Fractional Statistics in terms of the r-Generalized Fibonacci Sequences

TL;DR

This paper introduces a new framework for fractional quantum statistics using r-generalized Fibonacci sequences, unifying Fermi and Bose statistics and extending Haldane's interpolation method.

Contribution

It develops a basis for 2D generalized quantum statistical systems using r-generalized Fibonacci sequences, classifying quasiparticles by spin and linking statistical weights to Fibonacci hierarchies.

Findings

Statistical weights for certain quasiparticles are given by Fibonacci sequences.

The framework unifies Fermi and Bose statistics.

Provides an alternative to Haldane's interpolating statistics.

Abstract

We develop the basis of the two dimensional generalized quantum statistical systems by using results on $r$-generalized Fibonacci sequences. According to the spin value $s$ of the 2d-quasiparticles, we distinguish four classes of quantum statistical systems indexed by $ s=0,1/2:mod(1)$, $s=1/M:mod(1)$, $s=n/M:mod(1)$ and $0\leq s\leq 1:mod(1)$. For quantum gases of quasiparticles with $s=1/M:mod(1)$, $M\geq 2,$, we show that the statistical weights densities $\rho_M$ are given by the integer hierarchies of Fibonacci sequences. This is a remarkable result which envelopes naturally the Fermi and Bose statistics and may be thought of as an alternative way to the Haldane interpolating statistical method.