We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.
We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.
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 D 1 evolves smoothly from its d = 2 form, in which β 2 ≤ 0 , to its β ∞ ≥ 0 form, which is represented by the random regular 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 nonequilibrium quantum systems.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions Dq. The RG flows recover the qualitative structure previously reported for the information dimension D1, while their extension to other Dq with smaller moment orders q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1 need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2 and small q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
Anderson localization on random regular graphs (RRG) attracts much attention as a toy model of many-body localization. Here, the critical behavior at the localization transition on RRG, related to that on infinite Bethe lattices, is studied numerically by means of a population dynamics approach. Proceeding up to correlation volumes as large as 1019, the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder W_{E} W E which is strictly less than the Anderson localization critical disorder W_C W C , and that of other works which suggest that there is no such region. We find that, although no separate W_{E} W E exists from W_C W C , the length scale at which fully developed ergodicity is found diverges like |W-W_C|^{-1} | W − W C | − 1 , while the critical length over which delocalization develops is \sim |W-W_C|^{-1/2} ∼ | W − W C | − 1 / 2 . The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder WE which is strictly less than the Anderson localization critical disorder WC, and that of other works which suggest that there is no such region. We find that, although no separate WE exists from WC, the length scale at which fully developed ergodicity is found diverges like ∣W−WC∣−1, while the critical length over which delocalization develops is ∼∣W−WC∣−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.