Polar unidirectional magnetotransport in $p-$type tellurene from quantum geometry

Unidirectional magnetoresistance, or electric magnetochiral anisotropy (eMChA), is a nonlinear magnetotransport phenomenon that arises in noncentrosymmetric conductors , where changes in resistance $R(B)$ are: (i) chiral, $\Delta R(B)/R(0)=2\,\chi\, {\bf I}\cdot{\bf B}$, or (ii) polar, $\Delta R(B)/R(0)=2\,\gamma\, {\bf I}\cdot({\bf P}\times{\bf B})$, with eMChA coefficients $\chi$ and $\gamma$. In [Phys. Rev. Lett. 135, 106602 (2025)], we showed that the eMChA in the conduction band of tellurene is polar ($\chi=0$, $\gamma\neq 0$) and emerges from the quantum metric dipole due to its Weyl node and from the lone pair polarization ${\bf P}$. Here, we extend our work to the valence band of tellurene, where the eMChA is usually said to be chiral ($\chi \neq 0, \gamma = 0$). We show that also a polar coefficient $\gamma \neq 0$ emerges naturally through a downfolding procedure, in which remote Weyl-node containing bands induce momentum-space gradients of the quantum metric in the low-energy levels, activating finite metric dipoles. Combining semiclassical Boltzmann transport with a ${\bf k}\cdot{\bf p}$ description of tellurene, our numerical calculations agree quantitatively with doping ($\mu$) dependent second-harmonic measurements of the longitudinal voltage $V^{2\omega}_\parallel(\mu)$ in perpendicular field. The combined chiral and polar characters ($\chi\neq0, \gamma\neq 0)$ of the eMChA in tellurene also explains the shift in the angular ($\phi$) dependence of $V^{2\omega}_\parallel(\phi)$ for in plane fields. Our results demonstrate that the polar eMChA can arise in topologically trivial bands through multiband effects and establishes tellurene as a platform for quantum-geometric rectification in both electron and hole regimes.