Field-theoretic methods for systems of particles with exotic exclusion statistics

TL;DR

This paper develops field-theoretic methods to compute the partition function of particles with exotic exclusion statistics, exploring different Hilbert space constructions and their consistency at low densities.

Contribution

It introduces a framework using fractional-dimensional Hilbert spaces and coherent states to analyze particles obeying Haldane exclusion statistics, highlighting multiple possible generalizations.

Findings

Multiple generalizations of the exclusion principle are possible.

Different Hilbert space constructions can be consistent with Haldane's definition.

At low densities, these different definitions converge.

Abstract

We calculate the partition function of a gas of particles obeying Haldane exclusion statistics, using a definition of a Hilbert space having a `fractional dimension' and constructing appropriate coherent states. The fractional dimension is expressed though the form of the identity operator in the Hilbert space. We find that there many possible generalisations of the Pauli exclusion principle, with particular choices of the scalar product leading to consistency either with Haldane's original definition of the effective dimensionality of the Hilbert space or with the combinatorial procedure invoked by Haldane and Wu. We explicitly demonstrate that at low particle densities these definitions are equivalent.