Photonic Floquet topological insulators with fluctuations and disorders

We study how the feature of chiral edge transport changes in photonic Floquet topological insulators when the driving is not perfectly periodic and the system is under the effect of disorders. Assuming that the aperiodicity is caused by fluctuations in the driving, we find that the fluctuations with finite correlation time can cause leakage of chiral edge excitations into the bulk of the system. For fluctuations with short correlation time, an effective master equation that captures the leakage of edge excitations is derived. When the correlation time of the fluctuations is longer than the typical time of the system, the lifetime of chiral edge excitations increases as a power of the correlation time. The lifetime of chiral edge excitations is insensitive to disorders with short correlation time, though when the disorder is strong enough, the lifetime of pointlike edge excitations can be several orders of magnitude longer than the lifetime of chiral edge excitations. The loss of photons into environment cannot stop the leakage of photons into the bulk of the system, but it eventually determines the lifetime of the excitation in the system. On the contrary, the chiral edge transport is robust against the quasirandom fluctuations.

Figure 1

Lieb lattice with nearest-neighbor (NN) and next-nearest-neighbor (NNN) hoppings. Three sites in a unit shell are labeled by A, B, and C, respectively.

Figure 2

Three bands as a function of $\beta \mathrm{\Delta }/\alpha$. Panels (a), (b), and (c) are for three typical values of $\lambda$ with $g=1$ and $\mathrm{\Delta }=15g$. The number in the band is the Chern number of that band. The dashed and thin lines that indicate the topological phase transition points are given by $\pm 2\sqrt{2}\lambda$ and $\pm \sqrt{2}g$, respectively. The results are obtained by numerically diagonalizing the Floquet unitary operator. The results agree well with the results obtained by the effective time-independent Hamiltonian in Eq. (6) (not shown here).

Figure 3

Panels (a), (b), and (c) show the disorder averaged Bott index of the three quasienergy bands as a function of $\beta \mathrm{\Delta }/\alpha$ and disorder strength $\sigma$. Panels (d), (e), and (f) show the disorder averaged normalized participation ratio of the three quasienergy bands as a function of $\beta \mathrm{\Delta }/\alpha$ and disorder strength $\sigma$. Panels (a), (b), and (c) are for the Bott index of lowest, middle, and highest band, respectively. Panels (d), (e), and (f) are for the normalized participation ratio of lowest, middle, and highest band, respectively. We set $g=1, \mathrm{\Delta }=15g$, and $\lambda =0.7$ and consider square wave drive. The results are obtained numerically for a lattice of size $3\times L_x\times L_y=3\times 12\times 12$ and are an average over 100 random choices of $H_{\mathrm{dis}}$ (will be referred to as random realizations in later discussion).

Figure 4

Panels (a) and (c) show the disorder averaged Bott index and disorder averaged normalized participation ratio of two quasienergy bands as functions of disorder strength $\sigma$, respectively. Panel (b) shows the disorder averaged quasienergy gap between lowest and middle band. Panel (d) shows the scaling of disorder averaged normalized participation ratio of two quasienergy bands for two different disorder strengths marked in (c). Solid and hollow markers in (a), (c), and (d) are for the lowest and middle band, respectively. The dashed black line in (d) indicates $R\sim 30/N_{\mathrm{total}}$. We set $g=1, \mathrm{\Delta }=15g, \lambda =0.7$, and $\beta =0$ and consider square wave drive. The results are obtained numerically on a lattice of size $3\times L\times L$ with $L=6,8,12,18,20$ and averaged over 300 disorder realizations for each $L$ [$L=18$ only shows in (d)].

Figure 5

Evolution of normalized edge photon density $n_{\mathrm{edge}}\left(t\right)/[n_{\mathrm{edge}}\left(\infty \right)+n_{\mathrm{bulk}}\left(\infty \right)]$. Panel (a) for fluctuations with correlation time ${\tau }_s=\overline{T}/2$ and two different strengths $\overline{\delta }=0.08,0.15$. Panel (b) for fluctuations with fixed strength $\overline{\delta }=0.1$ and two different correlation times ${\tau }_s=\overline{T},110\overline{T}$. The lines with empty and filled markers are for two different sizes of lattice $3\times L\times L$ with $L=9$ and 5, respectively. The lines with up and down triangles are for numerical results. We have averaged over 50 realizations for fluctuation. The lines with circles and squares are for exponential fitting. The other parameters chosen are $g=1, \mathrm{\Delta }=15g, \lambda =0.7, \beta =0, \omega =-\lambda \alpha , A=1, t_0=3\sqrt{2}\nu$, and $\nu =5\sqrt{2}$. The lines with star and diamond markers in (a) are obtained for comparison by solving Eq. (28) with $\rho \left(2t_0\right)=|\varphi \left(2t_0\right)\rangle \langle \varphi \left(2t_0\right)|$ and $L=5$. Here $\left|\varphi \right(2t_0)\rangle$ is the single particle wave function given by $\langle vac|c_i|\varphi \left(2t_0\right)\rangle =\langle c_i\left(2t_0\right)\rangle$. Star and diamond is for $\overline{\delta }=0.08,0.15$, respectively.

