  Mathematically, topological invariants arise from the parallel transport of
eigenstates on the energy bands, which, in physics, correspond to the adiabatic
dynamical evolution of transient states. It determines the presence of boundary
states, while lacking direct measurements. Here, we develop time-varying
programmable coupling circuits between acoustic cavities to mimic the
Hamiltonians in the Brillouin zone, with which excitation and adiabatic
evolution of bulk states are realized in a unit cell. By extracting the Berry
phases of the bulk band, topological invariants, including the Zak phase for
the SSH model and the Chern number for the AAH model, are obtained
convincingly. The bulk state evolution also provides insight into the
topological charges of our newly developed non-Abelian models, which are also
verified by observing the adiabatic eigenframe rotation. Our work not only
provides a general recipe for telling various topological invariants but also
sheds light on transient acoustic wave manipulations.
