The Weyl-Heisenberg ensemble: hyperuniformity and higher Landau levels

TL;DR

This paper proves that Weyl-Heisenberg ensembles are hyperuniform, including models for electrons in higher Landau levels and new anisotropic processes, advancing understanding of spatial point distributions and directional hyperuniformity.

Contribution

It establishes hyperuniformity for Weyl-Heisenberg ensembles and introduces new anisotropic models with directional dependence in point statistics.

Findings

Weyl-Heisenberg ensembles are hyperuniform.

Includes models for higher Landau levels.

Introduces anisotropic point processes.

Abstract

Weyl-Heisenberg ensembles are a class of determinantal point processes associated with the Schr\"odinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl-Heisenberg ensembles are hyperuniform. Weyl-Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl-Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.