Phase vortices of the quenched Haldane Model

TL;DR

This paper demonstrates how quench dynamics in the Haldane model reveal topological properties through vortex linking in phase space, enabling direct mapping of topological phase boundaries.

Contribution

It introduces a novel vortex-based method to determine the Chern number in the quenched Haldane model using Bloch-state tomography.

Findings

Vortices in phase space correspond to topological invariants.

Linking number of vortex trajectories equals the Chern number.

Method allows direct mapping of topological phase boundaries.

Abstract

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.