Renormalization group for Anderson localization on high-dimensional lattices

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 [Formula: see text] evolves smoothly from its [Formula: see text] form, in which [Formula: see text], to its [Formula: see text] form, which is represented by the random regular graph (RRG) result. We show how the [Formula: see text] expansion and the [Formula: see text] 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.