A Dynamical Bulk-Boundary Correspondence in Two Dimensional Topological Matter

Dynamical quantum phase transitions occur when a dynamical free energy becomes non-analytic at critical \emph{times}. They have been shown to exist in, among other systems, topological insulators and superconductors. Additionally in both one dimensional systems and two dimensional higher order topological matter a dynamical analogue of the bulk-boundary correspondence has been observed which is suggestive of a dynamical switching between different phases. If the time evolving Hamiltonian is topologically non-trivial, zeroes of the Loschmidt matrix appear between critical times, leading to significant, periodically occurring contributions to the boundary dynamical free energy. In this article we extend these ideas to two dimensional topological matter by showing that in-gap bands in the spectrum of the Loschmidt matrix between successive critical times exist if the time evolving Hamiltonian is topologically non-trivial. We further show that these in-gap bands are directly responsible for a boundary contribution to the dynamical free energy, thus establishing a dynamical bulk-boundary correspondence in two dimensional topological matter.