# Spectral and transport properties of a $\mathcal{PT}$-symmetric tight-binding chain with gain and loss

## TL;DR

This paper investigates the transport properties of a $ ext{PT}$-symmetric tight-binding chain with gain and loss, deriving a continuity equation and identifying states with distinct transport behaviors, including opaque and transparent states.

## Contribution

It introduces a continuity equation approach to analyze transport in $ ext{PT}$-symmetric systems and identifies eigenstates with coupling-independent properties, revealing complex state behaviors.

## Key findings

- Broken $ ext{PT}$-symmetry states show inefficient transport.
- Existence of opaque and transparent eigenstates with coupling-independent eigenvalues.
- Number of opaque and transparent states varies irregularly with system parameters.

## Abstract

We derive a continuity equation to study transport properties in a $\mathcal{PT}$-symmetric tight-binding chain with gain and loss in symmetric configurations. This allows us to identify the density fluxes in the system, and to define a transport coefficient to characterize the efficiency of transport of each state. These quantities are studied explicitly using analytical expressions for the eigenvalues and eigenvectors of the system. We find that in states with broken $\mathcal{PT}$-symmetry, transport is inefficient, in the sense that either inflow exceeds outflow and density accumulates within the system, or outflow exceeds inflow, and the system becomes depleted. We also report the appearance of two subsets of interesting eigenstates whose eigenvalues are independent on the strength of the coupling to gain and loss. We call these opaque and transparent states. Opaque states are decoupled from the contacts and there is no transport; transparent states exhibit always efficient transport. Interestingly, the appearance of such eigenstates is connected with the divisors of the length of the system plus one and the position of the contacts. Thus the number of opaque and transparent states varies very irregularly.