Anisotropic conductivity for the type-I and type-II phases of Weyl/multi-Weyl semimetals in planar Hall set-ups

We compute the non-Drude part of the conductivity tensor in planar Hall set-ups, for tilted Weyl and multi-Weyl semimetals, considering both the type-I and type-II phases. We do so in three distinct set-ups, taking into account the possible relative orientations of the plane spanned by the electric and magnetic fields ($\mathbf E $ and $\mathbf B $) and the direction of the tilt-axis. We derive the analytical expressions for the response tensor, including the effects of the Berry curvature (BC) and the orbital magnetic moment (OMM), both of which arise due to a nontrivial topology of the three-dimensional manifold defined by the Brillouin zone. We exhibit the interplay of the BC-only and the OMM-dependent parts in the nonzero components of the magnetoelectric conductivity, and outline whether the contributions from the former or the latter dominate the overall response. Our results also show that, depending on the configuration of the planar Hall set-up, one may or may not get terms which have a linear-in-$ B$ dependence.