Critical State of the Anderson Transition: Between a Metal and an Insulator

Using a three-frequency one-dimensional kicked rotor experimentally realized with a cold atomic gas, we study the transport properties at the critical point of the metal-insulator Anderson transition. We accurately measure the time evolution of an initially localized wave packet and show that it displays at the critical point a scaling invariance characteristic of this second-order phase transition. The shape of the momentum distribution at the critical point is found to be in excellent agreement with the analytical form deduced from the self-consistent theory of localization.

Figure 1

First row: Measured AIGF of the quasiperiodic atomic kicked rotor at different times $t$ in the (left to right) localized $(K,\epsilon )=(4.5,0.1)$, critical $(K=K_c,\epsilon )=(6.3,0.38)$ and diffusive $(K,\epsilon )=(9.0,0.8)$ regimes. Second row: Appropriate rescalings of the momentum by $t^0$ (localized), $t^{1/3}$ (critical), or $t^{1/2}$ (diffusive), bring the curves at different times into coincidence (the vertical scales are also rescaled in order to preserve normalization). The shapes are different in the three regimes: exponential localization, Eq. (1), in the localized regime, Gaussian shape, Eq. (2), in the diffusive regime, and the new “Airy shape,” Eq. (6), at the critical point. Parameters are $\mathrm{k̵}=2.89$, ${\omega }_2=2\pi \sqrt{7}$, ${\omega }_3=2\pi \sqrt{17}$. Time is measured in units of number of kicks and momentum in units of $2\hslash k_L$.

Figure 2

Numerically simulated time evolution of an initially localized momentum distribution (log scale), for the quasiperiodic kicked rotor at the critical point of the Anderson transition. The spreading follows an anomalous diffusion, with $⟨p^2(t)⟩\propto t^{2/3}$, and the shape is identical at all times, being neither exponential (as it is in the localized regime), nor Gaussian (as it is in the diffusive regime). The analytic prediction, Eq. (6), is shown as the red thick curve at $1\times {10}^6$ kicks. The agreement is excellent, without any adjustable parameter. Parameters are those of Fig. 1. Time is measured in millions of kicks and momentum in units of two recoil momenta $2\hslash k_L$.

Figure 3

(a) Experimental data for the rescaled critical AIGF (see Fig. 1) averaged over time (black circles with error bars) and a fit given by Eq. (6), with $\rho$ as the only fitting parameter. The agreement is clearly excellent. The residual does not significantly differ from zero [panel (b)]. Fits by an exponentially localized (c) or a Gaussian (d) distribution show significant deviations.