Field theory approach to quantum interference in chaotic systems

Müller, Jan; Altland, Alexander

doi:10.1088/0305-4470/38/14/003

Cited by 11 publications

(14 citation statements)

References 25 publications

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“…Interestingly, while we confirm the field-theoretic result for the O(t 2 ) term in δK 1 (t) [5,21], our result for the leading O(t 3 ) contribution to δK 2 (t) differs from that of Ref. 5,…”

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confidence: 78%

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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confidence: 78%

Spectral form factor near the Ehrenfest time

Brouwer

Rahav²,

Tian³

2006

Phys. Rev. E

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“…The techniques here elaborated (in particular the analogy between orbit-based semiclassics and the sigma model) will be, and in fact are already helpful in tackling further challenges in the field. Examples are (i) localization effects in long thin wires [38], (ii) non-universal effects related to the Ehrenfest time (which has no counterpart in RMT) [39], (iii) a semiclassical treatment of arithmetic billiards [40] (where intrinsically quantum Hecke symmetries have strong action degeneracies as classical counterparts), (iv) spectral correlations of higher order [41] and the density of nearest-neighbor spacings, (v) transport through ballistic chaotic conductors [32,42] including the "full counting statistics" [43], (vi) further symmetry classes, (vii) transitions between symmetry classes [44], and (viii) last but not least, the relative status of the ballistic sigma model and periodic-orbit theory [23].…”

Section: Discussionmentioning

confidence: 99%

Periodic-orbit theory of universal level correlations in quantum chaos

et al. 2009

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Using Gutzwiller's semiclassical periodic-orbit theory we demonstrate universal behavior of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the periodic-orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.

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“…Further possible applications concern localization, a clarification of open problems in the nonlinear sigma model [40], and the crossover between universality classes [16,41].…”

Section: Discussionmentioning

confidence: 99%

Periodic-orbit theory of universality in quantum chaos

et al. 2005

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We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor K(τ ) as power series in the time τ . Each term τ n of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.

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Field theory approach to quantum interference in chaotic systems

Cited by 11 publications

References 25 publications

Spectral form factor near the Ehrenfest time

Spectral form factor near the Ehrenfest time

Periodic-orbit theory of universal level correlations in quantum chaos

Periodic-orbit theory of universality in quantum chaos

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