Nearest-neighbor Spacing Distribution Research Articles

U(1) staggered dirac operator and random matrix theory

We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge theory and its relationship to random matrix theory. In the confined as well as in the Coulomb phase the nearest-neighbor spacing distribution of the unfolded eigenvalues is well described by the chiral unitary ensemble. The same is true for the distribution of the smallest eigenvalue and the microscopic spectral density in the confined phase. The physical origin of the chiral condensate in this phase deserves further study.

The quantum-classical crossover in the adiabatic response of chaotic systems

The autocorrelation function of the force acting on a slow classical system, resulting from interaction with a fast quantum system is calculated following Berry-Robbins, Wilkinson and Jarzynski within the leading order correction to the adiabatic approximation. The time integral of the autocorrelation function is often proportional to the rate of energy transfer between the systems. The fast quantum system is assumed to be chaotic in the classical limit for each configuration of the slow system. An analytic formula is obtained for the finite-time integral of the correlation function, in the framework of random matrix theory (RMT), for a specific dependence on the adiabatically varying parameter. Extension to a wider class of RMT models is discussed. For the Gaussian unitary and symplectic ensembles for long times the time integral of the correlation function vanishes or falls off as a Gaussian with a characteristic time that is proportional to the Heisenberg time, depending on the details of the model. The fall-off is inversely proportional to time for the Gaussian orthogonal ensemble. The correlation function is found to be dominated by the nearest-neighbour level spacings. It was calculated for a variety of nearest-neighbour level spacing distributions, including ones that do not originate from RMT ensembles. The various approximate formulae obtained are tested numerically in RMT. The results shed light on the quantum to classical crossover for chaotic systems. The implications on the possibility to experimentally observe deterministic friction are discussed.

Beyond nearest-neighbor spacing distributions: A new indicator for core-induced chaos

We present a method for the statistical analysis of quantum spectra which goes beyond the standard nearest-neighbor spacing distribution. The method exploits a peculiar correlation of oscillator strength and level spacing distributions. For diamagnetic Rydberg spectra we demonstrate that core-induced chaos and generic chaos can be distinguished in an experimentally accessible regime far below the semiclassical limit and with already a few hundred lines per spectrum. We present two hypotheses for a generalization of our findings.

Transition curves for the variance of the nearest neighbor spacing distribution for Poisson to Gaussian orthogonal and unitary ensemble transitions.

Using an appropriate 2x2 random matrix ensemble, transition curves for the variance of the nearest neighbor spacing distribution are constructed for the Poisson to Gaussian orthogonal and unitary ensemble transitions in terms of an easily identifiable transition parameter.

Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.

Quantum chaos in compact lattice QED

Complete eigenvalue spectra of the staggered Dirac operator in quenched 4D compact QED are studied on ${8}^{3}\ifmmode\times\else\texttimes\fi{}4$ and ${8}^{3}\ifmmode\times\else\texttimes\fi{}6$ lattices. We investigate the behavior of the nearest-neighbor spacing distribution $P(s)$ as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble of random-matrix theory indicating quantum chaos. Combining this with previous results on QCD, we conjecture that quite generally the non-linear couplings of quantum field theories lead to a chaotic behavior of the eigenvalues of the Dirac operator.

Quantum chaos and QCD at finite chemical potential

We investigate the distribution of the spacings of adjacent eigenvalues of the lattice Dirac operator. At zero chemical potential μ, the nearest-neighbor spacing distribution P( s) follows the Wigner surmise of random matrix theory both in the confinement and in the deconfinement phase. This is indicative of quantum chaos. At nonzero chemical potential, the eigenvalues of the Dirac operator become complex. We discuss how P( s) can be defined in the complex plane. Numerical results from an SU(3) simulation with staggered fermions are compared with predictions from non-hermitian random matrix theory, and agreement with the Ginibre ensemble is found for μ ≈ 0.7.

Intramolecular vibrational energy redistribution and conformational isomerization in vibrationally excited 2-fluoroethanol: High-resolution, microwave-infrared double-resonance spectroscopy investigation of the asymmetric –CH2(F) stretch near 2980 cm−1

