We study quantum ergodicity on Erdős–Rényi (ER) random graphs by coupling a U(1) gauge field to a tight-binding Hamiltonian with random on-site disorder. Using the adjacent gap ratio (Oganesyan–Huse), we map the complete dual phase diagram of the system in the plane of mean degree d and disorder strength W. Two complementary transition paths are established: (1) tuning d at fixed W drives a sharp Poisson-to-GUE crossover, with the transition point d_c ≈ 1.96 obtained from finite-size scaling at N = 64–512; (2) tuning W at fixed d reproduces the same transition, with critical disorder scaling linearly as W_c ∝ d at large connectivity. The universality class is set by the U(1) phase: complex couplings saturate at GUE, real couplings at GOE. A null test on 2-regular ring graphs shows that the mere presence of cycles is insufficient for quantum ergodicity. Remarkably, the spectral transition point coincides, within one error bar, with the inflection point of the 2-core coverage on the ER graph (d_0 = 1.965 ± 0.053), establishing a dictionary between graph-theoretic percolation transitions and quantum spectral universality. All results are fully reproducible from the included code and data. This work connects Anderson localization on random graphs to many-body localization physics and opens applications in quantum transport on disordered networks.
We study quantum ergodicity on Erdős–Rényi (ER) random graphs by coupling a U(1) gauge field to a tight-binding Hamiltonian with random on-site disorder. Using the adjacent gap ratio (Oganesyan–Huse), we map the complete dual phase diagram of the system in the plane of mean degree d and disorder strength W. Two complementary transition paths are established: (1) tuning d at fixed W drives a sharp Poisson-to-GUE crossover, with the transition point d_c ≈ 1.96 obtained from finite-size scaling at N = 64–512; (2) tuning W at fixed d reproduces the same transition, with critical disorder scaling linearly as W_c ∝ d at large connectivity. The universality class is set by the U(1) phase: complex couplings saturate at GUE, real couplings at GOE. A null test on 2-regular ring graphs shows that the mere presence of cycles is insufficient for quantum ergodicity. Remarkably, the spectral transition point coincides, within one error bar, with the inflection point of the 2-core coverage on the ER graph (d_0 = 1.965 ± 0.053), establishing a dictionary between graph-theoretic percolation transitions and quantum spectral universality. All results are fully reproducible from the included code and data. This work connects Anderson localization on random graphs to many-body localization physics and opens applications in quantum transport on disordered networks.
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 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.
Quantum marked vertex search algorithms are known to outperform their classical counterparts, yet their behavior in the presence of disorder remains largely unexplored. Here, we address this gap by studying marked vertex search on disordered random graphs. To introduce disorder, we implement the Rosenzweig-Porter (RP) model, a random matrix ensemble with tunable ergodic, non-ergodic extended, and localized phases, on Erdős-Rényi (ER) graphs. This produces a doubly random system where ER graph connectivity randomizes which interactions exist, while RP disorder controls their strength and `on-site' potentials, providing a two-parameter framework to study quantum dynamics on disordered networks. First, we show that the characteristic Wigner-Dyson-to-Poisson spectral crossover of the RP ensemble survives under graph constraints across the sparse-to-dense range, and we derive an analytical estimate for the finite-size localization boundary that shifts systematically with the graph edge probability p, consistent with a resonant-hybridization argument. Thereafter, using this disordered graph ensemble, we study the marked vertex search problem and find that search performance tracks the underlying quantum phase directly. Counterintuitively, the ergodic phase, despite supporting fast transport, yields lower success probability than the localized phase, which achieves high success probability at the cost of significantly longer search times. These results establish a direct and quantitative link between random matrix disorder on graphs and the performance of continuous-time quantum walk search, and suggest that disorder, rather than being merely an obstacle, can be exploited as a tunable parameter in quantum search protocols.
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.
The Anderson transition on random graphs is a paradigm for understanding high-dimensional quantum phase transitions driven by disorder and closely mirrors several features of many-body localization. In this Letter, we introduce a unitary Anderson model on small-world graphs, enabling large-scale, long-time simulations of wave-packet dynamics. This allows us to uncover logarithmically slow critical dynamics, two distinct localization times, and a crossover to ergodic diffusion. Finite-time scaling reveals distinct critical exponents, establishing a dynamical universality class for this exotic transition. Our model opens the way to an experimental realization of the Anderson transition on random graphs in quantum simulators. By combining disorder, high dimensionality, and dynamics, our approach provides a general framework for probing universal features of out-of-equilibrium quantum phase transitions, including those relevant to many-body localization.
Anderson localization on random regular graphs (RRG) serves as a toy-model of many-body localization (MBL). We explore the transition from ergodicity to localization on RRG with large connectivity m. In the analytical part, we focus on the inverse participation ratio of eigenstates and identify several regimes on the ergodic side of the transition---self-consistent golden rule, golden rule, precritical, and critical---that the system consecutively goes through when the disorder increases toward the point Wc of the localization transition. We also perform exact-diagonalization numerics as well as a population-dynamics analysis combining analytical and numerical techniques. Results of all the approaches are in excellent mutual agreement. We further explore the evolution from ergodicity to localization in two models of Fock-space MBL: fermionic and spin quantum dots. Large-connectivity RRG models serve as an approximation for this class of many-body problems; one of our central goals is to better understand the status of this approximation. Our numerical simulations support the conjecture that, in the limit of a large system, there is a sharp ergodicity-to-localization transition in the quantum-dot models. Furthermore, our results are consistent with the RRG-like scaling of the critical disorder, ${W}_{c}\ensuremath{\sim}mlnm$. While in the golden-rule range the behavior of quantum-dot models is in agreement with analytical predictions based on the RRG model, substantial deviations occur in the precritical regime. Our results indicate that the precritical and critical behavior (as well as the numerical coefficient in the formula for Wc) in the quantum-dot models may be different from what one would expect from the RRG-like approximation.