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International Journal of Quantum Information

The open XXZ spin chain model and the topological basis realization

Qingyong Wang, Yangyang Du, Chunfeng Wu, Gangcheng Wang, Chunfang Sun, Kang Xue

Published Apr 1, 2016• 1 citation
Publisher

Abstract

In this paper, it is shown that the Hamiltonian of the open spin-1 XXZ chain model can be constructed from the generators of the Birman–Murakami–Wenzl (B–M–W) algebra. Without the topological parameter d (describing the unknotted loop [Formula: see text] in topology) reducing to a fixed value, the topological basis states can be connected with the open XXZ spin chain. Then some particular properties of the topological basis states in this system have been investigated. We find that the topological basis states are the three eigenstates of a four-spin-1 XXZ chain model without boundary term. Specifically, all the spin single states of the system fall on the topological basis subspace. And the number of the spin single states of the system is equal to that of the topological basis states.

DOI
10.1142/s0219749916500180

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In this paper, it is shown that the Hamiltonian of the open spin-1 XXZ chain model can be constructed from the generators of the Birman–Murakami–Wenzl (B–M–W) algebra. Without the topological parameter d (describing the unknotted loop [Formula: see text] in topology) reducing to a fixed value, the topological basis states can be connected with the open XXZ spin chain. Then some particular properties of the topological basis states in this system have been investigated. We find that the topological basis states are the three eigenstates of a four-spin-1 XXZ chain model without boundary term. Specifically, all the spin single states of the system fall on the topological basis subspace. And the number of the spin single states of the system is equal to that of the topological basis states.
In this paper, it is shown that the XXX model can be constructed from the Temperley-Lieb algebra (TLA) generator. We find that the topological basis states are the two eigenstaes of a closed four-qubit Heisenberg XXX spin chain. Specifically, the spin single states and the energy single state of the system all fall on the topological basis states. It is worth mentioning that for the closed 2 N -qubit ( N =2, 3, 4, ...) Heisenberg XXX spin chain, all the topological basis states for 2 N particles are the spin single states of the system. And the number of the topological basis states is equal to the number of the spin single states of the system, which is .
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.
We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.
The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both S+=SS^+=SS+=S and S+=S−1S^+=S^{-1}S+=S−1. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points (θ=±π/2(θ=\pmπ/2(θ=±π/2 and ±π)\pmπ)±π) of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.