An arXiv preprint predicts a quantum ground state that sits between two familiar descriptions: it is neither fully extended nor fully localized. The paper labels the pattern semi-localized. For an infinitely long chain, the predicted generalized fractal dimensions are Dq=1 for 0<q<1, Dq=0 for q>1, and D1=M∞/(1+M∞) at q=1. These quantities track how different moments of the wavefunction spread as the system grows.
The model is a single-particle, one-dimensional system with algebraic long-range hopping. The hopping strength is J=1/2, and each site carries uncorrelated onsite disorder drawn uniformly from U(-W,W). For 1<a<3/2, the paper describes a disorder-driven transition at the band edge; high-energy states remain localized at any disorder strength.
A peak and a statistical tail
In momentum space, the predicted ground-state profile combines a dominant zero-momentum component with nonzero-momentum tails. After normalization, the tail intensities are exponentially distributed.
The analytical treatment uses momentum-space perturbation theory in the weak-disorder limit, W≪1, and includes second-order corrections. A supersymmetry, or SUSY, construction maps all ground-state moments to those of the E=0 state of an associated chiral Hamiltonian. The stated moment correspondence is for the ground state rather than generic excited states.
A numerical check
Exact diagonalization was used at a=1.48, with system length L=20,000 and ND=6,000 disorder realizations. For mean normalized tail intensities, first-order perturbation theory agreed with exact diagonalization across the full momentum range, with deviations of a few percent.
At a=1.48, finite-size results for the fractal dimensions move toward the predicted spectrum as system size increases: Dq approaches 1 for q<1 and tends to 0 for q>1. The comparison remains a finite-size check on an asymptotic prediction; it does not settle the full infinite-size picture by itself.
The boundary depends on the hopping exponent
The proposed boundary between regimes is described by a critical disorder Wc(a). The first-order estimate gives β=1/2 as a approaches 3/2 and δ=−1 as a approaches 1 from above. The paper reports reasonable correspondence with numerical exponents of about β≈0.67 and δ=−1.
At a≥3/2, the perturbative normalization no longer remains finite, the zero-momentum weight tends to zero and the semi-localized phase breaks down. At a≤1, the accumulated tail weight vanishes and the normalized ground state concentrates at zero momentum; at a=1, the tails disappear logarithmically rather than algebraically.
Beyond the boundary, the picture remains unresolved
Beyond Wc(a), limited exact-diagonalization information suggests an exponential momentum-space profile, while the authors conjecture y=2a−3. The paper leaves this post-critical behavior unresolved.
The evidence is limited to the specified single-particle Hamiltonian, its ground state, momentum-space statistics, weak-disorder perturbation theory and finite-size exact-diagonalization checks. The SUSY moment correspondence is stated for the ground state, not generic excited states, and the analytical treatment assumes W≪1.
The manuscript is identified as arXiv:2608.25806v2 and dated 27 Aug 2026. Its supplementary material details the momentum-space perturbation theory, wavefunction statistics, moments, fractal dimensions and energies, and compares them with exact diagonalization. The work was financially supported by the Russian Science Foundation, project No. 25-72-00135, with additional support from HSE University HPC facilities.
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
Original paper · Full text