Semiclassical analysis of spectral correlations in mesoscopic systems

Argaman, Nathan; Imry, Y.; Smilansky, Uzy

doi:10.1103/physrevb.47.4440

Cited by 162 publications

(143 citation statements)

References 50 publications

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“…The asymptotic results (9.8) and (9.9) have recently been rederived by Argaman, Imry and Smilansky [42] using a more intuitive semiclassical method for electrons in the diffusive regime. The smallenergy universal regime is obtained for times long enough to allow a diffusing electron to ergodically explore all the sample, while for energy intervals larger than the Thouless energy (short times) the correlations depend on the diffusing, unbounded dynamics of the electron, which is dimensionality and disorder dependent.…”

Section: Deviations From the Universal Rmt Behaviormentioning

confidence: 99%

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

Jalabert

Pichard

1995

J. Phys. I France

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We consider elastic reflection and transmission of electrons by a disordered system characterized by a 2N × 2N scattering matrix S. Expressing S in terms of the N radial parameters and of the four N×N unitary matrices used for the standard transfer matrix parametrization, we calculate their probability distributions for the circular orthogonal (COE) and unitary (CUE) Dyson ensembles. In this parametrization, we explicitely compare the COE-CUE distributions with those suitable for quasi-1d conductors and insulators. Then, returning to the usual eigenvalue-eigenvector parametrization of S, we study the distributions of the scattering phase shifts. For a quasi-1d metallic system, microscopic simulations show that the phase shift density and correlation functions are close to those of the circular ensembles. When quasi-1d longitudinal localization breaks S into two uncorrelated reflection matrices, the phase shift form factor b(k) exhibits a crossover from a behavior characteristic of two uncoupled COE-CUE (small k) to a single COE-CUE behavior (large k). Outside quasi-one dimension, we find that the phase shift density is no longer uniform and S remains nonzero after disorder averaging. We use perturbation theory to calculate the deviations to the isotropic Dyson distributions. When the electron dynamics is no longer zero dimensional in the transverse directions, small-k corrections to the COE-CUE behavior of b(k) appear, which are reminiscent of the dimensionality dependent non universal regime of energy level statistics. Using a known relation between the scattering phase shifts and the system energy levels, we analyse those corrections to the universal random matrix behavior of S which result from d-dimensional diffusion on short time scales.

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Section: Deviations From the Universal Rmt Behaviormentioning

confidence: 99%

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

Jalabert

Pichard

1995

J. Phys. I France

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“…Not every stochastic matrix T fulfilling the conditions (3) can be associated with a unitary matrix as defined in (2). Unitarity indeed requires that besides (3), also the condition…”

Section: A Unitary Stochastic Matricesmentioning

confidence: 99%

“…Not every doubly-stochastic matrix can be associated with a unitary matrix as defined in (2). The rows and columns of a unitary matrix have to obey orthogonality conditions which impose further restrictions on the matrix elements t ij .…”

Section: A Unitary Stochastic Matricesmentioning

confidence: 99%

Unitary-stochastic matrix ensembles and spectral statistics

Tanner

2001

J. Phys. A: Math. Gen.

149

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We propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with Markov processes on graphs. We argue that the spectral statistics of such an ensemble (after ensemble averaging) depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix. It is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size N → ∞ if the spectral gap of the corresponding transition matrices closes slower than 1/N . The hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non-uniformly connected star graphs and diffusive networks in arbitrary dimensions.

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“…Even calculation of the first nonvanishing non-oscillating terms for the Wigner-Dyson statistics is not simple 23 and difficulties grow when calculating next orders 24 . On the other hand, the WignerDyson statistics can easily be obtained from the zerodimensional σ-model and the only question is when this 0D σ-model description is valid.…”

Section: Periodic Orbitsmentioning

confidence: 99%

Nonlinearσmodel for long-range disorder and quantum chaos

Efetov

Kogan

2003

Phys. Rev. B

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We suggest a new scheme of derivation of a non-linear ballistic σ-model for a long range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long range potential and exact represention of their solutions in terms of functional integrals over supermatrices Q with the constraint Q 2 = 1. Averaging over the long range disorder or energy we are able to write a ballistic σ-model for all distances exceeding the electron wavelength (Eq. 3.13). Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a course graining procedure that allows us to get rid off scales in the Lapunov region we come to a reduced σ-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the σ-model to a one-dimensional σ-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.PACS: 05.45. Mt, 73.23.Ad

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Semiclassical analysis of spectral correlations in mesoscopic systems

Cited by 162 publications

References 50 publications

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

Unitary-stochastic matrix ensembles and spectral statistics

Nonlinearσmodel for long-range disorder and quantum chaos

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