Nearest-neighbor Spacing Distribution Research Articles

Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition

The nearest-neighbor level spacing distribution is numerically investigated by directly diagonalizing disordered Anderson Hamiltonians for systems of sizes up to 100 x 100 x 100 lattice sites. The scaling behavior of the level statistics is examined for large spacings near the delocalization-localization transition and the correlation length exponent is found. By using high-precision calculations we conjecture a new interpolation of the critical cumulative probability, which has size-independent asymptotic form \ln I(s) \propto -s^{\alpha} with \alpha = 1.0 \pm 0.1.

Quantum Chaos in a Yang–Mills–Higgs System

We study the energy fluctuations of a spatially homogeneous SU(2) Yang–Mills–Higgs system. In particular, we analyze the nearest-neighbour spacing distribution which shows a Wigner–Poisson transition by increasing the value of the Higgs field in the vacuum. This transition is a clear quantum signature of the classical chaos–order transition of the system.

From Landau levels to universal fluctuations: Level statistics for lateral superlattices

We study the energy-level statistics for electrons in a lateral two-dimensional superlattice in dependence on the strength of the modulation potential at a given perpendicular magnetic field. The interminiband statistics at some selected point in the magnetic Brillouin zone shows a transition from degenerate Landau levels to universal spectral correlations while the corresponding classical dynamics displays a crossover from regular cyclotron orbits to chaotic motion. In terms of the nearest-neighbor spacing distribution, we observe the occurrence of level repulsion for all nonzero modulation amplitudes, which indicates a discontinuous transition to universal fluctuations for the smallest level spacings when the rotational invariance of the system is broken by a finite superlattice potential. In order to test the universality of the transition on larger scales, we propose a random matrix model with one transition parameter for comparison. Confirmed by two different types of potential shapes, we detect an explicit influence of the potential shape on the level spacing distributions, which hence cannot be described by the one-parameter model. Using the spectral rigidity, however, universal long-range fluctuations in the regime of negligible Landau-level mixing are found, in agreement with our random matrix model.

Chaos in intrinsic motion of nuclei and its role in dissipative collective dynamics

We shall discuss the role of chaotic intrinsic motion in dissipative dynamics of the collective coordinates for nuclear systems. Using the formalism of linear response theory, it will be shown that the dissipation in adiabatic collective motion depends on the degree of chaos in the intrinsic dynamics of a system. This gives rise to a shape dependent dissipation rate for collective coordinates when the intrinsic motion is described by the independent particle model in a nucleus. The shape dependent chaos parameter measuring the degree of chaos in the intrinsic dynamics of the nuclear system will be obtained using the interpolating Brody distribution of nearest neighbour spacings in the single particle energy spectrum. A similar shape dependence is also found to be essential for phenomenological dissipation rates used in fission dynamics calculations.

Nearest neighbor spacing distributions of low-lying levels of vibrational nuclei.

Energy-level statistics are considered for nuclei whose Hamiltonian is divided into intrinsic and collective-vibrational terms. The levels are described as a random superposition of independent sequences, each corresponding to a given number of phonons. The intrinsic motion is assumed chaotic. The level spacing distribution is found to be intermediate between the Wigner and Poisson distributions and similar in form to the spacing distribution of a system with classical phase space divided into separate regular and chaotic domains. We have obtained approximate expressions for the nearest neighbor spacing and cumulative spacing distribution valid when the level density is described by a constant-temperature formula and not involving additional free parameters. These expressions have been able to achieve good agreement with the experimental spacing distributions. {copyright} {ital 1996 The American Physical Society.}

Microwave billiards with broken time reversal invariance

We consider a microwave resonator with three single-channel waveguides attached. One of these serves to couple waves into and out of the resonator; the remaining two are connected to form a one-way handle so as to break time reversal invariance. The poles of the input-output scattering coefficient of such a resonator are shown to be the eigenvalues of a non-Hermitian effective 'Hamiltonian' Heff D H i0, the anti-Hermitian part 0 of which has rank 1 and is responsible for the breaking of time reversal invariance. All of the spectral statistics recently observed for such a microwave billiard are reproduced quantitatively by taking H and 0 as random matrices. In particular, the distribution of nearest-neighbour spacings of the resonances is close to that of the GUE when H belongs to the GOE corresponding to a Sinai shape of the resonator; linear level repulsion results when H belongs to the Poissonian ensemble as it corresponds to a rectangular resonator.

