John Chalker

Abstract:

We review the network model for the integer quantum Hall effect. The model provides a simplified description of Anderson localization in this context. It represents non-interacting electrons moving in two dimensions under the combined influence of a strong magnetic field and a smooth disordered potential. In this setting, electron eigenstates form disorder-broadened Landau levels and their character varies with energy across the Landau level. States in both the low-energy and the high-energy tails of the Landau level are localized, with a spatial extent characterized by the localization length. At the center of the Landau level there is a transition between phases with different quantized values of the Hall conductance and the localization length is divergent. The network model captures universal features of this transition.

Abstract:

By varying the disorder realization in the many-body localized (MBL) phase, we investigate the influence of resonances on spectral properties. The standard theory of the MBL phase is based on the existence of local integrals of motion (LIOM), and eigenstates of the time evolution operator can be described as LIOM configurations. We show that smooth variations of the disorder give rise to avoided level crossings, and we identify these with resonances between LIOM configurations. Through this parametric approach, we develop a theory for resonances in terms of standard properties of nonresonant LIOM. This framework describes resonances that are locally pairwise, and is appropriate in arbitrarily large systems deep within the MBL phase. We show that resonances are associated with large level curvatures on paths through the ensemble of disorder realizations, and we determine the curvature distribution. By considering the level repulsion associated with resonances, we calculate the two-point correlator of the level density. We also find the distributions of matrix elements of local observables and discuss implications for low-frequency dynamics.