Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics

We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the $[k_x,k_y,t]$ space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the $[k_x,k_y,t]$ space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.

Figure 1

(a) Schematic of hopping in the Haldane model in a honeycomb lattice. (b) Phase diagram of the Haldane model. The arrow indicates a quench from a topologically trivial regime to a topologically nontrivial regime.

Figure 2

(a),(b) Inverse images of two vectors $\mathbf{n}$ and $-\mathbf{n}$ on the equator, when the Hamiltonian is quenched from ${\mathbf{h}}^i(\mathbf{k})$ with $M=-\infty$ (topologically trivial regime) to ${\mathbf{h}}^f(\mathbf{k})$ with $\varphi =0.1$ and $M=1$ (topologically trivial regime) (a), and to ${\mathbf{h}}^f(\mathbf{k})$ with $\varphi =\pi /2$ and $M=0$ (topologically nontrivial regime) (b), respectively. (c),(d) Inverse images of the north and the south poles, when the Hamiltonian is quenched from ${\mathbf{h}}^i(\mathbf{k})$ with $M=-1$ and $\varphi =\pi /2$ to ${\mathbf{h}}^f(\mathbf{k})$ with $M=0.33\sqrt{3}$ and $\varphi =\pi /2$ (topologically trivial regime) (c), and to ${\mathbf{h}}^f(\mathbf{k})$ with $M=0.27\sqrt{3}$ and $\varphi =\pi /2$ (topologically nontrivial regime) (d). For all plots we have taken $J_0=1$ and $J_1=0.1$.

Figure 3

Schematic of the mapping $f_1$ from $[k_x,k_y,t]$ ($[k_x,k_y]$ forms a torus) to $[\mathbf{h},t]$, and mapping $f_2$ from $[\mathbf{h},t]$ to $\mathbf{n}$ on a Bloch sphere.