Interplay of Power-Law correlated Disorder and Long-Range Hopping in One Dimension: Mobility Edges, Criticality, and ML-Based Phase Identification

TL;DR

This work studies a one-dimensional tight-binding model where onsite potentials have power-law spatial correlations with exponent $α$ and hopping amplitudes decay as $|i-j|^{-β}$, aiming to map mobility edges and spectral criticality in a two-parameter phase diagram. The authors employ large-scale exact diagonalization and a suite of energy-resolved diagnostics — including level statistics, participation ratio, multifractal spectra, entanglement entropy, and the LDOS-based $\rho$-ratio — complemented by a finite-size scaling cost function to extract critical behavior. They introduce four phase-indicator functions $f_1$–$f_4$ to classify energy bins into localized, extended, resonant, or critical sectors, and validate these with a supervised autoencoder that learns latent representations aligned with the physics-driven classification. The results reveal robust mobility edges across substantial regions of $(α,β)$, with distinct spectral sectors and consistent cross-validation between traditional diagnostics and ML. This unified framework advances understanding of localization-delocalization transitions in long-range, correlated disordered 1D systems and provides a blueprint for applying ML to phase identification in quantum matter.

Abstract

We investigate a one-dimensional tight-binding model in which onsite potentials $\{\varepsilon_i\}$ exhibit power-law spatial correlations (with exponent $α$) and the hopping amplitudes decay as $t_{ij}\sim |i-j|^{-β}$. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the $(α,β)$ plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio $r$, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behaviour across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the $(α,β)$ parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.

Figures (10)

• Figure 1: Real-space onsite potentials $\varepsilon_j$ for a chain of length $N=200$, obtained by superposing Fourier modes with power-law amplitudes [Eq. \ref{['eq:eps_fourier_real']}]. (Left panel) $\alpha=0.2$: weakly correlated, rapidly varying (rough) disorder. (Right panel) $\alpha=2$: enhanced low-$k$ weight produces a much smoother, long-range correlated potential. These representative values illustrate the crossover from rough, short-range disorder at small $\alpha$ to smooth, long-range-correlated landscapes at large $\alpha$.
• Figure 2: Level–spacing distributions $P(s)$ and number variances $\Sigma^{2}(L)$ are shown for four corner points in the $(\alpha, \beta)$ parameter space at system size $N = 256$. Each panel compares the numerical spacing distribution with standard theoretical benchmarks: Poisson, Wigner–Dyson (WD), semi–Poisson, and Brody forms. The insets display $\Sigma^{2}(L)$, which further substantiates the evolution from localized to delocalized spectral statistics as functions of the disorder–correlation exponent $\alpha$ and the hopping–range exponent $\beta$. For each parameter set, the results are averaged over $200$ disorder realizations.
• Figure 3: Energy-resolved behavior of (left) average participation ratio $\langle \mathrm{PR} \rangle$ and (right) multifractal dimension $\langle \mathrm{D_q} \rangle$ for selected $(\alpha, \beta)$ pairs.
• Figure 4: Composite local diagnostics for representative parameter points. Each panel contains: Top-left: the normalized participation ratio $\mathrm{PR}/N$, plotted together with the single-particle entanglement entropy and $s_{\mathrm{proxy}}$, each resolved by energy. Top-right: the binned spectral ratio $r$ and the density-of-states ratio $\rho_{\mathrm{typ}}/\rho_{\mathrm{ave}}$, both highlighting spectral regions that deviate sharply from the expected localized or extended limits. Bottom row: three representative eigenstate intensities $|\psi(i)|^{2}$ taken from energies near the band center, mid-band, and band edge. Each intensity profile is colored according to the classifier label associated with its energy bin (color scheme consistent with Fig. \ref{['fig:classification_maps']}).
• Figure 5: Classification maps in the energy--parameter plane. Each panel shows the classifier output at $N=256$, averaged over $\sim 200$ disorder realizations. Color code: blue = localized, yellow = resonant, green = critical/multifractal, red = extended.