The Magnetized Electron Gas in terms of Hurwitz Zeta Functions

TL;DR

This paper derives explicit formulas for the thermodynamics of a relativistic degenerate electron gas in a magnetic field using Hurwitz Zeta functions, enabling systematic analysis across different regimes.

Contribution

It introduces a novel formulation expressing thermodynamic quantities in terms of Hurwitz Zeta functions, allowing comprehensive analysis of the electron gas in magnetic fields.

Findings

Reproduces magnetization oscillations at low temperatures.

Shows dilution of oscillations at higher temperatures.

Provides explicit formulas applicable across regimes.

Abstract

We obtain explicit expressions for thermodynamic quantities of a relativistic degenerate free electron gas in a magnetic field in terms of Hurwitz Zeta functions. The formulation allows for systematic expansion in all regimes. Three energy scales appear naturally in the degenerate relativistic gas: the Fermi energy Ef, the temperature T and an energy related to the magnetic field or Landau level spacing, eB/Ef. We study the cold and warm scenarios, T << eB/Ef and eB/Ef << T, respectively. We reproduce the oscillations of the magnetization as a function of the field in the cold regime and the dilution of them in the warm regime.