We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams. Fully chaotic dynamics enjoy ergodicity and thus visit everywhere in the accessible space with uniform likelihood, over long periods of time. Even long periodic orbits bring about such uniform coverage. Moreover, classical ergodicity provides quantum chaos with universal characteristics.Given chaos, quantum energy levels are correlated within local few-level clusters but become statistically independent as their distance grows much larger than the mean level spacing ∆. The decay of correlations on the scale ∆ is empirically found system independent, within universality classes distinguished by presence or absence of time-reversal (T ) invariance [1,2]. The corresponding universal long-time characteristics act on the Heisenberg scale T H = 2πh/∆, withh Planck's constant.Universal spectral fluctuations were conjectured as a manifestation of quantum chaos two decades ago [3]. Now, the semiclassical core of a proof can be given. Based on Gutzwiller's periodic-orbit theory [4], our progress comes with two surprises: one lies in its simplicity, the other in the appearance of interesting mathematics (nontrivial properties of permutations). Moreover, the often disputed intimate relation between periodic orbits and quantum field theory is confirmed for good. We thus expect the underlying ideas to radiate beyond spectral fluctuations, like to transport and localization.Technically speaking, we want to show that each completely hyberbolic classical dynamics has a quantum energy spectrum with the same fluctuations as a randommatrix caricature H RMT of its Hamiltonian, even though that caricature has nothing in common with the Hamiltonian but symmetry (absence or presence of T invariance). The theory of random matrices (RMT) [1, 2, 5], developed by Wigner and Dyson to account for fluctuations in nuclear spectra yields analytic results for correlators of the level density ρ(E), by averaging over suitable ensembles of random matrices. Simplest is the two-point correlator ρ(E)ρ(E ′ ) − ρ(E) ρ(E ′ ), where the overlines denote ensemble average. Its Fourier transform with respect to the energy difference E − E ′ , called spectral form factor K(τ ), is predicted by RMT for systems without time reversal invariance (unitary class) and with that symmetry (orthogonal class) asrespectively; here τ is a...
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.
We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or "clean," agrees with predictions of random-matrix theory.
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...
We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered. PACS numbers: 73.23.-b, 72.20.My, 72.15.Rn, 05.45.Mt, 03.65.Sq t a1a2 ∼ 1 √ T H α:a1→a2 A α e iSα/h .
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