Entanglement entropy and the Berry phase in solid states

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.