Renormalization group for Anderson localization on high-dimensional lattices

• URL: http://arxiv.org/abs/2403.01974v3
• Date: Mon, 30 Sep 2024 14:33:20 GMT
• Title: Renormalization group for Anderson localization on high-dimensional lattices
• Authors: Boris L. Altshuler, Vladimir E. Kravtsov, Antonello Scardicchio, Piotr Sierant, Carlo Vanoni,
• Abstract summary: We show how in the delocalized region, including the transition point, the $beta$-function for the fractal dimension $D_1$ evolves smoothly. We put forward a conjecture about a lower bound for the fractal dimension.
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• Abstract: We discuss the dependence of the critical properties of the Anderson model on the dimension $d$ in the language of $\beta$-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 $\beta$-function for the fractal dimension $D_{1}$ evolves smoothly from its $d=2$ form, in which $\beta_2\leq 0$, to its $\beta_\infty\geq 0$ form, which is represented by the regular random graph (RRG) result. We show how the $\epsilon=d-2$ 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.

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