  The entanglement entropy (von Neumann entropy) has been used to characterize
the complexity of many-body ground states in strongly correlated systems. In
this paper, we try to establish a connection between the lower bound of the von
Neumann entropy and the Berry phase defined for quantum ground states. As an
example, a family of translational invariant lattice free fermion systems with
two bands separated by a finite gap is investigated. We argue that, for one
dimensional (1D) cases, when the Berry phase (Zak's phase) of the occupied band
is equal to $\pi \times ({odd integer})$ and when the ground state respects a
discrete unitary particle-hole symmetry (chiral symmetry), the entanglement
entropy in the thermodynamic limit is at least larger than $\ln 2$ (per
boundary), i.e., the entanglement entropy that corresponds to a maximally
entangled pair of two qubits. We also discuss this lower bound is related to
vanishing of the expectation value of a certain non-local operator which
creates a kink in 1D systems.
