Singularities in the Loschmidt echo of quenched topological superconductors

We study the Loschmidt echo in the quenched two-dimensional $p$-wave topological superconductor. We find that if this superconductor is quenched out of the critical point separating its topological and non-topological phases into either of the two gapful phases, its Loschmidt echo features singularities occurring periodically in time where the second derivative of the Loschmidt echo over time diverges logarithmically. Conversely, we give arguments towards $s$-wave superconductors not having singularities in their Loschmidt echo regardless of the quench. We also demonstrate that the conventional mean field theory calculates classical echo instead of its quantum counterpart, and show how it should be modified to capture the full quantum Loschmidt echo.