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 Ref.[arXiv:2306.14965] 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 regular random 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 non-equilibrium quantum systems.