Hall Effect in Two-Dimensional Disorderd Systems

The Hall conductivity, σ x y , is evaluated in two-dimensional random systems to the leading order of singular behavior at low frequency, ω. At weak magnetic field the result is σ x y =σ x y 0 1-1/π ε F τln 1/ωτ where σ x y 0 , ε F and τ are ordinary Hall conductivity, the Fermi energy, and the relaxation time due to impurity scattering, respectively. This result together with that for σ x x indicates that the effective carrier number deduced from the Hall coefficient does not contain leading singular terms and that only the relaxation time is logarithmically reduced.