Longitudinal conductivity of massless fermions with tilted Dirac cone in magnetic field

We investigate a longitudinal conductivity of a two-dimensional relativistic electron gas with a tilted Dirac cone in magnetic field. It is demonstrated that the conductivity behaves differently in the directions parallel and perpendicular to the tilting of the cone. At high magnetic fields the conductivity at non-zero Landau levels in the direction perpendicular to the tilting modifies non-trivially, in contrast to the parallel case. At zero temperature the crossover of the conductivity at the Dirac point from high to low magnetic field is studied numerically. It is found that that the tilting produces anisotropy of the conductivity which changes with the magnetic field which is different from the anisotropy coming from the Fermi velocity. We also discuss the conductivity at finite temperatures and finite magnetic fields which can be directly compared with the experiments in $\alpha$-(BEDT-TTF)$_2$I$_3$ organic conductor. We find that the tilting does not affect so much the magnetic-filed dependence of the conductivity except for the prefactor. We discuss the interpretation of recent experimental data and make some proposals to detect the effect of the tilting in future experiments.