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XXZ MODEL AND THE STRUCTURE OF SYMMETRICAL ALGEBRA

马中骐, Zhong-Qi Ma, By Hou

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL(q) (2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyze the finite-dimensional representations of the quantum SL(q) (2) with q(p) = +/- 1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type I representation), their properties, and the complete set of the states related to a definite energy where the new states are obtained by an appropriate limiting process.

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In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL(q) (2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyze the finite-dimensional representations of the quantum SL(q) (2) with q(p) = +/- 1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type I representation), their properties, and the complete set of the states related to a definite energy where the new states are obtained by an appropriate limiting process.
In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL_q(2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyse the finite-dimensional representations of the quantum SL_q(2) with q~p=±1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type Ⅰ representation), their properties, and the completeset of the states related to a definite energy where the new states are obtained by an appropriare limit process.
We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley–Lieb algebra. For a system of size L these representations are all of dimension 2 L and, for generic points of the algebra, equivalent. However, at exceptional points they can possess different indecomposable structures. We study a centralizer of the one-boundary Temperley–Lieb algebra in the 'non-diagonal' spin- representation and find its eigenvalues and eigenvectors. In the exceptional cases this centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.
We investigate the Temperley–Lieb–Jones algebraic structures in isomorphic higher spin chain models with nearest neighbour interactions and symmetry. The Temperley–Lieb–Jones algebraic constructions for such chains of spins from 1 to are presented. For the case of spin-1 we also give the Temperley–Lieb–Jones algebraic representations related to the anisomorphic spin chain with -deformed symmetry. Their related spin chain Hamiltonians with and symmetries as well as the corresponding solutions of Yang–Baxter equation are also discussed.