Non-hermitean hamiltonians with unitary and antiunitary symmetry

TL;DR

This paper investigates non-Hermitian Hamiltonians with antiunitary symmetry, using point-group symmetry to predict energy degeneracies, classify solutions, and analyze PT phase transitions, revealing cases where complex eigenvalues occur universally.

Contribution

It introduces a symmetry-based framework to analyze non-Hermitian Hamiltonians, explaining anomalous PT-symmetry breaking and classifying solutions by point-group irreducible representations.

Findings

Some PT-symmetric Hamiltonians with $C_{2v}$ symmetry have complex eigenvalues for all parameters.

PT phase transition can occur at the Hermitian limit, indicating non-robustness.

Point-group symmetry helps identify suitable antiunitary operators for PT symmetry.

Abstract

We analyze several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyze their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry $C_{2v}$ exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry.