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?
Sorry for unclearly description. This formula is used in a two dimensional system. v1 and v2 are velocity operators in x and y direction
Which part is unfamiliar to you? The general linear response formula involves the time correlation function of the observable and the applied field. In a Hall measurement, the current is measured in the direction orthogonal to the applied field, which is why vx and vy show up. Writing the field in terms of the current density then gives the time integral of a current-current (or velocity-velocity) correlation function and performing the time evolution in the basis of eigenstates (Lehmann representation) should give the final result.