Exact Solutions and Degenerate Properties of Spin Chains with Reducible Hamiltonians

Abstract

The Jordan--Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of $N$ spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the $N-1$ bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan--Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model (${\sigma }_x{\sigma }_y-{\sigma }_x{\sigma }_y$) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.

9 pages, 3 figures

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• Topological characterization of quantum phase transitions in a S=1/2 spin model
• Loschmidt Echo Revivals: Critical and Noncritical
• Quench dynamics and ground state fidelity of the one-dimensional extended quantum compass model in a transverse field
• Hartree-Fock-Bogoliubov Theory of Polarized Fermi Systems
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Cited by in corpus (1)

• State selection in frustrated magnets