Complex random matrix models with possible applications to spin-impurity scattering in quantum Hall fluids

Hikami, Shinobu; Zee, A.

doi:10.1016/0550-3213(95)00269-x

Cited by 16 publications

(13 citation statements)

References 32 publications

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“…As already remarked in [4], one particular simple way to test for the universal oscillation studied here is to compare the height of the first peak in the density of states to the height of the second peak. According (7.17), the density of states is proportional to the universal function r(y) = y(J 2 0 (y) + J 2 1 (y)) (9.1)…”

Section: Discussionmentioning

confidence: 92%

“…In this work we consider instead an ensemble of random hermitian matrices made of complex blocks. These matrices have been discussed recently for its application to impurity scattering in the presence of a magnetic field [4,5] and to a study of the zero modes of a Dirac operator [6]. In the large N limit the average density of eigenvalues is again a semi-circle for Gaussian ensembles.…”

Section: Introductionmentioning

confidence: 99%

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Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

1996

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We consider random hermitian matrices made of complex blocks. The symmetries of these matrices force them to have pairs of opposite real eigenvalues, so that the average density of eigenvalues must vanish at the origin. These densities are studied for finite N × N matrices in the Gaussian ensemble. In the large N limit the density of eigenvalues is given by a semi-circle law. However, near the origin there is a region of size 1 N in which this density rises from zero to the semi-circle, going through an oscillatory behavior. This cross-over is calculated explicitly by various techniques. We then show to first order in the non-Gaussian character of the probability distribution that this oscillatory behavior is universal, i.e. independent of the probability distribution. We conjecture that this universality holds to all orders. We then extend our consideration to the more complicated block matrices which arise from lattices of matrices considered in our

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Section: Discussionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

1996

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“…The oscillating behavior of (4.6) is similar to that of S(τ ). The oscillating behavior of the density of state for the Laguerre ensemble near the origin can be seen in the Fig.2 of [10]. In the large N limit, we know that the oscillations of the density of state near zero energy become universal, and are given in terms of Bessel functions.…”

Section: Average Of the Form Factormentioning

confidence: 90%

Spectral form factor in a random matrix theory

1997

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In the theory of disordered systems the spectral form factor S(τ ), the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for τ < τ c and constant for τ > τ c . Near zero and near τ c its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function ρ(λ 1 , λ 2 ) exhibits several well-known universal properties in the large N limit. Its Fourier transform is linear as a consequence of the short distance universality of ρ(λ 1 , λ 2 ). However the cross-over near zero and τ c requires to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these cross-over oscillatory properties. The method is also extended to the time-dependent case.

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“…Recently, in a series of papers [1,2,3,10,9,30], we, and with the collaboration of J. D'Anna and of S. Hikami, have studied the correlation between the density of eigenvalues of large random matrices.…”

Section: Introductionmentioning

confidence: 99%

Universal relation between Green functions in random matrix theory

Zee

Brézin

1995

Nuclear Physics B

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We prove that in random matrix theory there exists a universal relation between the one-point Green's function G and the connected two-point Green's function G c given byThis relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though G is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered a couple of 1 years ago, which states that a 2 G c (az, aw) is independent of the probability distribution, where a denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a Hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively

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Complex random matrix models with possible applications to spin-impurity scattering in quantum Hall fluids

Cited by 16 publications

References 32 publications

Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

Spectral form factor in a random matrix theory

Universal relation between Green functions in random matrix theory

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