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Journal of Mathematical Physics

Construction of quasi-two- and higher-dimensional quantum integrable models

Anjan Kundu

Published Feb 1, 2000• 8 citations
Publisher

Abstract

A class of quantum integrable quasi-two- and higher-dimensional quantum spin as well as strongly correlated electron systems with localized interactions are proposed. The basic idea of construction is to introduce interchain interactions in an array of spin chains or one-dimensional Hubbard models through twisting transformation. The models allow explicit quantum R-matrix, Lax operator, and exact eigenvalue solution.

DOI
10.1063/1.533161

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A class of quantum integrable quasi-two- and higher-dimensional quantum spin as well as strongly correlated electron systems with localized interactions are proposed. The basic idea of construction is to introduce interchain interactions in an array of spin chains or one-dimensional Hubbard models through twisting transformation. The models allow explicit quantum R-matrix, Lax operator, and exact eigenvalue solution.
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Within the framework of quantum groups and additive R -matrices, the fusion procedure allows one to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here fusion is shown to generalize naturally to non-additive R -matrices, which therefore do not have a quantum group symmetry. This method is then applied to the generalized Hubbard models. Although the resulting integrable models are not as simple as the starting ones, the general structure is that of two spin-( s × s ´) sl (2) models coupled at the free-fermion point. An important issue is the probable lack of regular points which give local Hamiltonians. This problem is related to the existence of second-order zeros in the unitarity equation, and arises for the XX models of higher spins, the building blocks of the Hubbard models. A possible connection between some Lax operators L and R -matrices is noted.
Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two dimensions involving interacting fields. The Yang-Baxter integrability is proved for the model by finding a new kind of commutation rule for its basic fields, representing nonstandard scalar fields along the transverse direction. In spite of a close link with the quantum Landau-Lifshitz equation, the present model differs widely from it, in its content and the result obtained. Using further the algebraic Bethe ansatz we solve exactly the eigenvalue problem of this quantum field model for all its higher conserved operators. The idea presented here should instigate the construction of a novel class of integrable field and lattice models and exploration of a new type of underlying algebras.
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.