Closed-form evaluation of the missing yy conductivity component

Determine the explicit closed-form expression for the conductivity component \(\sigma_{yy}(B_x,0,B_z)\) of the vortex nodal-ring and three-band Hopf-semimetal models within the semiclassical Boltzmann formalism, completing the conductivity tensor on the \(xz\)-plane.

Background

The paper shows that axial rotational covariance and the antiunitary symmetry $\Theta$ determine eight of the nine conductivity components on the $xz$-plane. The only component not obtained from the three analyzed planar-Hall configurations or symmetry relations is $\sigma_{yy}(B_x,0,B_z)$.

The missing component is not physically undetermined: its non-anomalous-Hall and Lorentz-force contributions are fixed by the microscopic conductivity formulas already derived in the paper. The authors state that obtaining its explicit form would require a direct evaluation with an additional configuration, namely $E_y$ in the $xz$-plane, and leave this calculation for future work. Completing it would provide the explicit full three-dimensional conductivity tensor after propagation using rotational covariance.