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, for three inequivalent planar-Hall configurations, with the electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields oriented differently relative to the nodal-ring plane and the Berry-curvature (BC) dipole axis. Unlike Weyl and multifold semimetals with point-monopole BC flux, the HSM carries a single, point-like BC-dipole fixed at its node, while the VNR hosts a continuous ring of such sources around the nodal loop. Both have vanishing Chern numbers. Working consistently to cubic order in $\mathbf{B}$, we evaluate the Drude, BC, orbital-magnetic-moment (OMM), anomalous-Hall, and Lorentz-force contributions. In both systems, the dipolar BC and OMM activate out-of-plane and odd-in-$\mathbf{B}$ in-plane response-channels that are symmetry-forbidden for a BC-monopole. Mirror symmetry confines the odd longitudinal terms to $B_z$, the field component along the dipole axis. The internode-scattering conductivity of the HSM vanishes identically, since the dipolar BC and OMM are even in momentum and no charge is pumped between two conjugate Hopf nodes, which carry oppositely oriented BC-dipoles. Comparisons with a $\mathcal{P}\mathcal{T}$-broken gapped nodal-ring and a pseudospin-1 triple-point semimetal confirm the dipolar-BC origin of these features.

arXiv commentrevised