Chiral vortical conductivities and the moment of inertia of a rigidly rotating Fermi gas

We determine the chiral vortical conductivities, as well as the orbital and spin moment of inertia of a charged, chiral, and rigidly rotating free Fermi gas. To this purpose, we begin by calculating the vacuum expectation values of a vector and axial vector current using the free fermion propagator in this medium. This propagator is derived by employing the Fock-Schwinger method based on the solutions of the Dirac equation in the presence of rotation and finite axial chemical potential. We present a complete derivation of these solutions. We demonstrate that in the first approximation, the chiral vortical conductivity associated with the vector current is proportional to the product of the vector and axial chemical potentials. In contrast, the chiral vortical conductivity related to the axial vector current depends on the temperature, the vector, and the axial vector chemical potential squares. We use the relation between the axial vector current and the angular momentum density associated with spin to determine the spin and the orbital moment of inertia of a rigidly rotating Fermi gas in a charged and chirally imbalanced medium separately. In addition, we compute the total moment of inertia by utilizing the methods presented in the first part of the paper and show that the orbital moment of inertia of a free fermion gas vanishes.