Eigenfunction distribution for the Rosenzweig-Porter model

Bogomolny, E.; Sieber, Martin

doi:10.1103/physreve.98.032139

Cited by 53 publications

(82 citation statements)

References 24 publications

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“…In order to uncover the origin of this counterintuitive result we first use the matrix inversion trick suggested in [19] to rewrite the eigenproblem in the coordinate basis in an alternative way. Furthermore we develop the self-consistent method of eigenvector calculation based on the averaging over off-diagonal matrix elements, allowing one to access wave-function statistics and, in particular, confirming the phenomenological ansatz known in the literature for RP ensemble [42][43][44] (see also [38]). Unlike the standard renormalization group analysis [22,49] or the Wigner-Weisskopf approximation [42] used in the literature before this self-consistent method is sensitive to the hopping correlations.…”

Section: Introductionsupporting

confidence: 56%

Robustness of delocalization to the inclusion of soft constraints in long-range random models

Nosov

Khaymovich

2019

Phys. Rev. B

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Motivated by the constrained many-body dynamics, the stability of the localization-delocalization properties to the inclusion of the soft constraints is addressed in random matrix models. These constraints are modeled by correlations in long-ranged hopping with Pearson correlation coefficient different from zero or unity. Counterintuitive robustness of delocalized phases, both ergodic and (multi)fractal, in these models is numerically observed and confirmed by the analytical calculations. First, matrix inversion trick is used to uncover the origin of such robustness. Next, to characterize delocalized phases a method of eigenstate calculation, sensitive to correlations in long-ranged hopping terms, is developed for random matrix models and approved by numerical calculations and previous analytical ansatz. The effect of the robustness of states in the bulk of the spectrum the inclusion of to soft constraints is generally discussed for single-particle and many-body systems.

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Section: Introductionsupporting

confidence: 56%

Robustness of delocalization to the inclusion of soft constraints in long-range random models

Nosov

Khaymovich

2019

Phys. Rev. B

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“…(4) was mentioned in Ref. [64]. For correlated long-range hopping, specifically for the translation-invariant hopping described in Section III, the spectrum ofĵ is often non-compact, with an infinite support set in the energy space.…”

Section: Localization Criteria For Models Withmentioning

confidence: 99%

“…coordinate) basis does not imply invariance of wave function statistics under basis rotation. In some models (see, e.g., [64]) weak ergodicity may survive beyond the condition (3), showing that (3) is only the sufficient but not the necessary condition of weak ergodicity and, thus, ∆ = ∆ p is the lower bound for the ergodic transition between the weakly ergodic extended phase and the non-ergodic phases (localized or extended).…”

Section: Localization Criteria For Models Withmentioning

confidence: 99%

Correlation-induced localization

2019

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A new paradigm of Anderson localization caused by correlations in the long-range hopping along with uncorrelated on-site disorder is considered which requires a more precise formulation of the basic localizationdelocalization principles. A new class of random Hamiltonians with translation-invariant hopping integrals is suggested and the localization properties of such models are established both in the coordinate and in the momentum spaces alongside with the corresponding level statistics. Duality of translation-invariant models in the momentum and coordinate space is uncovered and exploited to find a full localization-delocalization phase diagram for such models. The crucial role of the spectral properties of hopping matrix is established and a new matrix inversion trick is suggested to generate a one-parameter family of equivalent localization/delocalization problems. Optimization over the free parameter in such a transformation together with the localization/delocalization principles allows to establish exact bounds for the localized and ergodic states in long-range hopping models. When applied to the random matrix models with deterministic power-law hopping this transformation allows to confirm localization of states at all values of the exponent in power-law hopping and to prove analytically the symmetry of the exponent in the power-law localized wave functions.

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“…The statistical properties of eigenstates have been characterized and studied in multiple ways. The distributions of eigenstates have been examined directly, e.g, for quantum billiards [4][5][6][7][8][9][10][32][33][34][35], for many-body systems [34,36], and for quantum maps [11][12][13][14][15][16] and random-matrix ensembles [37][38][39]. The maxima of random waves and chaotic eigenstates have also been considered [15,40].…”

Section: Introductionmentioning

confidence: 99%

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

2019

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Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems -the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction -the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic", in the sense that their eigenfunction statistics deviate from random matrix theory. arXiv:1905.03099v2 [cond-mat.stat-mech]

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Eigenfunction distribution for the Rosenzweig-Porter model

Cited by 53 publications

References 24 publications

Robustness of delocalization to the inclusion of soft constraints in long-range random models

Robustness of delocalization to the inclusion of soft constraints in long-range random models

Correlation-induced localization

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

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