Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes

TL;DR

This paper investigates how localized modes in PT-symmetric lattices lead to an exponentially fragile exact PT phase, with a cascade of bifurcations and a narrow symmetry-breaking threshold that varies with system size and localization.

Contribution

It introduces a theoretical framework describing the effect of localized modes on PT symmetry breaking, revealing exponential fragility and phase transition characteristics.

Findings

PT symmetry breaking threshold is exponentially small in system size.

Localized modes cause a cascade of bifurcations in eigenvalue evolution.

Distribution of the PT breaking parameter shifts from log-normal to semi-Gaussian as parameters change.

Abstract

We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasi-degenerate levels with splitting delta exp{-N/xi}, where \xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gamma min(delta), thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gamma (1/N)^2 and the distribution P(gamma) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.