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We investigate topological signatures in the short-time non-equilibriumdynamics of symmetry protected topological (SPT) systems starting from initialstates which break the protecting symmetry. Na\"ively, one might expect thattopology loses meaning when a protecting symmetry is broken. Defying thisintuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model,how this combination of symmetry breaking and quench dynamics can give rise toboth single-particle and many-body signatures of topology. From the dynamics ofthe symmetry broken state, we find that we are able to dynamically probe theequilibrium topological phase diagram of a symmetry respecting projection ofthe post-quench Hamiltonian. In the ensemble dynamics, we demonstrate howspontaneous symmetry breaking (SSB) of the protecting symmetry can result in aquantized many-body topological `invariant' which is not pinned under unitarytime evolution. We dub this `dynamical many-body topology' (DMBT). We shownumerically that both the pure state and ensemble signatures are remarkablyrobust, and argue that these non-equilibrium signatures should be quite genericin SPT systems, regardless of protecting symmetries or spatial dimension.

Dynamical signatures of symmetry protected topology following symmetry breaking