Anyonic $\mathcal{P}\mathcal{T}$ symmetry, drifting potentials and non-Hermitian delocalization

Abstract

We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian $\mathcal{H}$ satisfying the generalized (anyonic) parity-time symmetry $\mathcal{P}\mathcal{T}\mathcal{H}=\exp (2i\phi )\mathcal{H}\mathcal{P}\mathcal{T}$, where $\mathcal{P}$ and $\mathcal{T}$ are the parity and time-reversal operators. For a stationary potential, the anyonic phase $\phi$ just rotates the energy spectrum of $\mathcal{H}$ in complex plane, however for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when $\phi \ne 0$. In particular, in the unbroken $\mathcal{P}\mathcal{T}$ phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.

7 pages, 5 figures