Spontaneous PT-symmetry breaking for systems of noncommutative Euclidean Lie algebraic type

TL;DR

This paper introduces a noncommutative Euclidean Lie algebra $E_2$ and investigates non-Hermitian Hamiltonians, identifying parameter regions with real spectra and exceptional points where PT-symmetry breaks.

Contribution

It develops a noncommutative version of the Euclidean Lie algebra $E_2$ and analyzes PT-symmetry breaking in associated Hamiltonian systems.

Findings

Identified domains with real energy spectra in parameter space.

Determined exceptional points indicating PT-symmetry breaking.

Found invariant and deformation-dependent exceptional points.

Abstract

We propose a noncommutative version of the Euclidean Lie algebra $E_2$. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.