Renormalization group for Anderson localization on high-dimensional lattices

Quantum particles can localize when the surrounding environment is sufficiently disordered, a phenomenon named Anderson localization. In this work, we discuss how the properties of the localization transition depend on the dimensionality of real space, showing how to take properly the limit of infinite dimensions. To do so, we use the renormalization group for quantities that are easily accessible numerically via exact diagonalization. The results presented here are the bridge connecting low dimensional systems, amenable to controllable analytical treatment, and infinite dimensional systems, relevant for interacting disordered systems. Moreover, the method developed in this work allows for a quantitative analysis of numerical data for the Many-Body Localization problem and can help resolve discrepancies still existing in this problem. We discuss the dependence of the critical properties of the Anderson model on the dimension d in the language of β-function and renormalization group recently introduced in Vanoni et al. [C. Vanoni et al., Proc. Natl. Acad. Sci. U.S.A. 121, e2401955121 (2024)] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the β-function for the fractal dimension D1 evolves smoothly from its d=2 form, in which β2≤0, to its β∞≥0 form, which is represented by the random regular graph (RRG) result. We show how the ϵ=d−2 expansion and the 1/d expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent y depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general nonequilibrium quantum systems.

Recommended citation: Altshuler, B. L., Kravtsov, V. E., Scardicchio, A., Sierant, P., and Vanoni, C. (2025). "Renormalization group for Anderson localization on high-dimensional lattices." Proceedings of the National Academy of Sciences 122(35), e2423763122.
Download Paper