Quenched disorder in a solid state system can result in Anderson localization, where electrons are exponentially localized and the system behaves like an insulator. By solving exactly a disordered electronic lattice model out of equilibrium, we investigate the effect of a DC electric field on Anderson localization in an open system, and provide a minimal platform to study disorder-nonequilibrium interplay in electronic lattice systems. We perform steady-state Keldysh Green's function calculations on an infinite lattice with a finite-range of disorder-active region that are coupled to fermion reservoirs. Our solutions out of a fully electronic model verify Mott's temperature scaling of the variable-range-hopping transport and the Lifshitz tail, well-corroborated by the coherent-potential approximation. We further reveal that a nonequilibrium electronic lattice creates a statistical evolution that shows a counterintuitive shift of the distribution edge in the opposite direction of the band edge. The rich evolution of non-thermal statistics highlights the importance of an explicit band structure and the impurity correlations in strong nonequilibrium theories.
We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short-range hopping in higher dimensions, and also because the renormalization group can be applied exactly without the approximations that generally are required in other models. In the hierarchical Anderson model, we find a finite critical disorder strength
Wc where the average inverse participation ratio goes to zero; at small disorder,
W<Wc, the model lies in a delocalized phase. This result is based on numerical calculation of the inverse participation ratio in the infinite volume limit using an exact renormalization group approach facilitated by the model's hierarchical structure. Our results are consistent with the presence of an Anderson transition in short-range models with
D>2 dimensions, which was predicted using renormalization group arguments. Our finding should stimulate interest in the hierarchical Anderson model as a simplified and tractable model of the Anderson localization transition, which occurs in finite-dimensional systems with short-range hopping.