Interplay of fractionality and $\cal{PT}$- symmetry on a 1D lattice

TL;DR

This paper investigates how fractional calculus and $ ext{PT}$ symmetry influence the stability and localization of eigenmodes in a 1D discrete Schr"odinger system, revealing critical thresholds and mode behavior changes.

Contribution

It introduces a combined analysis of fractional $ ext{PT}$-symmetric systems, highlighting the effects of fractional order and gain/loss on stability and mode localization.

Findings

Instability gain increases abruptly past a critical fractional exponent.

Stable fraction decreases with system size and gain/loss.

Participation ratio decreases, indicating mode localization.

Abstract

We examine the stability domains of a 1D discrete Schr\"{o}dinger equation in the simultaneous presence of parity-time ($\cal{PT}$) symmetry and fractionality. Direct numerical examination of the eigenvalues of the system reveals that, as the fractional exponent is decreased away from unity (the standard case), the instability gain increases abruptly past a critical value. Also, as the length of the system increases, the stable fraction decreases as well. Also, for a fixed fractional exponent and lattice size, an increase in gain/loss also brings about an abrupt increase in the instability gain. Finally, the participation ratio of the modes is seen to decrease with an increase of the gain/loss parameter and with a decrease of the fractional exponent, evidencing a tendency towards localization.