Quantum annealing and nonequilibrium dynamics of Floquet Chern insulators

Inducing topological transitions by a time-periodic perturbation offers a route to controlling the properties of materials. Here, we show that the adiabatic preparation of a nontrivial state involves a selective population of edge states, due to exponentially small gaps preventing adiabaticity. We illustrate this by studying graphenelike ribbons with hopping's phases of slowly increasing amplitude, as, e.g., for a circularly polarized laser slowly turned on. The induced currents have large periodic oscillations, but flow solely at the edges upon time averaging, and can be controlled by focusing the laser on either edge. The bulk undergoes a nonequilibrium topological transition, as signaled by a local Hall conductivity, the Chern marker introduced by Bianco and Resta in equilibrium. The breakdown of this adiabatic picture in the presence of intraband resonances is discussed.

Figure 1

(a) Phase diagram of the Haldane model, with three representative zigzag strip spectra, along the path of the QA evolution (arrow). (b), $\left({\mathrm{b}}^{\prime }\right)$ The mechanism by which right-edge states get selectively populated as the exponentially small LZ gap sweeps to larger values of $k$ during the QA evolution. The (equilibrium) edge states shown refer to ${\mathrm{\Delta }}_{AB}/t_2=2.5$ [for (b)] and ${\mathrm{\Delta }}_{AB}/t_2=2.4$ [for $\left({\mathrm{b}}^{\prime }\right)$]. Solid/open circles denote occupied/empty edge states. The blue vertical arrow in (b) points to an occupied right-edge electron that remains occupied [red vertical arrow in $\left({\mathrm{b}}^{\prime }\right)$] after the Landau-Zener event. (c) Residual energy (stars), separated into bulk (squares) and edge (circles) contributions, vs the annealing time ${\tau }_{\mathrm{QA}}$. Here, data with $L=N_x=N_y=6n$ from 18 to 102 were used to get ${\varepsilon }_{\text{bulk/edge}}$ for each ${\tau }_{\mathrm{QA}}$.

Figure 2

(a) Final Floquet quasienergy bands for a uniform driving with $\varphi =-\pi /2,\hslash \omega =7|t_1|>W,{\mathrm{\Delta }}_{AB}={10}^{-3}\left|t_1\right|$ (effectively representing graphene), and $\lambda \left(t\right)$ linearly ramped up to ${\lambda }_{\mathrm{f}}=1$ in ${\tau }_{\mathrm{QA}}=100\tau$, with $\tau =2\pi /\omega$. Here, the topological transition occurs at ${\lambda }_{\mathrm{cr}}\approx 0$. Valence states (in red) are filled, conduction states (in black) empty. Solid circles denote occupied edge states. Inset: The initial graphene spectrum at $\lambda \left(0\right)=0$. $\left({\mathrm{a}}^{\prime }\right)$ Time-averaged bond currents calculated with a periodic evolution at constant ${\lambda }_{\mathrm{f}}$ for ${\tau }_{\mathrm{f}}=100\tau$. The inset shows the large intraperiod oscillations. (b), $\left({\mathrm{b}}^{\prime }\right)$ Same as (a) and $\left({\mathrm{a}}^{\prime }\right)$ for an inhomogeneous driving focused on the right edge $\left(x_c=L_x\right)$ of width $\sigma =0.4L_x$. In the inset, a sketch of the zigzag strip.

Figure 3

The bulk average $C_{\mathrm{bulk}}\left(t\right)$ of the local Chern marker $\mathcal{C}(\mathbf{r},t)$ for a uniform driving with $\varphi =-\pi /2,\hslash \omega =7|t_1|>W,{\mathrm{\Delta }}_{AB}=0.1\left|t_1\right|$, and $\lambda \left(t\right)$ linearly ramped up to ${\lambda }_{\mathrm{f}}=1$ in ${\tau }_{\mathrm{QA}}=300\tau$, followed by a constant-${\lambda }_{\mathrm{f}}$ evolution for ${\tau }_{\mathrm{f}}=220\tau$, with $\tau =2\pi /\omega$. The topological transition occurs at ${\lambda }_{\mathrm{cr}}\approx 0.57$. Here, $L=N_x=N_y=48$, and we average on a central square of size $12\times 12$. The horizontal line at $\approx 0.96$ is the time average $C_{\mathrm{bulk}}^{\mathrm{av}}$, calculated from $t={\tau }_{\mathrm{QA}}$ to $t={\tau }_{\mathrm{QA}}+{\tau }_{\mathrm{f}}$. The upper inset shows the saturation of $C_{\mathrm{bulk}}^{\mathrm{av}}(L,{\tau }_{\mathrm{QA}}\to \infty )$ to a limiting value that (see the lower inset, where we fit points with standard power-law corrections $1+c_1/L+c_2/L^2)$ goes to 1 for $L\to \infty$.

Figure 4

Final Floquet quasienergies $E_{k,\alpha }$ and occupations $n_{k,\alpha }$ (dot size proportional to $n_{k,\alpha }$) for $\varphi =-\pi /2,\hslash \omega =4|t_1|<W,{\mathrm{\Delta }}_{AB}=0.1\left|t_1\right|$ (constant), and $\lambda \left(t\right)$ linearly ramped up to ${\lambda }_{\mathrm{f}}=1$ in ${\tau }_{\mathrm{QA}}=300\tau$.