Anderson Model on Bethe Lattices: Density of States, Localization Properties and Isolated Eigenvalue

Biroli, Giulio; Semerjian, Guilhem; Tarzia, Marco

doi:10.1143/ptps.184.187

Cited by 66 publications

(113 citation statements)

References 22 publications

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“…It is known from previous studies that the RRG ensemble of sparse random matrices belongs to the GOE universality class (with Wigner-Dyson-level statistics and fully delocalized eigenvectors) [49,50]. Hence, in absence of the hopping rates connecting sites on adjacent layers (t = 0), each layer i corresponds to a M × M GOElike block, with energy spectra akin to semicircle laws of width 4γ √ k [51] and centered around i . When the interlayer hopping matrix elements is turned on (t > 0), these GOE-like blocks become then coupled along the chain.…”

Section: The Modelmentioning

confidence: 99%

“…The model (1) allows, in principle, for an exact solution which yield the probability distribution function of the diagonal elements of the resolvent matrix, defined as G(z) = (H − zI) −1 [51,53]. In order to obtain the recursive equations, the key objects are the so-called cavity Green's functions, i.e., the diagonal elements on a given site i of the resolvent matrix of the modified Hamiltonian where the edge between the site (i, p) and one of its neighbors has been removed.…”

Section: Exact Recursion Relationsmentioning

confidence: 99%

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Toy model for anomalous transport and Griffiths effects near the many-body localization transition

Schiró

Tarzia

2020

Phys. Rev. B

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We introduce and study a toy model for anomalous transport and Griffiths effects in one dimensional quantum disordered isolated systems near the Many-Body Localization (MBL) transitions. The model is constituted by a collection of 1d tight-binding chains with on-site random energies, locally coupled to a weak GOE-like perturbation, which mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in absence of such a coupling the model is localized as expected for the one dimensional Anderson model, increasing the coupling with the GOE perturbation we find a delocalization transition to a conducting one driven by the proliferation of quantum avalanches which does not fit the standard paradigm of Anderson localization. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few, rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponential and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find sub-diffusive transport and power-laws decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region.Concomitantly, the a.c. conductivity vanishes near zero frequency with an anomalous power-law.

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Section: The Modelmentioning

confidence: 99%

Section: Exact Recursion Relationsmentioning

confidence: 99%

Toy model for anomalous transport and Griffiths effects near the many-body localization transition

Schiró

Tarzia

2020

Phys. Rev. B

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“…For the explicit form of this distribution, we refer the reader to 29 . For k = 2, where each node is connected precisely to three neighbors, all eigenfunctions become localized provided W > W c 17.4 3,11 . The value of W c is the same for the infinitely large Cayley tree and the RRG.…”

Section: The Anderson Model On a Regular Randommentioning

confidence: 99%

Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

Metz

Castillo

2017

Phys. Rev. B

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We calculate the level compressibility χ(W, L) of the energy levels inside [−L/2, L/2] for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in [−W/2, W/2]. We show that χ(W, L) approaches the limit lim L→0 + χ(W, L) = 0 for a broad interval of the disorder strength W within the extended phase, including the region of W close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erdös-Rényi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.

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“…The resolvent elements at z and −z * are both calculated on the same cavity graph [22,35], defined as the graph in which an arbitrary node and all its edges are deleted. Equation (34) has a simpler form when compared to Eq.…”

Section: The Statistical Properties Of the Indexmentioning

confidence: 99%

“…The quantity g represents the diagonal elements of the resolvent on the cavity graph [22,35]. Substituting Eq.…”

Section: B Random Regular Graphsmentioning

confidence: 99%

Index statistical properties of sparse random graphs

Metz

Stariolo

2015

Phys. Rev. E

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Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P N (K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P N (K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as ln N , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

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Anderson Model on Bethe Lattices: Density of States, Localization Properties and Isolated Eigenvalue

Cited by 66 publications

References 22 publications

Toy model for anomalous transport and Griffiths effects near the many-body localization transition

Toy model for anomalous transport and Griffiths effects near the many-body localization transition

Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

Index statistical properties of sparse random graphs

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