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arXiv:q-alg/9711018 · Journal of Mathematical Physics

A one-dimensional many-body integrable model from Zn Belavin model with open boundary conditions

Heng Fan, Bo-Yu Hou, Guang-Liang Li, Kang-Jie Shi, Yan-Shen Wang

Published Sep 1, 1998• 1 citation
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Abstract

We use a factorized L operator to construct an integrable model with open boundary conditions. By taking the trigonometric (τ→−1∞) and scaling limits (w→0), we get a Hamiltonian of a classical integrable system. It shows that this integrable system is similar to those found by Calogero et al. [J. Math. Phys. 10, 2191 (1969); 12, 419 (1971)].

DOI
10.1063/1.532533

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We use a factorized L operator to construct an integrable model with open boundary conditions. By taking the trigonometric (τ→−1∞) and scaling limits (w→0), we get a Hamiltonian of a classical integrable system. It shows that this integrable system is similar to those found by Calogero et al. [J. Math. Phys. 10, 2191 (1969); 12, 419 (1971)].
The Inozemtsev limit (IL) or the scaling limit is known to be a procedure applied to the elliptic Calogero Model. It is a combination of the trigonometric limit, infinite shifts of particles coordinates and rescalings of the coupling constants. As a result, one obtains an exponential type of interaction. In the recent paper it is shown that the IL applied to the sl(N,C)sl(N,\bf C)sl(N,C) elliptic Euler-Calogero Model and the elliptic Gaudin Model produces new Toda-like systems of NNN interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit in sl(n,C)sl(n,\bf C)sl(n,C). The limits corresponding to the complete degeneration of the orbital degrees provide only ordinary periodic and non periodic Toda systems. We introduce a classification of the systems appearing in the sl(3,C)sl(3,\bf C)sl(3,C) case via IL. The classification is represented on two-dimensional space of parameters describing the infinite shifts of the coordinates. This space is subdivided into symmetric domains. The mixture of the Toda and the trigonometric Calogero-Sutherland potentials emerges on the low dimensional domain walls of this picture. Due to obvious symmetries this classification can be generalized to the arbitrary number of particles. We also apply IL to sl(2,C)sl(2,\bf C)sl(2,C) elliptic Gaudin Model with two marked points on the elliptic curve and discuss main features of its possible limits. The limits of Lax matrices are also considered.
The Inozemtsev limit (IL) or the scaling limit is known to be a procedure applied to the elliptic Calogero Model. It is a combination of the trigonometric limit, infinite shifts of particles coordinates and rescalings of the coupling constants. As a result, one obtains an exponential type of interaction. In the recent paper it is shown that the IL applied to the sl(N,C)sl(N,\bf C)sl(N,C) elliptic Euler-Calogero Model and the elliptic Gaudin Model produces new Toda-like systems of NNN interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit in sl(n,C)sl(n,\bf C)sl(n,C). The limits corresponding to the complete degeneration of the orbital degrees provide only ordinary periodic and non periodic Toda systems. We introduce a classification of the systems appearing in the sl(3,C)sl(3,\bf C)sl(3,C) case via IL. The classification is represented on two-dimensional space of parameters describing the infinite shifts of the coordinates. This space is subdivided into symmetric domains. The mixture of the Toda and the trigonometric Calogero-Sutherland potentials emerges on the low dimensional domain walls of this picture. Due to obvious symmetries this classification can be generalized to the arbitrary number of particles. We also apply IL to sl(2,C)sl(2,\bf C)sl(2,C) elliptic Gaudin Model with two marked points on the elliptic curve and discuss main features of its possible limits. The limits of Lax matrices are also considered.
The Hamiltonian of a 1D quantum chain corresponding to Belavin's z n × z n symmetric model is derived. This Hamiltonian is a multicomponent generalisation of the XYZ model with n 2 coupling constants in general. In the limit tau to i infinity it reduces to the generalised XXZ model with (n-1) n-fold-degenerate coupling constants and n non-degenerate ones. The Hamiltonian is Hermitian only for n=2 or for n > 2 with a crossing parameter w restricted to integers. There exists a domain of parameters in which the Boltzmann weights are positive but the corresponding Hamiltonian is non-Hermitian. The relations of the Hamiltonian to other models are discussed.