  Three decades ago, in a celebrated work, Zak found an expression for the
geometric phase acquired by an electron in a one-dimensional periodic lattice
as it traverses the Bloch band. Such a geometric phase is useful in
characterizing the topological properties and the electric polarization of the
periodic system. Unfortunately Zak's expression suffers from two flaws: its
value depends upon the choice of origin of the unit cell, and is gauge
dependent. Here we explain that these flaws in Zak's expression arise from the
assumption that the electron's adiabatic motion is cyclic in the sense of
recurrence of the density matrix in course of time evolution. We find through a
careful investigation that the system displays cyclicity in a generalized sense
wherein the physical observables return in the course of evolution. This notion
of generalized cyclicity paves the way for a correct and consistent expression
for the geometric phase in this system, christened as Pancharatnam-Zak phase.
Pancharatnam-Zak geometric phase does not suffer from the flaws inherent in
Zak's expression, and correctly classifies the Bloch bands of the lattice. A
natural filled band extension of the Pancharatnam-Zak phase is also constructed
and studied.
