Influence of boundary conditions on level statistics and eigenstates at the metal–insulator transition

Schweitzer, L.; Potempa, H.

doi:10.1016/s0378-4371(98)00636-0

Cited by 19 publications

(17 citation statements)

References 35 publications

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“…δ is the mean level spacing and E 0 is the upper cutoff of multifractality. Numerics on the integer Quantum Hall systems and in the critical point of the 3D Anderson model was consistent 22,23 with this scaling. An important feature of Eq.…”

Section: Multifractality Of Critical Eigenfunctionssupporting

confidence: 54%

Two-eigenfunction correlation in a multifractal metal and insulator

2007

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We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.Comment: 16 pages, 18 figure

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Section: Multifractality Of Critical Eigenfunctionssupporting

confidence: 54%

Two-eigenfunction correlation in a multifractal metal and insulator

2007

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“…This fact, however, does not reflect on the accuracy of the RG approach. Indeed, as it was demonstrated recently, the critical ELS -although being system size independent -nevertheless depends on the geometry of the samples [31] and on the specific choice of boundary conditions [32,33]. Sensitivity to the boundary conditions does not affect the asymptotics of the critical distribution, but rather manifests itself in the shape of the "body" of the ELS.…”

Section: P(s)mentioning

confidence: 90%

Real-Space Renormalization and Energy-Level Statistics at the Quantum Hall Transition

Röemer¹,

Cain²

2003

Advances in Solid State Physics

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Abstract. We review recent applications of the real-space renormalization group (RG) approach to the integer quantum Hall (QH) transition. The RG approach, applied to the Chalker-Coddington network model, 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 yields a value of the critical exponent, νG = 2.39 ± 0.01, that agrees with most accurate large-size lattice simulations. Analyzing the evolution of the distribution of phases of the transmission coefficients upon a step of the RG transformation, we obtain information about the energy-level statistics (ELS). From the fixed point of the RG transformation we extract a critical ELS. Away from the transition the ELS crosses over towards a Poisson distribution. Studying the scaling behavior of the ELS around the QH transition, we extract the critical exponent νELS = 2.37 ± 0.02.The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electronic wavefunctions. In contrast to a usual metal-insulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center ǫ = 0 of each Landau band [1]. When approaching ǫ = 0, the localization length ξ of the electron wavefunction diverges according to a power law ǫ −ν , where ǫ defines the distance to the MIT for a suitable control parameter, e.g., the electron energy. On the theoretical side, the value of ν has been extracted from various numerical simulations, e.g., ν = 2.5 ± 0.5 [2], 2.4 ± 0.2 [3], 2.35 ± 0.03 [4], and 2.39 ± 0.01 [5]. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency [6] or the sample size [7] dependence of the critical behavior of the resistance in the transition region at strong magnetic field.Recently, a semianalytical description of the integer QH transition, based on the extension of the scaling ideas for the classical percolation [8] to the Chalker-Coddington (CC) model of the quantum percolation [2], has been developed [9,10]. The key idea of this description, a real-space-renormalization group (RG) approach, is the following. Each RG step corresponds to a doubling of the system size. The RG transformation relates the conductance distribution of the sample at the next step to the conductance distribution at the previous step. The fixed point (FP) of this transformation, yields the distribution of the conductance, P c (G), of a macroscopic sample at the QH

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“…As the disorder strength is increased, the boundary of two regions, called mobility edge, move toward the band center and disappear alongside the extended eigenstates between two edges, leading to the metal-insulator transition [18,19] Being an unstable boundary region separating Wigner-Dyson and Poisson statistics, the mobility edge is expected to exhibit an intermediate level statistics associated with fractal wave functions, corresponding to the NLσ M precisely at the IR-unstable fixed point. Although the width of the mobility edge (in the physical unit) shrinks under an increment of the lattice size, such 'critical' statistics [20] should be stable and depend only on the fixed point value of the conductance (which in turn depends on the dimensionality d = 2+ε), and possibly on the boundary condition and the aspect ratio of the lattice [21]. It should otherwise be universal in a sense that it originates from fine-tuning of a single relevant coupling constant (conductance) and all other irrelevant couplings should play no role [22].…”

Section: Critical Statistics At the Mobility Edgementioning

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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Influence of boundary conditions on level statistics and eigenstates at the metal–insulator transition

Cited by 19 publications

References 35 publications

Two-eigenfunction correlation in a multifractal metal and insulator

Two-eigenfunction correlation in a multifractal metal and insulator

Real-Space Renormalization and Energy-Level Statistics at the Quantum Hall Transition

Critical statistics at the mobility edge of QCD Dirac spectra

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