Full counting statistics for noninteracting fermions: Exact results and the Levitov-Lesovik formula

Exact numerical results for the full counting statistics (FCS) for a one-dimensional tight-binding model of noninteracting electrons are presented without using an idealized measuring device. The two initially separate subsystems are connected at t=0 and the exact time evolution for the large but finite combined system is obtained numerically. At zero temperature the trace formula derived by Klich is used to to calculate the FCS via a finite dimensional determinant. Even for surprisingly short times the approximate description of the time evolution with the help of scattering states agrees well with the exact result for the local current matrix elements. An additional approximation has to be made to recover the Levitov-Lesovik formula in the limit where the system size becomes infinite and afterwards the long time limit is addressed. The new derivation of the Levitov-Lesovik formula is generalized to more general geometries like a Y-junction enclosing a magnetic flux.