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Universality in Anderson localization on random graphs with varying connectivity
Disordered Systems and Neural Networks · Submitted 29 May 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. | contribution 0.94 | 0.96 | 0.04 | asserted 0.96 |
| s1 | 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. | result 0.80 | 0.97 | 0.68 | asserted 1.00 |
| s2 | 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 $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. | result 0.47 | 0.45 | 0.11 | asserted 0.85 |
| s3 | We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. | result 1.00 | 0.97 | 0.30 | asserted 1.00 |
| s4 | The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. | interpretation 0.87 | 0.80 | 0.26 | asserted 1.00 |
| s5 | 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. | result 0.83 | 0.97 | 0.07 | asserted 1.00 |
| s6 | We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition. | contribution 0.58 | 0.63 | 0.04 | asserted 0.61 |
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The paper
Original abstract
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 $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. 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.
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How the literature uses it
12 arXiv papers cite this in 46 sentences
Papers citing it · 27 sentences without a confident label
Contests it 1 paper
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This confirms that there is a unique critical exponent $ν≈0.5$ in the delocalized phase and that $Λ= 𝒱(ξ)$ is the correlation volume associated with $ξ$ (contrary to what is reported in [ref] ).
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization · Delocalized phase $W<W_c$ · contest 0.88
Uses its data or values 3 papers
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Where the RG $β$-function cannot be completely fixed by theoretical arguments we rely on the state-of-the-art numerical results, presented in Ref. [ref] , to extract the missing information we need.
Renormalization Group Analysis of the Anderson Model on Random Regular Graphs · Introduction · 0.99
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We can fit the correlation volume by [equation] with $W_c=1.8$ fixed (the value for $p=0.06$ given in [ref] ), $a_1$ and $ν$ two free parameters, and $a_0$ fixed by the fact that $Λ=1$ for the smallest disorder value considered.
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization · Delocalized phase $W<W_c$ · 1.00
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For $K_0 = 2$, which will be assumed in the rest of the paper, the critical value of the disorder is known to be $W_c ≃18.17$ [ref] .
An analysis of localization transitions using non-parametric unsupervised learning · Model and results · 0.93
Compares with it 4 papers
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Here, we show that the first principal component stores the information on the localization transition critical point and can be employed to estimate it with remarkable accuracy in agreement with recent results [ref] .
An analysis of localization transitions using non-parametric unsupervised learning · Introduction · 0.95
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In contrast, Ref. [ref] focuses on the metal side and finds two length scales with power-law criticality (not KT).
Wegner model on a tree graph: U(1) symmetry breaking and a non-standard phase of disordered electronic matter · Conjectured RG Flow Diagram · 0.93
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These values are compatible with the theoretical values recently obtained in [ref] .
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization · Determination of $W_c$ and of the type of scaling · 0.98
Cites it as background 5 papers
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The Anderson localization on RRG (including some variations of this model) was addressed in a considerable number of recent works [ref] , see also earlier papers [ref] on a closely related sparse random matrix model.
Ergodicity-to-localization transition on random regular graphs with large connectivity and in many-body quantum dots · MBL and Anderson localization on RRG · 0.99
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This problem has raised a very strong interest recently [ref] , in particular because of its analogy with the MBL transition [ref] .
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization · Conclusion · 0.94
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Conventionally, in condensed matter settings, the system size dependence of the participation entropy is parametrized as [ref] [equation] where $D_q$ is the multifractal dimension and $c_q$ is a sub-leading term.
Measuring nonstabilizerness via multifractal flatness · Stabilizer and participation entropies · 0.97
What it cites · 12 sentences without a confident label
Uses the method or tool of 1 paper
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In order to analyze the crossover between delocalized and localized regimes, we consider, following [ref] , two system-size dependent disorder strengths: i) $W^T_r(L)$ – the disorder strength for which, at a given system size $L$, the average gap ratio $r$ deviates by a small parameter $p_r$ from the value $r_GOE$ characteristic for delocalized system; ii) $W^*_r(L)$ – the disorder strength at which the curves $r(W)$ cross for the system sizes $L-ΔL$ and $L+ΔL$, where $ΔL ≪L$.
Stability of many-body localization in Floquet systems · Analysis of the crossover between delocalized and localized regimes · 0.96
Compares with 2 papers
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Moreover, the behavior of $W^T_r(L)$ and $W^*_r(L)$ can be directly compared with for a direct comparison with similar disorder strengths computed at the many-body localization crossover [ref] . * In [ref] we propose a scaling theory of the Anderson transition on random graphs that is consistent with the observed finite size drifts at the delocalization/localization crossover.
Stability of many-body localization in Floquet systems · Outline of the work and main results · 0.94
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Fig. [ref] shows the average gap ratio for Anderson model on random graphs, demonstrating that it crossovers between $r = r_GOE$, for small disorder strengths $W$, and $r = r_P$ in the strong disorder limit, consistenly with the earlier observations for RRG [ref] .
Difference between level statistics, ergodicity and localization transitions on the Bethe lattice · Average gap ratio · 0.94
Cites as background 2 papers
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The Anderson model on RRG has been a subject of intense studies [ref] aimed at understanding features of the transition between delocalized and localized phases of the model.
Difference between level statistics, ergodicity and localization transitions on the Bethe lattice · Introduction · 0.96
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The analysis of the crossover between delocalized and localized phases is considerably more complicated for disordered interacting quantum many-body systems which recently has lead to controversies around the MBL transition [ref] .
Thermalization of dilute impurities in one dimensional spin chains · Analysis of the crossover between delocalized and localized regimes · 0.99
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