Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization

García-Mata, Ignacio; Martin, John P.; Giraud, Olivier; Georgeot, Bertrand; Dubertrand, Rémy; Lemarié, Gabriel

doi:10.1103/physrevb.106.214202

Cited by 27 publications

(2 citation statements)

References 158 publications

(511 reference statements)

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“…Note that here and further, we focus mostly on the localization, D 2 = 0, versus the delocalization, D 2 > 0, but not on the ergodicity, D 2 = 1, versus the non-ergodicity, 0 < D 2 < 1. Already in a fully disordered RRG at β = 1, the question of the existence of a non-ergodic phase in RRG has been a discussion point for years [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][27][28][29][30][31][32], and even now the maximal system sizes of a few millions, N ∼ 10 6 , do not resolve this issue [14,31,32]. Therefore, in this work we calculate the fractal dimensions D 2 (and their generalization D q together with the singularity spectrum f (α) with the definitions given below) in the Appendix A only of finite sizes up to N ∼ 30000 and do not claim any ergodicity or non-ergodicity.…”

Section: Robustness Of Delocalization and Fractal Dimension Dmentioning

confidence: 99%

Robust extended states in Anderson model on partially disordered random regular graphs

Kochergin,

Khaymovich,

Valba

et al. 2024

SciPost Phys.

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In this work we analytically explain the origin of the mobility edge in the partially disordered random regular graphs of degree d, i.e., with a fraction \betaβ of the sites being disordered, while the rest remain clean. It is shown that the mobility edge in the spectrum survives in a certain range of parameters (d,\betaβ) at infinitely large uniformly distributed disorder. The critical curve separating extended and localized states is derived analytically and confirmed numerically. The duality in the localization properties between the sparse and extremely dense RRG has been found and understood.

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Section: Robustness Of Delocalization and Fractal Dimension Dmentioning

confidence: 99%

Robust extended states in Anderson model on partially disordered random regular graphs

Kochergin,

Khaymovich,

Valba

et al. 2024

SciPost Phys.

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“…A paradigmatic example of the latter are spectral statistics, for which a connection between the 3D Anderson model below the transition (i.e., at W < W c ) and the predictions of the random matrix theory (RMT) were made already in the mid-eighties [69], shortly after the formulation of the quantum chaos conjecture [70,71]. The transition in finite systems then corresponds to a crossover between a quantum chaotic regime at W W c , in which spectral statistics follow predictions from the Gaussian orthogonal ensemble (GOE), and a localized regime at W W c , in which spectral statistics are Poissonian [66,[72][73][74][75][76].…”

Section: A Anderson Modelmentioning

confidence: 99%

Localization challenges quantum chaos in the finite two-dimensional Anderson model

Šuntajs

Prosen²,

Vidmar³

2023

Phys. Rev. B

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It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum-chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter u = (W ln V ) −1 where W is the disorder strength and V is the number of lattice sites.

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$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

Poudevigne–Auboiron,

Wildemann

2024

Commun. Math. Phys.

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We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $$\mathbb {H}^{2|2}$$ H 2 | 2 -model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $$\beta > 0$$ β > 0 . Approaching the critical point from the transient regime, $$\beta \searrow \beta _{\textrm{c}}$$ β ↘ β c , we show that the expected total time spent at the starting vertex diverges as $$\sim \exp (c/\sqrt{\beta - \beta _{\textrm{c}}})$$ ∼ exp ( c / β - β c ) . Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $$\beta _{\textrm{c}}< \beta < \beta _{\textrm{c}}^{\textrm{erg}}$$ β c < β < β c erg . In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model.

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Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization

Cited by 27 publications

References 158 publications

Robust extended states in Anderson model on partially disordered random regular graphs

Robust extended states in Anderson model on partially disordered random regular graphs

Localization challenges quantum chaos in the finite two-dimensional Anderson model

$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

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