Relativistic Barnett effect and Curie law in a rigidly rotating free Fermi gas

TL;DR

This paper investigates the relativistic Barnett effect in a rotating Fermi gas, revealing how spin polarization, magnetic susceptibility, and moment of inertia depend on temperature and rotation, with a focus on thermodynamic and quantum effects.

Contribution

It introduces a regularization scheme to analyze spin polarization in a rotating Fermi gas and demonstrates a Curie-like law for the moment of inertia at high temperatures.

Findings

Spin polarization depends on spin-rotation coupling and effective chemical potential.

Magnetic susceptibility is proportional to the moment of inertia.

Moment of inertia exhibits a 1/T behavior at high temperatures, akin to Curie law.

Abstract

By combining methods from thermal field theory and statistical mechanics, we reexamine the spin polarization caused by the relativistic Barnett effect in a rigidly rotating Fermi gas. We determine the pressure of this medium and show that it depends on an effective chemical potential, which includes contributions from orbital angular momentum-rotation and spin-rotation coupling. We introduce a specific regularization scheme to sum over the angular momentum quantum numbers. As a result, the thermal pressure and all thermodynamic quantities are separated into two parts that differ only in the spin fugacities of spin-up and spin-down fermions. We calculate the Fermi energy for both components and show that the Fermi energy of the spin-down fermions is lower than that of the spin-up ones. This difference arises from the spin-rotation coupling and leads to a spin polarization consistent with the Barnett effect. In particular, we introduce the spin-chemicorotational ratio $\eta\equiv \Omega^{(0)}/2\mu^{(0)}$, which adjusts the spin polarization of the Fermi gas. Here, $\Omega^{(0)}$ and $\mu^{(0)}$ represent the angular velocity and chemical potential at zero temperature, respectively. The factor $1/2$ accounts for the fermion's spin. We explore the temperature dependence of $\mu$ and $\Omega$, while assuming that the number of spin-up and spin-down fermions remains temperature independent. Our findings indicate that the spin-down component of the rotating Fermi gas dilutes at lower temperatures compared to the spin-up component. Additionally, we calculate the magnetic susceptibility arising from the Barnett magnetization and demonstrate that it is proportional to the moment of inertia $I$ of the rotating Fermi gas. Finally, we prove that $I$ exhibits a $1/T$ behavior in the high-temperature limit, similar to the Curie law of paramagnetism.