Universal Fluctuations of Floquet Topological Invariants at Low Frequencies

We study the low-frequency dynamics of periodically driven one-dimensional systems hosting Floquet topological phases. We show, both analytically and numerically, in the low frequency limit $\mathrm{\Omega }\to 0$, the topological invariants of a chirally symmetric driven system exhibit universal fluctuations. While the topological invariants in this limit nearly vanish on average over a small range of frequencies, we find that they follow a universal Gaussian distribution with a width that scales as $1/\sqrt{\mathrm{\Omega }}$. We explain this scaling based on a diffusive structure of the winding numbers of the Floquet-Bloch evolution operator at low frequency. We also find that the maximum quasienergy gap remains finite and scales as ${\mathrm{\Omega }}^2$. Thus, we argue that the adiabatic limit of a Floquet topological insulator is highly structured, with universal fluctuations persisting down to very low frequencies.

Figure 1

The topological invariant ${\nu }^{♭}$ is the winding number of the complex function $D_k$, $k\in [-\pi ,\pi )$ (left), computed from the crossing points on the real axis. The inversion symmetry of the SSH model yields a reflection symmetric contour around the real axis, the solid (dashed) line designating the portion corresponding to $k\in [-\pi ,0]$ ($[0,\pi ]$). As the frequency is lowered, new crossing points are emitted from $D_0$ when the contour twists (top right) and absorbed into $D_{\pm \pi }$ when it untwists (bottom right).

Figure 2

(a) The quasienergy gap (top panels) and the topological invariant (bottom panels) at the Floquet zone edge as a function of the frequency. The two-step drive protocols are critical (${\delta }_1/w=-0.5$, ${\delta }_2/w=0$, left), asymmetric (${\delta }_1/w=-0.5$, ${\delta }_2/w=0.495$, center), and trivial (${\delta }_1/w=-0.5$, ${\delta }_2/w=-0.3$, right). In all cases $p_1=p_2=0.5$. The (orange) markers indicate analytically calculated gap closings at the symmetric points $k_s=0$ (square), and $k_s=\pm \pi$ (triangle) and nonsymmetric points $k_c^{♭}\ne k_s$ (circle). (b) Probability distribution of ${\nu }^{♭}$ for the critical protocol in (a) and frequency range $\mathrm{\Omega }/w\in (0.5,1)\times {10}^{-3}$. The solid line is a Gaussian fit. (c) Statistics of crossing points and winding number for the three drive protocols in (a). From top to bottom in each panel: the total number of crossing points $N^{♭}$, the positively oriented crossing points $N_{+}^{♭}$, the ratio ${\alpha }_{+}^{♭}=N_{+}^{♭}/N^{♭}$, and the winding number $|{\nu }^{♭}|$ are calculated numerically at 65 000 frequencies. At this resolution, fluctuations in the winding numbers render their graphs random. The vertical dashed line in the center panel marks twice the value of the asymmetry parameter. The insets show the probability distribution $P({\nu }^{♭})$ over the shaded range as in (b). The solid line shows $\sqrt{{\mathcal{D}}^{♭}/\mathrm{\Omega }}$ with ${\mathcal{D}}^{♭}=||{\delta }_1|-|{\delta }_2||$. (d) Standard deviation ${\sigma }^{♭}(\mathrm{\Omega })$ for various two-step drive protocols found by a Gaussian fit to $P({\nu }^{♭})$. The horizontal (vertical) line at each point indicates the range of frequencies (fitting error). The legend shows the values (${\delta }_1/w$, ${\delta }_2/w$). The solid line is ${\sigma }^{♭}=\sqrt{{\mathcal{D}}^{♭}/\mathrm{\Omega }}$.

Figure 3

(a) Winding number $|{\nu }^{♭}|$ and the ratio ${\alpha }_{+}^{♭}$ in a multistep protocol as a function of the number of steps, $n$. The two frequencies are ${\mathrm{\Omega }}_1/w=1.09\times {10}^{-3}$ and ${\mathrm{\Omega }}_2/w=4.15\times {10}^{-3}$, with the mean value $\overline{\delta }/w=-0.4$ and amplitude $A/w=0.1$. (b) The root-mean-square $\delta {\nu }^{♭}$ averaged over $n$ as a function frequency. The dashed line corresponds to $\sqrt{{\mathcal{D}}^{♭}/\mathrm{\Omega }}$, fitted with ${\mathcal{D}}^{♭}/w=0.35$.