Simple mechanism that breaks the Hall effect linearity at low temperatures

Hall resistance $R_{\text{xy}}$ is commonly suggested to be linear-in-magnetic-field $B$, provided the field is small. We argue here that at low temperatures this linearity is broken due to weak localization/antilocalization phenomena in inhomogeneous systems, while in a uniform medium these effects do not affect the linear field dependence of $R_{\text{xy}}(B)$. We calculate the Hall resistance for different two-component media using a mean-field approach and show that this non-linearity is experimentally observable.