Direction-dependent linear response for gapped nodal-line semimetals in planar-Hall configurations

We compute the magnetoelectric conductivity for ideal nodal-line semimetals (NLSMs), with a finite but tiny mass-gap, in distinct planar-Hall set-ups. Each differing configuration results from the relative orientation of the nodal-ring's plane with respect to the plane spanned by the electric ($\mathbf E$) and magnetic ($\mathbf B$) fields. The net conductivity tensor has components comprising the Drude, anomalous-Hall, in-plane (with $\mathbf E$ and $\mathbf B$) longitudinal and transverse, and Lorentz-force-operator-induced parts. Our results feature the signatures of the inherent topology of a gapped NLSM, revealed through nonzero values of the Berry curvature and the orbital magnetic moment. In particular, we show that both of these vector fields, arising in the momentum space, give rise to terms of comparable magnitudes in the resulting response. Our explicit theoretical expressions will help identify unique signatures of NLSMs in contemporary experiments.