Berry phases near degeneracies: Beyond the simplest case

The Berry phase is reviewed with emphasis on the Berry curvature and the Chern number. The behavior of these quantities and the analytic properties of adiabatically continued wave functions in the vicinity of degeneracies are discussed. An example of a spin Hamiltonian is given in which the Chern numbers associated with the states involved in a double degeneracy are ±2 rather than ±1, as is usually the case. Degeneracies in the spectrum of magnetic molecular solids are discussed.