Coherent forward scattering as a signature of Anderson metal-insulator transitions

We show that the coherent forward scattering (CFS) interference peak amplitude sharply jumps from zero to a finite value upon crossing a metal-insulator transition. Extensive numerical simulations reveal that the CFS peak contrast obeys the one-parameter scaling hypothesis and gives access to the critical exponents of the transition. We also discover that the critical CFS peak directly controls the spectral compressibility at the transition where eigenfunctions are multifractal, and we demonstrate the universality of this property with respect to various types of disorder.

Figure 1

Left: Long-time limit of the disorder-averaged momentum distribution $n(\mathit{k},t)$ obtained in the insulating (top, $W=20J$), critical (middle, $W=16.5J$), and metallic (bottom, $W=12J$) phases of the cubic 3D Anderson model (lattice constant $a$, tunneling rate $J$) when an initial plane wave $\left|{\mathit{k}}_0\right.〉$ is numerically propagated up to time $t=8000\hslash /J$. The energy is set to $E=J$ and ${\mathit{k}}_0=\pi /\left(3a\right){\widehat{\mathit{e}}}_x$. The CFS and CBS peaks are visible at ${\mathit{k}}_0$ and $-{\mathit{k}}_0$, respectively. The solid curve is a cut along $k_x$. Right: Time evolution of the normalized CFS contrast $\mathrm{\Lambda }$ in the three phases.

Figure 2

(a) Normalized contrast $\mathrm{\Lambda }$ as a function of disorder strength $W$ for different propagation times (at energy $E=J$). As time grows, $\mathrm{\Lambda }$ converges toward a step function. All curves cross at $W\approx 16.5J$, marking the critical point $W_c$ of the Anderson MIT. The corresponding contrast ${\mathrm{\Lambda }}_c\approx 0.34$ is time independent. (b) Universality of the CFS contrast at the critical point. Uncorrelated disorder (red points): On-site box distribution (P1) and on-site Gaussian distribution (P2). Correlated disorder (blue points): On-site Gaussian distribution with Gaussian correlation (P3), blue-detuned speckle (P4), and red-detuned speckle (P5). Error bars are discussed in the main text.

Figure 3

Time dependence of the normalized contrast $\mathrm{\Lambda }$. (a) In the metallic phase, the numerical points are well fitted by $1/\sqrt{t}$ (solid curve), in agreement with the theoretical prediction Eq. (1). (b) In the insulating phase, the theoretical prediction Eq. (4), obtained in the limit $t\gg {\tau }_H$, is confirmed with $\alpha =3.72$ and $\eta =0.129$ (solid curve). Here, the Heisenberg time ${\tau }_H$ has been computed independently by using the dynamics of the CBS peak width [20].

Figure 4

CFS scaling function $\beta \left(\mathrm{\Lambda }\right)$ obtained by compiling data for $\mathrm{\Lambda }$. Each color corresponds to a given value of $W$ and, for each $W$, each point corresponds to a different propagation time $t$. All points fall on a single curve. The function changes sign at the critical point where $\mathrm{\Lambda }={\mathrm{\Lambda }}_c\simeq 0.34$. Equations (1) and (4) give the asymptotic behaviors $\beta (\mathrm{\Lambda }\to 0)=-\frac{3}{2}$ in the metallic phase and $\beta \approx 3(1-\mathrm{\Lambda })$ in the insulating phase.