Hardy and Lieb-Thirring inequalities for anyons

TL;DR

This paper establishes Hardy and Lieb-Thirring inequalities for anyons, particles in two dimensions with fractional statistics, extending fundamental quantum inequalities to this intermediate particle type.

Contribution

It introduces Hardy and Lieb-Thirring inequalities for anyons, providing new bounds for their kinetic energy and stability properties in quantum mechanics.

Findings

Proves a magnetic Hardy inequality for anyons.

Derives a local exclusion principle for odd numerator fractions of the statistics parameter.

Establishes a Lieb-Thirring inequality for anyons.

Abstract

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter $\alpha \in [0,1]$ ranging from bosons ($\alpha=0$) to fermions ($\alpha=1$). We prove a (magnetic) Hardy inequality for anyons, which in the case that $\alpha$ is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard's original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.