Floquet topological transition by unpolarized light

We study Floquet topological transition in irradiated graphene when the polarization of incident light changes randomly with time. We numerically confirm that the noise-averaged time-evolution operator approaches a steady value in the limit of exact Trotter decomposition of the whole period during which incident light has a different polarization at each interval of the decomposition. This steady limit is found to coincide with the time-evolution operator calculated from the noise-averaged Hamiltonian. We observe that at the six corners (Dirac $K$ point) of the hexagonal Brillouin zone of graphene random Gaussian noise strongly modifies the phase band structure induced by circularly polarized light, whereas in the zone center ($\mathrm{\Gamma }$ point) even a strong noise is not able to do the same. This can be understood by analyzing the deterministic noise-averaged Hamiltonian, which has a different Fourier structure as well as a lower number of symmetries compared to the noise-free one. In one-dimensional systems noise is found to renormalize only the drive amplitude.

Figure 1

Noise-averaged phase bands vs $T$ for the $\mathrm{\Gamma }$ point (left; at $\alpha =1.5$) and for the $K$ point (right; at $\alpha =2.0$) for various values of standard deviation $\sigma$. $N=1000$, the number of samples is 1000, and $\alpha =e{A}_0/c$.

Figure 2

Chern number of the noise-averaged lower Floquet band, showing the transitions through the $\mathrm{\Gamma }$ (left) and $K$ (right) points. Others parameters are the same as in Fig. 1.

Figure 3

Decrease of $D_N$ (top left panel) and $S_N$ (top right panel) with $N$ for the $\mathrm{\Gamma }$ point at $\alpha =1.5$ and $T=4.0$. The same is shown for the Dirac point in the lower left and right panels at $\alpha =2$ and $T=4.0$. The number of samples is 1000, and $\sigma$ is $\pi /10$ and $\pi /3$ for the thick blue and think red curves, respectively. The slope of the linear fit is mentioned in the insets.

Figure 4

Comparison of phase bands obtained by numerically disorder averaging the $U(T,0)$ operator (black solid line) and by using the ensemble-averaged $H\left(t\right)$ to calculate the $U(T,0)$ operator (red dashed line) for the $\mathrm{\Gamma }$ point (left panel) and for the $K$ point (right panel). The relevant parameters are the same as in Fig. 3.

Figure 5

Comparison of cosines of Eqs. (11) (black solid line) and (15) (red dashed line) for the $\mathrm{\Gamma }$ point (left panel) and of Eqs. (12) (black solid line) and (16) (red dashed line) for the $K$ point (right panel). All parameters are the same as before.

Figure 6

Decrease of $D_N$ (top left panel) and $S_N$ (top right panel) with $N$ for the $\mathrm{\Gamma }$ point at $\alpha =2.0$ and $T=60$. $\sigma$ for the blue solid, red dotted, and green dashed curves is $\pi /50,\pi /10$, and $\pi /3$ respectively. $N^{*}$ (for which $D_{N^{*}}$ fall below ${10}^{-4}$) vs $T$ is shown in the bottom left panel for $\sigma =\pi /3$. The slope of the linear fit in the log-log plot is 1.83 (inset). The bottom right panel shows the matching of the phase band from $\langle U\left(T\right)\rangle$ (black solid curve) and $U\left(T\right)$ calculated from $\langle H\left(t\right)\rangle$ (red dashed curve) for the $\mathrm{\Gamma }$ point at $\alpha =2.2$ and $\sigma =\pi /10$.

Figure 7

(a) Instantaneous energies vs $t/T$ for CP (top panel) and for unpolarized ($\sigma =\pi /8$; bottom panel) light at the $\mathrm{\Gamma }$ point for $\alpha =2.3$. (b) $cos\left[\mathrm{\Phi }\left(\frac{T}{3}\right)\right]$ vs $T$. The red curve is achieved by exact numerical averaging with 1000 samples, $N=10000$ for $\alpha =2.35$. (c) $cos\left[\mathrm{\Phi }\left(\frac{T}{3}\right)\right]$ vs $T$ for $\alpha =2.35$. (d) $cos\left[\mathrm{\Phi }\left(\frac{2T}{3}\right)\right]$ vs $T$ for $\alpha =2.28$. Phase bands in (c) and (d) are calculated using the ensemble-averaged Hamiltonian.

Figure 8

Left: Phase bands from the numerically averaged $U$ operator (lines) and from the averaged Hamiltonians (dots) for the 1D model [in Eq. (17)]. $A=1.5$, $\omega =1.0$, $r=3$. Right: The change in instantaneous energies with the insertion of noise.