The asymmetric –CH2(F) stretch spectrum of 2-fluoroethanol near 2980 cm−1 has been rotationally assigned using microwave-infrared double-resonance spectroscopy methods in an electric-resonance optothermal molecular-beam spectrometer. The eigenstate-resolved infrared spectrum shows the effects of intramolecular vibrational energy redistribution (IVR) through the fragmentation of each rotational level of the vibrationally excited state into a set of transitions. From the spectrum we determine the IVR lifetime of the asymmetric –CH2(F) stretch to be 275 ps. The measured vibrational state density at 2980 cm−1 is 44 states/cm−1, and matches the value for the total state density obtained from a direct count. This agreement suggests that vibrational states of both the Gg′ and Tt conformers are coupled by the intramolecular dynamics. From measurements of the c-type pure rotational transitions of the Gg′ conformer we determine that the tunneling splitting for the Gg′ ground state is less than 35 kHz. The infrared spectrum is characterized by a large number of closely spaced infrared transitions. The clustering of vibrational energy levels is attributed to the weak interaction between the degenerate Gg′ and nondegenerate Tt vibrational states. This lifting of the effective Gg′ parity degeneracy is quantitatively investigated through the shape of the nearest-neighbor level spacing distribution. From this analysis we estimate that the isomerization lifetime for the Gg′ conformer of 2-fluoroethanol is 2 ns.

Deformed Gaussian orthogonal ensemble and the statistical fluctuations in the spectra of the quartic oscillator

The nearest-neighbor spacing distributions proposed by four models, namely, the Berry-Robnik, Caurier-Grammaticos-Ramani, Lenz-Haake, and the deformed Gaussian orthogonal ensemble, as well as the ansatz by Brody, are applied to the transition between chaos and order that occurs in the isotropic quartic oscillator. The advantages and disadvantages of these five descriptions are discussed. In addition, the results of a simple extension of the expression for the Dyson-Mehta statistic ${\ensuremath{\Delta}}_{3}$ are compared with those of a more popular one, usually associated with the Berry-Robnik formalism.

Classical and quantum chaos in the generalized parabolic lemon-shaped billiard

Two-dimensional billiards of a generalized parabolic lemonlike shape are investigated classically and quantum mechanically depending on the shape parameter δ. Quantal spectra are analyzed by means of the nearest-neighbor spacing distribution method. Calculated results are well accounted for by the proposed new two-parameter distribution function P(s), which is a generalization of Brody and Berry-Robnik distributions. Classically, Poincare diagrams are shown and interpreted in terms of the lowest periodic orbits. For δ=2, the billiard has some unique characteristics resulting from the focusing property of the parabolic mirror. Comparison of the classical and quantal results shows an accordance with the Bohigas, Giannoni, and Schmit conjecture and confirms the relevance of the new distribution for the analysis of realistic spectral data.

Energy level statistics at the metal-insulator transition in the Anderson model of localization with anisotropic hopping

Recently, a metal-insulator transition (MIT) was found in the anisotropic Anderson model of localization by transfer-matrix methods (TMM). This MIT has been also investigated by multifractal analysis (MFA) and the same critical disorders $W_c$ have been obtained within the accuracy of the data. We now employ energy level statistics (ELS) to further characterize the MIT. We find a crossover of the nearest-neighbor level spacing distribution $P(s)$ from GOE statistics at small disorder indicating metallic behavior to the Poisson distribution at large disorder characteristic for localized states. An analysis of the system size dependence of the spectral rigidity $\Delta_3(L)$ confirms the values of $W_c$ from TMM and MFA.

Fluctuation properties of the Dirac spectrum at finite temperature and density

The eigenvalue spectrum of the staggered Dirac matrix on a 6 3 × 4 lattice is analyzed in full QCD at finite temperature and in the presence of a chemical potential μ We construct the nearest-neighbor spacing distribution P( s) for the real eigenvalues (if μ = 0) and for the complex eigenvalues (if μ ≠ 0). At zero chemical potential, we find the expected agreement with the Wigner distribution of random matrix theory, both in the confinement phase and in the deconfinement phase. As μ is increased, we observe a crossover from Wigner statistics to Ginibre statistics, in accordance with random matrix predictions for complex non-hermitian matrices.

Statistical analysis of the computed X̃ 2A1/Ã 2B2 spectrum of NO2: Some insights into the causes of its irregularity

We present a statistical analysis of the B2 symmetry lines of the nonadiabatic X̃ 2A1/à 2B2 spectrum of NO2 in the energy range 10 000–22 000 cm−1, obtained by ab initio diabatic potentials refined with respect to the experimental lines until 10 000 cm−1. The two most common statistical measures, the nearest neighbor spacing distribution and the Δ̄3 are taken into account. Both the statistics show a high degree of irregularity of the spectrum. The analysis is repeated in restricted ranges for which analogous investigations on experimental data are reported in the literature. The comparison is always satisfactory. Comparison is also discussed with respect to previous theoretical results obtained by other authors. We investigate the causes of the spectrum irregularity by removing systematically some selected couplings from the full nonadiabatic Hamiltonian. We find that the irregularity is due to two main causes; the sensible vibronic coupling between the two diabatic states X̃ 2A1 and à 2B2 and the stretch–bend couplings on the diabatic ground surface X̃ 2A1.

