Nonlinear planar Hall effect induced by interband transitions: Application to surface states of topological insulators

Motivated by recent experiments observing the nonlinear planar Hall effect (NPHE) in nonmagnetic topological materials, we employ the density matrix method to consider all the intraband and interband transitions. This gives a deeper insight for the different mechanisms of NPHE on the same footing beyond the semiclassical theory. Under broken time-reversal symmetry, besides the usual Berry curvature dipole (BCD) contribution, there exists the quantum metric (QM) induced NPHE, which includes the intrinsic and extrinsic components, and exists even within the band gap. This QM term extends the Berry-connection polarizability (BCP) theory which captures only the intrinsic contribution and cannot be applied to the case with finite scattering and nonzero frequency. Moreover, we reveal that the underlying physics of BCP originates essentially from the combination of three interband transitions (injection, shift, and anomalous), very differently from the BCD which is only contributed by an anomalous mechanism. We compare different mechanisms by calculating the NPHE on the surface states of topological insulators and find that the NPHE from different mechanisms exhibits different dependence on the in-plane magnetic field and the chemical potential. Our theory provides an alternative perspective to understand the complicated lineshapes of the NPHE observed near the Dirac point.