Level Spacing Distribution Research Articles

Quantum chaos effects in the axial channeling of electrons

The statistical properties of the energy levels of electrons performing finite motion in a system of [110] silicon crystal chains are studied in the parameter range where quantum chaos appears in the classical limit. The resulting level-spacing distributions are in good agreement with predictions of quantum chaos theory.

Repulsively interacting fermions in a two-dimensional deformed trap with spin-orbit coupling

We investigate a two-dimensional system of with two values of the internal (spin) degree of freedom. It is confined by a deformed harmonic trap and subject to a Zeeman field, Rashba or Dresselhaus one-body spin-orbit couplings and two-body short range repulsion. We obtain self-consistent mean-field $N$-body solutions as functions of the interaction parameters. Single-particle Spectra and total energies are computed and compared to the results without interaction. We perform a statistical analysis for the distributions of nearest neighbor energy level spacings and show that quantum signatures of chaos are seen in certain parameters regimes. Furthermore, the effects of two-body repulsion on the nearest neighbor distributions are investigated. This repulsion can either promote or destroy the signatures of potential chaotic behavior depending on relative strengths of parameters. Our findings support the suggestion that cold atoms may be used to study quantum chaos both in the presence and absence of interactions.

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

Level Spacing Distribution for the Prototype of the Bianchi IX Model

Our results concern quantum chaos of the vacuum Bianchi IX model. We apply the equilateral triangle potential well approximation to the potential of the Bianchi IX model to solve the eigenvalue problem for the physical Hamiltonian. Such approximation is well satisfied in vicinity of the cosmic singularity. Level spacing distribution of the eigenvalues is studied with and without applying the unfolding procedure. In both cases, the obtained distributions are qualitatively described by Brody's distribution with the parameter $\beta\approx0.3$, revealing some sort of the level repulsion. The observed repulsion may reflect chaotic nature of the classical dynamics of the Bianchi IX universe. However, full understanding of this effects will require examination of the Bianchi IX model with the exact potential.

Two dimensional kicked quantum Ising model: dynamical phase transitions

Using an efficient one and two qubit gate simulator operating on graphical processing units, we investigate ergodic properties of a quantum Ising spin model on a two-dimensional lattice, which is periodically driven by a δ-pulsed transverse magnetic field. We consider three different dynamical properties: (i) level density, (ii) level spacing distribution of the Floquet quasienergy spectrum, and (iii) time-averaged autocorrelation function of magnetization components. Varying the parameters of the model, we found transitions between ordered (non-ergodic) and quantum chaotic (ergodic) phases, but the transitions between flat and non-flat spectral density do not correspond to transitions between ergodic and non-ergodic local observables. Even more surprisingly, we found good agreement of level spacing distribution with the Wigner surmise of random matrix theory for almost all values of parameters except where the model is essentially non-interacting, even in regions where local observables are not ergodic or where spectral density is non-flat. These findings question the versatility of the interpretation of level spacing distribution in many-body systems and stress the importance of the concept of locality.

Statistical properties of spectral fluctuations ofNinteracting bosons in a harmonic trap

Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.

Spectral statistics for random Schrödinger operators in the localized regime

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E. Restrict it to some large cube . Consider now I, a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive • uniform Poisson behavior of the locally unfolded eigenvalues, • a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales, • the distribution of the unfolded level spacings, locally and globally, • the distribution of the unfolded localization centers, locally and globally.

Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin–orbit coupling. We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.

Chaos in Jahn–Teller Rattling

We unveil chaotic behavior hidden in the energy spectrum of a Jahn-Teller ion vibrating in a cubic anharmonic potential as a typical model for rattling in cage-structure materials. When we evaluate the nearest-neighbor level-spacing distribution $P(s)$ of eigenenergies of the present oscillator system, we observe the transition of $P(s)$ from the Poisson to the Wigner distribution with the increase of cubic anharmonicity, showing the occurrence of chaos in the anharmonic Jahn-Teller vibration. The energy scale of the chaotic region is specified from the analysis of $P(s)$ and we discuss a possible way to observe chaotic behavior in the experiment of specific heat. It is an intriguing possibility that chaos in nonlinear physics could be detected by a standard experiment in condensed matter physics.

E(K, L) level statistics of classically integrable quantum systems based on the Berry–Robnik approach

Theory of the quantal level statistics of classically integrable system, developed by Makino et al. in order to investigate the non-Poissonian behaviors of level-spacing distribution (LSD) and level-number variance (LNV)\cite{MT03,MMT09}, is successfully extended to the study of $E(K,L)$ function which constitutes a fundamental measure to determine most statistical observables of quantal levels in addition to LSD and LNV. In the theory of Makino et al., the eigenenergy level is regarded as a superposition of infinitely many components whose formation is supported by the Berry-Robnik approach in the far semiclassical limit\cite{Robn1998}. We derive the limiting $E(K,L)$ function in the limit of infinitely many components and elucidates its properties when energy levels show deviations from the Poisson statistics.

