Orbital and spin bilinear magnetotransport effect in Weyl/Dirac semimetal

We theoretically investigate the bilinear current, scaling as $j\sim EB$, in two- and three-dimensional systems. Based on the extended semiclassical theory, we develop a unified theory including both longitudinal and transverse currents. We classify all contributions according to their different scaling relations with the relaxation time. We reveal the distinct contributions to the ordinary Hall effect, planar Hall effect, and magnetoresistance. We further report an intrinsic ordinary Hall current, which has a geometric origin and has not been discussed previously. Our theory is explicitly applied to studying a massive Dirac model and a $\mathcal{PT}$-symmetric system. Our work presents a general theory of electric transport under a magnetic field, potentially laying the groundwork for future experimental studies or device fabrications.