Periodic-Orbit Theory of Level Correlations

Abstract

We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the non-oscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.PACS numbers: 05.45. Mt, 03.65.Sq Quantum spectra of individual chaotic systems can be phenomenologically described in terms of random-matrix theory (RMT) [1,2]. This universality -asserted by the celebrated Bohigas-Giannoni-Schmit conjecture (BGS) [3] -is an empirical fact, supported by a huge body of experimental and numerical data. Proving its conceptual origin remains one of the fundamental challenges in quantum or wave chaos.Spectral fluctuations are conveniently characterized in terms of the two-point correlation function, R(ǫ) = ∆ 2 ρ(E + ǫ∆ 2π )ρ(E − ǫ∆ 2π ) − 1, where ρ(E) is the energydependent density of states, ∆ ≡ 1/ ρ the mean level spacing and · denotes averaging over the energy E. Predictions made by RMT for the two-point correlation function are fully universal in that they depend only on the parameter ǫ, and the fundamental symmetries of the system under consideration. Specifically, the complex representation R(ǫ) = lim γ→0 Re C(ǫ + ) where ǫ ± = ǫ ± iγ andis employed, with G ± (E) = (E ± iγ∆/2π − H) −1 and H the Hamiltonian. The Wigner-Dyson unitary and orthogonal symmetry classes (the only ones to be considered here) of RMT afford the asymptotic seriesIn either case, C(ǫ + ) is a sum of a non-oscillatory part (power series in 1/ǫ + ) and an oscillatory one (e 2iǫ + times a series in 1/ǫ + ). Resumming the series by Borel summation techniques and extrapolating to small positive values of ǫ, one obtains R(ǫ) + 1 ∝ ǫ β for ǫ → 0, a signature of the level repulsion symptomatic for chaos (β = 1, 2 for the case of orthogonal, unitary symmetry, resp.) The question to be addressed below is how to obtain the RMT prediction (1) for a concrete chaotic (globally hyperbolic) quantum system. A step in this direction was recently made [4] on the basis of Gutzwiller's semiclassical periodic-orbit theory [5]. Gutzwiller represents the level density ρ(E) as a sum over periodic orbits, whereupon the function R(ǫ) becomes a sum over orbit pairs [6,7,8]. Relevant contributions to that double sum were shown [4] to originate from orbit pairs which are identical, mutually time-reversed, or differ only by connections in certain close self-encounters. By summing over all distinct families of orbit pairs, the Fourier transform of R(ǫ), the spectral form factor K(τ ), was found to coincide with the RMT prediction for times t = τ T H smaller than the Heisenberg time T H = 2π /∆, the time needed to resolve the mean level spacing. The behavior of K(τ ) for τ > 1, also known from RMT, was left unexplained.We now want to fill the gap...

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

“…In the present case of autonomous flows, only the limit of small offsets, e A/B/C/D /N → 0 yielding (33), is of physical interest. Two significant advantages over the derivation in [1,2,6] are worth noting. First, we now need only infinitely small imaginary parts of the offset phases e C/D .…”

“…The phenomenological description previously given by random-matrix theory has been fully recovered for individual (rather than ensemble averages of) systems from the unitary, orthogonal [1,2], and symplectic [3] symmetry classes.…”

“…We Taylor expand all four pertinent exponentials and so obtain a fourfold sum over pseudo-orbits. Neglecting orbit repetitions and again representing the four phases ϕ A/B/C/D as in (17) we find [1,2,6]…”

Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

“…Strictly speaking, the semiclassical approximation and the Bohigas-Giannoni-Schmit [12,37,38] conjecture, which states, that the spectral fluctuation properties in quantum systems with a chaotic classical dynamics coincide with those of random matrices drawn from the Gaussian random-matrix ensembles, apply in the regime when the wave length is the smallest length scale of the system, i.e. kr 2 ≫ 1 must hold.…”

Chaos in fermionic many-body systems and the metal-insulator transition

“…This fact is commonly taken as evidence for chaotic motion [2,3]. In RMT, all pairs of states are coupled by independent matrix elements.…”