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:1904.09282

From algebraic to coordinate Bethe ansatz for square ice

Silvère Gangloff

Published Apr 16, 2019• 0 citations
arXivHTMLPDF

Abstract

In this text, we provide a detailed exposition of the Algebraic Bethe ansatz for square ice (or six vertex model), which allows the construction of candidate eigenvectors for the transfer matrices of this model. We also prove some formula of V.E. Korepin for these vectors, which leads to an identification, up to a non-zero complex factor, with the vector obtained by coordinate Bethe ansatz.

DOI
10.48550/arxiv.1904.09282

Related papers

The Bethe ansatz for the six-vertex and XXZ models: An exposition

Hugo Duminil‐Copin, Maxime Gagnebin, Matan Harel +2 more

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where $V$ is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a=b=1$ and $c>0$. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on $\mathbb{Z}^{2}$ with cluster weight $q>4$ exhibits a first-order phase transition.

arXiv:1611.09909•Probability Surveys•Jan 1, 2018•16 citations

Bethe ansätze for 19-vertex models

A. Lima-Santos

The 19-vertex models of Zalomodchikov-Fateev, Izergin-Korepin and the supersymmetric with periodic boundary conditions are studied. We find the spectrum of these quantum spin chains using the coordinate Bethe ansatz. The approach is a suitable parametrization of their wavefunctions. We also applied the algebraic Bethe ansatz in order to obtain the eigenvalues and eigenvectors of the corresponding transfer matrices.

arXiv:hep-th/9807219•Journal of Physics A Mathematical and General•Jan 1, 1999•25 citations

The nineteen-vertex model and alternating sign matrices

Christian Hagendorf

It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.

arXiv:1405.4726•Journal of Statistical Mechanics Theory and Experiment•Jan 19, 2015•10 citations

Algebraic Bethe ansatz for 19-vertex models with upper triangularK-matrices

Rodrigo A. Pimenta, A. Lima-Santos

By means of an algebraic Bethe ansatz approach, we study the Zamolodchikov–Fateev and Izergin–Korepin vertex models with non-diagonal boundaries, characterized by reflection matrices with an upper triangular form. Generalized Bethe vectors are used to diagonalize the associated transfer matrix. The eigenvalues as well as the Bethe equations are presented.

arXiv:1406.5757•Journal of Statistical Mechanics Theory and Experiment•Nov 6, 2014•2 citations

Algebraic Bethe ansatz for the six vertex model with upper triangularK-matrices

Rodrigo A. Pimenta, A Lima-Santos

We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K -matrices have an upper triangular form. We show that the main difficulty entailed by those forms of the K -matrices is the construction of the excited states. However, it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator.

arXiv:1308.4446•Journal of Physics A Mathematical and Theoretical•Oct 24, 2013•24 citations

Bethe Ansatze for Nineteen-vertex Models

A. Lima-Santos

The nineteen-vertex models of Zamolodchikov-Fateev, Izergin-Korepin and the supersymmetric osp(1|2) with periodic boundary conditions are studied. We find the spectrum of these quantum spin chains using the Coordinate Bethe Ansatz. The approche is a suitable parametrization of their wavefunctions. We also applied the Algebraic Bethe Ansatz in order to obtain the eigenvalues and eigenvectors of the corresponding transfer matrices.

•Jul 29, 1998•4 citations
In this text, we provide a detailed exposition of the Algebraic Bethe ansatz for square ice (or six vertex model), which allows the construction of candidate eigenvectors for the transfer matrices of this model. We also prove some formula of V.E. Korepin for these vectors, which leads to an identification, up to a non-zero complex factor, with the vector obtained by coordinate Bethe ansatz.
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.