Is space-time symmetry a suitable generalization of parity-time symmetry?

TL;DR

This paper explores the conditions under which space-time symmetric Hamiltonians have real or complex eigenvalues, emphasizing the role of point-group symmetry and perturbation theory, with illustrative examples in various dimensions.

Contribution

It provides a theoretical framework linking point-group symmetry to the spectral properties of space-time symmetric Hamiltonians, extending the understanding of parity-time symmetry.

Findings

Point-group symmetry influences eigenvalue reality in space-time symmetric Hamiltonians.

Perturbation theory can predict when eigenvalues are real or complex.

Examples demonstrate the theoretical predictions across different symmetries.

Abstract

We discuss space-time symmetric Hamiltonian operators of the form $% H=H_{0}+igH^{\prime}$, where $H_{0}$ is Hermitian and $g$ real. $H_{0}$ is invariant under the unitary operations of a point group $G$ while $H^{\prime}$ is invariant under transformation by elements of a subgroup $G^{\prime}$ of $G$. If $G$ exhibits irreducible representations of dimension greater than unity, then it is possible that $H$ has complex eigenvalues for sufficiently small nonzero values of $g$. In the particular case that $H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all $% 0<g<g_{c}$, where $g_{c}$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether $% H $ may exhibit real or complex eigenvalues for $g>0$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.