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

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.     

Generalized coordinate Bethe ansatz for non-diagonal boundaries

We compute the spectrum and the eigenstates of the open XXX model with non-diagonal (triangular) boundary matrices. Since the boundary matrices are not diagonal, the usual coordinate Bethe ansatz does not work anymore, and we use a generalization of it to solve the problem.     

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

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.     

The algebraic Bethe ansatz for the Osp(2|2) model with open boundary conditions

We obtain four different diagonal reflecting matrices by solving the reflection equation of the Osp(2|2) model. At the same time, we solve the model with open boundary condition by using the algebraic Bethe ansatz. The procedure of constructing the multi-particle state and achieving the eigenvalue of the transfer matrix and corresponding Bethe equations is presented in detail.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

Algebraic Bethe ansatz for gl(2, 1) invariant 36-vertex models

Four-dimensional irreducible representations of the superalgebra gl(2, 1) carry a freee parameter. We compute the spectra of the corresponding transfer matrices by means of the nested algebraic Bethe ansatz together with a generalized fusion procedure.     

Nested bethe ansatz for <Emphasis Type="Italic">y</Emphasis>(<Emphasis Type="Italic">gl</Emphasis>(n)) open spin chains with diagonal boundary conditions

In this proceeding we present the nested Bethe ansatz for open spin chains of XXX-type, with arbitrary representations (i.e. “spins”) on each site of the chain and diagonal boundary matrices (K ⁺(u), K ⁻(u)). The nested Bethe ansatz applies for a general K ⁻(u), but a particular form of the K ⁺(u) matrix. We give the eigenvalues, Bethe equations and the form of the Bethe vectors for the corresponding models. The Bethe vectors are expressed using a trace formula     

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S.     

A supersymmetric Uq[osp(2|2)]-extended Hubbard model with boundary fields

A strongly correlated electron system associated with the quantum superalgebra U_q[osp(2|2)] is studied in the framework of the quantum inverse scattering method. By solving the graded reflection equation, two classes of boundary-reflection K-matrices leading to four kinds of possible boundary interaction terms are found. Performing the algebraic Bethe ansatz, we diagonalize the two-level transfer matrices which characterize the charge and the spin degrees of freedom, respectively. The Bethe-ansatz equations, the eigenvalues of the transfer matrices and the energy spectrum are presented explicitly. We also construct two impurities coupled to the boundaries. In the thermodynamic limit, the ground state properties and impurity effects are discussed.     

Noncompact <Emphasis Type="Italic">sl</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) Spin Chains: BGG-Resolution,Q-Operators and Alternating Sum Representation for Finite-Dimensional Transfer Matrices

We study properties of transfer matrices in the sl(N) spin chain models. The transfer matrices with an infinite-dimensional auxiliary space are factorized into the product of N commuting Baxter Q{\mathcal{Q}}-operators. We consider the transfer matrices with auxiliary spaces of a special type (including the finite-dimensional ones). It is shown that they can be represented as the alternating sum over the transfer matrices with infinite- dimensional auxiliary spaces. We show that certain combinations of the Baxter Q{\mathcal{Q}}-operators can be identified with the Q-functions, which appear in the Nested Bethe Ansatz.     

O(n) model on the honeycomb lattice via reflection matrices: Surface critical behaviour

We study the O(n) loop model on the honeycomb lattice with open boundary conditions. Reflection matrices for the underlying Izergin-Korepin R-matrix lead to three inequivalent sets of integrable boundary weights. One set, which has previously been considered, gives rise to the ordinary surface transition. The other two sets correspond respectively to the special surface transition and the mixed ordinary-special transition. We analyse the Bethe ansatz equations derived for these integrable cases and obtain the surface energies together with the central charges and scaling dimensions characterizing the corresponding phase transitions.     

Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions

By using the intertwiner and face-vertex correspondence relation, we obtain the Bethe ansatz equation of the eight-vertex model with open boundary conditions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for the eight-vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries.     

Non-abelian exclusion statistics

We introduce the notion of ‘order-k non-abelian exclusion statistics’. We derive the associated thermodynamic equations by employing the thermodynamic Bethe ansatz for specific non-diagonal scattering matrices. We make contact with results obtained by different methods and we point out connections with ‘fermionic sum formulas’ for characters in a conformal field theory. As an application, we derive thermodynamic distribution functions for quasi-holes over a class of non-abelian quantum Hall states recently proposed by Read and Rezayi.     

Exact solution for the spin-s XXZ quantum chain with non-diagonal twists

We study integrable vertex models and quantum spin chains with toroidal boundary conditions. An interesting class of such boundaries is associated with non-diagonal twist matrices. For such models there are no trivial reference states upon which a Bethe ansatz calculation can be constructed, in contrast to the well-known case of periodic boundary conditions. In this paper we show how the transfer matrix eigenvalue expression for the spin-s XXZ chain twisted by the charge-conjugation matrix can in fact be obtained. The technique used is the generalization to spin-s of the functional relation method based on “pair propagation through a vertex”. The Bethe ansatz-type equations obtained reduce, in the case of lattice size N = 1, to those recently found for the Hofstadter problem of Bloch electrons on a square lattice in a magnetic field.     

The spectrum of the transfer matrices connected with Kac-Moody algebras

The spectrum of the transfer matrices corresponding to trigonometrical Bazhanov-Jimbo R matrices is found. The Bethe equations characterizing the eigenvalues of the transfer matrices are written down in terms of root systems. Using the generalization of the Bethe equations for Kac-Moody algebras D _inf4 ^sup(3) , G _inf2 ^sup(1) , E _inf6 ^sup(1) and E _inf6 ^sup(2) , we give conjectures for the eigenvalues of the corresponding transfer matrices.     

Algebraic Bethe ansatz for integrable Kondo impurities in the one-dimensional supersymmetric t-J model

An integrable Kondo problem in the one-dimensional supersymmetric t-J model is studied by means of the boundary supersymmetric quantum inverse scattering method. the boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.     

Elliptic Quantum Groups and Ruijsenaars Models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groupsE _Τ,η(slN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.     

Two-Term Dilogarithm Identities Related to Conformal Field Theory

We study 2 × 2 matrices A such that the corresponding thermodynamic Bethe ansatz (TBA) equations yield in the form of the effective central charge of a minimal Virasoro model. Certain properties of such matrices and the corresponding solutions of the TBA equations are established. Several continuous families and a discrete set of admissible matrices A are found. The corresponding two-term dilogarithm identities (some of which appear to be new) are obtained. Most of them are proven or shown to be equivalent to previously known identities.     

Classification of reflection matrices related to (super-)Yangians and application to open spin chain models

We present a classification of diagonal, antidiagonal and mixed reflection matrices related to Yangian and super-Yangian R matrices associated to the infinite series so(m), sp(n) and osp(m|n). We formulate the analytical Bethe ansatz resolution for the so(m) and sp(n) open spin chains with boundary conditions described by the diagonal solutions.     

Anyonic interpretation of Virasoro characters and the thermodynamic Bethe ansatz

Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless.