Spectra, eigenstates and transport properties of a $\mathcal{PT}$-symmetric ring

TL;DR

This paper analyzes a $ ext{PT}$-symmetric ring's spectral and transport properties, revealing unique phenomena like diabolical points, backflow, and singular eigenstates, with implications for non-Hermitian quantum systems.

Contribution

It provides a comprehensive analytical and numerical study of the spectral singularities and transport behavior in a $ ext{PT}$-symmetric ring, including the discovery of diabolical points and classification of eigenstates.

Findings

Identification of diabolical and inverse exceptional points.

Observation of stationary backflow in density transport.

Classification of singular eigenstates with distinct transport properties.

Abstract

We study, analytically and numerically, a simple $\mathcal{PT}$-symmetric tight-binding ring with an onsite energy $a$ at the gain and loss sites. We show that if $a\neq 0$, the system generically exhibits an unbroken $\mathcal{PT}$-symmetric phase. We study the nature of the spectrum in terms of the singularities in the complex parameter space as well as the behavior of the eigenstates at large values of the gain and loss strength. We find that in addition to the usual exceptional points, there are "diabolical points", and inverse exceptional points at which complex eigenvalues reconvert into real eigenvalues. We also study the transport through the system. We calculate the total flux from the source to the drain, and how it splits along the branches of the ring. We find that while usually the density flows from the source to the drain, for certain eigenstates a stationary "backflow" of density from the drain to the source along one of the branches can occur. We also identify two types of singular eigenstates, i.e. states that do not depend on the strength of the gain and loss, and classify them in terms of their transport properties.