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

Abstract

The Aubry-André-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 Vc 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 δ of wave-packet spreading, with δ=1 in the delocalized phase V<Vc (ballistic transport), δ≃1/2 at the critical point V=Vc (diffusive transport), and δ=0 in the localized phase V>Vc (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-André-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 δ, 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.