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International Journal of Quantum Information

THE TOPOLOGICAL BASIS REALIZATION OF FOUR-QUBIT XXZ SPIN CHAIN

Guijiao Du, Kang Xue, Chunfang Sun, Gangcheng Wang, CHENGCHENG ZHOU

Published Mar 1, 2012• 1 citation
Publisher

Abstract

In this paper, a set of topological basis realization is presented. By acting on the different subspaces, we obtain the new nontrivial six-dimensional (6D) and four-dimensional Temperley–Lieb matrix representations. Then it is shown that XXZ model can be constructed from the Temperley–Lieb algebra (TLA) generators. The eigenstates of XXZ model can be denoted by combinating the set of topological bases. It is interesting that the ground state is closely related to parameter Δ.

DOI
10.1142/s0219749912500219

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We investigate the topological basis realization associated with the Hermitian and non-Hermitian Heisenberg XXZ model based on the matrix representation of the Temperley-Lieb algebra. The results show that the topological space is spanned by the topological basis associated with the Temperley-Lieb algebra. When the q -deformed parameter q is a complex number and | q | = 1 , the corresponding model is the non-Hermitian XXZ model. The real q corresponds to the Hermitan case. The topological basis realization for these two cases is constructed by means of matrix representation of the Temperley-Lieb algebra. Then the properties of the topological basis and ground state are studied. Particularly, the four-qubit Heisenberg XXZ model is investigated in detail. We also give some discussions on the case in which q is a root of unity.
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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.