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Investigation of Magnetic Transport Properties by Wannier Interpolation

We present an efficient {\it ab initio} approach for the study of magnetic transport properties based on the Boltzmann equation with the Wannier interpolation scheme. Within the relaxation time approximation, band-resolved electric conductivity under a finite magnetic field is obtained and the historical motion of the electron wave packet in reciprocal space is determined. As a typical application of this method, we have calculated the electric conductivities of MgB$_2$ under finite magnetic fields. Multiband characters for the individual bands are revealed, and the field dependence of the conductivity tensor is studied systematically with the field orientated parallel and normal to the $c$-axis, respectively. The obtained historical motion is employed to simulate directly the cyclotron motion in the extremal orbit and determine the corresponding effective mass. Moreover, This approach is further exploited to calculate the Hall coefficient in the low-field limit, without the complicated computation for the second ${\mathbf k}$ derivative of the band.