Machine-learning-guided inverse design of Anderson localization in one-dimensional chains with multi-scale correlated disorder

Abstract

We study localization control in a one-dimensional tight-binding chain with diagonal disorder whose spatial correlations are governed by one or two exponential length scales. The main objective is not only to characterize localization, but to develop an interpretable machine-learning framework for inverse design: given a desired transport or localization behavior, the method identifies disorder profiles that realize it. We compare three disorder ensembles: uncorrelated disorder, single-scale exponentially correlated disorder, and a two-scale correlated ensemble with correlation function \(C(r)=a_1 e^{-r/\xi _1}+a_2 e^{-r/\xi _2}\), where \(a_1+a_2=1\). Exact diagonalization is used to compute central-window eigenstate observables, including the participation ratio normalized by system size, \(\textrm{PR}/N\), and the localization length normalized by system size, \(\ell _{\textrm{loc}}/N\). Even with a second, independently tunable correlation scale, we find no signature of a genuine localization-delocalization transition: dedicated finite-size scaling up to \(N=1024\) shows no size-independent crossing point, so the correlation-length and weight dependence remains a smooth, monotonic crossover throughout the exponential-correlation family rather than a critical phenomenon. A central outcome of the work is a machine-learning surrogate and inverse-design workflow. An ensemble ExtraTrees regressor learns the mapping from disorder parameters and system size to the normalized logarithmic observables, achieving \(R^2\approx 0.98\) on held-out data for the two targets (\(R^2\approx 0.980\) for the normalized participation ratio and \(R^2\approx 0.979\) for the normalized localization length; Sect. 3). The surrogate provides smooth interpolation of the high-dimensional parameter space, uncertainty estimates, SHAP-based feature attribution, and a fast inverse-design search for disorder profiles that maximize localization (the framework is equally applicable to a delocalization objective, though only the localization-maximizing search is reported here). The analysis shows that the scaled correlation lengths and the disorder-weight asymmetry are the dominant controls on transport, and that the strongest localization is realized by short-range-dominated disorder at the edge of the sampled correlation-length range, consistent with the general expectation that weakly correlated disorder localizes most strongly. This combination of exact diagonalization, interpretable machine learning, and inverse design establishes a practical framework for designing and controlling localization in low-dimensional quantum systems.