  Dynamical characterization of equilibrium topological phases has attracted
considerable attention in recent years. In this paper, we make a thorough
exploration of the non-adiabatic characterization of topological phases under
slow quench protocol. We first propose an exactly solvable multi-state
Landau-Zener model that can be directly applied to the non-adiabatic slow
quench dynamics of topological systems. Then we present two different schemes
to characterize the bulk topology of the system based on the so called spin
inversion surface. The first one needs least number of quenching processes, but
requires to measure the gradients of time-averaged spin-polarization on the
SIS. The second one only needs to measure the value of time-averaged
spin-polarization on the SIS, thus makes it possible to directly characterize
the topological phases by introducing an extra quenching process. Moreover,
high-order SIS or band inversion surface (BIS) relying on the dimension
reduction approach, is also generalized to the above two different
characterization schemes. One can extract the topological invariant from pairs
of points with opposite signs both on the 0D highest order BIS and on the 0D
highest order SIS, which greatly simplifies the measurement strategy and
characterization process. In a word, superior to the sudden quench protocol, we
demonstrate that the topological invariant can be captured not only by the
topological information on BIS, but also on the SIS. In particular, direct
characterization of topological phases based on BIS and SIS can be realized.
