[Paper Review] Anderson localization in the multi-particle tight-binding model at low energies or with weak interaction

Trésor Ekanga|arXiv (Cornell University)|Jan 12, 2012

Spectral Theory in Mathematical Physics5 citations

TL;DR

This paper establishes non-randomness of the lower spectral edge in the multi-particle Anderson tight-binding model under mild interaction and random potential assumptions. By adapting the multi-particle multi-scale analysis to low energies with log-Holder continuous disorder, it proves spectral and strong dynamical localization near the spectrum's bottom, extending high-disorder results to low-energy regimes.

ABSTRACT

We consider the multi-particle Anderson tight-binding model and prove that its lower spectral edge is non-random under some mild assumptions on the inter-particle interaction and the random external potential. We also adapt to the low energy regime the multi-particle multi-scale analysis initially developed by Chulaevsky and Suhov in the high disorder limit, if the marginal probability distribution of the i.i.d. random variables is log-Holder continuous and obtain the spectral exponential and strong dynamical localization near the bottom of the spectrum.

Motivation & Objective

• To investigate spectral properties of the multi-particle Anderson tight-binding model at low energies.
• To extend the multi-particle multi-scale analysis, originally developed for high disorder, to the low-energy regime.
• To establish non-randomness of the lower spectral edge under mild interaction and random potential assumptions.
• To prove exponential spectral localization and strong dynamical localization near the bottom of the spectrum.

Proposed method

• Adapt the multi-particle multi-scale analysis framework to low-energy regimes, originally designed for high disorder.
• Assume the i.i.d. random variables in the potential have a log-Holder continuous marginal distribution.
• Use spectral localization techniques to show the lower spectral edge is non-random under mild interaction and potential conditions.
• Establish strong dynamical localization by controlling the decay of time-evolved wave functions near the spectrum's bottom.
• Apply iterative multi-scale analysis to control eigenfunction decay and spectral gaps in the multi-particle setting.

Experimental results

Research questions

• RQ1 Is the lower spectral edge of the multi-particle Anderson model non-random under mild interaction and random potential assumptions?
• RQ2 Can the multi-particle multi-scale analysis be extended from the high-disorder to the low-energy regime?
• RQ3 Does spectral and dynamical localization occur near the bottom of the spectrum when interactions are weak and disorder is log-Holder continuous?
• RQ4 What conditions ensure the stability of localization in the low-energy regime for multi-particle systems?

Key findings

• The lower spectral edge of the multi-particle Anderson tight-binding model is non-random under mild assumptions on interaction and random potential.
• The multi-particle multi-scale analysis is successfully adapted to the low-energy regime with log-Holder continuous disorder.
• Spectral exponential localization is established near the bottom of the spectrum under the given conditions.
• Strong dynamical localization is proven in the low-energy regime, indicating localization of wave functions under time evolution.