Intermediate statistics in quantum maps

We present a one-parameter family of quantum maps whose spectral statistics are of the same intermediate type as observed in polygonal quantum billiards. Our central result is the evaluation of the spectral two-point correlation form factor at small argument, which in turn yields the asymptotic level compressibility for macroscopic correlation lengths

The spin contribution to the form factor of quantum graphs

Following the quantisation of a graph with the Dirac operator (spin-1/2) we explain how additional weights in the spectral form factor K(\tau) due to spin propagation around orbits produce higher order terms in the small-\tau asymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.Comment: 9 page

Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.Comment: 8 pages, no figure

Spectral Statistics for the Dirac Operator on Graphs

We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit

We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost integrable map, where we compare to Semi-Poissonian statistics.Comment: 29 pages, 20 figure

Reactivity of diatomics and of ethylene on zeolite-supported 13-atom platinum nanoclusters

Monodisperse Pt clusters of 132 atoms, supported on the zeolites NaY and KL and saturated with chemisorbed hydrogen, are investigated as well-defined model catalysts for reactions of CO, NO, O2 and ethene. CO reacts within <10 min, leading to the formation of dinuclear Pt carbonyl molecular clusters. The similar behaviour of NO suggests an analogous reaction. In stark contrast, O2 reveals very sluggish reaction on a timescale of days although the reaction with chemisorbed hydrogen to H2O is thermodynamically still favoured. This is ascribed to the inability of O2 to adsorb atop of Pt when all neighbouring sites are blocked by chemisorbed hydrogen. The hydrogenation reaction of ethene yields ethane as the only product. The turnover frequency at room temperature is somewhat lower than the one reported for the same reaction on Pt(111) single crystal surfaces or on Pt nanoparticles, but its activation energy is double of that typically found in the other systems. This means that the reaction which has been known to be structure-insensitive becomes structure-sensitive for catalyst clusters as small as 13 atoms. This fact is ascribed to a significantly larger binding energy of H on Pt as a consequence of the small cluster size and the influence of the support.http://pubs.rsc.org/en/Journals/JournalIssues/CY#!recentarticles&advhb2016Chemistr

Spin Orientation and Spin Precession in Inversion-Asymmetric Quasi Two-Dimensional Electron Systems

Inversion asymmetry induced spin splitting of the electron states in quasi two-dimensional (2D) systems can be attributed to an effective magnetic field B which varies in magnitude and orientation as a function of the in-plane wave vector k||. Using a realistic 8x8 Kane model that fully takes into account spin splitting because of both bulk inversion asymmetry and structure inversion asymmetry we investigate the spin orientation and the effective field B for different configurations of a quasi 2D electron system. It is shown that these quantities depend sensitively on the crystallographic direction in which the quasi 2D system was grown as well as on the magnitude and orientation of the in-plane wave vector k||. These results are used to discuss how spin-polarized electrons can precess in the field B(k||). As a specific example we consider GaInAs-InP quantum wells.Comment: 10 pages, 6 figure

Zitterbewegung and semiclassical observables for the Dirac equation

In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in \hbar by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac-Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity

Magnetotransport in Two-Dimensional Electron Systems with Spin-Orbit Interaction

We present magnetotransport calculations for homogeneous two-dimensional electron systems including the Rashba spin-orbit interaction, which mixes the spin-eigenstates and leads to a modified fan-chart with crossing Landau levels. The quantum mechanical Kubo formula is evaluated by taking into account spin-conserving scatterers in an extension of the self-consistent Born approximation that considers the spin degree of freedom. The calculated conductivity exhibits besides the well-known beating in the Shubnikov-de Haas (SdH) oscillations a modulation which is due to a suppression of scattering away from the crossing points of Landau levels and does not show up in the density of states. This modulation, surviving even at elevated temperatures when the SdH oscillations are damped out, could serve to identify spin-orbit coupling in magnetotransport experiments. Our magnetotransport calculations are extended also to lateral superlattices and predictions are made with respect to 1/B periodic oscillations in dependence on carrier density and strength of the spin-orbit coupling.Comment: 8 pages including 8 figures; submitted to PR