  Characterization of equilibrium topological quantum phases by non-equilibrium
quench dynamics provides a novel and efficient scheme in detecting topological
invariants defined in equilibrium. Nevertheless, most of the previous studies
have focused on the ideal sudden quench regime. Here we provide a generic
non-adiabatic protocol of slowly quenching the system Hamiltonian, and
investigate the non-adiabatic dynamical characterization scheme of topological
phase. The {\it slow} quench protocol is realized by introducing a Coulomb-like
Landau-Zener problem, and it can describe, in a unified way, the crossover from
sudden quench regime (deep non-adiabatic limit) to adiabatic regime. By
analytically obtaining the final state vector after non-adiabatic evolution, we
can calculate the time-averaged spin polarization and the corresponding
topological spin texture. We find that the topological invariants of the
post-quench Hamiltonian are characterized directly by the values of spin
texture on the band inversion surfaces. Compared to the sudden quench regime,
where one has to take an additional step to calculate the {\it gradients} of
spin polarization, this non-adiabatic characterization provides a {\it minimal}
scheme in detecting the topological invariants. Our findings are not restricted
to 1D and 2D topological phases under Coulomb-like quench protocol, but are
also valid for higher dimensional system or different quench protocol.
