The entanglement entropy and the Berry phase in solid states

The entanglement entropy (von-Neumann entropy) has been used to characterize the quantum entanglement of many-body ground states in strongly correlated systems. In this talk, 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 1D Hamiltonians with two bands separated by a finite gap is investigated. We argue that when the Berry phase (Zak's phase) of the occupied band is equal to $\pm \pi \times (\mbox{odd integer})$ 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 spins. We also discuss it is related to vanishing of the expectation value of a certain non-local operator which creates a kink in 1D systems.