Effects of dissipation and de-phasing on the Aubry- Andre-Haper model

The Aubry-Andre-Harper model of spinless fermions hopping in an on-site quasiperiodic potential exhibits a delocalization-localization transition at a critical value of the strength of the on-site potential. We couple this model to the environment and study the effects of random on-site dissipation and de-phasing. To do this, we examine the time evolution of the density matrix (as described by the GKLS master equation) and thereby the Liouvillian operator. In this work, we describe the non-equilibrium steady state (NESS) and some computational methods to study it. We also describe diagnostics that can be used to study the entanglement spectrum associated with the NESS such as the inverse participation ratio, von Neumann entropy, logarithmic negativity and the ratio of level spacings. We see how the behavior of these quantities changes with the strength of the onsite potential, dissipation and de-phasing and compare the behavior to that in the closed system.