Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization

García-Mata, I., Martin, J., Giraud, O., Georgeot, B., Dubertrand, Remy and Lemarié, G. (2022) Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization. Physical Review B, 106 (21). p. 214202. ISSN 2469-9950

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The Anderson transition in random graphs has raised great interest, partly out of the hope that its analogy with the many-body localization (MBL) transition might lead to a better understanding of this hotly debated phenomenon. Unlike the latter, many results for random graphs are now well established, in particular, the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. In this paper, we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length ζ than the one in the transverse direction ζ. Importantly, these two lengths have different critical behaviors: ζ diverges with a critical exponent ν=1, while ζ reaches a finite universal value ζc at the transition point Wc. Indeed, ζ-1≈ζc-1+ζ-1, with ζ∼(W-Wc)-ν associated with a new critical exponent ν=12, where exp(ζ) controls finite-size effects. The delocalized phase inherits the strongly nonergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent ν=12. This shows a very strong analogy with the MBL transition: the behavior of ζ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a small-world complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

Item Type: Article
Additional Information: Funding information: The authors thank D. Huse, V. Kravtsov, A. Mirlin, A. Scardicchio, P. Sierant, and K. Tikhonov for interesting discussions. This study has been (partially) supported through the EUR Grant No. NanoX ANR-17-EURE-0009 in the framework of the “Programme des Investissements d'Avenir,” the French-Argentinian LIA LICOQ, and also by research funding Grants No. ANR-17-CE30-0024, No. ANR-18-CE30-0017, and No. ANR-19-CE30-0013. We thank Calcul en Midi-Pyrénées (CALMIP) and the Consortium des Equipements de Calcul Intensif (CECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.- FNRS) under Grant No. 2.5020.11, for computational resources and assistance. I.G.-M. received funding from CONICET (Grant No. PIP 11220150100493CO) and ANCyPT (Grants No. PICT-2020-SERIEA-00740 and PICT-2020-SERIEA-01082).
Subjects: F300 Physics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
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Depositing User: Elena Carlaw
Date Deposited: 04 Jan 2023 12:43
Last Modified: 04 Jan 2023 12:45

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