Connecting finite-size scaling and renormalization-group flows in Anderson localization on random graphs

Abstract:We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions $D_q$. The RG flows recover the qualitative structure previously reported for the information dimension $D_1$, while their extension to other $D_q$ with smaller moment orders $q<1/2$ confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension $D_1$ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large $q>1/2$ and small $q<1/2$ eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.

Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)

arXiv:2609.31872 [cond-mat.dis-nn]

(or arXiv:2609.31872v1 [cond-mat.dis-nn] for this version)

https://doi.org/10.48550/arXiv.2609.31872 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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