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:1009.4119 · Journal of Statistical Mechanics Theory and Experiment

Eigenvectors of openXXZand ASEP models for a class of non-diagonal boundary conditions

Nicolas Crampé, E. Ragoucy, Damien Simon

Published Nov 23, 2010• 74 citations
arXivHTMLPDF

Abstract

We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of the algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the BC n Weyl group.

DOI
10.1088/1742-5468/2010/11/p11038

Related papers

Matrix coordinate Bethe Ansatz: applications to XXZ and ASEP models

Nicolas Crampé, E. Ragoucy, Damien Simon

We present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process (ASEP) and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This 'matrix coordinate Bethe Ansatz' can be viewed as a non-commutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz.

arXiv:1106.4712•Journal of Physics A Mathematical and Theoretical•Sep 14, 2011•60 citations

Algebraic Bethe Ansatz Solution to C N Vertex Model with Open Boundary Conditions

Guang-Liang Li, Kangjie Shi, Rui‐Hong Yue

We present three diagonal reflecting matrices for the C N vertex model with open boundary conditions and exactly solve the model by using the algebraic Bethe ansatz. The eigenvector is constructed and the eigenvalue and the associated Bethe equations are achieved. All the unwanted terms are cancelled out by three kinds of identities.

Communications in Theoretical Physics•Jul 1, 2005•6 citations

Generalized Coordinate Bethe Ansatz for open spin chains with non-diagonal boundaries

E. Ragoucy

We introduce a generalization of the original Coordinate Bethe Ansatz that allows to treat the case of open spin chains with non-diagonal boundary matrices. We illustrate it on two cases: the XXX and XXZ chains. Short review on a joint work with N. Crampe (L2C) and D. Simon (LPMA), see arXiv:1009.4119, arXiv:1105.4119 and arXiv:1106.3264.

arXiv:1109.6189•Journal of Physics Conference Series•Feb 8, 2012•3 citations

Algebraic Bethe Ansatz Solution to CN Vertex Model with Open Boundary Conditions

LIGuang-Liang, Guangliang, Shi +3 more

We present three diagonal reflecting matrices for the CN vertex model with open boundary conditions and exactly solve the model by using the algebraic Bethe ansatz. The eigenvector is constructed and the eigenvalue and the associated Bethe equations are achieved. All the unwanted terms are cancelled out by three kinds of identities.

理论物理通讯：英文版•Jan 1, 2005•0 citations

Construction of a coordinate Bethe ansatz for the asymmetric simple exclusion process with open boundaries

Damien Simon

The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

arXiv:0903.4968•Journal of Statistical Mechanics Theory and Experiment•Jul 8, 2009•109 citations

Exact solution of the XXZ alternating spin chain with generic non-diagonal boundaries

Cao, Junpeng, Wen‐Li Yang, Kangjie Shi +1 more

The integrable XXZ alternating spin chain with generic non-diagonal boundary terms specified by the most general non-diagonal K-matrices is studied via the off-diagonal Bethe Ansatz method. Based on the intrinsic properties of the fused R-matrices and K-matrices, we obtain certain closed operator identities and conditions, which allow us to construct an inhomogeneous T-Q relation and the associated Bethe Ansatz equations accounting for the eigenvalues of the transfer matrix.

arXiv:1409.3646•Sep 12, 2014•5 citations
We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of the algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the BC n Weyl group.
The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.