Berry curvature where S is any surface whose boundary is the loop C. Two useful formula: Bj=ϵjkl∂kAl=−Imϵjkl∂k⟨n|∂ln⟩=−Imϵjkl⟨∂kn|∂ln⟩, that is B(n)=−Im∑n′≠n⟨∇n|n′⟩×⟨n′|∇n⟩.
Zak phase, which refers to the Berry's phase picked up by a particle moving across the Brillouin zone, characterizes the topological properties of Bloch bands in one-dimensional periodic system.