Phase transitions in a non-Hermitian Aubry-André-Harper model

Abstract

The Aubry-Andr\'e-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization phase transition at a finite critical value ${V}_{c}$ of the quasiperiodic potential amplitude $V$. In terms of the dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent $\ensuremath{\delta}$ of wave-packet spreading, with $\ensuremath{\delta}=1$ in the delocalized phase $V<{V}_{c}$ (ballistic transport), $\ensuremath{\delta}\ensuremath{\simeq}1/2$ at the critical point $V={V}_{c}$ (diffusive transport), and $\ensuremath{\delta}=0$ in the localized phase $V>{V}_{c}$ (dynamical localization). However, the phase transition turns out to be smooth when one measures, as a dynamical variable, the speed $v(V)$ of excitation transport in the lattice, which is a continuous function of potential amplitude $V$ and vanishes as the localized phase is approached. Here we consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, in which hopping along the lattice is asymmetric, and show that the dynamical localization-delocalization transition is discontinuous, not only in the diffusion exponent $\ensuremath{\delta}$, but also in the speed $v$ of ballistic transport. This means that even very close to the spectral phase transition point, rather counterintuitively, ballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballistic velocity can increase as $V$ is increased above zero, i.e., surprisingly, disorder in the lattice can result in an enhancement of transport.