Dynamical quantum anomalous Hall effect in strong optical fields

Topological insulators are characterized by the quantum anomalous Hall effect (QAHE) on the topological surface states under time-reversal symmetry breaking. Motivated by recent experiments on the magneto-optical effects induced by the QAHE, we develop a theory for the dynamical Hall conductivity for subgap optical frequency and intense optical fields using the Keldysh-Floquet Green's function formalism. Our theory reveals a nonlinear regime in which the Hall conductivity remains close to $e^2/2h$ at low frequencies. At higher optical fields, we find that the subsequent collapse of the half quantization is accompanied by coherent oscillations of the dynamical Hall conductivity as a function of field strength, triggered by the formation of Floquet subbands and the concomitant intersubband transitions.

Figure 1

Comparison of the dynamical Hall conductivity obtained by our theory with the counterpart (a) by the Kubo theory in the linear-response regime and (b) by the Bloch theory in the rotating wave approximation (RWA) near resonance. In (a), we set $\eta /\mathrm{\Delta }=0.01$ with varying $E$; in (b), $\hslash \mathrm{\Omega }/\mathrm{\Delta }=0.99$, or equivalently the detuning $\delta /\mathrm{\Delta }=1-\hslash \mathrm{\Omega }/\mathrm{\Delta }=0.01$, with varying $\eta$. In both (a) and (b), we have common parameters $k_{B}T/\mathrm{\Delta }=0.02$ and ${\mathcal{E}}_c/\mathrm{\Delta }=10$.

Figure 2

Dynamical breakdown of the one-half QAHE. Each panel shows (a) the dynamical longitudinal conductivity, (b) the dynamical Hall conductivity, and (c) the generalized Hall angle as a function of optical field strength $E$ for some values of optical frequency $\mathrm{\Omega }$. Here, we used the parameters $k_{B}T/\mathrm{\Delta }=0.02, \eta /\mathrm{\Delta }=0.01$, and ${\mathcal{E}}_c/\mathrm{\Delta }=10$.

Figure 3

Dynamical Hall conductivity as a function of the optical frequency $\mathrm{\Omega }$ and the optical field strength $E$. The right panel indicates the color legend for the dynamical Hall conductivity. “QAHE” indicates the sector exhibiting the robust QAHE, and “${\mathcal{F}}_n$” ($n\in \mathbb{N}$) denotes the sector populated predominantly by the leading Floquet state with $n$-photon excitations. The contribution due to subleading $m$-photon excitations with $1\le m<n$, although present, is small compared to the $n$-photon contribution. Unassigned parameters are the same as in Fig. 2.