  Dynamical phase transitions (DPT) are characterized by nonanalytical time
evolution of the dynamical free energy. For general 2-band systems in one and
two dimensions (eg. SSH model, Kitaev-chain, Haldane model, p+ip
superconductor, etc.), we show that the time evolution of the dynamical free
energy is crucially affected by the ground state topology of both the initial
and final Hamiltonians, implying DPTs when the topology is changed under the
quench. Similarly to edge states in topological insulators, DPTs can be
classified as being topologically protected or not. In 1D systems the number of
topologically protected non-equilibrium time scales are determined by the
difference between the initial and final winding numbers, while in 2D no such
relation exists for the Chern numbers. The singularities of dynamical free
energy in the 2D case are qualitatively different from those of the 1D case,
the cusps appear only in the first time derivative.
