Quantum metric induced nonlinear anomalous Hall effect and nonreciprocal longitudinal response in a topological antiferromagnet

The Berry curvature and quantum metric are the imaginary part and real part, respectively, of the quantum geometric tensor which characterizes the topology of quantum states. The former is known to generate a zoo of important discoveries such as quantum Hall effect and anomalous Hall effect (AHE), while the consequences of the quantum metric have rarely been probed by transport. In this work, we observed quantum metric induced nonlinear transport, including both nonlinear AHE and diode-like nonreciprocal longitudinal response, in thin films of a topological antiferromagnet, MnBi$_2$Te$_4$. Our observation reveals that the transverse and longitudinal nonlinear conductivities reverse signs when reversing the antiferromagnetic order, diminish above the N\'eel temperature, and are insensitive to disorder scattering, thus verifying their origin in the band structure topology. They also flip signs between electron and hole-doped regions, in agreement with theoretical calculations. Our work provides a pathway to probe the quantum metric through nonlinear transport and to design magnetic nonlinear devices.