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Journal of Physics A Mathematical and General

Equivalence and solution of anisotropic spin-1 models and generalized t-J fermion models in one dimension

Andreas Klümper, Andreas Schadschneider, J. Zittartz

Published Aug 21, 1991• 231 citations
Publisher

Abstract

The authors study the relationship of two 'q-deformed' spin-1 chains-both of them are solvable models-with a generalized supersymmetric t-J fermion model in one dimension. One of the spin-1 chains is an anisotropic VBS model for which they calculate ground state and ground-state properties. The other spin-1 chain corresponds to the Zamolodchikov-Fateev model which is solvable by Bethe ansatz and is equivalent to a certain t-J model. The two spin-1 models intersect for a certain value of the 'deformation' parameter q in a second-order phase transition.

DOI
10.1088/0305-4470/24/16/012

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The authors study the relationship of two 'q-deformed' spin-1 chains-both of them are solvable models-with a generalized supersymmetric t-J fermion model in one dimension. One of the spin-1 chains is an anisotropic VBS model for which they calculate ground state and ground-state properties. The other spin-1 chain corresponds to the Zamolodchikov-Fateev model which is solvable by Bethe ansatz and is equivalent to a certain t-J model. The two spin-1 models intersect for a certain value of the 'deformation' parameter q in a second-order phase transition.
In this work we investigate the supersymmetric t-J model in one dimension. The spectrum of the hamiltonian is obtained by means of the algebraic nested Bethe-ansatz method. Furthermore, we present a detailed analysis of the algebraic structure of the states. By combining the Bethe ansatz with the spl(2,1) supersymmetry of the t-J hamiltonian a complete set of eigenstates is constructed. The ground state and the elementary excitations of the model are investigated. We also introduce a new integrable vertex model, the anisotropic t-J model, which possesses splq(2, 1) "quantum supergroup" invariance.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe 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 antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities 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 behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
An exact solvable model of the coupled supersymmetric t - J chains interacting via the spin exchange interaction is proposed and solved by the means of the nested Bethe ansatz. It is shown, that in addition to the direct spin-exchange interaction an effective interaction between fermions of different chains called as "an effective attractive hard-core interaction" is realized in the model. The critical exponents describing asymptotic behavior of the correlation functions are calculated using the Bethe ansatz equations and the finite-state scaling analysis in conformal field theory. It is shown that the two-component Tomonaga-Luttinger liquid with the dominating correlations of fermion pairs is realized in the low-density region. This non-traditional behavior of the correlation functions is realized due to the effective attractive hard-core interaction between particles.
Quantum spin chains with exact valence-bond ground states are of great interest in condensed-matter physics. A class of such models was proposed by Affleck et al., each of which is su(2)-invariant and constructed as a sum of projectors onto definite total spin states at neighboring sites. We propose to use the machinery of the q-deformation of su(2) to obtain generalisations of such models, and work out explicitly the two simplest examples. In one case we recover the known anisotropic spin-1 VBS model while in the other we obtain a new anisotropic generalisation of the spin-½ Majumdar-Ghosh model.