Haldane-Inspired Generalized Statistics

TL;DR

This paper introduces a generalized quantum statistics framework called alpha statistics, interpolating between Bose and Fermi statistics, and analyzes its thermodynamic properties and phase behavior using geometric methods.

Contribution

It extends Haldane's exclusion statistics to a continuous parameter alpha, exploring thermodynamic geometry and phase transitions in this generalized setting.

Findings

Identification of a crossover temperature T* where thermodynamic curvature changes sign.

Discovery of curvature singularities for negative alpha indicating modified condensation phenomena.

Universal dependence of T*/Tc ratio on alpha, linking fractional exclusion to thermodynamic geometry.

Abstract

We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for $\alpha < 0$. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter $\alpha$. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature $T^{*}$, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature $T_{c}$, the ratio $T^{*}/T_{c}$ depends universally on $\alpha$. For negative $\alpha$, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.