Effective delocalization in the one-dimensional Anderson model with stealthy disorder

TL;DR

This work analyzes Anderson localization in a 1D tight-binding chain with stealthy (spectrally gapped) disorder, showing that spectral constraints controlled by the stealthiness parameter $\chi$ can dramatically alter localization, producing effective delocalization in finite systems and allowing $\xi$ to scale as $\xi\sim W^{-2n}$ with arbitrarily large $n$ as $\chi$ increases. Using a perturbative expansion of the self-energy, the authors identify the leading nonzero contributions to $1/\xi$ and map a phase diagram in $(k,\chi)$ that delineates regimes where higher-order processes dominate backscattering. They validate the theory with large-scale numerical diagonalization and fractal-dimension analysis, observing consistent $\xi$ scaling ($\xi\sim W^{-2}$ for small $\chi$, $\xi\sim W^{-4}$ for larger $\chi$) and transitions that align with the predicted separatrices. The results generalize stealthy disorder to quantum tight-binding models and connect to optical transparency phenomena in 1D layered media, with implications for photonic and phononic wave systems and programmable quantum simulators. Nonperturbative effects near $\chi\approx 1/2$ remain for future study, but the work establishes stealthy hyperuniformity as a tunable route to modulate localization in disordered quantum systems.

Abstract

We study analytically and numerically the Anderson model in one dimension with "stealthy" disorder, defined as having a power spectrum that vanishes in a continuous band of wave numbers. Motivated by recent studies on the optical transparency properties of stealthy hyperuniform layered media, we compute the localization length using a perturbative expansion of the self-energy. We find that, for fixed energy and small but finite disorder strength $W$, there exists for any finite length system a range of stealthiness $χ$ for which the localization length exceeds the system size. This kind of "effective delocalization" is the result of the novel kind of correlated disorder that spans a continuous range of length scales, a defining characteristic of stealthy systems. Unlike uncorrelated disorder, for which the localization length $ξ$ scales as $W^{-2}$ to leading order for small W, the leading order terms in the perturbation expansion of $ξ$ for stealthy disordered systems vanish identically for a progressively large number of terms as $χ$ increases such that $ξ$ scales as $W^{-2n}$ with arbitrarily large $n$. Moreover, we support our analytical results with numerical simulations. Our results introduce stealthy disorder into quantum tight-binding models and show that enforcing a low-$k$ spectral gap markedly alters the scattering landscape, enabling localization lengths that exceed the system size at fixed disorder strength. Since this mechanism relies only on the spectral properties of the disorder, it carries over directly to photonic and phononic wave systems.

Figures (9)

• Figure 1: Graphical representation of Dyson equation, see Eq. \ref{['eq:DysonEq']} in the main text.
• Figure 2: Self-energy $\Sigma(k,E)$ contribution up to second order in $S(q)$: (left) first-order contribution $\Sigma^{(1)}(k,E)$, (center) second order contribution with crossing disorder lines $\Sigma^{(2)}_{\mathrm{cross}}(k,E)$, (right) second order contribution without crossing disorder lines $\Sigma^{(2)}_{\mathrm{non-cross}}(k,E)$.
• Figure 3: Phase diagram showing the dependence of the localization length $\xi(k,k_0)$ on the disorder strength $W$ at the leading non-vanishing order in perturbation theory. We also indicate the expression for the separatrices between different regions, whose general expression is given in Eq. \ref{['eq:separatixes']}. The red dotted line indicates the energy for which we will present numerical results.
• Figure 4: Localization length as a function of $\chi$ for different values of $W$. The vertical lines are positioned where the perturbation theory predicts a change in the first non-vanishing contribution (corresponding to the first two red dots from the left in Fig. \ref{['fig:phasediagr']}). Further transitions require system sizes larger than the numerically accessible one.
• Figure 5: Localization length as a function of $\log W$ for different values of $\chi$, as shown in the legend, obtained for eigenstates at energy $E = -1.8$, corresponding to the horizontal red line in Fig. \ref{['fig:phasediagr']}. For $\chi=0.1$, the perturbation theory predicts that $\xi \sim W^{-2}$, as confirmed by the numerical result (see red dashed line, which is a guide for the eye). For $\chi=0.2, \, 0.25,\, 0.3$ the perturbation theory predicts $\xi \sim W^{-4}$, as confirmed by the numerical result.