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.     

An open-boundary integrable model of three coupled XY spin chains

The integrable open-boundary conditions for the model of three coupled one-dimensional XY spin chains are considered in the framework of the quantum inverse scattering method. The diagonal boundary K-matrices are found and a class of integrable boundary terms is determined. The boundary model Hamiltonian is solved by using the coordinate space Bethe ansatz technique and Bethe ansatz equations are derived.     

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.     

Integrable open boundary conditions for the Bariev model of three coupled XY spin chains

The integrable open-boundary conditions for the Bariev model of three coupled one-dimensional XY spin chains are studied in the framework of the boundary quantum inverse scattering method. Three kinds of diagonal boundary K-matrices leading to nine classes of possible choices of boundary fields are found and the corresponding integrable boundary terms are presented explicitly. The boundary Hamiltonian is solved by using the coordinate Bethe ansatz technique and the Bethe ansatz equations are derived.     

Exact thermodynamics of the Hubbard chain: free energy and correlation lengths

We study the one-dimensional Hubbard model at finite temperatures in the quantum transfer matrix approach. The eigenvalue equations of this matrix are obtained by a nested Bethe ansatz. The largest and next-largest eigenvalues yield the free energy as well as the correlation lengths of the system. An equivalent set of four integral equations is derived from the Bethe ansatz equations. The limit of Trotter-Suzuki number N → ∞ is taken analytically. For half-filling the final equations are studied in the low-temperature limit yielding analytic expressions for the free energy and spin-spin correlation length. Numerical results are presented for intermediate temperatures.     

Algebraic properties of an integrable t-J model with impurities

We investigate the algebraic structure of a recently proposed integrable t-J model with impurities. Three forms of the Bethe ansatz equations are presented corresponding to the three choices for the grading. We prove that the Bethe ansatz states are highest weight vectors of the underlying gl(2′1) supersymmetry algebra. By acting with the gl(2′1) generators we construct a complete set of states for the model.     

Exact excitation spectrum of the Zn + 1 × Zn + 1 generalized Heisenberg model

We describe the excitations of the generalization of the Heisenberg anisotropic hamiltonian to Z_{n + 1} spins. We analyse the Bethe ansatz equations without assuming the existence of strings. Excited states above an antiferromagnetic ground state are described by a finite number of parameters, which verify system of equations. We give the energy and momentum of the excitations. In a suitable limit we recover the spectrum of a relativistically invariant theory.     

Two magnetic impurities with arbitrary spins in open boundary t-J model

From the open boundary t-J model an impurity model is constructed in which magnetic impurities of arbitrary spins are coupled to the edges of the strongly correlated electron system. The boundary R matrices are given explicitly. The interaction parameters between magnetic impurities and electrons are related to the potentials of the impurities to preserve the integrability of the system. The Hamiltonian of the impurity model is diagonalized exactly. The integral equations of the ground state are derived and the ground state properties are discussed in detail. We discuss also the string solutions of the Bethe ansatz equations, which describe the bound states of the charges and spins. By minimizing the thermodynamic potential we get the thermodynamic Bethe ansatz equations. The finite size correction of the free energy contributed by the magnetic impurities is obtained explicitly. The properties of the system at some special limits are discussed and the boundary bound states are obtained.     

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property.     

Algebraic Bethe ansatz for the gl(1|2) generalized model and Lieb–Wu equations

We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang's model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang–Baxter algebra generated by the gl(1|2) invariant R-matrix.     

The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite-dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.     

Exact solution of the XXX Gaudin model with generic open boundaries

The XXX Gaudin model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices is studied. Besides the inhomogeneous parameters, the associated Gaudin operators have six free parameters which break the U(1)$U\left(1\right)$-symmetry. With the help of the off-diagonal Bethe ansatz, we successfully obtained the eigenvalues of these Gaudin operators and the corresponding Bethe ansatz equations.     

A Note on the Bethe Ansatz Solution of the Supersymmetric t-J Model

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.     

Algebraic Bethe ansatz for the one-dimensional Hubbard model with chemical potential

The integrability and the algebraic Bethe ansatz approach for the one-dimensional (1D) Hubbard model with chemical potential are studied in the framework of the quantum inverse scattering method. We also investigate the hidden local gauge invariance for the model. It is found that the R-matrix only permits Abelian $\text{U(1) ⋇}{}_{\mathrm{s}}\text{U(1)}$ gauge transformations, and it is shown that the energy spectrum is gauge invariant whereas the eigenvectors and the Bethe ansatz equations are explicitly gauge dependent.     

Generic boundary scattering in the open XXZ chain

The open critical XXZ spin chain with a general right boundary and a trivial diagonal left boundary is considered. Within this framework we propose a simple computation of the exact generic boundary S-matrix (with diagonal and non-diagonal entries), starting from the ‘bare’ Bethe ansatz equations. Our results as anticipated coincide with the ones obtained by Ghoshal and Zamolodchikov, after assuming suitable identifications of the bulk and boundary parameters.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q$T\text{–}Q$ relation and the Bethe ansatz equations are derived.     

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.     

Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.     

Separation of variables for integrable spin–boson models

We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang–Baxter algebra. The main deviation from the standard approach consists in a half infinite Sklyanin lattice made of the eigenvalues of the operator zeros of the Bethe annihilation operator. By a separation of variables, functional TQ-equations are obtained for this half infinite lattice. They provide valuable information about the spectrum of a given Hamiltonian model. We apply this procedure to integrable spin–boson models subject to both twisted and open boundary conditions. In the case of general twisted and certain open boundary conditions polynomial solutions to these TQ-equations are found and we compute the spectrum of both the full transfer matrix and its quasi-classical limit. For generic open boundaries we present a two-parameter family of Bethe equations, derived from TQ-equations that are compatible with polynomial solutions for Q. A connection of these parameters to the boundary fields is still missing.     

Asymptotic Bethe ansatz for the Sutherland–Römer model with open boundary conditions

By constructing the reflection spin-Dunkl operators, the integrable Sutherland–Römer model (SRM) with open boundary condition is established, which describes a one-dimensional, two-component, quantum many-particle system in which like particles interact with a pair potential g(g+1)/sinh²(r), while unlike particles interact with a pair potential −g(g+1)/cosh²(r). By solving the Schrödinger equation and using the properties of the hypergeometric functions and gamma functions, the two-particle scattering matrix and the reflection matrix are obtained in the framework of the asymptotic Bethe ansatz method. The Bethe ansatz equations of the system are obtained. The Hamiltonians of SRM with some other open boundary conditions are expressed explicitly. Our method can be generalized, as a example, to the boundary Calogero–Sutherland model which is also constructed by the reflection spin-Dunkl operators.