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Journal of the Physical Society of Japan

Quantum Integrable Model on (2+1)-D Lattice

Rei Inoue, Kazuhiro Hikami

Published Jun 15, 1999• 1 citation
Publisher

Abstract

We construct the quantum integrable dynamical system on a (2+1)-dimensional lattice. We consider the q -commuting operators on 2-D lattice as a discretization of the current algebra, and two evolution operators are explicitly defined in terms of the quantum dilogarithm function.
DOI
10.1143/jpsj.68.1843

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In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.
We study the quantum group symmetry of the \({\bb Z}_{N}\) symmetric lattice field theory, which is viewed as a discretization of the sℓ(N)^\widehat{s\ell (N)}sℓ(N)​ affine Toda field theory. We reveal the quantum group structure, and construct explicitly the quantum local integrals of motion by use of the quantum dilogarithm function.
We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.
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