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

Brézin, E.; Hikami, Shinobu; Zee, A.

doi:10.1016/0550-3213(96)00063-6

Cited by 69 publications

(78 citation statements)

References 34 publications

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“…It is easy to see, by rescaling τ in the expansion of S ef f into T = ( √ r − 1)τ , (5.28) that the Airy function behavior of ∂ρ(λ 2 ) ∂λ 2 near λ = ±b breaks down as r → 1. Indeed, from previous work [1,2,3,4,5,6,7,8] we know that the oscillations near the origin in the density of the eigenvalues of matrices built out of square blocks (r = 1) are governed by the Bessel function and not by the Airy function.…”

Section: The Edges Of the Eigenvalue Distributionmentioning

confidence: 99%

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Renormalizing rectangles and other topics in random matrix theory

Feinberg

Zee

1997

J Stat Phys

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We consider random Hermitian matrices made of complex or real M × N rectangular blocks, where the blocks are drawn from various ensembles. These matrices have N pairs of opposite real nonvanishing eigenvalues, as well as M − N zero eigenvalues (for M > N .) These zero eigenvalues are "kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large N, M limit, in which the "rectangularity" r = M N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r − 1| is held fixed as N → ∞, the N non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As r → 1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that r → 1 drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

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Section: The Edges Of the Eigenvalue Distributionmentioning

confidence: 99%

“…Hermitian matrices made of square blocks [7]. Here we generalize it to random Hermitian matrices made of rectangular blocks.…”

Section: Contour Integralmentioning

confidence: 99%

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Renormalizing rectangles and other topics in random matrix theory

Feinberg

Zee

1997

J Stat Phys

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“…The formalism is of course indifferent to the method one may choose to use to determine G µ ν (η; z, z * ). The problem of determining the Green's function of chiral matrices such as H has been discussed by numerous authors [16,20,22,23].…”

Section: Some Basic Formalismmentioning

confidence: 99%

Non-hermitian random matrix theory: Method of hermitian reduction

1997

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We consider random non-hermitean matrices in the large N limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitean random matrices, in contrast to hermitean random matrices. To overcome this difficulty, we show that associated to each ensemble of nonhermitean matrices there is an auxiliary ensemble of random hermitean matrices which can be analyzed by the usual methods. We then extract the Green's function and the density of eigenvalues of the non-hermitean ensemble from those of the auxiliary ensemble. We apply this "method of hermitization" to several examples, and discuss a number of related issues.

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“…As an analogous matrix model to HSW model, a complex block matrix model has been studied and the universal oscillation of the density of state near E = 0 has been obtained in the large N limit, where N is a size of the matrix [9,10]. The simple matrix model is given by…”

Section: Diagrammatic Analysis Of the Two-state Quantum Hall Systemmentioning

confidence: 99%

Diagrammatic analysis of the two-state quantum Hall system with chiral invariance

Hikami

Minakuchi

1997

Phys. Rev. B

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The quantum Hall system in the lowest Landau level with Zeeman term is studied by a two-state model, which has a chiral invariance. Using a diagrammatic analysis, we examine this two-state model with random impurity scattering, and find the exact value of the conductivity at the Zeeman energy E = ∆. We further study the conductivity at the another extended state E = E 1 (E 1 > ∆). We find that the values of the conductivities at E = 0 and E = E 1 do not depend upon the value of the Zeeman energy ∆. We discuss also the case where the Zeeman energy ∆ becomes a random field.

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

Cited by 69 publications

References 34 publications

Renormalizing rectangles and other topics in random matrix theory

Renormalizing rectangles and other topics in random matrix theory

Non-hermitian random matrix theory: Method of hermitian reduction

Diagrammatic analysis of the two-state quantum Hall system with chiral invariance

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