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