Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations

Smolyarenko, Igor; Simons, Benjamin D.

doi:10.1088/0305-4470/36/12/339

Cited by 12 publications

(25 citation statements)

References 31 publications

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“…We remark that the present study can be considered an instance of the so-called parametric statistics used to characterize the evolution of individual eigenvalues as a function of an external parameter [16]. In the case of complex eigenvalues, parametric evolution has been experimentally investigated by considering resonances trapped in an open microwave cavity in which the slit width can be varied [17].…”

Section: Discussionmentioning

confidence: 99%

Structure of trajectories of complex-matrix eigenvalues in the Hermitian–non-Hermitian transition

Bohigas¹,

Carvalho²,

Pato³

2012

Phys. Rev. E

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Section: Discussionmentioning

confidence: 99%

Structure of trajectories of complex-matrix eigenvalues in the Hermitian–non-Hermitian transition

Bohigas¹,

Carvalho²,

Pato³

2012

Phys. Rev. E

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“…This dependence is thus manifested only on time scales much shorter than t H , which is of interest here. In invariant unitary RMT ensembles universality under shifts was shown in [14].…”

Section: Prl 100 190404 (2008) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 98%

Surprising Relations between Parametric Level Correlations and Fidelity Decay

Kohler

Smolyarenko²,

Pineda³

et al. 2008

Phys. Rev. Lett.

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Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain. DOI: 10.1103/PhysRevLett.100.190404 PACS numbers: 03.65.Sq, 03.65.Yz, 05.30.Ch, 05.45.Mt Fidelity decay presently attracts considerable attention [1]. It measures the change of quantum dynamics of a state under a modification of the Hamiltonian. In quantum information, fidelity measures the deviation between a mathematical algorithm and its physical implementation. From a different point of view, important insight into the properties of the underlying systems is provided by the studies of correlations between spectra of random and/or chaotic Hamiltonians which differ by a parameterdependent perturbation [2]. Since statistical properties of fidelity decay in random or chaotic systems involve both spectra and eigenfunctions of the original and perturbed Hamiltonians, the existence of any connections between fidelity and purely spectral correlations is not a priori obvious.Random matrix theory (RMT) has been successful in describing quantum many-body systems and as a model for the spectral properties of single particle systems whose classical analogue is chaotic [3]. Within RMT fidelity was analyzed in linear response approximation [4] and both fidelity [5][6][7] and parametric correlations [8] were calculated exactly using the supersymmetry method. An unexpected fidelity revival at Heisenberg time was encountered [5] within RMT and confirmed in a dynamical coupled spin chain model [9].Earlier, differential relations between parametric spectral correlations and parametric density correlations were established [10,11]. By relating the latter to the fidelity amplitude via Fourier transform, we show in this Letter that the existence of these relations opens a crucial insight into the properties of fidelity decay. By analyzing the characteristic features of the parametric correlations in the time domain, the cross-form-factor, we discover a new, simple interpretation of the previously puzzling phenomenon of revival [5]. These relations follow directly from the basic definitions and symmetries of the underlying matrix models, being essentially Ward identities. We show that they are valid under very general assumptions. No explicit (e.g., supersymmetric) calculation is required; however, they rely on the universality of the parametric spectral correlations at the scale of mean level spacing. We thus explain the origin of various relations connecting spectral and wave-function correlations, and establish a unified framework for their analysis and generalizations. A relat...

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“…19 Finally, it is interesting to note that the correlation field is related to the amount of disorder in the grains when the electron motion is diffusive. It is straightforward to show that at so → ϱ,…”

Section: ͑21͒mentioning

confidence: 99%

g-factors and discrete energy level velocities in nanoparticles

2006

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We establish relations between the statistics of g factors and the fluctuations of energy in metallic nanoparticles where spin-orbit coupling is present. These relations assume that the electron dynamics in the grain is chaotic. The expressions we provide connect the second moment of the g factor to the root-mean square "level velocity" ͑the derivative of the energy with respect to magnetic field͒ calculated at magnetic fields larger than a characteristic correlation field. Our predictions relate readily observable quantities and allow for a parameterfree comparison with experiments.

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Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations

Cited by 12 publications

References 31 publications

Structure of trajectories of complex-matrix eigenvalues in the Hermitian–non-Hermitian transition

Structure of trajectories of complex-matrix eigenvalues in the Hermitian–non-Hermitian transition

Surprising Relations between Parametric Level Correlations and Fidelity Decay

g-factors and discrete energy level velocities in nanoparticles

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