Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.     

Distribution of the local density of states, reflection coefficient, and Wigner delay time in absorbing ergodic systems at the point of chiral symmetry

Employing the chiral Gaussian unitary ensemble of random matrices, we calculate the probability distribution of the local density of states for zero-dimensional ("quantum chaotic") two-sublattice systems at the point of chiral symmetry E=0 and in the presence of uniform absorption. The obtained result can be used to find the distributions of the reflection coefficient and of the Wigner time delay for such systems.     

Finite element distributions in statistical theory of energy levels in quantum systems

The probability density functions of the three-point finite elements of the three adjacent energy levels for the three-level quantum system are introduced as a supplementary characteristics of quantum chaos. The three-level quantum system is studied. The probability density functions of the second difference and asymmetrical three-point first finite element are computed for the three-dimensional Gaussian orthogonal ensemble GOE(3), the three-dimensional Gaussian unitary ensemble GUE(3), the three-dimensional Gaussian symplectic ensemble GSE(3), as well as for the Poisson ensemble PE.     

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Recently, a new approach, called a non-parametric model of random uncertainties, has been introduced for modelling random uncertainties in linear and non-linear elastodynamics in the low-frequency range. This non-parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive-definite real random matrices. This ensemble differs from the Gaussian orthogonal ensemble (GOE) and from the other known ensembles of the random matrix theory. The present paper has three main objectives. The first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the GOE. The second one is to compare this new ensemble of random matrices with the GOE in the context of the non-parametric approach of random uncertainties in structural dynamics for the low-frequency range. The last objective is to give a new validation for the non-parametric model of random uncertainties in structural dynamics in comparing, in the low-frequency range, the dynamical response of a simple system having random uncertainties modelled by the parametric and the non-parametric methods. These three objectives will allow us to conclude about the validity of the different theories.     

Quantum Quenches with Random Matrix Hamiltonians and Disordered Potentials

We numerically investigate statistical ensembles for the occupations of eigenstates of an isolated quantum system emerging as a result of quantum quenches. The systems investigated are sparse random matrix Hamiltonians and disordered lattices. In the former case, the quench consists of sudden switching‐on the off‐diagonal elements of the Hamiltonian. In the latter case, it is sudden switching‐on of the hopping between adjacent lattice sites. The quench‐induced ensembles are compared with the so‐called “quantum micro‐canonical” (QMC) ensemble describing quantum superpositions with fixed energy expectation values. Our main finding is that quantum quenches with sparse random matrices having one special diagonal element lead to the condensation phenomenon predicted for the QMC ensemble. Away from the QMC condensation regime, the overall agreement with the QMC predictions is only qualitative for both random matrices and disordered lattices but with some cases of a very good quantitative agreement. In the case of disordered lattices, the QMC ensemble can be used to estimate the probability of finding a particle in a localized or delocalized eigenstate.     

Interaction of regular and chaotic states

Modelling the chaotic states in terms of the Gaussian Orthogonal Ensemble of random matrices (GOE), we investigate the interaction of the GOE with regular bound states. The eigenvalues of the latter may or may not be embedded in the GOE spectrum. We derive a generalized form of the Pastur equation for the average Green’s function. We use that equation to study the average and the variance of the shift of the regular states, their spreading width, and the deformation of the GOE spectrum non-perturbatively. We compare our results with various perturbative approaches.     

Eigenvalue density of cross-correlations in Sri Lankan financial market

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.     

Universality of “level curvature” distribution for large random matrices: systematic analytical approaches

We perform an extensive analytical study of distributions of “level curvatures” (the second derivatives of eigenvalues with respect to a perturbation parameter) for different classers of random matrice. First, we consider the case of three Gaussian ensembles: GUE, GOE and GSE. This part of our calculation is complementary to that done recently by von Oppen [22, 23], but evaluation goes along different lines and allows to treat all the three cases uniformly. In the second part of the paper we exploit completely another method allowing to treat the problem analytically for the broad class of disordered systems subject to time-reversal symmetry breaking perturbation. That gives us a possibility to prove the conjecture by Zakrzewski and Delande [17] for the ensemble of symmetric sparse random matrices.     

