Zak-Pancharatnam phase.

The geometric phase gained by an electron in a one-dimensional periodic lattice due to weak electric perturbation, referred to as the Zak-Pancharatnam phase, is found and the underlying structure is unveiled. As opposed to the well known Zak phase, the Zak-Pancharatnam phase is a genuine observable phase, and correctly characterizes the energy bands of the lattice. A filled band generalization of this geometric phase is constructed and is observed to be sensitive to the Fermi-Dirac statistics of the band electrons.