Level-spacing statistics and spectral correlations in diffuse van der Waals clusters

We present a statistical analysis of eigenenergies and discuss several measures of spectral fluctuations and spectral correlations for the van der Waals clusters of different sizes. We show that the clusters become more and more complex with increase in cluster size. We study nearest-neighbour level spacing distribution $P(s)$, the level number variance $\Sigma^2(L)$, and the Dyson-Mehta $\Delta_3-$statistics for various cluster sizes. For large clusters we find that although the Bohigas-Giannoni-Schmit (BGS) conjecture seems to be valid, it does not exhibit true signatures of quantum chaos. However contrasting conjecture of Berry and Tabor is observed with smaller cluster size. For small number of bosons, we observe the existence of large number of quasi-degenerate states in low-lying excitation which exhibits the Shnirelman peak in $P(s)$ distribution. We also find a narrow region of intermediate spectrum which can be described by semi-Poisson statistics whereas the higher levels are regular and exhibit Poisson statistics. These observations are further supported by the analysis of the distribution of the ratio of consecutive level spacings $P(r)$ which is independent of unfolding procedure and thereby provides a tool for more transparent comparison with experimental findings than $P(s)$. Thus our detail numerical study clearly shows that the van der Waals clusters become more correlated with the increase in cluster size.

Universality and the QCD Anderson Transition

We study the Anderson-type transition previously found in the spectrum of the QCD quark Dirac operator in the high-temperature, quark-gluon plasma phase. Using finite size scaling for the unfolded level spacing distribution, we show that in the thermodynamic limit there is a genuine mobility edge, where the spectral statistics changes from Poisson to Wigner-Dyson statistics in a nonanalytic way. We determine the correlation length critical exponent ν and find that it is compatible with that of the unitary Anderson model.

Spectral properties of microwave graphs with local absorption

The influence of absorption on the spectra of microwave graphs has been studied experimentally. The microwave networks were made up of coaxial cables and T junctions. First, absorption was introduced by attaching a 50Ω load to an additional vertex for graphs with and without time-reversal symmetry. The resulting level-spacing distributions were compared with a generalization of the Wigner surmise in the presence of open channels proposed recently by Poli et al. [Phys. Rev. Lett. 108, 174101 (2012)]. Good agreement was found using an effective coupling parameter. Second, absorption was introduced along one individual bond via a variable microwave attenuator, and the influence of absorption on the length spectrum was studied. The peak heights in the length spectra corresponding to orbits avoiding the absorber were found to be independent of the attenuation, whereas, the heights of the peaks belonging to orbits passing the absorber once or twice showed the expected decrease with increasing attenuation.

ABSORPTION SPECTRA AND LEVEL SPACING STATISTICS IN A TERNARY ALLOY WITH AN ORNSTEIN–UHLENBECK DISORDER DISTRIBUTION

In this paper, we study the dynamics of a one-electron in a one-dimensional (1d) alloy with a correlated Ornstein–Uhlenbeck (OU) disorder distribution. The model considered here corresponds to an alloy with three types of atoms where the position of each atom is obtained using a stochastic rule based on the OU process. We analyze in detail the effect of this correlated disorder in the optical absorption spectrum and the level spacing statistics near the band center. Our results reveal a new collection of optical absorption peaks. We explain in details the appearance of each peak. Our calculations about the level spacing's distribution reveals a Poisson distribution thus contradicting previous statements about the existence of extended states in ternary electronic models with correlated disorder distribution.

Quantum chaos for two-dimensional Sinai billiard

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

Confining backgrounds and quantum chaos in holography

Classical world-sheet string theory has recently been shown to be nonintegrable and chaotic in various confining string theory backgrounds – the AdS soliton background in particular. In this Letter we study a minisuperspace quantization of the theory and look at properties of the spectrum like the distribution of level spacing, which are indicative of quantum order or chaos. In the quantum spectrum we find a gradual transition from chaotic (Wigner GOE) to integrable (Poisson) regime as we look at higher energies. This is expected since our system is integrable asymptotically, and at higher energies, the dynamics is entirely dominated by the kinetic terms.

Spectral properties of a two dimensional photonic crystal with quasi-integrable geometry

In this paper we study the statistical properties of the allowed frequencies for electromagnetic waves propagating in two-dimensional photonic crystals with quasi-integrable geometry. We compute the level spacing, group velocity, and curvature distributions (P(s), P(v), and P(c), respectively) and compare them with the corresponding random matrix theory predictions. Due to the quasi-integrability of the crystal we observe signatures of intermediate statistics in P(s) and P(c) for high refractive index contrasts.

Quantum localization of chaotic eigenstates and the level spacing distribution

The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance S goes like [proportionality]S(β) for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system [M. Robnik, J. Phys. A: Math. Gen. 16, 3971 (1983)], in which the separation of regular and chaotic eigenstates is performed.

Eigenvalue Statistics as an Indicator of Integrability of Nonequilibrium Density Operators

We propose to quantify the complexity of nonequilibrium steady state density operators, as well as of long-lived Liouvillian decay modes, in terms of the level spacing distribution of their spectra. Based on extensive numerical studies in a variety of models, some solvable and some unsolved, we conjecture that the integrability of density operators (e.g., the existence of an algebraic procedure for their construction in finitely many steps) is signaled by a Poissonian level statistics, whereas in the generic nonintegrable cases one finds level statistics of a Gaussian unitary ensemble of random matrices. Eigenvalue statistics can therefore be used as an efficient tool to identify integrable quantum nonequilibrium systems.

Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates

We study the quantum mechanics of a billiard (Robnik 1983 J. Phys. A: Math. Gen. 16 3971) in the regime of mixed-type classical phase space (the shape parameter λ = 0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré–Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not uniformly occupy the entire available chaotic classical phase space component, but are localized on a proper subset. We find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010 J. Phys. A: Math. Theor. 43 215101), where the entire spectrum was found to obey the Berry–Robnik–Brody statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.