Spectral spacing correlations for chaotic and disordered systems

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron-Frobenius operator, is derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear replicated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing ``data''. The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros.Comment: 28 pages, 6 figures, 1 table, To appear in the Gutzwiller Festschrift, a special Issue of Foundations of Physic

On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates

We consider a single particle spectrum as given by the eigenvalues of the Wigner-Dyson ensembles of random matrices, and fill consecutive single particle levels with n fermions. Assuming that the fermions are non-interacting, we show that the distribution of the total energy is Gaussian and its variance grows as n^2 log n in the large-n limit. Next to leading order corrections are computed. Some related quantities are discussed, in particular the nearest neighbor spacing autocorrelation function. Canonical and gran canonical approaches are considered and compared in detail. A semiclassical formula describing, as a function of n, a non-universal behavior of the variance of the total energy starting at a critical number of particles is also obtained. It is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the Riemann zeta function on the critical line.Comment: 28 pages in Latex format, 5 figures, submitted for publication to Physica

ESR study of the single-ion anisotropy in the pyrochlore antiferromagnet Gd2Sn2O7

Single-ion anisotropy is of importance for the magnetic ordering of the frustrated pyrochlore antiferromagnets Gd2Ti2O7 and Gd2Sn2O7. The anisotropy parameters for the Gd2Sn2O7 were measured using the electron spin resonance (ESR) technique. The anisotropy was found to be of the easy plane type, with the main constant D=140mK. This value is 35% smaller than the value of the corresponding anisotropy constant in the related compound Gd2Ti2O7.Comment: 8 pages, 3 figure

Black and gray Helmholtz Kerr soliton refraction

efraction of black and gray solitons at boundaries separating different defocusing Kerr media is analyzed within a Helmholtz framework. A universal nonlinear Snell’s law is derived that describes gray soliton refraction, in addition to capturing the behavior of bright and black Kerr solitons at interfaces. Key regimes, defined by beam and interface characteristics, are identified and predictions are verified by full numerical simulations. The existence of a unique total non-refraction angle for gray solitons is reported; both internal and external refraction at a single interface is shown possible (dependent only on incidence angle). This, in turn, leads to the proposal of positive or negative lensing operations on soliton arrays at planar boundaries