Multi-Particle Anderson Localisation: Induction on the Number of Particles

TL;DR

This paper proves Anderson localisation for multi-particle quantum systems on a lattice using an adapted multi-scale analysis and induction on the number of particles, extending previous two-particle results to N particles.

Contribution

It introduces a novel multi-scale analysis approach combined with induction on particle number for proving localisation in multi-particle systems.

Findings

Established Anderson localisation for N-particle lattice systems.

Extended multi-scale analysis techniques to multi-particle contexts.

Provided a framework adaptable to continuous systems and correlated potentials.

Abstract

This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09} covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum $N$-particle system on a lattice $\Z^d$ with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS}, \cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript \cite{CS07}. Similar results have been recently obtained in an independent work by Aizenman and Warzel \cite{AW08}: they proposed an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in \cite{AM93} and \cite{ASFH01} (see also references therein) which is also combined with an induction on the number of particles. An important role in our proof is played by a variant of Stollmann's eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved earlier in \cite{C08}, admits a straightforward extension covering the case of multi-particle systems with correlated external random potentials: a subject of our future work. We also stress that the scheme of our proof is \textit{not} specific to lattice systems, since our main method, the MSA, admits a continuous version. A proof of multi-particle Anderson localization in continuous interacting systems with various types of external random potentials will be published in a separate papers.