  Two-dimensional topologically distinct insulators are separated by
topological gapless points, which exist as Weyl points in three-dimensional
momentum space. Slowly varying parameters in the two-dimensional Hamiltonian
across two distinct phases therefore necessarily experiences the gap closing
process, which prevents the intrinsic physical observable, the Hall response,
from equilibrating. To equilibrate the Hall response, engineered laser noises
were introduced at the price of destroying the quantum coherence. Here we
demonstrate a new scheme to equilibrate the quantized Hall response from pure
coherent dynamics as the Hamiltonian is slowly tuned from the topologically
trivial to nontrivial regimes. We show the elements that affect the process of
equilibration including the sequence when the electric field is switched on,
its strength and the band dispersion of the final Hamiltonian. We further apply
our method to Weyl semimetals in three dimensions and find the equilibrated
Hall response despite the underlying gapless band structure. Our finding not
only lays the theoretical foundation for observing the two-dimensional
topological phase transition but also for observing and controlling Weyl
semimetals in ultracold atomic gases.
