Renormalization group for Anderson localization on high-dimensional lattices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916918595682304 |
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| author | Altshuler, Boris L. Kravtsov, Vladimir E. Scardicchio, Antonello Sierant, Piotr Vanoni, Carlo |
| author_facet | Altshuler, Boris L. Kravtsov, Vladimir E. Scardicchio, Antonello Sierant, Piotr Vanoni, Carlo |
| contents | 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 $D_{1}$ evolves smoothly from its $d=2$ form, in which $β_2\leq 0$, to its $β_\infty\geq 0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01974 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renormalization group for Anderson localization on high-dimensional lattices Altshuler, Boris L. Kravtsov, Vladimir E. Scardicchio, Antonello Sierant, Piotr Vanoni, Carlo Disordered Systems and Neural Networks Statistical Mechanics Quantum Physics 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 $D_{1}$ evolves smoothly from its $d=2$ form, in which $β_2\leq 0$, to its $β_\infty\geq 0$ 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. |
| title | Renormalization group for Anderson localization on high-dimensional lattices |
| topic | Disordered Systems and Neural Networks Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2403.01974 |