Asymptotic level spacing distribution for aq-deformed random matrix ensemble

We obtain the asymptotic behaviour of the nearest-neighbour level spacing distribution for a q-deformed unitary random matrix ensemble.

Universal fluctuations of zeros of chaotic wavefunctions

Wavefunctions of one and two-dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space. We study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials. Excellent agreement is found for the two-point correlation function and nearest-neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems. We conjecture that these correlation functions are valid for any chaotic system after rescaling the phase-space distances (unfolding). Some consequences for the distribution of zeros due to time-reversal symmetry are also discussed.

Level statistics for asymmetric-top rotors in strong electric fields

Spectral fluctuations of energy levels of an asymmetric-top rotor in strong electric fields are studied. Transition from a Wigner distribution for low excitation levels to a Poisson distribution for high excitation levels is found in an analysis of the nearest neighbour spacing distribution. Spectral rigidity undergoes an analogous type of transition. Consequences on the underlying classi- cal motion are discussed.

Time domain analysis of the bound states and resonances of the hydroperoxy radical HO2: Signature of quantum chaos

The nature of the bound states and resonances of the hydroperoxy radical HO2 was analysed by computing ensemble averaged survival probabilities, 〈〈P(t)〉〉, from the Fourier transformation of the spectral autocorrelation function followed by averaging over the initial states and the Hamiltonian. The results are fitted with the 〈〈P(t)〉〉 values derived from the random matrix theory for an intermediate situation (partly regular and partly chaotic) to estimate the fractional irregularity of the states in the spectrum. The estimate is found to be in agreement with the result obtained from an analysis of the nearest-neighbor spacing distribution.

Quantum chaos in collinear (He, H2+) collisions

The quasibound spectrum of the transition state in collinear (He, H2+) collisions is obtained from time-dependent wave packet calculations. Examination of short- and long-range correlations in the eigenvalue spectra through a study of the nearest neighbor spacing distribution, P(s), and the spectral rigidity, Δ3(L), reveals signatures of quantum chaotic behavior. Analysis in the time domain is carried out by computing the survival probability 〈〈P(t)〉〉 averaged over initial states and Hamiltonian. All these indicators show intermediate behavior between regular and chaotic. A quantitative comparison of 〈〈P(t)〉〉 with the results of random matrix theory provides an estimate of the fraction of phase space exhibiting chaotic behavior, in reasonable agreement with the classical dynamics. We also analyse the dynamical evolution of coherent Gaussian wave packets located initially in different regions of phase space and compute the survival probability, power spectrum and the volume of phase space over which the wave packet spreads and illustrate the different behaviors.

Level density fluctuations and characterization of chaos in the realistic model spectra for odd-odd nuclei

Statistical properties of the realistic energy spectra of the odd-odd nuclei 106Ag, 198Au, 134Cs, 40K and 94Rb, calculated within the Interacting Boson Fermion Fermion Model, are investigated by means of the Δ3 statistics and the Nearest Neighbor Spacing Distribution method. New probability distribution function, which describes well the calculated results and enables the characterization of chaos with a physically meaningfull parameter, is proposed. Level spacing fluctuations of the examined nuclei exhibit the transitional behavior between Poisson and GOE limits, revealing different degrees of chaoticity in their dynamics.

Nearest-neighbour spacing distribution of energy levels in the region between integrability and chaos

We derive an expression for the level spacing distribution for a quantal system displaying both regular and chaotic classical motion, assuming that the level-repulsion function for the mixed system is obtained by averaging the corresponding functions for the regular and chaotic regimes with weights given by their fractional phase-space volumes. The Liouville measures of the regular region obtained from comparing this expression with the level spacing distributions of a hydrogen atom in a uniform magnetic field agree with the corresponding values obtained in the classical-mechanical analysis of this system.

Quantum chaotic transport in double barrier tunnel structures

The transition from regular to chaotic classical electron dynamics in a wide potential well with a high tilted magnetic field is investigated using Poincaré sections. The corresponding quantized energy level spectrum for the well is calculated as a function of the tilt angle π. For values of π where the system exhibits strong classical chaos, the distribution of nearest-neighbor level spacings obeys universal Wigner statistics. Regular long-range fluctuations in the density of levels are identified and related to distinct unstable closed classical orbits in accordance with the Gutzwiller trace formula. These orbits are found to produce regions of high probability density (scars) in the wavefunctions associated with subsets of almost equally-spaced energy levels. The energies of these scarred states can be located using a simple semiclassical quantization rule. This periodic scarring of individual wavefunctions is shown to have a pronounced influence on the tunneling characteristics of double barrier structures. Tunneling transitions into the scarred states dominate the current-voltage curves and generate a series of resonant peaks as observed in recent magnetotunneling experiments. Regimes in which resonant tunneling spectroscopy might provide experimental evidence for the existence of scarred states are identified.

