Bethe subalgebras in twisted Yangians

We study analogues of the Yangian of the Lie algebra for the other classical Lie algebras and . We call them twisted Yangians. They are coideal subalgebras in the Yangian of and admit homomorphisms onto the universal enveloping algebras U( ) and U( ) respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in U( ) and U( ) of these subalgebras are also maximal commutative.     

Gaudin model,Bethe Ansatz and critical level

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensoproducts of Wakimoto modules. This gives explicit formulas for the eigenvectors via bosonic correlation functions. Analogues of the Bethe Ansatz equations naturally appear as equations on the existence of singular vectors in Wakimoto modules. We use this construction to explain the connection between Gaudin's model and correlation functios of WZNW models.     

Y-System and Deformed Thermodynamic Bethe Ansatz

We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed TBA is a system of five coupled nonlinear integral equations, which in a particular case reduces to the Zamolodchikov TBA equation for the three-state Potts model. Our method generalizes the Dorey–Tateo analysis of the (monomial) cubic oscillator. We introduce a Y-system corresponding to the Deformed TBA and give it an elegant geometric interpretation.     

Drinfel'd Twists and Functional Bethe Ansatz

Using the Functional Bethe Ansatz technique, factorizing Drinfel'd twists for any finite dimensional irreducible representations of the Yangian Y(sl₂) are constructed.     

Analytic Bethe ansatz for fundamental representations of Yangians

We study the analytic Bethe, ansatz in solvable vertex models associated with the YangianY(X _r ) or its quantum affine analogueU _q (X _r ⁽¹⁾ ) forX _r =B _r ,C _r andD _r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations ofY(X _r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying theT-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.     

Extending the Bethe Ansatz: The Quantum Three-Particle Ring

The quantum problem of three impenetrable particles of arbitrary mass confined to a ring is solved by the Bethe ansatz. The solution of this problem is intimately related to the solution a Helmholtz equation in the interior of an arbitrary acute triangle, a problem thought insoluble by Bethe ansatz methods.     

Algebraic Bethe Ansatz for Generalized t-J Model

In this paper we show the generalized t-J model with periodic condition integrable. By using the QISM, we exactly diagonalize the model in three different backgrounds. The energy spectrum and Bethe ansatz equations are obtained.     

推广t-J模型的坐标Bethe Ansatz方法

利用坐标Bethe Ansatz方法,研究了推广的t—J模型的精确解,导出了两组Bethe Ansatz方程;同时证明了在两体散射问题中的散射矩阵正是三角型的非对称六顶角R矩阵.     

Cyclotomic Gaudin Models: Construction and Bethe Ansatz

To any finite-dimensional simple Lie algebra \({\mathfrak{g}}\) and automorphism \({\sigma: \mathfrak{g}\to \mathfrak{g}}\) we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of \({U(\mathfrak{g})^{\otimes N}}\) generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case \({\sigma ={\rm id}}\).     

Airy Functions in the Thermodynamic Bethe Ansatz

Thermodynamic Bethe ansatz equations are coupled nonlinear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painlevé III and the Toda hierarchy. In the simplest such case, the massless limit of these nonlinear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrödinger equation in a linear potential.     

Functional Relations and Bethe Ansatz for the XXZ Chain

There is an approach due to Bazhanov and Reshetikhin for solving integrable RSOS models which consists of solving the functional relations which result from the truncation of the fusion hierarchy. We demonstrate that this is also an effective means of solving integrable vertex models. Indeed, we use this method to recover the known Bethe Ansatz solutions of both the closed and open XXZ quantum spin chains with U(1) symmetry. Moreover, since this method does not rely on the existence of a pseudovacuum state, we also use this method to solve a special case of the open XXZ chain with nondiagonal boundary terms.     

Completeness of the Bethe Ansatz on Weyl Alcoves

We prove the completeness of the Bethe ansatz eigenfunctions of the Laplacian on a Weyl alcove with repulsive boundary condition at the walls. For the root system of type A this amounts to the result of Dorlas of the completeness of the Bethe ansatz eigenfunctions of the quantum Bose gas on the circle with repulsive delta-function interaction.     

A New Asymmetric Long-Range Model and Algebraic Bethe Ansatz

A new integrable long-range model is derived from a new asymmetric R-matrix recently discussed by Bibikov in relation to a XXZ spin chain in an external magnetic field. The algebraic Bethe Ansatz is used to derive the eigenvalues and equations for the eigen momenta both for the usual and long-range model.     

The Heun Equation and the Calogero-Moser-Sutherland System I: The Bethe Ansatz Method

 We propose and develop the Bethe Ansatz method for the Heun equation. As an application, holomorphy of the perturbation for the BC ₁ Inozemtsev model from the trigonometric model is proved. Received: 28 September 2001 / Accepted: 31 October 2002 Published online: 31 January 2003 Communicated by L. Takhtajan     

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper.For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY=0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.Supported in part by NSF grant DMS-0140460Supported in part by NSF grant DMS-0244579     

Thermodynamic Bethe Ansatz for N = 1 supersymmetric theories

We study a series of N = 1 supersymmetric integrable particle theories in d = 1 + 1 dimensions. These theories are represented as integrable perturbations of specific N = 1 superconformal field theories. Starting from the conjectured S-matrices for these theories, we develop the Thermodynamic Bethe Ansatz (TBA), where we use that the 2-particle S-matrices satisfy a free fermion condition. Our analysis proves a conjecture by E. Melzer, who proposed that these N = 1 supersymmetric. TBA systems are “folded” versions of N = 2 supersymmetric TBA systems that were first studied by P. Fendley and K. Intriligator.     

The Statistical Interaction and the Bethe Ansatz of the XXZ Chain Corresponding Potts Case

After introducing briefly the basic concept of statistical interaction, we illustrate it on integrable XXZ chain corresponding Potts case by the Bethe ansatz and point out that the nontrivial part of this statistical interaction comes from the rates of change of phase shifts with respect to momentum.     

Combinatorial Bethe Ansatz and Ultradiscrete Riemann Theta Function with Rational Characteristics

The vertex model at q = 0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann theta function with rational characteristics.