Dynamically characterizing topological phases by high-order topological charges

We propose a new theory to characterize equilibrium topological phase with non-equilibrium quantum dynamics by introducing the concept of high-order charges, novel phenomena being predicted. Through dimension reduction approach, we can $d$-dimensional ($d$D) integer-invariant lower-dimensional number quantified which $s$th-order charges denote monopoles confined on $(s-1)$th-order band inversion surfaces (BISs) that are $(d-s+1)$D momentum subspaces. The bulk topology is determined $s$th order enclosed BISs. By quenching system from trivial regime, show post-quench Hamiltonian be detected through dynamical bulk-surface correspondence, in both and BISs identified quench dynamics. This characterization has essential advantages two aspects. First, highest ($d$th) characterized only discrete signs spin-polarization zero (i.e. $0$th Chern numbers), whose measurement much easier than $1$st-order continuous charge-related spin texture higher dimensional space. Secondly, more striking result first-order high integer-valued charge always reduces multiple highest-order unit value, latter readily experiment. fundamental features greatly simplify detection also phases, shall advance experimental studies near future.