Spectral form factor near the Ehrenfest time

Abstract

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 ...

Particle-time duality in the kicked Ising spin chain

“…From the semiclassical point of view this can be attributed to diagonal correlations between periodic orbits in the double sum representing K 2 (T ), see [5]. While non diagonal correlations can arise only for time scales larger than the Ehrenfest time [18,19,20]. Interestingly, similar considerations give rise to the linear growth for the KIC as well.…”

Semiclassical theory of the Ehrenfest-time dependence of quantum transport

“…͑This is different for quantum corrections to K that are of higher order in ប. 64,65 Such quantum corrections are not considered here.͒ In an open quantum system, the density of states is formally expressed in terms of the Wigner-Smith time-delay matrix,…”

Ehrenfest-time dependence of quantum transport corrections and spectral statistics

“…As without time reversal symmetry there are no contributions at the order 1/N considered here, current conservation, already fulfilled at the diagonal level, is thus not violated. In the case of time reversal symmetry the contributions to T and R at the considered order are given in (22) and (23) and sum to zero. Current conservation is thus again fulfilled.…”

“…Naturally we only want to include one copy later so we subtract them all here. All told, the contribution resulting from the periodic orbit encounters, δK 2d (ω) was calculated in [23] to be…”