Quantized nonlinear transport and its breakdown in Fermi gases with Berry curvature

Quantized transport not only exist in gapped topological states but also in metallic states. Recently, Kane proposed a quantized nonlinear conductance in ballistic metals whose value is determined by the Euler characteristic of the Fermi sea [Phys. Rev. Lett. 128, 076801 (2022)]. In this paper, we consider two-dimensional noninteracting fermionic systems whose Fermi surface has nonvanishing Berry curvature. We find that the Berry curvature at the Fermi surface does not affect the quantized nonlinear transport for translationally invariant systems. When spatial inhomogeneity is introduced, such quantization breaks down due to the combined effect of Berry curvature and the gradient of local potential. Such breakdown of quantization can be observed in trapped ultracold atoms with topological bands.