A Scattering Formulation Of Linear Response Theory

We develop a novel exact eigenstate formulation of linear response theory for the DC current or voltage response of a phase-coherent conductor with multiple leads, valid both in zero and in arbitrarily strong magnetic field. We are able to express the total current response I sub m through lead m completely in terms of the voltages V sub n applied at the leads, independent of the electric field in the material, i.e. I sub m = sum above n gmnVn; and then show that the conductance coefficients g sub mn depend on the wave-functions at the fermi surface as T->0. This yields a new and useful Green function expression for the g sub mn and the Hall resistance. When transformed by appropriate applications of scattering theory, these expression are shown to be equivalent to the relation g sub mn =T sub mn where T sub mn is the sum of all the transmission coefficients between lead m and n, as first proposed by Buttiker on the basis of Landauer-type arguments. This formulation is very useful for calculations of the Quantum Hall Effect, both in mesoscopic and macroscopic systems.