Abstract
We extend the techniques and results of the multi-particle variant of the Fractional Moment Method, developed by Aizenman and Warzel, to disordered quantum systems in general finite or countable graphs with polynomial growth of balls, in presence of an exponentially decaying interaction. In the strong disorder regime, we prove complete exponential multi-particle strong localization. Prior results, obtained with the help of the multi-scale analysis, proved only a sub-exponential decay of eigenfunction correlators.
Highlights
The motivation and the modelThe rigorous multi-particle Anderson localization theory is a relatively recent direction in the spectral theory of disordered media
In the multi-particle models with finite-range interaction, the MPFMM, when applicable, provides the strongest decay bounds upon the eigenfunction correlators (EFC), as does its original, single-particle variant
Such bounds are stronger than those proved with the help of the multiparticle MSA (MPMSA), provided both methods apply to the same model
Summary
The rigorous multi-particle Anderson localization theory is a relatively recent direction in the spectral theory of disordered media. The first results in this direction, establishing the stability of Anderson localization in a two-particle system in Zd with respect to a short-range interaction [16], have been immediately followed by the proofs of exponential spectral localization (cf [5, 17]) and exponential strong dynamical localization (cf [5]) in N-particle systems, for any fixed N ≥ 2. In the multi-particle models with finite-range interaction, the MPFMM, when applicable, provides the strongest decay bounds upon the eigenfunction correlators (EFC), as does its original, single-particle variant. Such bounds are stronger than those proved with the help of the multiparticle MSA (MPMSA), provided both methods apply to the same model. In the situation where the interaction potential decays slower than exponentially, the existing techniques (based on the MPFMM or the MPMSA) allow one to prove
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