Electric and magnetic conductivities in magnetized fermion systems

In Wigner function approach with relaxation time approximation, we calculate electric and magnetic conductivities of a fermion system in the strong magnetic field. The linear response has been calculated to the perturbation of electromagnetic fields on the background constant magnetic field. The Wigner function is separated into an equilibrium part in the background magnetic field and an off-equilibrium part induced by perturbative fields. The analytical expression for the equilibrium part and the corresponding equilibrium conditions are given. For the off-equilibrium part, we obtain the kinetic equation at the leading order in $\hbar$ from the master equation of the Wigner function. When perturbative fields only depend on the proper time, the off-equilibrium part can be analytically solved from which the vector and axial vector currents are obtained. We obtain the longitudinal and transverse Ohm conductivities as well as Hall conductivity as the linear response of the vector current to the perturbative electric field. The behaviors of these conductivities as functions of the evolving time, relaxation time, particle mass, and strength of the background magnetic field are investigated both analytically and numerically.