The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids and in the theory of the quantum Hall effect. The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form ψ ψ n k ( r ) = e i k ⋅ ⋅ r u n k ( r ) , where n is a band index, k is a wavevector in the reciprocal-space (Brillouin zone), and u n k ( r ) is a periodic function of r . Due to translational symmetry, the momentum operator p ^ ^ i could be replaced with m ℏ ℏ ∂ ∂ H / ∂ ∂ k i by the Peierls substitution and the wavevector k plays the role of the parameter R . Thus, one can define Berry phases, connections, and curvatures in the reciprocal space. For example, in an N-band system, the Berry connection of the nth band in reciprocal space is A n ( k ) = i ⟨ ⟨ u n k | ∇ ∇ k | u n k ⟩ ⟩ . In the system, the Berry curvature of the nth band Ω Ω n , μ μ ν ν ( k ) is given by all the other N − 1 bands for each value of k . In a 2D crystal, the Berry curvature only has the component out of the plane and behaves as a pseudoscalar. It is because there only exists in-plane translational symmetry when translational symmetry is broken along z direction for a 2D crystal. Because the Bloch theorem also implies that the reciprocal space itself is closed, with the Brillouin zone having the topology of a 3-torus in three dimensions, the requirements of integrating over a closed loop or manifold can easily be satisfied. In this way, such properties as the electric polarization, orbital magnetization, anomalous Hall conductivity, and orbital magnetoelectric coupling can be expressed in terms of Berry phases, connections, and curvatures.