Researchers have predicted an unusual ground state in a one-dimensional disordered lattice: its wave function keeps a substantial peak at zero momentum while also having algebraic tails that carry a finite share of the normalization. The authors describe this combination as semi-localization in momentum space. The state is therefore neither fully concentrated in one mode nor spread evenly across all modes. The result comes from a theoretical single-particle model, with the main calculation carried out in the weak-disorder regime.
A peak with a tail
The model combines algebraically long-range hopping with uncorrelated onsite disorder. Each site energy is drawn uniformly between -W and W, with W setting the disorder amplitude. The calculation starts from the clean ground state at zero momentum and asks how a weak random potential changes it; the perturbative limit is the regime in which W is much smaller than 1.
Two different statistical descriptions are involved. In momentum space, the predicted tail falls algebraically away from the zero-momentum peak and retains finite normalization weight. Across those nonzero-momentum components, the normalized intensities are treated as exponentially distributed random variables. The algebraic decay describes where the weight sits, while the exponential law describes the spread of intensity values within the tail.
The split shows up in the scaling
To quantify the pattern as the lattice grows, the analysis uses wave-function moments and generalized fractal dimensions, which summarize how those moments scale with system size. In the large-system perturbative prediction, the generalized dimension is 1 for moment orders between 0 and 1, and 0 for moment orders above 1. In plain terms, changing the way the calculation weights the wave function changes the answer sharply at moment order 1. The authors interpret the split as the signature of a semi-localized state: probes emphasizing larger components behave as localized, while lower-order probes retain ergodic-like scaling. The information dimension is small and scales quadratically with weak disorder.
Where the calculation stops
The calculation also sets a boundary on its own perturbative description. For hopping exponents at or above 3/2, the normalization factor no longer remains finite as the system becomes indefinitely large. The authors estimate a critical disorder, written Wc(a), by taking the condensate-depletion correction to be of order one. On that estimate, the perturbatively stable semi-fractal phase shrinks continuously to zero as a approaches 3/2.
What happens beyond Wc(a) remains explicitly unresolved. The paper conjectures a post-transition scaling and a formally fractal, non-ergodic extended phase for hopping exponents below 2 at weak disorder, but presents that proposal as conjectural rather than established. Here y is the proposed scaling exponent and a is the long-range hopping exponent. The perturbative calculation is controlled only while disorder and condensate depletion remain small, so the stronger-disorder regime cannot be treated as settled by these results.
A narrow numerical check
The numerical check is narrower than the analytic claim. At disorder amplitude W=0.1, first-order perturbation theory reproduces the exact-diagonalization mean normalized tail intensities to within a few percent. A supplementary comparison used a=1.48, a system of 20,000 sites, and 6,000 disorder realizations. That supports the reported tail estimate at a selected finite-size setting, but it does not resolve behavior across all parameters.
A theoretical result, for now
This makes the manuscript a theoretical result about disordered quantum wave functions, not direct evidence about people or experiments. The unit of analysis is a single-particle one-dimensional lattice model, and the reported fractal dimensions describe wave-function structure rather than directly measuring transport or a real-space localization length. The supplied manuscript is a preprint, arXiv:2608.25806v3, dated 8 Sep 2026.
Paper data and sources
Original title: Semi-localized ground state in a 1D system with long-range hopping
Authors: Murod S. Bahovadinov, Faridun N. Jalolov, Vladimir E. Kravtsov et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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