Universality in the 2D localization problem

Fastenrath, Ulrich; Adams, Gunnar; Bundschuh, Ralf; Hermes, Thomas; Raab, Boris; Schlosser, Ingeborg; Wehner, Thomas; Wichmann, Thomas

doi:10.1016/0378-4371(91)90384-o

Physica A: Statistical Mechanics and its Applications

1991

DOI: 10.1016/0378-4371(91)90384-o

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Universality in the 2D localization problem

Ulrich Fastenrath

Gunnar Adams

Ralf Bundschuh

et al.

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Cited by 30 publications

(42 citation statements)

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“…The localization properties of symplectic models have previously been analyzed numerically [12,[15][16][17][18][19][20] from which a metal-insulator transition can be inferred. The most recent calculations of Fastenrath [20,21] report a critical exponent ν = 2.75 for the localization length at the band centre, E/V = 0, together with a critical disorder W c = 5.74 V for a constant probability distribution of the on-site disorder potentials {ε m } and a spin-orbit strength S = 0.5.…”

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Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Schweitzer¹

1995

J. Phys.: Condens. Matter

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The multifractal properties of electronic eigenstates at the metal-insulator transition of a two-dimensional disordered tight-binding model with spin-orbit interaction are investigated numerically. The correlation dimensions of the spectral measure D 2 and of the fractal eigenstate D 2 are calculated and shown to be related by D 2 = 2 D 2 . The exponent η = 0.35 ± 0.05 describing the energy correlations of the critical eigenstates is found to satisfy the relation η = 2 − D 2 .PACS numbers: 71.30.+h, 64.60.Ak, 72.15.Rn Typeset using REVT E X 1 Electronic states in disordered systems are known to be either localized or extended. At T = 0 the so called mobility edge separates insulating (localized) from current carrying (extended) states. Recently, the electronic properties directly at the critical point have received increasing attention [1][2][3][4][5][6]. Two-dimensional (2d) systems are very well suited for numerical investigations of the electronic eigenstates at a critical point. The only systems, however, that exhibit a complete Anderson transition in 2d are models with symplectic symmetry. This is in contrast to the orthogonal and the unitary case where all states are localized (weak localization) and the quantum Hall (QHE) systems where only localized states with a diverging localization length at some singular energies can be observed.In a recent paper, Chalker and coworkers [7] reported on numerical investigations of eigenstate fluctuations and correlations near the mobility edge of a two-dimensional tight-binding model with spin-orbit coupling. It was found that the probability amplitude distribution exhibits multifractal behavior which can be characterized by a set of generalized dimensions D q . The fractal spatial structure of the wavefunctions also shows up in the eigenfunction correlations between states close in energy. A similar behavior was found also in the quantum Hall model [1,[8][9][10], i.e. a two-dimensional disordered system of non-interacting electrons where a strong magnetic field causes the localization length to diverge at the centres of the Landau bands with a universal exponent [11]. For these critical states, characteristic quantities like the generalized correlation dimension of the wavefunction D 2 = 1.62 [8,9], the corresponding generalized dimension of the spectral measure D 2 = 0.81 [3], the exponent η = 0.38 [1,3] governing the energy eigenfunction correlations, and α 0 = 2.3 [9,10] which for −1 < ∼ q < ∼ 1 determines the so called f (α)-distribution are known, partly with sufficient precision. This is, however, not the case for the spin-orbit system mentioned above, mainly because of the small system sizes considered so far (M ≤ 18 a) [7]. The authors of Ref. [7] used a model proposed by Evangelou and Ziman [12] which represents a two-dimensional disordered electronic system with spin-orbit interaction.In this letter, a different lattice model with symplectic symmetry is investigated for system sizes up to 150 a × 150 a, where a is the lattice constant. It is fou...

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confidence: 99%

Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Schweitzer¹

1995

J. Phys.: Condens. Matter

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“…Anyway, in two dimensions, these two classes do not show the Anderson transition. In contrast, the symplectic class is known to exhibit the Anderson transition in two-dimensional systems [2][3][4][5][6].…”

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“…For the states at the center of the band (E = 0), the critical point of this model has been estimated as W c = 5.74V by the finite-size scaling method [3,5]. We confine ourselves in this letter to the case of the band center (E = 0) for simplicity.…”

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confidence: 99%

“…It has been proved that in two dimensions, such a system has localized states if the disorder is strong enough compared to the spin-orbit coupling [7]. Numerical and analytical studies including the finite-size scaling method [2][3][4][5][6] and the renormalization group theory [8][9][10][11] have been made on the critical behavior of the Anderson transition in two-dimensions. However, several important issues still remain to be clarified.…”

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confidence: 99%

“…Furthermore, we have made simulations for other types of distribution of site energies, namely, the Gaussian and the binary potential distributions [5,13]. The exponents at the critical points are estimated to be α = 0.85 ± 0.02 for the Gaussian distribution and α = 0.85 ±0.03 for the binary distribution.…”

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Diffusion of electrons in two-dimensional disordered symplectic systems

Kawarabayashi

Ohtsuki

1996

Phys. Rev. B

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Diffusion of electrons in two-dimensional disordered systems with spin-orbit interactions is investigated numerically. Asymptotic behaviors of the second moment of the wave packet and of the temporal auto-correlation function are examined. At the critical point, the auto-correlation function exhibits the power-law decay with a non-conventional exponent α which is related to the fractal structure in the energy spectrum and in the wave functions. In the metallic regime, the present results imply that transport properties can be described by the diffusion equation for normal metals. 71.30.+h, 71.55.J, 73.20.Dx, Typeset using REVT E X 1

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Review of recent progress on numerical studies of the Anderson transition

Ohtsuki¹,

Slevin²,

Kawarabayashi³

1999

Ann. Phys.

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A review of recent progress in numerical studies of the Anderson transition in three dimensional systems is presented. From high precision calculations the critical exponent ν for the divergence of the localization length is estimated to be ν = 1.57 ± 0.02 for the orthogonal universality class, which is clearly distinguished from ν = 1.43 ± 0.03 for the unitary universality class. The boundary condition dependences of some quantities at the Anderson transition are also discussed.

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Universality in the 2D localization problem

Cited by 30 publications

References 18 publications

Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Diffusion of electrons in two-dimensional disordered symplectic systems

Review of recent progress on numerical studies of the Anderson transition

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