Specific heat of a particle on the cone

TL;DR

This paper studies how a conical singularity influences the specific heat of a particle on a cone, revealing temperature-dependent effects and discontinuities related to the cone's sharpness and mathematical properties of Bessel functions.

Contribution

It introduces a model analyzing the specific heat of a particle on a conical surface, highlighting the impact of the deficit angle and uncovering discontinuities linked to Bessel function zeros.

Findings

Specific heat depends on the deficit angle and temperature.

Discontinuity in specific heat occurs at low temperatures as the cone sharpens.

Connections to Bessel function zeros explain the discontinuity.

Abstract

This work investigates how a conical singularity can affect the specific heat of systems. A free nonrelativistic particle confined to the lateral surface of a cone -- conical box -- is taken as a toy model. Its specific heat is determined as a function of the deficit angle and the temperature. For a vanishing deficit angle, the specific heat is that of a particle in a flat disk where a characteristic temperature separates quantum and classical behaviors, as usual. By increasing the deficit angle the characteristic temperature increases also, and eventually another characteristic temperature (which does not depend on the deficit angle) arises. When the cone gets sufficiently sharp, at low and intermediate temperatures the azimuthal degree of freedom is suppressed. At low temperatures the specific heat varies discontinuously with the deficit angle. Connections between certain theorems regarding common zeros of the Bessel functions and this discontinuity are reported.