Semi-localized ground state in a 1D system with long-range hopping

Murod S. Bahovadinov; Faridun N. Jalolov; Vladimir E. Kravtsov; Boris L. Altshuler; Georgy V. Shlyapnikov

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arXiv · 2608.25806

Semi-localized ground state in a 1D system with long-range hopping

Murod S. Bahovadinov·Faridun N. Jalolov·Vladimir E. Kravtsov·Boris L. Altshuler·Georgy V. Shlyapnikov

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Abstract

We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes $t(r)\propto r^{-a}$. In contrast to the standard one-dimensional Anderson model ($a\to\infty$), 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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Murod S. Bahovadinov, Faridun N. Jalolov, Vladimir E. Kravtsov, Boris L. Altshuler, Georgy V. Shlyapnikov. 2026-08-26. Semi-localized ground state in a 1D system with long-range hopping. https://arxiv.org/abs/2608.25806

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