Dissipative Floquet topological systems

Motivated by recent pump-probe spectroscopies, we study the effect of phonon dissipation and potential cooling on the nonequilibrium distribution function in a Floquet topological state. To this end, we apply a Floquet kinetic equation approach to study two-dimensional Dirac fermions irradiated by a circularly polarized laser, a system which is predicted to be in a laser-induced quantum Hall state. We find that the initial electron distribution shows an anisotropy with momentum-dependent spin textures whose properties are controlled by the switching-on protocol of the laser. The phonons then smoothen this out, leading to a nontrivial isotropic nonequilibrium distribution which has no memory of the initial state and initial switch-on protocol, and yet is distinct from a thermal state. An analytical expression for the distribution at the Dirac point is obtained that is relevant for observing quantized transport.

Figure 1

Contour plots for the time-averaged spin density $P_z(k_x,k_y)$. Left panel: without phonons and for a quench. Right panel: at steady state with phonons. $A_0/\Omega =0.5,{\lambda }^2{D}_{\mathrm{ph}}=0.1\Omega ,T=0.01\Omega ,\Omega =1$.

Figure 2

Spin density $P_z(k_x,k_y=0)$ for $A_0/\Omega =0.5,\Omega =1.0,{\lambda }^2{D}_{\mathrm{ph}}$=$0.1\Omega$ for three different cases: for the quench with no phonons, steady state with phonons at temperature $T=0.01\Omega$, and $T=\Omega$.

Figure 3

Intensity plot of the spin-averaged ARPES spectrum $i{G}_{\mathrm{tot}}^{<}(k,\omega )/2$ at $k_y={10}^{-4}$ for the quench with no phonons (upper panels) and at steady state with phonons at temperature $T=0.01\Omega ,1\Omega$ (middle and lower panels). $A_0/\Omega =0.5,{\lambda }^2{D}_{\mathrm{ph}}=0.1\Omega ,\Omega =1.0$.

Figure 4

$i{G}_{\mathrm{tot}}^{<}(k,\omega )$ for the quench with no phonons (upper panels) and at steady state with phonons at temperature $T=0.01\Omega ,1\Omega$ (middle and lower panels) for $k_y={10}^{-4}$ and $k_x=0.0,0.25,0.5$. $A_0/\Omega =0.5,{\lambda }^2{D}_{\mathrm{ph}}=0.1\Omega ,\Omega =1.0$. Normalization such that the peak heights equal the prefactor of the $\delta$ functions.

Figure 5

Time evolution of ${\rho }_{k,\alpha \alpha =\pm }$ from an initial state corresponding to a quench for $k_y=0$ and $k_x={10}^{-4},0.25,0.5$. $A_0/\Omega =0.5,{\lambda }^2{D}_{\mathrm{ph}}=0.1\Omega ,T=0.01\Omega ,\Omega =1.0$.