Random-Matrix Theory and Universal Statistics for Disordered Quantum Conductors

Muttalib, K. A.; Pichard, Jean‐Louis; Stone, A. Douglas

doi:10.1103/physrevlett.59.2475

Cited by 183 publications

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“…Shortly afterwards, a RMi of quantum transport was developed by Muttalib, Pichard, and Stone [11]. In this theory the role of the energy levels is played by the transmission eigenvalues T n , or more precisely by the ratio \ n = (l-T n }/T n of reflection and transmission coefficients.…”

Section: Universality In the Random-matrix Theory Of Quantum Transportmentioning

confidence: 99%

“…Note that V may be also a function of ß. The logarithmic interaction has a fundamental geometric origin: It is the Jacobian associated with the transformation from the space of scattering (or transfer) matrices to the smaller space of transmission eigenvalues [11][12][13]. The form (2) for the probability distribution is based on (a) an isotropy assumption, which implies that flux incident in one scattering channel is, on average, equally distributed among all outgoing channels; and (b) a maximum entropy hypothesis, which yields (2) äs the least restrictive distribution consistent with a given average eigenvalue density.…”

Section: Universality In the Random-matrix Theory Of Quantum Transportmentioning

confidence: 99%

“…The starting point of our analysis is the probability distribution [11] P({X n }) = Z' 1 exp [-ß-H({X n …”

mentioning

confidence: 99%

“…This constraint λ > 0 follows directly from unitarity of the scattering matrix. In contrast, the correlation functions in the RMT of energy levels are translationally invariant over the energy ränge of interest.Here we wish to announce that one can overcome this obstacle towards the establishment of Universality in the random-matrix theory of quantum transport.The starting point of our analysis is the probability distribution [11] P({X n }) = Z' 1 exp [-ß-H({X n where Z is a normalization constant. The variables X n (n = l, 2, .…”

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Universality in the random-matrix theory of quantum transport

1993

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A random-matrix formula is derived for the variance of an arbitrary linear statistic on the transmission eigenvalues. The variance is independent of the eigenvalue density and has a universal dependence on the symmetry of the matrix ensemble. The formula generalizes the Dyson-Mehta theorem in the statistical theory of energy levels. It demonstrates that the Universality of the conductance fluctuations is generic for a whole class of transport properties in mesoscopic Systems.PACS numbers: 72.10. Bg, 05.40,+j, 05.60.+w, 74.80.Fp In the sixties, Wigner, Dyson, Mehta, and others developed random-matrix theory (RMT) into a powerful tool to study the statistics of energy levels measured in nuclear reactions [1,2]. It was shown that the fluctuations in the energy level density are governed by level repulsion, which depends on the symmetry of the Hamiltonian ensemble-but is independent of the mean level density [3][4][5][6]. This Universality is at the origin of the remarkable success of RMT in nuclear physics. The universality of the level fluctuations is expressed by the celebrated Dyson-Mehta formula [7] for the variance of a linear statistic A = Σ η α(Ε η ) on the energy levels E n .[The quantity A is called a linear statistic because products of different E n do not appear, but the function a(E) may well depend nonlinearly on E.] The Dyson-Mehta formula readswhere a(k) = J^^dE e lkB a(E) is the Fourier transform of a(E), and β = 1,2, or 4 depending on whether the Hamiltonian ensemble belongs to the orthogonal, unitary, or symplectic symmetry class. Equation (1) shows that (1) the variance is independent of microscopic parameters; and (2) the variance has a universal l/β dependence on the symmetry parameter of the ensemble.In a seminal 1986 paper [8], Imry proposed to apply RMT to the phenomenon of universal conductance fluctuations (UCF), which was discovered diagrammatically by Al'tshuler [9] and Lee and Stone [10]. Shortly afterwards, a RMi of quantum transport was developed by Muttalib, Pichard, and Stone [11]. In this theory the role of the energy levels is played by the transmission eigenvalues T n , or more precisely by the ratio \ n = (l-T n }/T n of reflection and transmission coefficients. Their work is reviewed in Ref. [12], together with a closely related theory due to Mello, Pereyra, and Kumar [13]. (For still another approach, see Ref. [14].) Good agreement was obtained with the diagrammatic theory of UCF. However, it could not be shown that the Universality of the fluctuations is generic for arbitrary linear statistics on the transmission eigenvalues. In particular, no formula with the generality of the Dyson-Mehta theorem could be derived. The lack of such a general theory is being feit especially now that mesoscopic fluctuations in transport properties other than the conductance (both in conductors and superconductors) have become of interest [15][16][17]. The obstacle which prevents a straightforward generalization of the Dyson-Mehta formula was clearly identified by Stone et al. [12]: The cor...

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Section: Universality In the Random-matrix Theory Of Quantum Transportmentioning

confidence: 99%

Section: Universality In the Random-matrix Theory Of Quantum Transportmentioning

confidence: 99%

“…The starting point of our analysis is the probability distribution [11] P({X n }) = Z' 1 exp [-ß-H({X n …”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Universality in the random-matrix theory of quantum transport

1993

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“…The transmission matrix t can be expressed in terms of the transfer matrix M, for which a standard random matrix theory ("global approach") has been developed [14,15]. A set of N real positive parameters describing the radial part of the 2N×2N transfer matrix M (precise definitions are given in the next section) are the relevant "eigenvalues" in this approach, and their probability distribution is P ({λ a }) = exp (−βH({λ a })) , ( This is the usual RMT Coulomb gas analogy with logarithmic pairwise interaction, a system dependent confining potential V (λ), and an inverse temperature β = 1, 2, 4 depending on the system symmetries.…”

Section: Introductionmentioning

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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Light transmission properties in inhomogeneously‐disordered random media

Zhang

Lin

et al. 2017

Annalen der Physik

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Spatially inhomogeneous disorder exists widely in optical systems. We present a numerical study on the light transport properties and analysis of transmission channels in random media with inhomogeneous disorder. For the case of longitudinal inhomogeneity of disorder we find that the statistics of the transmission channels is independent of the inhomogeneity and the system can be equivalent to a counterpart with homogeneous disorder strength, both of which have the same statistical distribution of the transmission channels. However, for the case of transverse inhomogeneity of disorder, such equivalence does not exist. The distribution of the total transmission is broadened and one most transmitted incident channel emerges.

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Random-Matrix Theory and Universal Statistics for Disordered Quantum Conductors

Cited by 183 publications

References 15 publications

Universality in the random-matrix theory of quantum transport

Universality in the random-matrix theory of quantum transport

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

Light transmission properties in inhomogeneously‐disordered random media

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