Competing PT potentials and re-entrant PT symmetric phase for a particle in a box

TL;DR

This paper explores how competing complex $ ext{PT}$-symmetric potentials influence the phase behavior of a particle in a box, revealing conditions for phase strengthening and restoration, and comparing continuum and lattice models.

Contribution

It introduces a study of combined long-range and localized $ ext{PT}$-symmetric potentials, showing their effects on phase stability and symmetry breaking in quantum systems.

Findings

Strengthening of $ ext{PT}$ phase near the box edges with added losses.

Restoration of broken $ ext{PT}$ symmetry by increasing localized potential strength.

Continuum systems exhibit robust $ ext{PT}$ phase, unlike fragile lattice counterparts.

Abstract

We investigate the effects of competition between two complex, $\mathcal{PT}$-symmetric potentials on the $\mathcal{PT}$-symmetric phase of a "particle in a box". These potentials, given by $V_Z(x)=iZ\mathrm{sign}(x)$ and $V_\xi(x)=i\xi[\delta(x-a)-\delta(x+a)]$, represent long-range and localized gain/loss regions respectively. We obtain the $\mathcal{PT}$-symmetric phase in the $(Z,\xi)$ plane, and find that for locations $\pm a$ near the edge of the box, the $\mathcal{PT}$-symmetric phase is strengthened by additional losses to the loss region. We also predict that a broken $\mathcal{PT}$-symmetry will be restored by increasing the strength $\xi$ of the localized potential. By comparing the results for this problem and its lattice counterpart, we show that a robust $\mathcal{PT}$-symmetric phase in the continuum is consistent with the fragile phase on the lattice. Our results demonstrate that systems with multiple, $\mathcal{PT}$-symmetric potentials show unique, unexpected properties.