Multiparticle localization for disordered systems on continuous space via the fractional moment method

Abstract

We investigate the spectral and dynamical localization of a quantum system of n particles on R d which are subject to a random potential and interact through a pair potential which may have infinite range. We establish two conditions which ensure spectral and dynamical localization near the bottom of the spectrum of the n-particle system: (i) localization is established in the regime of weak interactions supposing one-particle localization, and (ii) localization is also established under a Lifshitz-tail type condition on the sparsity of the spectrum. In case of polynomially decaying interactions, we provide an upper bound on the number of particles up to which these conditions apply.

Efficient Localization Bounds in a Continuous N-Particle Anderson Model with Long-Range Interaction

“…Surprisingly, in the models with sub-exponential decay of the interaction potential, the MPMSA approach provides a qualitatively stronger decay rate (a genuine exponential one) of the localized EFs than the new variant of the MPFMM developed by Fauser and Warzel [25]. The decay rate of the EFs derived from the analysis of the EF correlators, naturally, cannot be stronger than that of the EFCs, while the (MP)MSA scheme is free from this logical dependence.…”

“…It is to be emphasized that the recent result by Fauser and Warzel [25] on exponential decay of the EFCs, for exponentially decaying interactions, remains the strongest one among those proved in terms of Hausdorff distance in a continuous space, hence, in an actually infinite configuration space. Due to some well-known limitations of the Multi-Scale Analysis (single-or multi-particle), proofs of exponential strong dynamical localization are still beyond the MSA' reach.…”

“…The first mathematical works considered a fixed number N > 1 of particles in an infinite configuration space: Z d in [6,[15][16][17], and later R d (cf. [14], [38], [34], [25]).…”

“…This is quite opposite to the usual situation where a finite-volume analysis is only a prelude for the rigorous study of the object inspiring mathematicians -an actually infinite system. If the results of [6,17] (or their continuous-space counterparts [14,25,34]) were to be applied to the physical models, they would be valid only if our Universe were found to be infinite. Otherwise, one could not be able to rule out some tunneling processes which might result in transfer of particles over arbitrarily large distances.…”

Exponential Scaling Limit of the Single-Particle Anderson Model Via Adaptive Feedback Scaling

“…• Secondly, the Fractional Moments Method ceases to be a "mono-scale" technology in the area of multi-particle Anderson localization. Indeed, both the paper by Aizenman and Warzel [7] and a recent work by Fauser and Warzel [24] are based on a scale induction for the fractional moments of the Green functions. This is unrelated to the regularity of the marginal distribution.…”