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

TL;DR

This study explores a one-dimensional model with power-law correlated disorder and long-range hopping, revealing complex phase behavior, mobility edges, and employing machine learning for phase classification.

Contribution

It introduces a comprehensive phase diagram for correlated disorder and long-range hopping, combining spectral analysis with machine learning to identify phases and transitions.

Findings

Robust mobility edges identified across parameter space

Multiple regimes of localized, extended, and critical states

Machine learning reliably classifies spectral phases

Abstract

We investigate a one-dimensional tight-binding model in which onsite potentials $\{\varepsilon_i\}$ exhibit power-law spatial correlations (with exponent $\alpha$) and the hopping amplitudes decay as $t_{ij}\sim |i-j|^{-\beta}$. 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 $(\alpha,\beta)$ 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 $(\alpha,\beta)$ 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.