Critical statistics in a power-law random-banded matrix ensemble

Varga, Imre; Braun, Daniel

doi:10.1103/physrevb.61.r11859

Cited by 60 publications

(63 citation statements)

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“…In order to fully represent the mesoscopic systems we introduce an explicit dependence on dimensionality d in the widely studied power-law random banded matrix (PRBM) ensemble [5,7,9,11,[26][27][28][29][30][31][32][33][34][35][36] (for closely related models see also Ref. [37]).…”

Section: The Modelmentioning

confidence: 99%

“…Although a great progress in understanding critical properties of the 1d long-range random hopping Hamiltonian has recently been made [5,7,9,11,[26][27][28][29][30][31][32][33][34][35][36][37] explicit results for the 2d and 3d systems are still lacking. Our aim was to investigate the two previously mentioned important quantities, the correlation dimension and nearest level spacing distribution, at the MIT, which have been left unexplored.…”

Section: Introductionmentioning

confidence: 99%

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Critical properties in long‐range hopping Hamiltonians

Cuevas¹

2004

Physica Status Solidi (b)

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Section: The Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Critical properties in long‐range hopping Hamiltonians

Cuevas¹

2004

Physica Status Solidi (b)

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confidence: 96%

“…At the critical value α = 1, we expect to have multifractal eigenstates and critical level statistics, intermediate between Poisson and Wigner-Dyson. Recent numerical calculations at criticality and for b = 1 have shown that the nearest level spacing distribution differs from the typical one at a metal-insulator transition [15].Our aim in this paper is to explore in detail the characteristics of the PRBM model in the interesting regime around α ≈ 1 and for the case b = 1 where theoretical treatments are not strictly applicable. We do so by studying systematically the nearest level spacing distribution function P (s), the behavior of the Green function (GF) as a function of distance and the fractal properties of the states.…”

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confidence: 99%

Anomalously Large Critical Regions in Power-Law Random Matrix Ensembles

2001

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We investigate numerically the power-law random matrix ensembles. Wavefunctions are fractal up to a characteristic length whose logarithm diverges asymmetrically with different exponents, 1 in the localized phase and 0.5 in the extended phase. The characteristic length is so anomalously large that for macroscopic samples there exists a finite critical region, in which this length is larger than the system size. The Green's functions decrease with distance as a power law with an exponent related to the correlation dimension. [5,6]. As the model describes a whole family of critical theories, it is also interesting for the study of intermediate spectral statistics [7][8][9].The model is represented by N ×N real symmetric matrices whose entries are randomly drawn from a normal distribution with zero mean and a variance depending on the distance of the matrix element from the diagonalFrom field theoretical considerations [1,5,10,11], the PRBM model was shown to undergo a sharp transition at α = 1 from localized states for α > 1 to delocalized states for α < 1. This transition is supposed to be similar to an Anderson metal-insulator transition, presenting multifractality of eigenfunctions and non-trivial spectral compressibility at criticality. The parameter b plays a role analogous to dimensionality establishing the character of the transition. From a perturbative treatment of the non-linear σ-model and from renormalization group calculations Mirlin considered different regimes depending on the exponent α. These models are only justified for b ≫ 1, although the other limiting case can also be studied following Levitov [12]. For α ≥ 3/2 the eigenstates are localized, but in contrast to usual tight-binding models they do not decay exponentially, but with a power-law with the same exponent α as the hopping matrix elements [4,13]. For 1 < α < 3/2 the states are still localized, but with a different typical length. In both cases, the wavefunctions are expected to have integrable powerlaw tails and the dimensionality to be equal to zero. For very large system sizes, the nearest neighbor normalized level spacings s should be distributed according to the Poisson law, P P (s) = exp(−s).For α < 1, the previous theories predict that all states should be delocalized, independently of the value of b. The inverse participation ratio (IPR) should then be proportional to the system size, as for Gaussian orthogonal ensembles (GOE), although higher order cumulants of the IPR should not necessarily behave in the same way as GOE [10]. For extended states, the nearest neighbor level spacing distribution approximately follows WignerDyson law [14] , P WD (s) = (πs/2) exp(−πs 2 /4) . At the critical value α = 1, we expect to have multifractal eigenstates and critical level statistics, intermediate between Poisson and Wigner-Dyson. Recent numerical calculations at criticality and for b = 1 have shown that the nearest level spacing distribution differs from the typical one at a metal-insulator transition [15].Our aim in this paper is to expl...

