Renormalization group analysis of the Anderson model on random regular graphs

A particle in quantum mechanics can escape thermalization in the presence of a random potential, a phenomenon known as Anderson localization. The properties of the Anderson localization transition depend on the dimensionality d of space. In this work, we consider Anderson localization on random regular graphs, a class of graphs that corresponds to infinite dimensions, therefore providing a starting point for expansions in 1/d. We propose a method for analyzing the spectral properties using the renormalization group. We recover the known results and make predictions, with the advantage that such scheme is suited for any geometry of the Anderson model, and for interacting quantum systems as well. We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

Recommended citation: Vanoni, C., Altshuler, B. L., Kravtsov, V. E., and Scardicchio, A. (2024). "Renormalization group analysis of the Anderson model on random regular graphs." Proceedings of the National Academy of Sciences 121(29), e2401955121.
Download Paper