Direction-dependent magnetoelectric conductivity from dipolar topological semimetals

Abstract: We compute the linear magnetoelectric conductivity of a two-band vortex nodal-ring (VNR) semimetal and a three-band Hopf semimetal (HSM) within the semiclassical Boltzmann formalism in the relaxation-time approximation. We consider three inequivalent planar-Hall configurations, in which the electric ($\boldsymbol E$) and magnetic ($\boldsymbol B$) fields are oriented differently with respect to the nodal-ring plane and the BC-dipole axis. Unlike Weyl and multifold semimetals, where Berry-curvature (BC) flux originates from monopole-like sources, both systems host dipole-like BC sources and vanishing Chern numbers. Working consistently to cubic order in the magnetic field, we evaluate the Drude, BC, orbital-magnetic-moment (OMM), anomalous-Hall, and Lorentz-force contributions to the conductivity. In both systems, the dipolar BC and OMM generate odd powers of $|\boldsymbol B|$ in in-plane response channels that are symmetry-forbidden in systems with monopole-like BC sources. For the HSM, the internode-scattering contribution to the conductivity vanishes identically. By comparing the VNR with a $\mathcal{PT}$-broken gapped nodal ring and the HSM with a pseudospin-1 triple-point semimetal, we demonstrate that the dipolar nature of the BC gives rise to distinctive features in magnetoelectric response that are absent in semimetals with monopole-like BC flux.