Time-dependent PT-symmetric phase transitions: exactly solvable models and their underlying geometry

Time-dependent 2×2 matrix Hamiltonians are designed which preserve formal PT-symmetry, pass through an exceptional point and whose evolution matrices can be obtained analytically. These exactly solvable models allow for a full analytical treatment of the underlying geometric deformation features of the eigenspaces (the structures of the eigenvector bundles) including their geometric and algebraic properties. It is shown that a passage through an exceptional point is not sufficient for a time-dependent Hamiltonian to describe a time-dependent PT phase transition. Rather additional singular gauge terms have to be added in order to force a time-dependent eigenvector coalescence of the models. This opens a way to new classes of exactly solvable models. First results in this direction are presented.