From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

Facoetti, Davide; Vivo, Pierpaolo; Biroli, Giulio

doi:10.1209/0295-5075/115/47003

Cited by 98 publications

(152 citation statements)

References 37 publications

(80 reference statements)

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“…Note that m 0 minimizing Λ(m) is 1/2 in the entire region of nonergodic extended states 1 < γ < 2 in the limit ln N → ∞. This implies that the exponent in the power-law dependence (34) is 3/2 for all values of γ > 1, in agreement with general expectations and the results of works 12,13 . We conclude that the exact result of the RSB theory in this case is associated with the value of the exponent m 0 = 1/2 in the entire region of non-ergodic states.…”

Section: Application To Rosenzweig-porter Modelsupporting

confidence: 85%

“…It was also suggested that the transition (referred to below as ergodic transition) from the extended ergodic (EE) to the non-ergodic extended (NEE) phases is a true transition as evidenced by the jump in the fractal dimensions rather than a crossover 11 . Existence of NEE phase and the transition from NEE to EE states has been recently proven 12,13 for an apparently different model, the random matrix theory with the special diagonal, suggested in 1960 by Rosenzweig and Porter (RP) 14 and generalized in Ref. 12 .…”

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confidence: 99%

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Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

2018

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We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graph models. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D1 that distinguishes between the extended ergodic, D1 = 1, and extended non-ergodic (multifractal), 0 < D1 < 1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression ln Sfor the typical imaginary part of self-energy Styp in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended nonergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization. Contents

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Section: Application To Rosenzweig-porter Modelsupporting

confidence: 85%

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confidence: 99%

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

2018

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“…30) has a well-defined scaling in N for each model as a function of its parameters. So this dynamical point of view is somewhat different from the closely recent studies based of the Green function G(z) as a function of the complex variable z = E + iη, where the imaginary part η introduced as a formal regularization can be chosen with various scalings with respect to the system size N in order to probe various regimes [26,32,50,72].…”

Section: Dynamics Within the Wigner-weisskopf Approximationmentioning

confidence: 95%

“…As recalled in the Introduction, the Generalized-Rosenzweig-Porter model is the simplest matrix model exhibiting a delocalized non-ergodic phase with an explicit multifractal spectrum for eigenvectors in [49], and has been analyzed recently from various points of view [32,50,51]. In this section, our goal is to show how the present dynamical approach is able to recover the multifractal spectrum obtained in [49].…”

Section: Generalized-rosenzweig-porter Matrix Modelmentioning

confidence: 97%

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models: Wigner–Weisskopf approach

Monthus¹

2017

J. Phys. A: Math. Theor.

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The delocalized non-ergodic phase existing in some random N × N matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site j0. The main output of this approach is the inverse Γj 0 (N ) of the characteristic time to leave the state j0 that provides some broadening Γj 0 (N ) for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening Γj 0 (N ) is smaller in scaling than the level spacing ∆j 0 (N ) ∝ 1 N , while the delocalized non-ergodic phase corresponds to the region where the broadening Γj 0 (N ) decays with N but is bigger in scaling than the level spacing ∆j 0 (N ). Then the numberof resonances grows only sub-extensively in N . This approach allows to recover the multifractal spectrum of the Generalized-Rosenzweig-Potter (GRP) Matrix model [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. We then consider the Lévy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of Lévy index 1 < µ < 2 : the dynamics is then governed by a stretched exponential of exponent β = 2(µ−1) µ and the multifractal properties of eigenstates are explicitly computed.

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“…Recent results [21,48] suggest that a non-ergodic delocalized phase could still arise at large K (see however [49]). It will thus be very interesting to investigate this regime K 2 with our approach.…”

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confidence: 99%

Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality

et al. 2017

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We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1 < K ≤ 2, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. In the critical regime, multifractal wavefunctions are located on few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.Ergodicity properties of quantum states are crucial to assess transport properties and thermalization processes. They are at the heart of the eigenstate thermalization hypothesis which has attracted enormous attention lately [1]. A paramount example of non-ergodicity is Anderson localization where the interplay between disorder and interference leads to exponentially localized states [2]. In 3D, a critical value of disorder separates a localized from an ergodic delocalized phase. At the critical point eigenfunctions are multifractal, another non trivial example of non-ergodicity [3,4]. Recently, those questions have been particularly highlighted in the problem of many-body localization [5][6][7][8][9]. Because Fock space has locally a tree-like structure, the problem of Anderson localization on different types of graphs [10][11][12][13][14][15] has attracted a renewed activity [16][17][18][19][20][21][22][23][24][25]. In particular, the existence of a delocalized phase with non-ergodic (multifractal) eigenfunctions lying on an algebraically vanishing fraction of the system sites is debated [19-21, 23, 25].The problem of non-ergodicity also arises in another context corresponding to glassy physics [26]. For directed polymers on the Bethe lattice [27], a glass transition leads to a phase where a few branches are explored among the exponential number available. As there is a mapping to directed polymer models in the Anderson-localized phase [10,[28][29][30], it has been recently proposed that this type of non-ergodicity (where the volume occupied by the states scales logarithmically with system volume) could also be relevant in the delocalized phase [18]. Note however that it has been envisioned that this picture could be valid only up to a finite but very large length scale [31].In this letter, we study the Anderson transition (AT) in a family of random graphs [32][33][34], where a tunable parameter p allows us to interpolate continuously between the 1D Anderson model and the random regular graph model of infinite dimensionality. Our main tool is the single parameter scaling theory of localization [35]. It has been used as a crucial tool to interpret the numerical simulations of Anderson localization in finite dimensions [3,[36][37][38] and to achieve...

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From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

Cited by 98 publications

References 37 publications

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models: Wigner–Weisskopf approach

Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality

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