The Bloch states of a crystal carry geometric information that goes beyond the energy bands. The Berry curvature measures how the phase of these states changes across the Brillouin zone, and its integrals give topological invariants such as Chern numbers and Z2 indices.
These quantities explain quantized transport, protected edge and surface states, and the anomalous Hall and related responses. We compute them from first principles and from tight-binding models, and we study how they behave when interactions, disorder or lattice distortions are present.