Figure 6

Lifetime $\gamma$ of edge states as a function of driving parameters. Panel (a) for fluctuations with fixed correlation time ${\tau }_s=\overline{T}/2$ and different strengths $\overline{\delta }$. The parameter $\beta \mathrm{\Delta }/\alpha$ and lattice size $3\times L\times L$ take different values for different lines. The gray line with circle denotes $1/({\overline{\delta }}^2{T}_0/2)$, where $T_0=2\pi /\mathrm{\Delta }$. Panel (c) is the same as (a), except for the fluctuation strength, $\overline{\delta }=0.1$. The gray lines with circles denote the power law fitting. Panel (b) shows the lifetime as a function of $\beta \mathrm{\Delta }/\alpha$ for fluctuations with correlation time ${\tau }_s=\overline{T}/2$ and three different strength $\overline{\delta }$. Panel (d) shows the lifetime as a function of $\beta \mathrm{\Delta }/\alpha$ for fluctuation with fixed strength $\overline{\delta }=0.1$ and three different correlation time ${\tau }_s$. The other parameters chosen are $g=1, \mathrm{\Delta }=15g, \lambda =0.7, \omega =-\lambda \alpha , A=1, t_0=3\sqrt{2}\nu$, and $\nu =5\sqrt{2}$ (the results are similar provided $\omega$ lies in the band gap). We average over 100 fluctuation realizations to obtain the lifetime $\gamma$ for each set of parameters above.

Figure 7

Evolution of normalized edge photon density $n_{\mathrm{edge}}\left(t\right)/[n_{\mathrm{edge}}\left(\infty \right)+n_{\mathrm{bulk}}\left(\infty \right)]$ for quasirandom fluctuations with two different strengths $\overline{\delta }=0.1,0.15$. The lines with up and down triangles are for numerical results with Hamiltonian Eq. (1). The lines with circle and square are for the results with effective Hamiltonian Eq. (19). The results have been averaged over 200 randomly chosen irrational numbers $r$. The other parameters chosen are $g=1, \mathrm{\Delta }=15g, \lambda =0.7, \beta =0, \omega =-\lambda \alpha , A=1, t_0=3\sqrt{2}\nu , \nu =5\sqrt{2}$ and the size of lattice $3\times L\times L$ with $L=10$. (We also check the result for the size of lattice $3\times L\times L$ with $L=5,6,7,8,9$ and do not find the decay behavior of normalized edge photon density.)

Figure 8

(a) Time average normalized edge photon density for a system with periodic drive as a function of disorder strength $\sigma /\alpha$. The shaded area denotes the standard deviation for 200 different realizations of disorder (i.e., 200 random choices of $H_{\mathrm{dis}}$). Lines with different markers are for different size $3\times L\times L$ of lattice with $L=6,8$, and 14. (b) The evolution of normalized edge photon density for a system with periodic drive and 10 different realizations of disorder. $I$ and $II$ are for disorder strengths $\sigma /\alpha =0.5$ and 5, respectively. The results are obtained with lattice size $3\times 14\times 14$. (c) The lifetime $\gamma$ of edge states as a function of disorder strength. The shaded area denotes the standard deviation for 50 different realizations of disorder. Different lines denote different fluctuation strengths and size of lattice. The correlation times of fluctuations are fixed ${\tau }_s=\overline{T}/2$. Panel (d) is the same as (c), except the fluctuation strengths are fixed to $\overline{\delta }=0.1$. The other parameters chosen are $g=1, \mathrm{\Delta }=15g, \lambda =0.7, \beta =0, A=1, t_0=3\sqrt{2}\nu , \nu =5\sqrt{2}$, and $\omega =(E_{N/3}+E_{N/3+1})/2$ with $N$ the total number of lattice sites and $E_i$ the $i\mathrm{th}$ quasienergy level (calculated numerically with periodic boundary condition). $\omega$ lies in the quasienergy gap between lowest and middle bulk band for weak disorder; the quasienergy gap gradually closes with the increase of disorder strength. Before the gap closes, all choices for $\omega$ that lie in the band gap would lead to a similar result. When the gap closes, the results remain unchanged for any $\omega$.

Figure 9

Evolution of participation ratio for three different disorder strengths $\sigma /\alpha$. The solid lines are for lattices of size $3\times 9\times 9$ and the shaded areas denote the standard deviation for 30 realizations of disorder. The dashed lines are for lattices of size $3\times 5\times 5$. The fluctuation strength $\overline{\delta }=0.1$ and correlation time ${\tau }_s=\overline{T}$ is used. The other parameters are the same as in Fig. 8.

Figure 10

(a) Normalized density for the steady edge photon as a function of fluctuation strength $\overline{\delta }$. The correlation time of fluctuation ${\tau }_s=\overline{T}/2$. The other parameters chosen are $g=1, \mathrm{\Delta }=15g, \lambda =0.7, \beta =0, \omega =-\lambda \alpha , A=1, \sigma =0$, and $\kappa =0.001$. Triangles are for lattice of size $3\times 5\times 5$ and squares are for lattice of size $3\times 9\times 9$. We have averaged over 100 fluctuations to obtain the steady photon density with specific system parameters. Panels (b) and (c) show the spatial distribution of normalized steady edge photon density for two different fluctuation strengths $\overline{\delta }=0.05$ and 0.15, respectively.