Several papers (eg. Di Xiao, et. al, Berry phase effects on electronic properties, RevModPhys, 82,2010)mentioned a formula to calculate the Hall conductivity(See the picture).This formula is used in an two dimensional system, v1 and v2 are velocity operators in x and y direction, Phi0 and PhiN are ground and excited state vector. The papers claim that this formula can be derived from the Kubo identity, but I am not sure how this can be done, since the form of Kubo formla is quite different from this expression. Could anyone help me with the derivation?