Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators

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.