Gauge-invariant formulation of electron linear transport

A general study of electron linear transport in systems with open boundary conditions is presented. It is shown that a consistent treatment of bulk and boundary contributions is only achieved in a strictly gauge-invariant formulation of the electomagnetic linear response. Invariance under large gauge transformations requires the use of a generalized version of the continuity equation in which current sources at infinity are explicitly included. These sources are described by delta functions located at the points of infinity and they guarantee global charge conservation. The formulations of steady-state transport in terms of global conductance coefficients and nonlocal conductivity tensors are shown to be strictly equivalent for interacting electron systems in the presence of a magnetic field. As a corollary, the reciprocity theorem for a four-lead resistance is proved in its full generality for systems with arbitrary many-body interactions.