  Quenching a quantum system involves three basic ingredients: the initial
phase, the post-quench target phase, and the non-equilibrium dynamics which
carries the information of the former two. Here we propose a dynamical theory
to characterize both the topology and symmetry-breaking order in correlated
quantum system, through quenching the Haldane-Hubbard model from an initial
magnetic phase to topologically nontrivial regime. The equation of motion for
the complex pseudospin dynamics is obtained with the flow equation method, with
the pseudospin evolution shown to obey a microscopic
Landau-Lifshitz-Gilbert-Bloch equation. We find that the correlated quench
dynamics exhibit robust universal behaviors on the so-called band-inversion
surfaces (BISs), from which the nontrivial topology and magnetic order can be
extracted. In particular, the topology of the post-quench regime can be
characterized by an emergent dynamical topological pattern of quench dynamics
on BISs, which is robust against dephasing and heating induced by interactions;
the pre-quench symmetry-breaking orders is read out from a universal scaling
behavior of the quench dynamics emerging on the BIS, which is valid beyond the
mean-field regime. This work opens a way to characterize both the topology and
symmetry-breaking orders by correlated quench dynamics.
