Coherent forward scattering peak and multifractality

It has recently been shown that interference effects in disordered systems give rise to two nontrivial structures: the coherent backscattering (CBS) peak, a well-known signature of interference effects in the presence of disorder, and the coherent forward scattering (CFS) peak, which emerges when Anderson localization sets in. We study here the CFS effect in the presence of quantum multifractality, a fundamental property of several systems, such as the Anderson model at the metal-insulator transition. We focus on Floquet systems, and find that the CFS peak shape and its peak height dynamics are generically controlled by the multifractal dimensions $D_1$ and $D_2$, and by the spectral form factor. We check our results using a one-dimensional Floquet system whose states have multifractal properties controlled by a single parameter. Our predictions are fully confirmed by numerical simulations and analytic perturbation expansions on this model. Our results, which we believe to be generic, provide an original and direct way to detect and characterize multifractality in experimental systems.

Figure 1

Contrast of the CFS peak, Eq. (2), with scaled time $\tau =2\pi t/N$, for the RS model with $N=4096$ and $a=0.322$. Left projection: Height of the CFS peak. The theoretical prediction (red solid line) [see text below Eq. (17)] matches perfectly the numerical data (black solid line). Right projection: Shape of the CFS peak at $\tau =150$. The black solid line represents the numerical data and the blue solid line is Eq. (9) rescaled (see text).

Figure 2

Contrast of the CFS peak at infinite time. (a) Finite-size scaling of ${\mathrm{\Lambda }}_{\infty }$ for different values of $a$. The dashed lines correspond to the nonlinear fit $1-{\mathrm{\Lambda }}_{\infty }=\alpha N^{-D_2^{{\mathrm{\Lambda }}_{\infty }}}$ with parameters $\alpha$ and $D_2^{{\mathrm{\Lambda }}_{\infty }}$. (b) Comparison between the values of $D_2$ extracted from a finite-size scaling of ${\mathrm{\Lambda }}_{\infty }$ in (a) and from the inverse participation ratio (IPR) defined by the right-hand side of Eq. (8). The two coincide very well. When not visible, error bars are smaller than the symbol size. (c) CFS peak contrast for the values of $a$ used in (a), with $N=4096$. Solid lines are numerical data, black dashed lines are Eq. (9) rescaled to minimize the difference with numerics on a spatial range $\mathrm{\Delta }x=0.5$ around $x=x_0$ (excluding the point at $x_0$), and the black dotted line is Eq. (10). (d) Zoom-in of (c) around $x_0$ and smoothed over $\delta x=0.007$.

Figure 3

Dynamics of the CFS peak at $x=x_0$. (a) Finite-size scaling of the difference between the numerically computed contrast and $K_{\text{reg}}$, Eq. (14), for $a=0.322$ and different $\tau =2\pi t/N$. Dashed lines correspond to the nonlinear fit $|\mathrm{\Lambda }-K_{\text{reg}}|=\alpha N^{-D_2^{\text{IPR}}}$, where $D_2^{\text{IPR}}$ is independently determined from a finite-size scaling of the IPR and $\alpha$ is the fitting parameter. (b) Dynamics of the contrast for different $N$. Solid lines are numerical data smoothed over $\mathrm{\Delta }\tau =0.3$. $C_0=0.55$ in $f_{★}\left(\tau \right)$ (see text) is numerically extracted from $\widehat{C}\left(\omega \right)$ (see SM [45]). ${\mathrm{\Lambda }}_{\infty }$ for large $N$ is extrapolated from Fig. 2. The black dashed line is $K_{\text{reg}}$ given by Eq. (14). (c) Finite-size scaling of the difference between the contrast at $\tau =0.5$ and the level compressibility ${\chi }_{\text{fit}}$ for different values of $a$. ${\chi }_{\text{fit}}$ is extracted from a nonlinear fit $\mathrm{\Lambda }={\chi }_{\text{fit}}+\alpha N^{-D_2^{\text{IPR}}}$. (d) $D_1=1-{\chi }_{\text{fit}}$ vs $D_1$ obtained from a finite-size scaling of the wave-function moments. When not visible, error bars are smaller than the symbol size.