Scheme to equilibrate the Hall response of topological systems from coherent dynamics

Two-dimensional topologically distinct Chern insulators are separated by gap closing points, which exist as Weyl points in three-dimensional momentum space. Slowly varying parameters in the two-dimensional Hamiltonian across two distinct phases therefore necessarily experience the gap closing process, which prevents the intrinsic physical observable, the Hall response, from equilibrating. Here we demonstrate a scheme to equilibrate the quantized Hall response from pure coherent dynamics as the Hamiltonian is slowly tuned from the topologically trivial to nontrivial regimes. We further apply our method to Weyl semimetals in three dimensions and find the equilibrated Hall response despite the underlying gapless band structure. Our finding not only lays the theoretical foundation for observing the Hall response in Chern insulators but also for observing and controlling Weyl semimetals in ultracold atomic gases.

Figure 1

Coherent dynamics of the Hall response for various strength of electric fields in two different protocols (see main text). Panels in the upper (lower) row correspond to the first (second) protocol. Here $m_i=-5J,v\hslash /\left(2J\right)=0.02$, and ${\tau }_e=10\hslash /\left(2J\right)$. We note that the results are similar for ${\tau }_e=20\hslash /\left(2J\right),30\hslash /\left(2J\right)$. For visuality, we have shifted the Hall response downward for different $E_0$ from an asymptotically quantized value. All curves are computed by the exact numerical method except for the pink curve in panel (a), which is calculated based on the perturbation approach. Here, $e_0=2J/\left(ea\right)$ is the unit of the electric field.

Figure 2

Energy spectra of the lower band of the Hamiltonian (1) with $m_z\left(t\right)=m_f$ for (a) $m_f=-2J$ and (b) $m_f=-3.4J$. The unit of energy is $2J$. The Berry curvature ${\mathrm{\Omega }}_{-}\left(\mathbf{k}\right)$ for (c) $m_f=-2J$ and (d) $m_f=-3.4J$.

Figure 3

Profiles of $\theta \left(\mathbf{k}\right)$ near the gap closing point ${\mathbf{k}}_c$ (for $\mathbf{k}\ne {\mathbf{k}}_c$) for (a) protocol (1) with $k_c=0$, and (b) protocol (2) with $k_c=-eA\left(t_c\right)/\hslash$ and $E_0=0.02e_0$. At momenta depicted by the dashed blue line, the corresponding population in the excited state is 0.1.

Figure 4

(a) The first Brillouin zone with a pair of Dirac points, one and two pairs of Weyl points, denoted by the solid red circles and green and blue squares, respectively. (b) Schematics illustrating that the Chern number of states remains unchanged on a closed surface when a Weyl point moves out of the surface. Evolution of the Hall response over time as $m_z$ varies slowly (c) from $-7J$ to $-5J$ and (d) from $-3J$ to $-J$.

Figure 5

Evolution of the Chern number of the Hamiltonian $H_{3\mathrm{D}}^{\prime }$ on a closed surface in the 3D momentum space: $C_H$ (denoted by the red line), and the Chern number of the time-evolved states on the same surface: $C_S$ (denoted by the black line). In (a), at $t=0$, we choose a closed surface (schematically denoted by a circle) such that there is a Weyl point (denoted by a red point) inside it. As time evolves, the Weyl point defined as the gap closing point in the instantaneous Hamiltonian moves out of the surface. In (b), the surface is chosen such that there are no Weyl points inside it at $t=0$. As time evolves, a Weyl point travels inside the surface. Note that our numerical calculation of $C_H$ around the sudden change region is not accurate due to the very small gap. Here the unit of time is $\hslash /\left(2J\right)$.