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We propose a new theory to characterize equilibrium topological phase withnon-equilibrium quantum dynamics by introducing the concept of high-ordertopological charges, with novel phenomena being predicted. Through a dimensionreduction approach, we can characterize a $d$-dimensional ($d$D)integer-invariant topological phase with lower-dimensional topological numberquantified by high-order topological charges, of which the $s$th-ordertopological charges denote the monopoles confined on the $(s-1)$th-order bandinversion surfaces (BISs) that are $(d-s+1)$D momentum subspaces. The bulktopology is determined by the $s$th order topological charges enclosed by the$s$th-order BISs. By quenching the system from trivial phase to topologicalregime, we show that the bulk topology of post-quench Hamiltonian can bedetected through a high-order dynamical bulk-surface correspondence, in whichboth the high-order topological charges and high-order BISs are identified fromquench dynamics. This characterization theory has essential advantages in twoaspects. First, the highest ($d$th) order topological charges are characterizedby only discrete signs of spin-polarization in zero dimension (i.e. the $0$thChern numbers), whose measurement is much easier than the $1$st-ordertopological charges that are characterized by the continuous charge-relatedspin texture in higher dimensional space. Secondly, a more striking result isthat a first-order high integer-valued topological charge always reduces tomultiple highest-order topological charges with unit charge value, and thelatter can be readily detected in experiment. The two fundamental featuresgreatly simplify the characterization and detection of the topological chargesand also topological phases, which shall advance the experimental studies inthe near future.

Dynamically characterizing topological phases by high-order topological charges