Characterizing spin-orbit torques by tensorial spin Hall magnetoresistance

Magnetoresistance (MR) provides a crucial tool for experimentally studying spin torques. While MR is well established in the device geometry of the spin Hall effect (SHE), as exemplified by the magnet/heavy-metal heterostructures, its role and manifestation beyond the SHE paradigm remain elusive. We propose a general form of MR where the underlying charge-to-spin conversion and its inverse process could violate the simple geometry of the SHE, calling for tensorial descriptions. This MR can generate a series of unique harmonic responses essential for the experimental characterization of unconventional spin-orbit torques in non-SHE materials. We demonstrate these harmonic signals with semimetal ${\mathrm{WTe}}_{2}$ in mind but the results are not restricted to specific materials.