  The Bloch theorem enables reduction of the eigenvalue problem of the
single-particle Hamiltonian that commutes with translational group. Based on a
group theory analysis we present generalization of the Bloch theorem that
incorporates all additional symmetries of a crystal. The generalized Bloch
theorem constrains the form of the Hamiltonian which becomes manifestly
invariant under additional symmetries. In the case of isotropic interactions
the generalized Bloch theorem gives a unique Hamiltonian. This Hamiltonian
coincides with the Hamiltonian in the periodic gauge. In the case of
anisotropic interactions the generalized Bloch theorem allows a family of
Hamiltonians. Due to the continuity argument we expect that even in this case
the Hamiltonian in the periodic gauge defines observables, such as Berry
curvature, in the inverse space. For both cases we present examples and
demonstrate that the average of the Berry curvatures of all possible
Hamiltonians in the Bloch gauge is the Berry curvature in the periodic gauge.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons