Anderson localization for two interacting quasiperiodic particles

TL;DR

This paper proves Anderson localization for two interacting quasiperiodic particles in one dimension at high disorder, considering symmetries and background potentials, with implications for quantum systems with complex potentials.

Contribution

It establishes localization results for two interacting quasiperiodic particles, including cases with symmetries and low-complexity background potentials, extending previous single-particle results.

Findings

Localization at large disorder for quasiperiodic particles

Localization outside finitely many energies with symmetries

Applicability to systems with periodic or finite-range background potentials

Abstract

We consider a system of two discrete quasiperiodic 1D particles as an operator on $\ell^2(\mathbb Z^2)$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values.