We propose a new viewpoint on the study of localization transitions in disordered quantum systems, showing how critical properties can be seen also as a geometric transition in the data space generated by the classically encoded configurations of the disordered quantum system. We showcase our approach to the Anderson model on regular random graphs, known for displaying features of interacting systems, despite being a single-particle problem. We estimate the transition point and verify the critical exponents in agreement with the best-known results in the literature. We provide a simple and coherent explanation of our findings, discussing the applicability of the method in real-world scenarios with a modest number of measurements.
We propose a new viewpoint on the study of localization transitions in disordered quantum systems, showing how critical properties can be seen also as a geometric transition in the data space generated by the classically encoded configurations of the disordered quantum system. We showcase our approach to the Anderson model on regular random graphs, known for displaying features of interacting systems, despite being a single-particle problem. We estimate the transition point and verify the critical exponents in agreement with the best-known results in the literature. We provide a simple and coherent explanation of our findings, discussing the applicability of the method in real-world scenarios with a modest number of measurements.
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.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.
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.
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.
Supervised and unsupervised machine-learning-based frameworks have been implemented in the past to classify various dynamical phases of disordered quantum systems using rather sophisticated (highly preprocessed or difficult to generate) input data. In this paper, we begin by questioning the existence of a nonergodic extended phase, namely, the many-body critical (MBC) phase in finite systems of an interacting quasiperiodic system. We find that this phase can be separately detected from the other phases, such as the many-body ergodic (ME) and many-body localized (MBL) phases, in the model through supervised neural networks made for both binary and multiclass classification tasks, utilizing, rather unpreprocessed, eigenvalue spacings and eigenvector probability densities as input features. Moreover, the output of our trained neural networks can also indicate the critical points separating ME, MBC, and MBL phases, which are consistent with the same obtained from other conventional methods. We also employ unsupervised learning techniques, particularly principal component analysis (PCA) of eigenvector probability densities, to investigate how this framework, without any training, captures the, rather unknown, many-body phases (ME, MBL, and MBC) and single-particle phases (delocalized, localized, and critical) of the interacting and noninteracting systems, respectively. Our findings reveal that PCA entropy serves as an effective indicator (order parameter) for detecting phase transitions in the single-particle systems. Moreover, this method proves applicable to many-body systems when the data undergoes a suitable preprocessing. Interestingly, when it comes to the extraction of critical (correlation length) exponents through a finite-size scaling, we find that for single-particle systems, scaling collapse of neural network outputs is obtained using components of inverse participation ratio (IPR) as input data. Remarkably, we observe identical critical exponents as obtained from the scaling collapse of the IPR directly for different single-particle phase transitions. However, the current finite-size scaling approach to the network output cannot reliably extract critical exponents for the many-body system that needs further investigation.
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.