  Dynamical quantum phase transitions (DQPTs) represent a counterpart in
non-equilibrium quantum time evolution of thermal phase transitions at
equilibrium, where real time becomes analogous to a control parameter such as
temperature. In quenched quantum systems, recently the occurrence of DQPTs has
been demonstrated, both with theory and experiment, to be intimately connected
to changes of topological properties. Here, we contribute to broadening the
systematic understanding of this relation between topology and DQPTs to
multi-orbital and disordered systems. Specifically, we provide a detailed
ergodicity analysis to derive criteria for DQPTs in all spatial dimensions, and
construct basic counter-examples to the occurrence of DQPTs in multi-band
topological insulator models. As a numerical case study illustrating our
results, we report on microscopic simulations of the quench dynamics in the
Harper-Hofstadter model. Furthermore, going gradually from multi-band to
disordered systems, we approach random disorder by increasing the (super) unit
cell within which random perturbations are switched on adiabatically. This
leads to an intriguing order of limits problem which we address by extensive
numerical calculations on quenched one-dimensional topological insulators and
superconductors with disorder.