Numerical simulation of level statistics

The statistics of energy eigenvalues for a class of Hamiltonians written as H = H 0+ tV is studied through the dynamics of levels regarding the perturbation parameter t as time. The statistical properties of eigenvalues are investigated by measuring quantities such as the nearest-neighbor level spacing distribution and the Dyson-Metha Δ 3 statistics for two types of ensembles known as orthogonal and unitary, and for three values of the system parameter controlling chaos.

Energy-level statistics and localization of 2d electrons in random magnetic fields

Using the method of energy-level statistics, the localization properties of electrons moving in two dimensions in the presence of a perpendicular random magnetic field and additional random disorder potentials are investigated. For this model, extended states have recently been proposed to exist in the middle of the band. In contrast, from our calculations of the large-s behavior of the nearest-neighbor level spacing distribution P( s) and from a finite-size scaling analysis we find only localized states in the suggested energy and disorder range.

Level-Spacing Distributions of Planar Quasiperiodic Tight-Binding Models

We study the statistical properties of energy spectra of two-dimensional quasiperiodic tight-binding models. We demonstrate that the nearest-neighbor level spacing distributions of these non-random systems are well described by random matrix theory. Properly taking into account the symmetries of models defined on various finite approximants of quasiperiodic tilings, we find that the underlying universal level-spacing distribution is given by the Wigner-Dyson distribution of the Gaussian orthogonal random matrix ensemble. Our data allow us to see the differences between the Wigner surmise and the exact level-spacing distribution. In particular, our result differs from the critical level-spacing distribution computed at the metal-insulator transition in the three-dimensional Anderson model of disorder.

Evidence for quantum chaos in the plasma phase of QCD

We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD on a $6^3\times 4$ lattice. As a measure of the fluctuation properties of the eigenvalues, we study the nearest-neighbor spacing distribution $P(s)$ for various values of $\beta$ both in the confinement and in the deconfinement phase. In both phases except far into the deconfinement region, the lattice data agree with the Wigner surmise of random-matrix theory which is indicative of quantum chaos. We do not find signs of a transition to Poisson regularity at the deconfinement phase transition.

Quantum chaos in QCD at finite temperature

We study complete eigenvalue spectra of the staggered Dirac matrix in quenched QCD on a 6 3 × 4 lattice. In particular, we investigate the nearest-neighbor spacing distribution P( s) for various values of β both in the confinement and deconfinement phase. In both phases except far into the deconfinement region, the data agree with the Wigner surmise of random matrix theory which is indicative of quantum chaos. No signs of a transition to Poisson regularity are found, and the reasons for this result are discussed.

Extracting the dynamics in classically chaotic quantum systems: Spectral analysis of the HO2 molecule

We present a scaling technique to analyze quantum spectra, i.e., to obtain from quantum calculations detailed information about the underlying important classical motions. The method can be applied to a general quantum system without a classical scaling property. A demonstration on the conventionally unassignable vibrational spectrum of the HO2 radical reveals remnants of classical broken tori embedded in the chaotic phase space and leads to a new assignment of spectral patterns in terms of classical Fermi resonances between the local mode motions. The scaling technique also allows to investigate the statistical properties of level spacings at fixed energies. The nearest neighbor spacing distribution of the HO2 molecule undergoes a transition from mixed phase space behavior at low energies to the Wigner distribution characteristic for chaotic systems at energies near the dissociation threshold.

The structure of 113Sn from proton and alpha- particle induced reactions

The results of in-beam investigations of 113 Sn using the (p,n), (p,3n), (α,n) and (α,2n) reactions are summarized. Excited states have been identified until E x = 4715 MeV and J π = (27/2t -). For a large number of levels mean lifetimes τ have been determined with the DSA method. For the J k = 25/2+ state at E x = 4059 MeV, τ = 1.0(4) ns has been measured with the γ-RF method. The experimental results are compared with the predictions of shell-model calculations. Most of the positive-parity states may be considered as one-or three-quasiparticle neutron excitations of the 2d5/2, 1g7/2, 3s1/2 and 2d3/2 shells, the negative-parity states as the coupling of one 1h11/2 neutron to the two- or four-quasiparticle neutron excitations in the even-mass 112Sn core. For the 25/2+ isomer the three-quasiparticle neutron configuration ν(h 11 2/2 g 7 2/−1 ) has been proposed on the basis of a shell-model analysis using the mass-formula formalism. The experimentally observed yrast states in 50 113 Sn63 are compared with the corresponding states in the valence mirror nucleus 63 145 Eu82 giving remarkable similarities although the parameters for the shell-model calculations differ considerably. The analysis of nearest-neighbour spacing distributions of experimentally obtained 5/2+ states in 113Sn does not allow definite conclusions about regularity or chaos.