXZ chain on the segment – I: Triangular cases

S. Belliard

The modified algebraic Bethe ansatz, introduced by Crampé and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-12 chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to N the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-12 chain on the segment with two upper triangular boundaries.

arXiv:1408.4840•Nuclear Physics B•Jan 8, 2015•90 citations

Modified algebraic Bethe ansatz for XXZ chain on the segment – III – Proof

Jean Avan, S. Belliard, Nicolas Grosjean +1 more

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-12 chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix.

arXiv:1506.02147•Nuclear Physics B•Aug 11, 2015•77 citations

Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz

S. Belliard

We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al.The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain.We prove this result for the chains with small length.We obtain also an off-shell equation (i.e.satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

arXiv:1309.6165•Symmetry Integrability and Geometry Methods and Applications•Nov 22, 2013•97 citations

A complete Bethe ansatz solution for the open spin-s XXZchain with general integrable boundary terms

L. Frappat, Rafael I. Nepomechie, E. Ragoucy

We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.

arXiv:0707.0653•Journal of Statistical Mechanics Theory and Experiment•Sep 14, 2007•55 citations

Bethe Ansatz and boundary energy of the open spin-1/2 XXZ chain

Rajan Murgan

We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{$\\alpha_-$,$\\alpha_+$,$\\beta_-$,$\\beta_+$}} are nonzero. A generalization of the Baxter $T-Q$ equation that involves more than one independent $Q$ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.

•Jan 1, 2006•0 citations

Modified Algebraic Bethe Ansatz: Twisted XXX Case

S. Belliard, Никита Андреевич Славнов, B. Vallet +1 more

We prove the modified algebraic Bethe Ansatz characterization of the spectral problem for the closed XXX Heisenberg spin chain with an arbitrary twist and arbitrary positive (half)-integer spin at each site of the chain.We provide two basis to characterize the spectral problem and two families of inhomogeneous Baxter T-Q equations.The two families satisfy an inhomogeneous quantum Wronskian equation.

arXiv:1804.00597•Symmetry Integrability and Geometry Methods and Applications•Jun 7, 2018•22 citations
The spectral problem of the Heisenberg XXZ spin-12 chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to N, the length of the chain, and which satisfies a set of Bethe equations with an additional term.
The modified algebraic Bethe ansatz, introduced by Crampé and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-12 chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to N the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-12 chain on the segment with two upper triangular boundaries.
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al.The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain.We prove this result for the chains with small length.We obtain also an off-shell equation (i.e.satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.