Colossal Spontaneous Hall Effect and Emergent Magnetism in KTaO$_3$ Two-Dimensional Electron Gases

There has been intense recent interest in the two-dimensional electron gases (2DEGs) that form at the surfaces and interfaces of KTaO$_3$ (KTO), with the discovery of superconductivity at temperatures significantly higher than those of similar 2DEGs based on SrTiO$_3$ (STO). Here we demonstrate that KTO 2DEGs fabricated under conditions that suppress the superconductivity show a large spontaneous Hall effect at low temperatures. The transverse response is asymmetric in an applied perpendicular magnetic field and becomes hysteretic at millikelvin temperatures. The hysteresis is due to long range magnetic order arising from local Ta$^{4+}$ moments. However, the most striking features of the data are the asymmetry of the transverse response and the large spontaneous transverse resistance at zero field, which can be a significant fraction of the longitudinal resistance and depends on crystal orientation. Both effects are due to the presence of a dominant contribution to the transverse response that is symmetric in perpendicular field, suggesting that its origin is topological in nature. We argue that this contribution arises from Berry curvature dipoles coupled with nonequilibrium conditions induced by the measuring current.