Direct measurement of topological invariants through temporal adiabatic evolution of bulk states in the synthetic Brillouin zone

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.