Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model

TL;DR

This paper explores how extending the XY spin chain model to non-Hermitian regimes reveals exceptional point rings and PT-symmetry breaking, linking spectral degeneracies to topological phase boundaries.

Contribution

It introduces the non-Hermitian XY model with complex anisotropy, demonstrating the emergence of exceptional point rings and their relation to topological phases and PT-symmetry.

Findings

Exceptional points form concentric rings in the complex plane.

EP rings converge to the unit circle at the topological boundary.

Finite systems show four EPs on the PT-broken line when size is multiple of 4.

Abstract

The XY spin chain is a paradigmatic example of a model solved by free fermions, in which the energy eigenspectrum is built from combinations of quasi-energies. In this article we show that by extending the XY model's anisotropy parameter $\lambda$ to complex values, it is possible for two of the quasi-energies to become degenerate. In the non-Hermitian XY model these quasi-energy degeneracies give rise to exceptional points (EPs) where two of the eigenvalues and their corresponding eigenvectors coalesce. The distinct $\lambda$ values at which EPs appear form concentric rings in the complex plane which are shown in the infinite system size limit to converge to the unit circle coinciding with the boundary between distinct topological phases. The non-Hermitian model is also seen to possess a line of broken $PT$ symmetry along the pure imaginary $\lambda$-axis. For finite systems, there are four EP values on this broken $PT$-symmetric line if the system size is a multiple of 4.