Does Broken Time Reversal Symmetry Modify the Critical Behavior at the Metal-Insulator Transition in 3-Dimensional Disordered Systems?

Hofstetter, E.; Schreiber, Michael

doi:10.1103/physrevlett.73.3137

Cited by 74 publications

(69 citation statements)

References 20 publications

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“…This is similar to the insulating regime, although the decay rate A c is larger than unity due to the level repulsion. The power law with the exponent α ≈ 1.2, which was recently obtained [17] by an analysis of the shape of P (s) in the range 0 < s < 5 for system sizes L ≤ 21 deviates from our results for s > ∼ 4. The linear asymptotic behavior of ln I c (s) is in contrast to the power law with α ≈ 1.31 obtained numerically [5] for smaller systems L ≤ 12.…”

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confidence: 55%

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Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition

1997

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The nearest-neighbor level spacing distribution is numerically investigated by directly diagonalizing disordered Anderson Hamiltonians for systems of sizes up to 100×100×100 lattice sites. The scaling behavior of the level statistics is examined for large spacings near the delocalization-localization transition and the correlation length exponent is found. By using high-precision calculations we conjecture a new interpolation of the critical cumulative probability, which has size-independent asymptotic form ln I(s) ∝ −s α with α = 1.0 ± 0.1. The statistical fluctuations in energy spectra of disordered quantum systems attract at present much attention [1,2,3,4,5]. It is known that by increasing the fluctuations of a random potential the one-electron states undergo a localization transition, which is the origin of the Anderson metal-insulator transition (MIT) [6]. The influence of the disorder on the wave functions is reflected by the mutual correlations between the corresponding energy levels, so that the statistics of energy levels is sensitive to the MIT. In the metallic limit the statistics of energy spectra can be described by the random-matrix theory (RMT) developed by Wigner and Dyson [7,8]. This was shown by solving the zero-mode nonlinear σ-model using the supersymmetric formalism [9]. Later, perturbative corrections to the two-level correlation function obtained in the RMT were evaluated in the diffusive regime by the impurity diagram technique [10]. In the insulating regime, when the degree of disorder W is much larger than the critical value W c , the energy levels of the strongly localized eigenstates fluctuate as independent random variables.An important quantity for analyzing the spectral fluctuations is the nearest-neighbor level spacing distribution P (s). It contains information about all of the n−level correlations. In the metallic regime P (s) is very close to the Wigner surmise P W (s) = π s/2 exp −π s 2 /4 [11] (s is measured in units of the mean level spacing ∆). In the localized regime the spacings are distributed according to the Poisson law, P P (s) = exp(−s), because the levels are completely uncorrelated. The study of the crossover of P (s) between the Wigner and the Poissonian limits which accompanies the disorder-induced MIT in threedimensional system (3D) was started in Ref. [12] and became the subject of several subsequent investigations [1,2,5,13,14].It was suggested earlier [1] that P (s) exhibits critical behavior and should be size-independent at the MIT. Investigating the finite-size scaling properties of P (s) provides not only an alternative method for locating the transition [2], but allows also to determine the critical behavior of the correlation length [14]. A technical advantage of the method is that one needs to compute only energy spectra and not eigenfunctions and/or the conductivity. On the other hand, a large number of realizations of the random potential has to be considered. In comparison with the well-established transfer-matrix method [15] by which one approache...

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“…The latter distribution decays faster than the Poissonian (α = 1), but slower than the Wigner surmise (α = 2). Several numerical calculations for the 3D Anderson model were recently performed [5,17] in order to analyze P c (s). The results were found to be consistent with the latter of the above suggestions with an exponent α ≈ 1.2−1.3 (ν ≈ 1.5).…”

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confidence: 99%

Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition

1997

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“…At the critical disorder W c = 16.5, the mobility edge occurs at E c = 0, all states with |E| > 0 are localized [3,4] and states with E = 0 are multifractal [3,17]. The value of ν has been computed from the non-linear sigma-model [19], transfermatrix methods [2,6], Green functions methods [2], and energy-level statistics [5,20]. Here we have chosen ν = 1.3, which is in agreement with experimental results in Si:P [9] and the numerical data of Ref.…”

Section: The Anderson Model Of Localizationmentioning

confidence: 99%

Thermoelectric transport properties in disordered systems near the Anderson transition

1999

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Abstract. We study the thermoelectric transport properties in the three-dimensional Anderson model of localization near the metal-insulator transition (MIT). In particular, we investigate the dependence of the thermoelectric power S, the thermal conductivity K, and the Lorenz number L0 on temperature T . We first calculate the T dependence of the chemical potential µ from the number density n of electrons at the MIT using an averaged density of states obtained by diagonalization. Without any additional approximation, we determine from µ(T ) the behavior of S, K and L0 at low T as the MIT is approached. We find that σ and K decrease to zero at the MIT as T → 0 and show that S does not diverge. Both S and L0 become temperature independent at the MIT and depend only on the critical behavior of the conductivity.

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“…Anderson tight-binding Hamiltonian on a d-dimensional lattice L d with or without a magnetic field is defined as [8,9,10],…”

Section: Anderson Hamiltonians and Random Matricesmentioning

confidence: 99%

Critical statistics at the mobility edge of QCD Dirac spectra

Nishigaki¹,

Giordano²,

Kovács³

et al. 2014

Proceedings of 31st International Symposium on Lattice Field Theory LATTICE 2013 — PoS(LATTICE 2013)

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We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear σ models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scaleinvariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices.

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Does Broken Time Reversal Symmetry Modify the Critical Behavior at the Metal-Insulator Transition in 3-Dimensional Disordered Systems?

Cited by 74 publications

References 20 publications

Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition

Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition

Thermoelectric transport properties in disordered systems near the Anderson transition

Critical statistics at the mobility edge of QCD Dirac spectra

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