Non-Hermitian Euclidean random matrix theory

Goetschy, Arthur; Skipetrov, S. E.

doi:10.1103/physreve.84.011150

Cited by 71 publications

(109 citation statements)

References 48 publications

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“…This suggests that in a cluster (or "cavity" [63]), i.e., in a dense agglomeration of two or more atoms, transport occurs much faster than in the more dilute parts of the system. As we discuss now, this implies that energy transport to and away from the cluster is slower than in the case where the cluster is replaced by a single atom.…”

Section: B Short Distances and Pair Localizationmentioning

confidence: 99%

“…While tempting, it is not possible to interpolate between the 1-stable and universal Gaussian RMT by introducing a high-energy cutoff to the density (17), since the statistics of (infinitely) large matrices with a truncated Lévy distribution always lie in the basin of attraction of the GOE and therefore make exactly the same predictions as universal Gaussian RMT. However, it might be possible to treat the intermediate case within the general theory of Euclidean random matrices (ERMT) [63,65,[93][94][95][96][97][98] that addresses all those very large N × N matrices M the elements M ij of which depend on pairs R i , R j of N randomly chosen coordinates-precisely as for the Rydberg Hamiltonian, Eq. (4).…”

Section: Stable Random Matricesmentioning

confidence: 99%

“…We specifically discuss the applicability of universal Gaussian orthogonal [53] and α-stable random matrix ensembles [49], the latter of which have been studied in the field of Lévy spin glasses [61,62]. Random matrix theory is typically restricted to matrices with independent and identically distributed entries [63], the Gaussian and α-stable ensembles being no exception. Spectral signatures of correlations between Hamiltonian matrix elements that are identified in our numerical reference can thus not be reproduced.…”

Section: Introductionmentioning

confidence: 99%

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Spectral backbone of excitation transport in ultracold Rydberg gases

2014

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The spectral structure underlying excitonic energy transfer in ultra-cold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics-in particular, the density of states and the nearest neighbor level spacing statistics-exhibits a transition from approximately a 1-stable Lévy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between inter-atomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.

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Section: B Short Distances and Pair Localizationmentioning

confidence: 99%

Section: Stable Random Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectral backbone of excitation transport in ultracold Rydberg gases

2014

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“…Indeed only statistical properties of this random matrix can be estimated [58]. A simplifying approach consists in expanding the problem in scattering orders.…”

Section: Coherent Backscatteringmentioning

confidence: 99%

Collective effects in the radiation pressure force

et al. 2016

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We discuss the role of diffuse, Mie and cooperative scattering on the radiation pressure force acting on the center of mass of a cloud of cold atoms. Even though a mean-field Ansatz (the 'timed Dicke state'), previously derived from a cooperative scattering approach, has been shown to agree satisfactorily with experiments, diffuse scattering also describes very well most features of the radiation pressure force on large atomic clouds. We compare in detail an incoherent, random walk model for photons and a diffraction approach to the more complete description based on coherently coupled dipoles. We show that a cooperative scattering approach, although it provides a quite complete description of the scattering process, is not necessary to explain the previous experiments on the radiation pressure force.

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“…It is usually analyzed in the frame of Hermitian or unitary matrices. Recently, the localization properties of nonunitary complex matrices has been analyzed for Euclidean matrices [4] in relation to light and wave localization [5].…”

Section: Introductionmentioning

confidence: 99%

Anderson transition for Google matrix eigenstates

Zhirov

Shepelyansky

2015

Annalen der Physik

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We introduce a number of random matrix models describing the Google matrix G of directed networks. The properties of their spectra and eigenstates are analyzed by numerical matrix diagonalization. We show that for certain models it is possible to have an algebraic decay of PageRank vector with the exponent similar to real directed networks. At the same time the spectrum has no spectral gap and a broad distribution of eigenvalues in the complex plain. The eigenstates of G are characterized by the Anderson transition from localized to delocalized states and a mobility edge curve in the complex plane of eigenvalues.

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Non-Hermitian Euclidean random matrix theory

Cited by 71 publications

References 48 publications

Spectral backbone of excitation transport in ultracold Rydberg gases

Spectral backbone of excitation transport in ultracold Rydberg gases

Collective effects in the radiation pressure force

Anderson transition for Google matrix eigenstates

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