Semi-localized ground state in a 1D system with long-range hopping
Abstract
We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes t(r)∝ r−a. In contrast to the standard one-dimensional Anderson model (a→∞), in which all states are localized and the localization length is minimal at the band edge, the long-range model with 1<a<3/2 exhibits a disorder-driven transition at the band edge, while high-energy states remain localized at arbitrary disorder strength. We investigate this transition for the ground state in momentum space. In the weak-disorder regime, we derive perturbative expressions for the characteristic functions and moments of the momentum-space wave function, as well as for its fractal dimensions. Our results demonstrate that the ground state exhibits it semilocalization rather than conventional localization, thereby extending the class of models displaying the unusual it semifractality of wave functions.
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BibTeX
@misc{bahovadinov2026semilocalized,
title = {{Semi-localized ground state in a 1D system with long-range hopping}},
author = {Murod S. Bahovadinov and Faridun N. Jalolov and Vladimir E. Kravtsov and Boris L. Altshuler and Georgy V. Shlyapnikov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25806},
eprint = {2608.25806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25806},
}
RIS
TY - UNPB
TI - Semi-localized ground state in a 1D system with long-range hopping
AU - Murod S. Bahovadinov
AU - Faridun N. Jalolov
AU - Vladimir E. Kravtsov
AU - Boris L. Altshuler
AU - Georgy V. Shlyapnikov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.25806
UR - https://arxiv.org/abs/2608.25806
AB - We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes t(r)∝ r^−a. In contrast to the standard one-dimensional Anderson model (a→∞), in which all states are localized and the localization length is minimal at the band edge, the long-range model with 1<a<3/2 exhibits a disorder-driven transition at the band edge, while high-energy states remain localized at arbitrary disorder strength. We investigate this transition for the ground state in momentum space. In the weak-disorder regime, we derive perturbative expressions for the characteristic functions and moments of the momentum-space wave function, as well as for its fractal dimensions. Our results demonstrate that the ground state exhibits it semilocalization rather than conventional localization, thereby extending the class of models displaying the unusual it semifractality of wave functions.
ER -