Some remarks on topological phases and topological insulators

The remarkable properties of topological phases and topological insulators are analyzed in very simple quantum mechanical models. A characteristic effect of magnetic fields on quantum systems is the appearance of degenerate ground states. The level of degeneracy of the corresponding quantum phases depends on the strength of the magnetic field and the topology of the configuration space. In topological insulators, in absence of magnetic fields, the essential ingredients are time reversal symmetry, the existence of edge states and a non-trivial topological structure of the Jacobian space (Brillouin zone). In both cases the main features are encoded by topological indices. In topological phases induced by magnetic fields, the index is a topological invariant that depends on the Chern class of the magnetic fields and the topology of the space. In topological insulators the topological index is given by the number of edge levels crossing the Fermi surface that remains invariant under time reversal invariant pe...