Shape analysis of the level-spacing distribution around the metal-insulator transition in the three-dimensional Anderson model

Varga, Imre; Hofstetter, E.; Schreiber, Michael; Pipek, János

doi:10.1103/physrevb.52.7783

Cited by 56 publications

(71 citation statements)

References 14 publications

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“…[23][24][25][26][27] Now, let us proceed with the shape analysis of the P (s) function using the other two quantities q and S str . As it has been shown in a number of cases 20,22) As we can see our data indicate a similar behavior as the random-Landau matrix model. This is another nice example of the presence of one-parameter scaling since all data can be described by a one-parameter family of P (s) functions.…”

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“…[23][24][25][26][27] Now, let us proceed with the shape analysis of the P (s) function using the other two quantities q and S str . As it has been shown in a number of cases 20,22) these parameters apart from showing scaling and fixed point in the critical regime, as a bonus enable us to determine the one-parameter family of P (s) functions describing the transition from e.g. GUE-like to Poissonian behavior.…”

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“…20, 21) Furthermore, we invoke the entropic moment of P (s), − s ln(s) and calculate, apart from − ln(q), another generalized Rényi-entropy S str = − s ln(s) + ln s 2 . [20][21][22] All of these quantities are numerically easily obtainable and they take all calculated neighboring level spacings into account. Since all of them show nice scaling behavior we present only that of J 0 .…”

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Spectral Properties of the Chalker-Coddington Network

Metzler

Varga

1998

J. Phys. Soc. Jpn.

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We numerically investigate the spectral statistics of pseudo-energies for the unitary network operator U of the Chalker-Coddington network. The shape of the level spacing distribution as well the scaling of its moments is compared to known results for quantum Hall systems. We also discuss the influence of multifractality on the tail of the spacing distribution.KEYWORDS: Chalker-Coddington model, level spacing distribution, scaling, quantum Hall systems, multifractality §1. IntroductionThe Chalker-Coddington network model, 1) representing systems of independent two-dimensional (2D) electrons in a strong perpendicular magnetic field and smooth disorder potential, is a convenient tool to investigate the quantum Hall universality class. 1-3) It has been used to determine various quantities characterizing the localization-delocalization (LD) transition taking place between the quantized plateaus of the Hall conductance. 4) The first approach was to use the transfer matrix method to numerically determine the characteristic exponent ν ≈ 2.35 for the divergence of the localization length ξ ∝ |E − E c | −ν when the energy of the electron E approaches the critical energy E c . 1, 5) At the critical energy itself the network model was used to calculate critical wave functions.Those were subjected to a multifractal analysis and the critical exponent α 0 ≈ 2.27 describing the scaling of typical value of their squared amplitude with the system size, exp |Ψ c | 2 ∝ L −α 0 , was determined. 2) The model can also very conveniently be used to simulate the diffusion of wave packets using a unitary time evolution operator U . By doing so it was possible to determine the scaling behavior of the local density of states. 6,7) Recently, a new approach in the investigation of network models was suggested. It was argued that network models can be used to determine the statistics of the eigenvalue spectrum of their underlying system by investigating the spectrum of the unitary network operator U . 3) So far * Permanent address: Department of Theoretical Physics, Institut of Physics, Technical University of Budapest, H-1521 Budapest, Hungary 1 the numerical investigation of the pseudo-energy statistics has been concentrating on the number variance Σ 2 (N ) = (n − n ) 2 of an energy interval containing on average N = n levels and the compressibility χ = lim N →∞ lim L→∞ dΣ 2 dN . It was found that the prediction by Chalker et al 8) that χ = (d − D 2 )/(2d), where D 2 is the correlation dimension, can be confirmed. Furthermore, it was seen that the level spacing distribution function P (s), denoting the probability to find two consecutive levels separated by an energy ǫ = s∆, ∆ being the average level spacing, deviates in its tail from the Wigner surmise. 3) This was expected at the critical energy 9-11) and was also seen in other numerical investigations. [12][13][14] In this paper we are going to take a closer look at the level spacing distribution P (s). We will show that by using the spectrum of the unitary network operator U all the kno...

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Spectral Properties of the Chalker-Coddington Network

Metzler

Varga

1998

J. Phys. Soc. Jpn.

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“…[34], in QCD the unfolded level spacing distributions P λ (s) found in different parts of the spectrum lie on a universal path in the space of probability distributions, a path that is independent of volume, temperature and lattice spacing. This can be seen by plotting two different parameters of P λ (s) against each other ("shape analysis" [50]), thus taking a two-dimensional projection of this path, which should therefore yield a universal curve. In figure 5 we plot the second moment of the unfolded spacing distribution, s 2 , against I s 0 , with each data point corresponding to a specific point in the spectrum, and to a given system size and value of the gauge coupling.…”

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Deconfinement, chiral transition and localisation in a QCD-like model

Giordano¹,

Katz²,

Kovács³

et al. 2017

J. High Energ. Phys.

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Abstract:We study the problems of deconfinement, chiral symmetry restoration and localisation of the low Dirac eigenmodes in a toy model of QCD, namely unimproved staggered fermions on lattices of temporal extension N T = 4. This model displays a genuine deconfining and chirally-restoring first-order phase transition at some critical value of the gauge coupling. Our results indicate that the onset of localisation of the lowest Dirac eigenmodes takes place at the same critical coupling where the system undergoes the first-order phase transition. This provides further evidence of the close relation between deconfinement, chiral symmetry restoration and localisation of the low modes of the Dirac operator on the lattice.

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“…Another reason why a large number of numerical simulations of the LSD at the transition 8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 were carried out during the past decade is the controversy that existed over the large-spacing tail of the critical LSD. Conclusive demonstration 12,13,21 that this tail is Poissonian, i.e., that there is no repulsion between the levels with spacings much larger than the mean value, 1 rather than super-Poissonian, 23 implying that repulsion is partially preserved, required a very high accuracy of the simulations.…”

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confidence: 99%

Renormalization group approach to energy level statistics at the integer quantum Hall transition

2003

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We extend the real-space renormalization group (RG) approach to the study of the energy level statistics at the integer quantum Hall (QH) transition. Previously it was demonstrated that the RG approach reproduces the critical distribution of the power transmission coefficients, i.e., two-terminal conductances, Pc(G), with very high accuracy. The RG flow of P (G) at energies away from the transition yielded the value of the critical exponent, ν, that agreed with most accurate large-size lattice simulations. To obtain the information about the level statistics from the RG approach, we analyze the evolution of the distribution of phases of the amplitude transmission coefficient upon a step of the RG transformation. From the fixed point of this transformation we extract the critical level spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that away from the transition the LSD crosses over towards the Poisson distribution. Studying the change of the LSD around the QH transition, we check that it indeed obeys scaling behavior. This enables us to use the alternative approach to extracting the critical exponent, based on the LSD, and to find ν = 2.37 ± 0.02 very close to the value established in the literature. This provides additional evidence for the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.

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Shape analysis of the level-spacing distribution around the metal-insulator transition in the three-dimensional Anderson model

Cited by 56 publications

References 14 publications

Spectral Properties of the Chalker-Coddington Network

Spectral Properties of the Chalker-Coddington Network

Deconfinement, chiral transition and localisation in a QCD-like model

Renormalization group approach to energy level statistics at the integer quantum Hall transition

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