The main subjects of this talk are topological Berry phases in solids, Bose-Einstein condensates in optical lattices and non-Hermitian systems. Topological Berry phases are Berry phases that are invariant under noise in parameters. One cause of such invariance can be crystallographic symmetries. I will explain that for cold atoms, differences in topological Berry phases could be inferred from topological Bloch oscillations. Another cause can be non-Hermiticities such as dissipation. Specifically, topological Berry phases result from adiabatic transport on manifolds of exceptional points--parameters at which not only the complex eigenvalues but also the eigenvectors of a non-Hermitian matrix coalesce. Topological Berry phases could thus be observed as discrete phase shifts in the oscillation amplitudes of a dielectric membrane in an optical cavity.
