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arXiv:solv-int/9801013 · Journal of Nonlinear Mathematical Physics

Particles and Strings in a 2 + 1-D Integrable Quantum Model

I. G. Korepanov

Published Jan 1, 2000• 4 citations
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Abstract

We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory.As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang-Baxter equation.We construct some eigenstates of the transfer matrix of that model.There arise, together with states composed of point-like particles analogous to those in the usual 1 + 1-dimensional Bethe ansatz, new string-like states and string-particle hybrids.

DOI
10.2991/jnmp.2000.7.1.7

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We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory.As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang-Baxter equation.We construct some eigenstates of the transfer matrix of that model.There arise, together with states composed of point-like particles analogous to those in the usual 1 + 1-dimensional Bethe ansatz, new string-like states and string-particle hybrids.
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.
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.