Phase vortices of the quenched Haldane Model

Using the recently developed Bloch-state tomography technique, the quasimomentum $\bf k$-dependent Bloch states ${\left( {\sin \left( {{\theta _{\mathbf{k}}}/2} \right),\; - \cos \left( {{\theta _{\mathbf{k}}}/2} \right){e^{i{\phi _{\mathbf{k}}}}}} \right)^T}$ of a two-band tight-binding model with two sublattices can be mapped out. We show that, if we prepare the initial Bloch state as the lower-band eigenstate of a topologically trivial Haldane Hamiltonian $H_i$, and then quench the Haldane Hamiltonian to $H_f$, the time-dependent azimuthal phase ${\phi _{\mathbf{k}}(t)}$ supports two types of vortices. The first type of vortices are static, with the corresponding Bloch vectors pointing to the north pole ($\theta_{\mathbf{k}}=0$). The second type of vortices are dynamical, with the corresponding Bloch vectors pointing to the south pole ($\theta_{\mathbf{k}}=\pi$). In the $(k_x,k_y,t)$ space, the linking number between the trajectories of these two types of vortices equals exactly to the Chern number of the lower band of $H_f$, which provides an alternative method to directly map out the topological phase boundaries of the Haldane model.