Universal spectral form factor for chaotic dynamics

Heusler, Stefan; Müller, Sebastian; Braun, P. A.; Haake, Fritz

doi:10.1088/0305-4470/37/3/l02

Cited by 57 publications

(95 citation statements)

References 20 publications

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“…The configurations of Fig. 1b and c were also considered by Heusler et al [6,7,19]. These two contributions cancel in the limit ατ E → 0, so that one finds δK 2 = 0 in that limit.…”

mentioning

confidence: 80%

“…For the calculation of the contribution of the trajectory of Fig. 1c one needs only one Poincaré surface of section, taken at a point where all three stretches of γ are within a phase space distance c. Labeling the phase space coordinates of the three piercings of γ through the surface of section as (s i , u i ), i = 1, 2, 3, the action difference is [7,19] …”

mentioning

confidence: 99%

“…In a parallel development, the full perturbation expansion for K(t) was derived from periodic-orbit theory [6,7,18,19]. The connection between K(t), which is a quantum-mechanical object, and classical periodic orbits follows from Gutzwiller's trace formula, which expresses K(t) as a double sum over periodic orbits γ and γ ′ [17,20],…”

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confidence: 99%

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Spectral form factor near the Ehrenfest time

Brouwer

Rahav²,

Tian³

2006

Phys. Rev. E

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We calculate the Ehrenfest-time dependence of the leading quantum correction to the spectral form factor of a ballistic chaotic cavity using periodic orbit theory. For the case of broken time-reversal symmetry, our result differs from that previously obtained using field-theoretic methods [Tian and Larkin, Phys. Rev. B 70, 035305 (2004)]. The discrepancy shows that short-time regularization procedures dramatically affect physics near the Ehrenfest-time.PACS numbers: 73.20.Fz, 05.45.Mt From a statistical point of view, the spectra of a quantum particle confined to a cavity with disorder or confined to a cavity with ballistic and chaotic classical dynamics are remarkably similar [1]. In both cases, the spectral statistics are independent of system specifics and follow the predictions of random matrix theory (RMT) on energy scales comparable to the mean spacing between energy levels ∆ or, equivalently, on time scales comparable to the Heisenberg time t H = 2πh/∆ [2,3], provided that the latter is much larger than the time τ erg needed for ergodic exploration of the phase space. Calculations of the spectral statistics have been the theoretical vehicle through which the profound correspondence between RMT, impurity diagrammatic perturbation theory, periodic-orbit theory, and the fieldtheoretic zero-dimensional σ model has been demonstrated [2,4,5,6,7].In the last decade, it has been understood that chaotic quantum systems are characterized by a time scale intermediate between τ erg and τ H that does not have its counterpart in diffusive systems. This time scale is the 'Ehrenfest time' τ E [8,9,10,11]. The Ehrenfest time is the time required for two classical trajectories initially a quantum distance (wavelength) apart to diverge and reach a classical separation (system size). It is expressed in terms of the Lyapunov exponent λ of the corresponding classical dynamics as τ E = λ −1 ln(c 2 /h), where c 2 is a classical (action) scale. The existence of the Ehrenfest time does not affect the universality of the spectral statistics. However, it is responsible for differences between otherwise universal properties of chaotic and disordered quantum systems for energies ∼h/τ E , which is well inside the universal range ifh/c 2 → 0 [5,11,12,13,14,15]. In this article we address the spectral statistics at this energy scale.Most of the literature on spectral statistics considers the spectral form factor K(t), which is the Fouriertransform of the two-point correlation function of the level density ρ(ε),Here the brackets . . . c denote the connected average obtained by varying the center energy ε and/or other parameters in the system. For τ erg ≪ t < t H , the RMT predicts that K(t) is dominated by a perturbative expansion in t/t H [1, 16, 17]where β = 1 (2) in the presence (absence) of time-reversal symmetry andFor times t > ∼ t H the perturbative expansion in t/t H breaks down, and K(t) is governed by non-perturbative contributions [16].The presence of the Ehrenfest time does not affect the leading contribution to K(t), but ...

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“…The configurations of Fig. 1b and c were also considered by Heusler et al [6,7,19]. These two contributions cancel in the limit ατ E → 0, so that one finds δK 2 = 0 in that limit.…”

mentioning

confidence: 80%

mentioning

confidence: 99%

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Spectral form factor near the Ehrenfest time

Brouwer

Rahav²,

Tian³

2006

Phys. Rev. E

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“…Yet in spite of its ubiquity, and notwithstanding a number of significant recent advances [5,6,7,8,9,10,11], the correspondence above is not yet fully understood theoretically. Specifically, the 'non-perturbative' aspects of the problem -which manifest themselves, e.g., in the low energy profile of spectral correlations -are not under quantitative control.…”

Section: Introductionmentioning

confidence: 99%

Field theory approach to quantum interference in chaotic systems

Müller¹,

Altland²

2005

J. Phys. A: Math. Gen.

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“…Certainly, if we send all individual bond lengths to 1 in a fixed graph with real S, it is clear from expressions (15) and (18) that K S (τ ) will converge to K U (τ ) is expected to converge to its limiting GOE form,…”

Section: Eigenvalue Vs Eigenphase Statisticsmentioning

confidence: 99%

Form factor expansion for large graphs: a diagrammatic approach

Berkolaiko¹

2006

Quantum Graphs and Their Applications

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Abstract. The form factor of a quantum graph is a function measuring correlations within the spectrum of the graph. It can be expressed as a double sum over the periodic orbits on the graph. We propose a scheme which allows one to evaluate the periodic orbit sum for a special family of graphs and thus to recover the expression for the form factor predicted by the Random Matrix Theory. The scheme, although producing the expected answer, undercounts orbits of a certain structure, raising doubts about an analogous summation recently proposed for quantum billiards.

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Universal spectral form factor for chaotic dynamics

Cited by 57 publications

References 20 publications

Spectral form factor near the Ehrenfest time

Spectral form factor near the Ehrenfest time

Field theory approach to quantum interference in chaotic systems

Form factor expansion for large graphs: a diagrammatic approach

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