Measuring Topological Number of a Chern-Insulator from Quench Dynamics

In this letter we address the issue that whether the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in twodimension, for instance, the Haldane model, whose quantum state is a pseudo-spin on the Bloch sphere. The dynamical process is described by a mapping from the [kx, ky, t] space to the Bloch sphere, characterized by the Hopf invariant. We show that, taking any two constant vectors on the Bloch sphere, the linking number of their inverse images of this mapping (i.e. two trajectories in the [kx, ky, t] space) exactly equals to the Chern number of the static Hamiltonian. Applying this result to a recent experiment from Hamburg group, we show that it is not the appearance of the phase vortices, but the linking number of the trajectories of these phase vortices that determines the phase boundary of the static Hamiltonian.