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International Journal of Theoretical Physics

Yangian and Topological Basis Expressions of Two Types of XXZ Spin Chains

Guijiao Du, Kang Xue, Chengcheng Zhou

Published May 4, 2019• 0 citations
Publisher
DOI
10.1007/s10773-019-04107-6

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International Journal of Quantum Information•Apr 1, 2016•1 citation
In this paper, we investigate the Yangian relations of Heisenberg spin chain systems. Firstly, we consider the closed XXZ spin chain model, through the Heisenberg spin XXZ model, we found the Hamiltonians for one kind system of three adjacent partial particles interaction systems. The model's constitution rules of energy levels and energy states which expand from the few-particle system to multi-particle system have good regularity. In this system, we found Yangian's law and illustrate it through graphs. Secondly, we further consider the closed XXZ spin chain's generalization of other three neighboring particles interaction systems from few-particle system to multi-particle system. Finally, we also discussed the laws of the three adjacent particles system of some models, they are the XXZ model with twist boundary condition, the open XXZ spin chain model and the XXZ model containing the next neighbor. In addition, not only XXZ model, XXX model, XY model and Ising model, but the relevant laws of spin-1 systems of these models were also discussed, they have similar rules to the XXZ model. Through calculation and research, the eigensystems of these models all have good Yangian and constitution laws.
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.