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Chinese Physics C

XXZ Model and the Structure of the Symmetric Algebra

Ma Z, Baolin Hou

Published Jan 1, 1991• 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 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.

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Communications in Theoretical Physics•Jan 1, 1993•1 citation
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.
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.
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.
The 1D Heisenberg spin model with an anisotropy of the XXZ type is analysed in terms of the symmetry given by the quantum Galilei group Gamma q (1). For a chain with an infinite number of sites the authors show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. In this case the Gamma q (1) symmetry provides a description naturally compatible with the Bethe ansatz. The recurrence relations determined by Gamma q (1) permit one to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.
We present a systematic method to construct the spin models of Heisenberg type in higher dimensions with nearest and non-nearest neighbour interactions. These models constructed in this way are of Temperley-Lieb-Jones (TLJ) algebraic structures and -invariances. The TLJ algebra is generalized to adjusting the lattice spin models. The Hamiltonians of Heisenberg spin models in one dimension (including second nearest neighbour interactions) and in two-dimensional triangular lattice (with nearest interaction) are constructed explicitly. The hidden symmetries are shown to be the group, and the terms in Hamiltonian for different lattice cells are explicitly shown to be the representations of elements of (TLJ) algebra.