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Memory-Critical Dynamical Buildup of Phonon-Dressed Majorana Fermions

We investigate the dynamical interplay between topological state of matter and a non-Markovian dissipation, which gives rise to a new and crucial time scale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed $p$-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of Majorana correlation.