Nonlinear planar magnetotransport as a probe of the topology of surface states

It has been recently established that transport measurements in the nonlinear regime can give direct access to the quantum metric (QM): the real part of the quantum geometric tensor characterizing the geometry of the electronic wave functions in a solid. In topological materials, the QM has been so far revealed in thin films of the topological antiferromagnet ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ where it provides a direct contribution to longitudinal currents quadratic in the driving electric field. Here, we show that the Dirac surface states of strong three-dimensional topological insulators have a QM that can be accessed from the nonlinear transport characteristics in the presence of an externally applied planar magnetic field. A previously unknown intrinsic part of the longitudinal magnetoconductivity carries the signature of the QM while coexisting with the extrinsic part responsible for the so-called bilinear magnetoelectric resistance. We prove that the QM-induced nonlinear magnetotransport carries specific signatures of single Dirac cones. This allows its use as an efficient diagnostic tool of the bulk topology of three-dimensional nonmagnetic insulators.