Bethe ansatz solvable multi-chain quantum systems

In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Be the ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example ('basic' model) of this class of models: the antiferromagne tic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-ne ighbour spin- frustrating interactions (zigzag chain). We show how the alge bra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topologica l properties of two dimensions together with the one-dimensional properties ). We consider the solution of the Bethe ansatz for the ground state (in pa rticular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct th e thermal Bethe ansatz (in the form of the 'quantum transfer matrix') for t his model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy. higher-spin representations (includ ing the important case of a quantum ferrimagnet), the multi-chain case, int ernal degrees of freedom of particles at each site, etc. We observe the sim ilarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behav iour of non-integrable (less constrained) multi-chain quantum models is dis cussed.