Out-of-equilibrium electrons and the Hall conductance of a Floquet topological insulator

Graphene irradiated by a circularly polarized laser has been predicted to be a Floquet topological insulator showing a laser-induced quantum Hall effect. A circularly polarized laser also drives the system out of equilibrium, resulting in nonthermal electron distribution functions that strongly affect transport properties. Results are presented for the Hall conductance for two different cases. One is for a closed system, such as a cold-atomic gas, where transverse drift due to nonzero Berry curvature can be measured in time-of-flight measurements. For this case the effect of a circularly polarized laser that has been suddenly switched on is studied. The second is for an open system coupled to an external reservoir of phonons. While for the former the Hall conductance is far from the quantized limit, for the latter, coupling to a sufficiently low temperature reservoir of phonons is found to produce effective cooling, and thus an approach to the quantum limit, provided the frequency of the laser is large as compared to the bandwidth. For laser frequencies comparable to the bandwidth, strong deviations from the quantum limit of conductance are found even for a very low temperature reservoir, with the precise value of the Hall conductance determined by a competition between reservoir-induced cooling and the excitation of photocarriers by the laser. For the closed system, the electron distribution function is determined by the overlap between the initial wave function and the Floquet states, which can result in a Hall conductance which is opposite in sign to that of the open system.

Figure 1

Hall conductance for the ideal case $\left({\sigma }_{xy}=C{e}^2/h\right)$ and for a closed system after a quench, for different strengths of the circularly polarized laser of frequency: (a) $\Omega =10t_h$. (b) $\Omega =5t_h$.

Figure 2

Excitation density ${\rho }_{kd}-{\rho }_{ku}$ in the closed system for a quench switch-on protocol for a laser of frequency $\Omega =10t_h$ and amplitude: (a) $A_{0}a=1.0$. (b) $A_{0}a=5.0$.

Figure 3

Hall conductance for the ideal case $\left({\sigma }_{xy}=C{e}^2/h\right)$ and at steady state with a phonon reservoir, for different strengths of the circularly polarized laser and for laser frequencies: (a) $\Omega =10t_h$. (b) $\Omega =5t_h$.

Figure 4

Excitation density ${\rho }_{kd}-{\rho }_{ku}$ at steady state with phonons. The parameters are $A_{0}a=1.0,\Omega =10t_h$, with the phonons at temperature (a) $T=0.01t_h$ and (b) $T=1.0t_h$.

Figure 5

Floquet spectrum over two Floquet BZs for laser frequency $\Omega =5.0t_h$ and for laser amplitudes and Chern numbers: (a) $A_{0}a=0.5,C=3$. (b) $A_{0}a=1.5,C=1$. Additional edge states at the Floquet zone boundaries appear for (a).

Figure 6

Hall conductance for the closed system after a quench, for the open system at steady state with phonons at $T=0.01t_h$, and for the ideal case $\left(C{e}^2/h\right)$, plotted for different laser frequencies and for the laser amplitudes: (a) $A_{0}a=1.0$. (b) $A_{0}a=5.0$.