Kohn's localisation in disordered fermionic systems with and without interactions

TL;DR

This paper explores the metal-insulator transition in disordered fermionic systems with and without interactions using the modern theory of the insulating state, demonstrating how the many-body localisation tensor can identify phases and how weak interactions influence the transition.

Contribution

It applies the modern theory of the insulating state to various disordered models and investigates the impact of weak interactions on the localization transition in a cold-atom relevant model.

Findings

The many-body localisation tensor effectively distinguishes insulating and metallic phases.

Disorder types include uncorrelated, deterministic, and long-range correlated potentials.

Weak repulsive interactions shift the transition point towards stronger disorder, indicating delocalization.

Abstract

Understanding the metal-insulator transition in disordered many-fermion systems, both with and without interactions, is one of the most challenging and consequential problems in condensed matter physics. In this paper we address this issue from the perspective of the modern theory of the insulating state (MTIS), which has already proven to be effective for band and Mott insulators in clean systems. First we consider noninteracting systems with different types of aperiodic external potentials: uncorrelated disorder (one-dimensional Anderson model), deterministic disorder (Aubry-Andr\'e Hamiltonian and its modification including next-nearest neighbour hopping), and disorder with long-range correlations (self-affine potential). We show how the many-body localisation tensor defined within the MTIS may be used as a powerful probe to discriminate the insulating and the metallic phases, and to locate the transition point. Then we investigate the effect of weak repulsive interactions in the Aubry-Andr\'{e} Hamiltonian, a model which describes a recent cold-atoms experiment. By treating the weak interactions within a mean-field approximation we obtain a linear shift of the transition point towards stronger disorder, providing evidence for delocalisation induced by interactions.