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

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 $η\equiv Ω^{(0)}/2μ^{(0)}$, which adjusts the spin polarization of the Fermi gas. Here, $Ω^{(0)}$ and $μ^{(0)}$ 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 $μ$ and $Ω$, 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.