Unusual Thermodynamics on the Fuzzy 2-Sphere

TL;DR

This paper explores the unusual thermodynamic properties of fermion-like spectra of higher spin Dirac operators on fuzzy and continuum 2-spheres, revealing non-standard specific heats and spectral behaviors that deviate from ideal gas laws.

Contribution

It introduces the assumption of Fermi-Dirac statistics for spin 1 particles on fuzzy spheres and analyzes their thermodynamics, zero modes, and spectral properties, highlighting novel features in higher spin Dirac operators.

Findings

Spin 1 systems have lower specific heat than spin 1/2 despite more degrees of freedom.

Zero modes of spin 1 Dirac operator follow a power law with respect to angular momentum cutoff.

Spectra deviate from standard ideal gas behavior in 2+1 dimensions.

Abstract

Higher spin Dirac operators on both the continuum sphere($S^2$) and its fuzzy analog($S^2_F$) come paired with anticommuting chirality operators. A consequence of this is seen in the fermion-like spectrum of these operators which is especially true even for the case of integer-spin Dirac operators. Motivated by this feature of the spectrum of a spin 1 Dirac operator on $S_F^2$, we assume the spin 1 particles obey Fermi-Dirac statistics. This choice is inspite of the lack of a well defined spin-statistics relation on a compact surface such as $S^2$. The specific heats are computed in the cases of the spin $\frac{1}{2}$ and spin 1 Dirac operators. Remarkably the specific heat for a system of spin $\frac{1}{2}$ particles is more than that of the spin 1 case, though the number of degrees of freedom is more in the case of spin 1 particles. The reason for this is inferred through a study of the spectrums of the Dirac operators in both the cases. The zero modes of the spin 1 Dirac operator is studied as a function of the cut-off angular momentum $L$ and is found to follow a simple power law. This number is such that the number of states with positive energy for the spin 1 and spin $\frac{1}{2}$ system become comparable. Remarks are made about the spectrums of higher spin Dirac operators as well through a study of their zero-modes and the variation of their spectrum with degeneracy. The mean energy as a function of temperature is studied in both the spin $\frac{1}{2}$ and spin 1 cases. They are found to deviate from the standard ideal gas law in 2+1 dimensions.