Interplay of strain-induced axial gauge fields and intrinsic band-topology in the magnetoelectric conductivity of gapped nodal rings

We compute the magnetoelectric conductivity of a semimetal hosting an ideal gapped nodal ring (GNR) in three distinct planar-Hall configurations, in the simultaneous presence of an external electric field $\boldsymbol{E}$, a magnetic field $\boldsymbol{B}$, and a strain-induced axial pseudomagnetic field $\boldsymbol{B}_5$. The latter arises from a nonuniform lattice deformation and couples to antipodal points on the toroidal Fermi surface with opposite signs, reflecting its chiral nature. Extending our earlier analysis to include $\boldsymbol{B}_5$, we demonstrate how its vortex-like field lines -- co-aligned with the Berry curvature (BC) and orbital magnetic moment (OMM) -- imprint qualitatively distinct signatures on the conductivity tensor. In particular, this alignment causes the dot product of $\boldsymbol{B}_5$ with the BC or OMM-induced quantities to be angle-independent on the Fermi surface, generating a nonvanishing integral linear-in-$B_5$, which is not possible for isotropic nodal points harbouring BC-monopoles. We show that a part of the planar-Hall conductivity in the first set-up remains completely immune to strain, providing a strain-insensitive internal reference for topological transport. Our explicit analytical expressions offer concrete and experimentally testable predictions for identifying strain-induced signatures in transport measurements on GNR materials.