Characterizing topological phases by quantum quenches: A general theory

We investigate a generic dynamical theory to characterize topological quantum phases by quantum quenches, and study the emergent topology of quantum dynamics when the quenches start from a deep or shallow trivial phase to topological regimes. Two dynamical schemes are examined: One is to characterize topological phases via quantum dynamics induced by a single quench along an arbitrary axis, and the other applies a sequence of quenches with respect to all (pseudo)spin axes. These two schemes are both built on the so-called dynamical bulk-surface correspondence, which shows that the $d$-dimensional ($d$D) topological phases with integer invariants can be characterized by the dynamical topological pattern emerging on $(d-1)$D band inversion surfaces (BISs). We show that the first dynamical scheme works for both deep and shallow quenches, the latter of which is initialized in an incompletely polarized trivial phase. For the second scheme, however, when the initial phase for the quench study varies from the deep trivial (fully polarized) to shallow trivial (incompletely polarized) regime, a new dynamical topological transition, associated with topological charges crossing BISs, is predicted in quench dynamics. A generic criterion of the dynamical topological transition is precisely obtained. Above the criterion, quantum dynamics on BISs well characterizes the topology of the post-quench Hamiltonian. Below the criterion, the quench dynamics may depict a new dynamical topology; the post-quench topology can be characterized by the emergent topological invariant plus the total charges moving outside the region enclosed by BISs. We illustrate our results by numerically calculating the 2D quantum anomalous Hall model. This work broadens the way to classify topological phases by non-equilibrium quantum dynamics, and has feasibility for experimental realization.