The single-point Berry phase in condensed-matter physics

The Berry phase plays an outstanding role in condensed-matter physics, where it provides the main entry for defining and computing a key observable: macroscopic electrical polarization. This phase obtains as a loop integral in reciprocal space, which has the topology of a torus. I show here that in some circumstances the loop may be contracted to a point, and polarization can be computed in practice from a single Hamiltonian diagonalization.