From percolation transition to Anderson localization in one-dimensional speckle potentials

TL;DR

This work addresses how a classical percolation transition in a 1D red-speckle potential connects to quantum Anderson localization as one moves away from the classical limit. It develops a semi-classical framework that links the algebraic divergence of the classical cluster size to a smooth, non-analytic growth of the localization length, revealing a bimodal transmission regime caused by speckle correlations and non-Gaussian statistics. Transfer-matrix numerics corroborate the theory, showing the breakdown of the conventional DMPK description in the semi-classical regime and the eventual restoration of universal localization in the deep quantum regime. The findings highlight non-universal effects due to correlated disorder, with implications for cold-atom experiments and the broader understanding of disorder-driven transport across classical-to-quantum crossovers.

Abstract

Classical particles in random potentials typically experience a percolation phase transition, being trapped in clusters of mean size $χ$ that diverges algebraically at a percolation threshold. In contrast, quantum transport in random potentials is controlled by the Anderson localization length, which shows no distinct feature at this classical critical point. Here, we present a comprehensive theoretical analysis of the semi-classical crossover between these two regimes by studying particle propagation in a one-dimensional, red speckle potential, which hosts a percolation transition at its upper bound. As the system deviates from the classical limit, we find that the algebraic divergence of $χ$ continuously connects to a smooth yet non-analytic increase of the localization length. We characterize this behavior both numerically and theoretically using a semi-classical approach. In this crossover regime, the correlated and non-Gaussian nature of the speckle potential becomes essential, causing the standard DMPK description for uncorrelated disorder to break down. Instead, we predict the emergence of a bimodal transmission distribution, a behavior normally absent in one dimension, which we capture within our semi-classical analysis. Deep in the quantum regime, the DMPK framework is recovered and the universal features of Anderson localization reappear.

Figures (6)

• Figure 1: (a) The model: we consider the transmission of a quantum particle of energy $E\sim V_0\gg E_\sigma$ through a 1D red-detuned speckle potential lying between $x=0$ and $x=L$. The potential $V(x)<V_0$ is bounded from above and has a correlation length $\sigma$. (b) For a large enough system size, a classical particle with energy $E<V_0$ always ends trapped within a potential valley (cluster). These clusters correspond to regions of space where $E-V(x)>0$, and have a characteristic size $\chi$. Near $E=V_0$, the speckle potential looks like a succession of inverted harmonic bumps of mean size $\sigma$.
• Figure 2: Localization length $\xi \equiv -L / \langle \ln T \rangle$, numerically computed from the average logarithmic transmission using the transfer-matrix method, as a function of $1/\hbar_\text{eff}$ for several energies below the percolation threshold (discrete points). The plot shows that $\xi \to 0$ as $\hbar_\text{eff} \to 0$, i.e., does not converge to the classical result (\ref{['eq:chiclassical']}). The solid curves represent the semi-classical prediction (\ref{['eq:xim1_SC']}), and the dotted lines the asymptotic formula (\ref{['Eq:SC_asymptotics']}). In the transfer-matrix simulations, $\xi$ is obtained from linear regressions of $\langle \ln T \rangle$ computed for 20 system sizes between $L=5 \sigma$ and $L=100\sigma$, each averaged over $10^4$ disorder realizations.
• Figure 3: (a) Average transmission $\langle T\rangle$ as a function of system size, computed numerically from the transfer-matrix method (black points) for $\hbar_\text{eff}=0.01$, for energies near the classical percolation threshold. Solid lines are fits to $\exp(-L/\chi)$. (b) Localization length $\chi(E)$ as a function of energy for several values of $\hbar_\text{eff}$, deduced from the transfer-matrix simulations (colored points). The solid curves are the theoretical prediction (\ref{['eq:chitheory']}) at finite $\hbar_{\text{eff}}$. The long-dashed curve is the classical limit (\ref{['eq:chiclassical']}), and the dashed and dotted curves the asymptotic formulas (\ref{['eq:chi_asymptote1']}) and (\ref{['eq:chi_asymptotics2']}), respectively, shown for $\hbar_\text{eff}=0.03$. Here $\chi$ is obtained from linear regressions of $\langle T \rangle$ computed for 20 system sizes between $L=5 \sigma$ and $L=100\sigma$, each averaged over $10^4$ disorder realizations.
• Figure 4: Transmission distribution $P(T)$ below, at and above the percolation threshold, computed numerically in the semi-classical regime with $\hbar_\text{eff}=0.005$. The simulations are performed for a system size $L=100\sigma$ and involve binning over $10^5$ disorder realizations. The dashed curves show the quasi-ballistic formula (\ref{['eq:qb']}) in the two cases $E=0$ and $E> V_0$.
• Figure 5: Transmission distribution $P(T)$ below the percolation threshold ($E-V_0=-0.005V_0$), in the semi-classical regime with $\hbar_\text{eff}=0.005$. A comparison between the numerical results and the DMPK prediction (\ref{['eq:abrikosov']}), using either $\xi$ or $\chi/4$ as the localization length (dotted and dashed curves), highlights the inadequacy of this approach. In contrast, the comparison with the bimodal law (\ref{['eq:bimodal']}) (solid green curve) shows excellent agreement. For reference, we also display $P(T)$ in the deep quantum regime ($\hbar_\text{eff}=1$), for which the DMPK equation becomes accurate again. All simulations are performed for a system size $L=100\sigma$ and involve binning over $10^5$ disorder realizations.