On the degree of level repulsion

We argue that the degree of level repulsion can be determined by the two-level correlation function R 2( s), by explicitly showing the identity of the small s-behaviors of R 2( s) and the nearest-neighbor spacing distribution p(0; s). This may provide a new channel for the theoretical explanation of the mystery of the fractional power law observed numerically in the transition between integrability and chaos. Using this result, we propose to justify the level repulsion law by means of the long-time behavior of the quantum decoherence of a quantum-nondemolition coupling system.

Grain-boundary atomic structure in nanocrystalline palladium from x-ray atomic distribution functions.

We discuss the scattering theory of nanocrystalline solids, based on an evaluation of the intragrain and intergrain parts of the atomic distribution function. The results are applied to experimental x-ray scattering data obtained on a set of nanocrystalline Pd samples, prepared by inert-gas condensation, with different consolidation, aging, and annealing parameters. The experimental results show that the number of atomic neighbors in the crystal lattice coordination shells of nanocrystalline Pd is substantially lower than the one in the coarse-grained polycrystal lattice. The findings suggest that as a function of their age and thermal treatment, the samples have two different atomic structures. In samples aged for several months at room temperature, and in annealed samples, practically all atoms are located on crystal lattice sites, and the reduction in nearest-neighbor coordination number is an effect of the finite size of the crystallites. In fine-grained samples examined within 10 days of preparation, about 10% of the atoms are located on nonlattice sites with little or no atomic short-range order. This corresponds to about two atomic monolayers of atoms on nonlattice sites at the grain boundaries in as-prepared samples, as compared to about one-quarter of a monolayer in aged or annealed samples. The distribution of nearest-neighbor interatomic spacings for atoms on crystal lattice sites in nanocrystalline Pd is not measurably widened compared to the one of coarse-grained reference samples, indicating that disorder in the crystal lattice, as probed by the x-ray Debye-Waller parameter, involves displacements correlated over several lattice parameters, rather than short-range, uncorrelated atomic displacements.

Fractal potentials from energy levels.

We analyze the reconstruction by Wu and Sprung [Phys. Rev. E 48, 2595 (1993)] of a fractal one-dimensional potential, the quantum spectrum of which reproduces the first 500 nontrivial zeros of the Riemann \ensuremath{\zeta} function. Our construction is based on a spectrum with Gaussian unitary ensemble statistics as far as the nearest-neighbor spacing distribution is concerned. Our results show that a reliable estimate of the fractal dimension of the potential necessitates a very large number of levels.

Mass dependence from statistical analysis of stretching vibrational levels. Application to the linear triatomic molecule CS 2

Vibrational spectra of polyatomic molecules depend on the atomic masses. Using the CS 2 molecule as an example, a statistical analysis is performed for the stretching vibrational energy levels as a function of the isotopic masses. It is found that both the nearest-neighbour level spacing distribution and Δ 3 statistics are modified by isotopic substitution which should have experimental consequences.

Effect of a magnetic flux line on the quantum beats in the Henon-Heiles level density.

The quantum density of states of the Henon-Heiles potential displays a pronounced beating pattern. This has been explained by the interference of three isolated classical periodic orbits with nearby actions and periods. A singular magnetic flux line, passing through the origin, drastically alters the beats even though the classical Lagrangian equations of motion remain unchanged. Some of the changes can be easily understood in terms of the Aharonov-Bohm effect. However, we find that the standard periodic orbit theory does not reproduce the diffraction-like quantum effects on those classical orbits which intersect the singular flux line, and argue that corrections of relative order variant Planck's over 2pi are necessary to describe these effects. We also discuss the changes in the distribution of nearest-neighbor spacings in the eigenvalue spectrum, brought about by the flux line. (c) 1995 American Institute of Physics.

Rectangular billiard in the presence of a flux line.

We analyze the quantum spectrum of a rectangular billiard coupled to flux line. The nearest neighbor spacing distribution shows a transition from Poisson to almost Wigner distribution as the flux parameter is varied even though classically the system remains pseudointegrable and nonchaotic. In fact, the level statistics do not display any generic behavior. The calculations presented here are simple and avoid the complexity involved in modeling the singular quantum billiard problem.