Exact Solutions and Degenerate Properties of Spin Chains with Reducible Hamiltonians

TL;DR

This paper identifies specific spin chains that can be exactly mapped to fermionic chains without redundancy, revealing high degeneracy in ground and excited states, exemplified by the compass and XY-XY models.

Contribution

It demonstrates that certain periodic spin chains can be exactly mapped to fermionic chains without redundancy, uncovering high degeneracy in their spectra.

Findings

High degeneracy in ground and excited states.

Exact mapping of specific spin chains to fermionic chains.

Degeneracy grows exponentially with system size.

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.