  Dynamical quantum phase transitions occur in dynamically evolving quantum
systems when non-analyticities occur at critical times in the return rate, a
dynamical analogue of the free energy. This extension of the concept of phase
transitions can be brought into contact with another, namely that of
topological phase transitions in which the phase transition is marked by a
change in a topological invariant. Following a quantum quench dynamical quantum
phase transitions can happen in topological matter, a fact which has already
been explored in one dimensional topological insulators and in two dimensional
Chern insulators. Additionally in one dimensional systems a dynamical bulk
boundary correspondence has been seen, related to the periodic appearance of
zero modes of the Loschmidt echo itself. Here we extend both of these concepts
to two dimensional higher order topological matter, in which the topologically
protected boundary modes are corner modes. We consider a minimal model which
encompasses all possible forms of higher order topology in two dimensional
topological band structures. We find that DQPTs can still occur, and can occur
for quenches which cross both bulk and boundary gap closings. Furthermore a
dynamical bulk boundary correspondence is also found, which takes a different
form to that in one dimension.
