Algebraic Bethe ansatz for invariant integrable models: Compact and non-compact applications

We apply the algebraic Bethe ansatz developed in our previous paper [C.S. Melo, M.J. Martins, Nucl. Phys. B 806 (2009) 567] to three different families of U(1) integrable vertex models with arbitrary N bond states. These statistical mechanics systems are based on the higher spin representations of the quantum group U_q[SU(2)] for both generic and non-generic values of q as well as on the non-compact discrete representation of the algebra. We present for all these models the explicit expressions for both the on-shell and the off-shell properties associated to the respective transfer matrices eigenvalue problems. The amplitudes governing the vectors not parallel to the Bethe states are shown to factorize in terms of elementary building blocks functions. The results for the non-compact model are argued to be derived from those obtained for the compact systems by taking suitable N→∞ limits. This permits us to study the properties of the non-compact model starting from systems with finite degrees of freedom.     

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.     

Analytical Bethe ansatz in gl(N) spin chains

We present a global treatment of the analytical Bethe ansatz for gl(N) spin chains admitting on each site an arbitrary representation. The method applies to closed and open spin chains, and also to the case of soliton non-preserving boundaries. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Unified algebraic Bethe ansatz for two-dimensional lattice models

We develop a unified formulation of the quantum inverse scattering method for lattice vertex models associated to the nonexceptional A(2)(2r), A(2)(2r-1), B(1)(r), C(1)(r), D(1)(r+1), and D(2)(r+1) Lie algebras. We recast the Yang-Baxter algebra in terms of different commutation relations between creation, annihilation, and diagonal fields. The solution of the D(2)(r+1) model is based on an interesting 16-vertex model, which is solvable without recourse to a Bethe ansatz.     

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.     

Intertwining relations,asymmetric face model,and algebraic Bethe ansatz forSU p,q (2) invariant spin chain

Intertwining relations for the quantumR-matrix of theSU _p,q (2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parametersp andq, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.     

One-parameter family of an integrable spl(2|1) vertex model: Algebraic Bethe ansatz and ground state structure

We formulate in terms of the quantum inverse scattering method the exact solution of a spl(2|1) invariant vertex model recently introduced in the literature. The corresponding transfer matrix is diagonalized by using the algebraic (nested) Bethe ansatz approach. The ground state structure is investigated and we argue that a Pokrovsky-Talapov transition is favored for a certain value of the 4-dimensional spl(2|1) parameter.     

The algebraic Bethe ansatz for rational braid-monoid lattice models

In this paper we study isotropic integrable systems based on the braid-monoid algebra. These systems constitute a large family of rational multistate vertex models and are realized in terms of the B_n, C_nand D_n Lie algebra and by the superalgebra Osp(n||2m). We present a unified formulation of the quantum inverse scattering method for many of these lattice models. The appropriate fundamental commutation rules are found, allowing us to construct the eigenvectors and the eigenvaluesof the transfer matrix associated to the B_n, C_n, D_n, Osp(2nt-1||2), Osp(2||2nt-2), Osp(2nt-2||2) and Osp(1||2n) models. The corresponding Bethe ansatz equations can be formulated in terms of the root structure of the underlying algebra.     

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.     

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.     

Algebraic Bethe Ansatz for the Osp(1|2) Model with Reflecting Boundaries

In the framework of graded quantum inverse scattering method, we obtain the eigenvalues and the eigenvectors of the Osp(l|2) model with reflecting boundary conditions in FBF background. The corresponding Bathe ansatz equations are obtained.     

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.     

Bethe ansatz solution for quantum spin-1 chains with boundary terms

The procedure for obtaining integrable open spin chain Hamiltonians via reflection matrices is explicitly carried out for some three-state vertex models. We have considered the 19-vertex models of Zamolodchikov–Fateev and Izergin–Korepin, and the Z₂-graded 19-vertex models with sl(2|1) and osp(1|2) invariances. In each case the eigenspectrum is determined by application of the coordinate Bethe ansatz.     

Nonequilibrium transport in quantum impurity models: the Bethe ansatz for open systems

We develop an exact nonperturbative framework to compute steady-state properties of quantum impurities subject to a finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates which satisfy an appropriate Lippman-Schwinger equation. Introducing a generalization of the equilibrium Bethe ansatz--the nonequilibrium Bethe ansatz--we explicitly construct the scattering eigenstates for the interacting resonance level model and derive exact, nonperturbative results for the steady-state properties of the system.     

Bethe ansatz and boundary energy of the open spin-1/2 XXZ chain

We review recent results on the Bethe ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters α_?, α₊, β_?, β₊ are nonzero. A generalization of the BaxterT-Q equation that involves more than one independentQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.     

Drinfeld Twists and Algebraic Bethe Ansatz of the Supersymmetric Model Associated with U q (gl(m|n))

We construct the Drinfeld twists (or factorizing F-matrices) of the super-symmetric model associated with quantum superalgebra U_q(gl(m|n)), and obtain the completely symmetric representations of the creation operators of the model in the F-basis provided by the F-matrix. As an application of our general results, we present the explicit expressions of the Bethe vectors in the F-basis for the U_q(gl(2|1))-model (the quantum t-J model).     

Pseudo-differential equations,and the Bethe ansatz for the classical Lie algebras

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.