Extremely broken generalized $\mathcal{PT}$ symmetry

TL;DR

This paper explores simple matrix models with generalized $\ ext{PT}$ symmetry that exhibit extremely broken antiunitary symmetry, leading to complex eigenvalues across all parameters, highlighting the role of point-group symmetry in non-Hermitian spectra.

Contribution

It introduces H"uckel-like matrix representations of non-Hermitian operators with generalized $\ ext{PT}$ symmetry demonstrating extremely broken antiunitary symmetry due to degeneracy.

Findings

Examples show complex eigenvalues for all parameter values.

Point-group symmetry influences the spectral properties.

Graphical methods help construct unitary matrices.

Abstract

We discuss some simple H\"uckel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized $\mathcal{PT}$ (parity transformation followed by time-reversal) symmetry. One of them exhibits extremely broken antiunitary symmetry (complex eigenvalues for all nontrivial values of the model parameter) because of the degeneracy of the operator in the Hermitian limit. These examples illustrate the effect of point-group symmetry on the spectrum of the non-Hermitian operators. We construct the necessary unitary matrices by means of simple graphical representations of the non-Hermitian operators.