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Quantum Information Processing

The quantum spin chains of Temperley–Lieb type and the topological basis states

Chunfang Sun, Kang Xue, Gangcheng Wang, Chengcheng Zhou, Guijiao Du

Published May 11, 2013• 2 citations
Publisher
DOI
10.1007/s11128-013-0542-8

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International Journal of Quantum Information•Mar 1, 2012•1 citation
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 determine the spectra of a class of quantum spin chains of Temperley–Lieb type by utilizing the concept of Temperley–Lieb equivalence with the S = 1/2 XXZ chain as a reference system. We consider open boundary conditions and, in particular, periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley–Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representative theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field .