Direction-dependent conductivity in planar Hall set-ups with tilted Weyl/multi-Weyl semimetals

We compute the magnetoelectric conductivity tensors in planar Hall set-ups, which are built with tilted Weyl semimetals (WSMs) and multi-Weyl semimetals (mWSMs), considering all possible relative orientations of the electromagnetic fields ($\mathbf E$ and $\mathbf B$) and the direction of the tilt. The non-Drude part of the response arises from a nonzero Berry curvature in the vicinity of the WSM/mWSM node under consideration. Only in the presence of a nonzero tilt do we find linear-in-$| \mathbf B|$ terms in set-ups where the tilt-axis is not perpendicular to the plane spanned by $\mathbf E$ and $\mathbf B$. The advantage of the emergence of the linear-in-$| \mathbf B|$ terms is that, unlike the various $| \mathbf B|^2$-dependent terms that can contribute to experimental observations, they have purely a topological origin, and they dominate the overall response-characteristics in the realistic parameter regimes. The important signatures of these terms are that they (1) change the periodicity of the response from $\pi$ to $2\pi$, when we consider their dependence on the angle $\theta$ between $\mathbf E$ and $\mathbf B$; and (2) lead to an overall change in sign of the conductivity depending on $\theta$, when measured with respect to the $\mathbf B =0$ case.