Spontaneous breakdown of PT symmetry in the solvable square-well model

TL;DR

This paper investigates the spontaneous PT-symmetry breaking in a solvable square-well model, revealing a gradual transition where only parts of the spectrum become complex, challenging previous oversimplified interpretations.

Contribution

It demonstrates that PT-symmetry breaking occurs stepwise in a solvable model, with some states remaining real while others become complex, contrasting with earlier all-or-nothing views.

Findings

Doublets of states with broken PT symmetry persist at complex conjugate energies.

Spectral complexification occurs gradually, starting from low-lying states.

The model challenges the notion of simultaneous spectrum complexification during PT-symmetry breaking.

Abstract

In many PT symmetric models with real spectra, apparently, energy levels "merge and disappear" at a point of the spontaneous PT-symmetry breaking. We argue that such an oversimplified and discontinuous physical interpretation of this mechanism as proposed, e.g., by one of us in Phys. Lett. A 285 (2001), p. 7 would be inappropriate. Using the elementary square-well model of the above reference in the strongly non-Hermitian regime we exemplify how the doublets of states with broken PT symmetry continue to exist at complex conjugate energies. In contrast to many other exactly solvable examples of such a mechanism (we listed some of them in quant-ph/0110064), our present model of symmetry breaking does not complexify all the spectrum at once. "Realistically", it rather proceeds step by step, starting from the low-lying part of the spectrum and involving more and more excited states with an increasing degree of non-Hermiticity of the underlying interaction.