Renormalization group for Anderson localization on high-dimensional lattices

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 ${\beta }_2\le 0$, to its ${\beta }_{\infty }\ge 0$ 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.

13 pages, 12 figures. Comments are welcome!

References in corpus (38)

• Many body localization and thermalization in quantum statistical mechanics
• Anderson Transitions
• Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states
• Localization of interacting fermions at high temperature
• Many-body localization, thermalization, and entanglement
• Integrals of motion in the Many-Body localized phase
• Quantum chaos challenges many-body localization
• Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system
• Anderson localization on the Bethe lattice: non-ergodicity of extended states
• Thouless time analysis of Anderson and many-body localization transitions
• Review: Local Integrals of Motion in Many-Body Localized systems
• A random matrix model with localization and ergodic transitions
• Distinguishing localization from chaos: challenges in finite-size systems
• Multifractal finite-size-scaling and universality at the Anderson transition
• Can we study the many-body localisation transition?
• Anderson localization on random regular graphs
• Challenges to observation of many-body localization
• Critical exponent for the Anderson transition in the three dimensional orthogonal universality class
• Many-Body Localization in the Age of Classical Computing
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• From Anderson localization on Random Regular Graphs to Many-Body localization
• Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph
• Return probability for the Anderson model on the random regular graph
• Two critical localization lengths in the Anderson transition on random graphs
• Critical properties of the Anderson localization transition and the high dimensional limit
• Eigenfunction entropy and spectral compressibility for critical random matrix ensembles
• Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model
• Dimensional dependence of the metal-insulator transition
• Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization
• Self-Consistent Theory of Anderson Localization: General Formalism and Applications
• Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions
• A semiclassical theory of the Anderson transition
• Universality in Anderson localization on random graphs with varying connectivity
• Dimensional Dependence of Critical Exponent of the Anderson Transition in the Orthogonal Universality Class
• Renormalization Group Analysis of the Anderson Model on Random Regular Graphs
• Termination of typical wavefunction multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group
• Integrable random matrix ensembles
• Investigating finite-size effects in random matrices by counting resonances

Cited by in corpus (6)

• Renormalization-Group Analysis of the Many-Body Localization Transition in the Random-Field XXZ Chain
• Effective delocalization in the one-dimensional Anderson model with stealthy disorder
• Scaling theory of fading ergodicity
• Multifractality in high-dimensional graphs induced by correlated radial disorder
• Unconventional Thermalization of a Localized Chain Interacting with an Ergodic Bath
• Universal Relation between Spectral and Wavefunction Properties at Criticality