Note on Formula of Weak-Field Hall Conductivity: Singular Behavior for Long-Range Scatterers

Two formulas are obtained for the weak-field Hall conductivity based on different configurations of a magnetic field in the long-wavelength limit: sigma((x))(xy) and sigma((y))(xy) for the field-variation in the x- and y-direction, respectively. These formulas are formally equivalent but can give different results in the case of long-range scatterers. The difference originates from a singular behavior of a double-vertex function appearing in sigma((x))(xy). This singularity is removed when an extra imaginary part is introduced in the energy variable and then let to be zero after calculations. This is demonstrated in analytical calculations in a simple two-dimensional system and in numerical calculations in monolayer graphene and a giant Rashba system.