Phase vortices of the quenched Haldane model

Abstract

Using the recently developed Bloch-state tomography technique, the quasimomentum $\mathbf{k}$-dependent Bloch states ${[sin\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right),\phantom{\rule{0.28em}{0ex}}\ensuremath{-}cos\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right){e}^{i{\ensuremath{\phi}}_{\mathbf{k}}}]}^{T}$ 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 ${\ensuremath{\phi}}_{\mathbf{k}}(t)$ supports two types of vortices. The first type of vortices is static, with the corresponding Bloch vectors pointing to the north pole (${\ensuremath{\theta}}_{\mathbf{k}}=0$). The second type of vortices is dynamical, with the corresponding Bloch vectors pointing to the south pole (${\ensuremath{\theta}}_{\mathbf{k}}=\ensuremath{\pi}$). In the $({k}_{x},{k}_{y},t)$ space, the linking number between the trajectories of these two types of vortices exactly equals 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.