Unconventional magnetoelectric conductivity and electrochemical response from dipole-like sources of Berry curvature

Abstract

We compute longitudinal magnetoelectric conductivity ($\sigma_{zz}$) and nonlinear electrochemical response (ECR), applying the semiclassical Boltzmann formalism, for three-dimensional nodal-ring semimetals (vortex nodal-rings and $\mathcal P \mathcal T$-symmetric nodal-rings) and three-band Hopf semimetals. While the nodal-curves of the former are taken to lie along the $k_z = 0$-plane, the nodal points of the latter harbour dipoles in their Berry-curvature (BC) profile, with the dipole's axis aligned along the $k_z$-axis. All these systems are topological and are unified on the aspect that their bands possess a vanishing Chern number. The linear response, $\sigma_{zz}$, is obtained from an exact solution when the systems are subjected to collinear electric and magnetic fields applied along the anisotropy axis, viz. $\boldsymbol{\hat z}$. The nonlinear part involves third-rank tensors representing second-order response coefficients, relating the electrical current to the combined effects of the gradient of the chemical potential and an external electric field. We analyse the similarities of the response arising from the vortex nodal-rings and the Hopf semimetals, which can be traced to the dipole-like sources in their BC fields.