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“…Ref. 17 ) typical for the spectra of matrices from the Gaussian orthogonal ensemble, which do not show any change in the level-spacing statistics (the Wigner surmise) across the band 18 . The inset in Fig.…”

Section: It Is Knownmentioning

confidence: 99%

Statistical properties of the critical eigenstates in power-law random banded matrices across the band

2005

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The level-spacing distribution in the tails of the eigenvalue bands of the power-law random banded matrix (PRBM) ensemble have been investigated numerically. The change of level-spacing statistics across the band is examined for different coupling strengths and compared to the density of states for the different systems. It is confirmed that, by varying the eigenvalue region, the same level-spacing statistics can be reached as by varying the coupling strength.PACS numbers: 71.30.+h, 72.15.Rn, 71.55.Jv The Anderson metal-insulator transition (MIT) 1 is a phenomenon of major physical importance that continues to attract a substantial research effort 2,3 . With just short-range interactions, localization-delocalization (LD) transitions are only found in systems with dimensionality, D, greater than two 4 . However, with the addition of long-range interactions, or correlations between the short-range interactions, it is possible to study the LD transition in systems with dimensionality less than two 5 . In this respect, power-law random-banded matrices (PRBMs), that exhibit this transition, have recently attracted much attention 6,7,8,9,10 . The PRBM ensemble was introduced by Mirlin et. al.11 and, in the real case, is defined as the ensemble of N × N random symmetric matrices,Ĥ. The PRBM elements, H ij , are randomly drawn from a Gaussian distribution, centred around zero, with a variance governed by a power-law decay:where α and b ∈ (0, ∞) are parameters. RegardingĤ as a Hamiltonian, the eigenvalues, E, are energies. For α = 1, it has been shown 11 that all the eigenstates of these matrices are critical (i.e. at the LD transition). The parameter b is inversely related to the coupling strength between the nodes. In the limit b ≫ 1 and α = 1, the PRBM critical states are analogous to the critical states at the Anderson transition with D = 2 + ǫ and ǫ ≪ 1, and for b ≪ 1 to those found in the Anderson model with D ≫ 1 12 . By varying b, it is possible to access a set of different critical theories parameterized by dimension (2 < D < ∞) in the conventional Anderson transition. This ability to examine Anderson transitions in different dimensions, in the same effectively one-dimensional model, makes the study of PRBMs a powerful method to make progress in this rich field. As well as being an analogue for the study of important transitions elsewhere, the PRBM is physically important in its own right and has been applied to the study of the finite-temperature Luttinger liquid 13 , the coherent propagation of two interacting particles in a 1D weak random potential 14 and other problems 7,15 . Recently, it has also been realized that, with the addition of chirality, the PRBM also describes the LD transition of quark zero modes in QCD 8 . In the QCD vacuum, the quark zeromode wavefunction decay has a power-law dependence and long-range hopping between sites is possible. In this model, the eigenvalues away from the centre of the spectral band are not affected by the chiral structure; this makes it relevant to the analysis ...

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Critical statistics in a power-law random-banded matrix ensemble

Cited by 60 publications

References 29 publications

Critical properties in long‐range hopping Hamiltonians

Critical properties in long‐range hopping Hamiltonians

Anomalously Large Critical Regions in Power-Law Random Matrix Ensembles

Statistical properties of the critical eigenstates in power-law random banded matrices across the band

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