Critical properties in long‐range hopping Hamiltonians

Cuevas, E.

doi:10.1002/pssb.200404783

Cited by 7 publications

(15 citation statements)

References 65 publications

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“…In order to reconcile our result with the apparent power-law scaling found in [15], we present in Fig. 2 the anomalous part of I 2 as a function of ln L and of ln ln L.…”

Section: Elements Reads [16]supporting

confidence: 52%

Criticality without self-similarity: a 2D system with random long-range hopping

Ossipov

Rushkin

Cuevas

2011

J. Phys.: Condens. Matter

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We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal-insulator transition -its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size. Instead, in one of the considered limits our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions. PACS numbers: 73.20.Fz, 72.15.Rn, 05.45.Df Critical wave functions have been a subject of intense theoretical research during the last two decades (see Ref.[1] for a recent review). Recently, they have also attracted a lot of attention from experimentalists [2][3][4]. Starting with the innovative work by Wegner [5], it has become customary to characterize the critical wave functions by their multifractal dimensions d q , which determine the scaling of the moments I q of the wave functions with system size L:Here the averaging is performed over different disorder realizations, as well as over a small energy window. The power-law dependence of the moments with exponents d q = 0 different from the dimensionality of the space d implies a self-similarity of fluctuations of the wave function amplitudes on different spatial scales. Among all quantum disordered systems showing critical behavior, two-dimensional (2D) systems occupy a special place. It is commonly believed that at the critical point such systems possess conformal invariance and should be described by conformal field theories (CFTs), similar to conventional phase transitions in 2D. Although no direct mapping of any disordered quantum system onto a CFT is available at the moment, this conjecture of conformal invariance imposes certain constraints on correlation functions and critical exponents. The validity of these predictions has been checked by numerical simulations for various 2D disordered systems [6,7], and no significant deviations have been found so far. One such prediction is a relation between the corner and the surface multifractalities [8]:where the exponents ∆ In the present work we study a random matrix model describing a long-range hopping of a particle on a two-dimensional lattice. Similar to its one-dimensional (1D) counterpart [9], the model undergoes the Anderson metal-insulator transition. At the transition point, we compute the fractal dimensions perturbatively in the regime of strong criticality and find thatThe dependence on θ is inverse to that predicted by CFT in Eq. (2). In the opposite regime of weak criticality, the scaling of the moments of the wave functions is found to be consistent not with the standard power-law (1), but with a logarithmic dependenceConsequently, our model is a rather surprising example of a disordered 2D critical system, whose...

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“…In order to reconcile our result with the apparent power-law scaling found in [15], we present in Fig. 2 the anomalous part of I 2 as a function of ln L and of ln ln L.…”

Section: Elements Reads [16]supporting

confidence: 52%

Criticality without self-similarity: a 2D system with random long-range hopping

Ossipov

Rushkin

Cuevas

2011

J. Phys.: Condens. Matter

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“…The statistics of scattering phases, Wigner delay times and resonance widths in the presence of one external wire have been discussed in [21,22]. Related studies concern dynamical aspects [23], the case with no on-site energies [24], and the case of power-law hopping terms in dimension d > 1 [25,26,27]. In this paper, we consider the PRBM in a ring geometry (dimension d = 1 with periodic boundary conditions) in the presence of two external wires to measure the transmission properties.…”

Section: Model and Observablesmentioning

confidence: 99%

Statistics of the two-point transmission at Anderson localization transitions

Monthus

Garel

2009

Phys. Rev. B

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At Anderson critical points, the statistics of the two-point transmission TL for disordered samples of linear size L is expected to be multifractal with the following properties [Janssen et al PRB 59, 15836 (1999)] : (i) the probability to have TL ∼ 1/L κ behaves as L Φ(κ) , where the multifractal spectrum Φ(κ) terminates at κ = 0 as a consequence of the physical bound TL ≤ 1; (ii) the exponents X(q) that govern the moments T q L ∼ 1/L X(q) become frozen above some threshold: X(q ≥ qsat) = −Φ(κ = 0), i.e. all moments of order q ≥ qsat are governed by the measure of the rare samples having a finite transmission (κ = 0). In the present paper, we test numerically these predictions for the ensemble of L×L power-law random banded matrices, where the random hopping Hi,j decays as a power-law (b/|i − j|) a . This model is known to present an Anderson transition at a = 1 between localized (a > 1) and extended (a < 1) states, with critical properties that depend continuously on the parameter b. Our numerical results for the multifractal spectra Φ b (κ) for various b are in agreement with the relation Φ(κ ≥ 0) = 2ˆf (α = d + κ 2 ) − d˜in terms of the singularity spectrum f (α) of individual critical eigenfunctions, in particular the typical exponents are related via the relation κtyp(b) = 2(αtyp(b) − d). We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.I.

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“…(16). At b = 5, for example, we find χ = 0.00012 ± 0.00016, a 1 = 1.007 ± 0.007 and a 2 = 0.1573 ± 0.003, while the standard random matrix theory prediction (16) corresponds to χ = 0, a 1 = 1 and a 2 = 0.1598. The values of χ, obtained in this way for different values of b, are presented in Fig.…”

Section: Level Number Variance At B ≫ 1 and The Transition Pointmentioning

confidence: 70%

“…The existence of the Anderson transition in such one-dimensional systems is related to the long-range nature of the hopping amplitudes. In our recent work [14], we studied the scaling of the moments of the eigenstates in a two-dimensional generalization of the powerlaw banded random matrix model [15,16]. In this ensemble, the matrix elements of the Hamiltonian H mn are complex independent Gaussian random variables, whose mean values are equal to zero and whose variances are determined by the distance between sites of a two-dimensional lattice:…”

Section: Introductionmentioning

confidence: 99%

Level-number variance and spectral compressibility in a critical two-dimensional random-matrix model

Ossipov

Rushkin²,

Cuevas³

2012

Phys. Rev. E

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We study level-number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter b. We find analytically that at small values of b the level number variance behaves linearly, with the compressibility χ between 0 and 1, which is typical for critical systems. For large values of b, we derive that χ=0, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of b at which the transition between these two phases occurs.

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Critical properties in long‐range hopping Hamiltonians

Abstract: Some properties of d-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2 (for d = 2) and the nearest level spacing distribution

Cited by 7 publications

References 65 publications

Criticality without self-similarity: a 2D system with random long-range hopping

Criticality without self-similarity: a 2D system with random long-range hopping

Statistics of the two-point transmission at Anderson localization transitions

Level-number variance and spectral compressibility in a critical two-dimensional random-matrix model

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