Sub-diffusion and non-equilibrium probes of phases in Aubry-Andre-Harper Model

We consider an open quantum system generalization of the well-known linear Aubry-Andre-Harper (AAH) model by putting it out-of-equilibrium with the aid of two baths (at opposite ends) at unequal temperatures and chemical potentials. Non-equilibrium steady state (NESS) properties are computed by a fully exact non-equilibrium Green's function method (valid for any strength of system-bath coupling). We find sub-diffusive scaling of NESS current with system-size at the critical point. We also show that the NESS spatial particle density profile provides a potential experimental probe of delocalized, localized and critical phases. Surprisingly, at the critical point, various closed system quantities such as wavefunction spread and dynamical susceptibility show anomalous transport which is `non-Gaussian but diffusive', but do not show any hints of sub-diffusive behavior seen in the open system. In fact, we find that Kubo formalism for closed system gives strong evidence of a finite conductivity in the thermodynamic limit, which is akin to a normal diffusive system. Thus, we find a dramatic difference between closed and open system transport properties. Our findings are valid for both the bosonic and fermionic versions. This is of special importance in light of recent experiments.