Universal distribution functions in two-dimensional localized systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.     

Level repulsion in the ground-state region of nuclei

Using the nuclei in the nuclear table as members of an ensemble of hamiltonianas we show that, once nuclei with a systematic behaviour are eliminated, the nearest-neighbour spacings between states with the same spin and parity follow, in the ground state region, a distribution which is roughly consistent with that predicted by the gaussian orthogonal ensemble (GOE). These is some indication, which however is not conclusive, that the two-body random hamiltonian ensemble fits the data somewhat better than the GOE.     

Large deviations of extreme eigenvalues of random matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.     

Donor ordering and the statistics of quantum dot level spacings and widths from full 3D self-consistent electronic structure

Using full 3D self-consistent electronic structure calculations of small (electron numberN 100) lateral quantum dots formed on GaAs–AlGaAs HEMT devices we calculate the statistics of level spacings Δε_pand tunneling coefficients Γ_pbetween leads and confined states of the dot. We employ random and ordered donor layer charge distributions, the latter generated through Monte Carlo variable range hopping simulations, as well as a homogeneous (jellium) ionic charge distribution, and examine the effects on these statistics.It has recently been argued that the statistics of the level spacings and widths follow from random matrix theory when the Hamiltonian is described by the Gaussian orthogonal ensemble (GOE) for zero magnetic fieldB, and by the Gaussian unitary ensemble (GUE) forBsufficiently large to break time reversal symmetry. Specifically it is argued that when the dot wave functions are expanded in an arbitrary basis the expansion coefficients, according to the postulate of Porter and Thomas, are uniformly distributed in Hilbert space.In our calculation we obtain statistics of level spacings and widths by generating many configurations of disordered and ordered donor charge. This corresponds to the experimental situation of thermal cycling of the device. We find that a pronounced transition occurs in the level spacing statistics between the completely disordered donor layer ensemble, which seems to be well described by random matrix theory, and the ordered ensemble which is dominated by secular variations in the coefficients. In particular, a shell structure in the levels, which results from approximate parabolicity in the self-consistent confining potential, is observed. This, and the effects of symmetry under inversion and azimuthal symmetry, are speculated to undermine level repulsion and result in Poisson statistics for the levels here at the band edge.Finally we find that distortions in the dot shape are markedly less significant in varying the widths (and level spacings) than calculations based on a hard wall potential for the dot predict. This suggests that the notion of invariant atomic structure for sufficiently small dots is not invalidated by the randomness inherent in donor positions and shape distortion but, on the contrary, a systematic study of dot structure is possible.     

Correlations in the adiabatic response of chaotic systems

Adiabatic variation of the parameters of a chaotic system results in a fluctuating reaction force. The quantum analog of a classical dissipative force, proportional to the time integral of the force-force correlation function, vanishes. We study this quantum-classical crossover for random matrix models. For the Gaussian unitary ensemble the crossover is found to take place on the Heisenberg time scale and the finite time integral practically vanishes for longer times. For the Gaussian orthogonal case, there is no such time scale and the integral falls off inversely proportional to time.     

Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.     

Analysis of symmetry breaking in quartz blocks using superstatistical random-matrix theory

We study the symmetry breaking of acoustic resonances measured by Ellegaard et al. (1996) [1] in quartz blocks. The observed resonance spectra show a gradual transition from a superposition of two uncoupled components, one for each symmetry realization, to a single component that is well represented by a Gaussian orthogonal ensemble (GOE) of random matrices. We discuss the applicability of superstatistical random-matrix theory to the final stages of the symmetry-breaking transition. A comparison is made between the formula from superstatistics and that from a previous work by Abd El-Hady et al. (2002) [7], which describes the same data by introducing a third GOE component. Our results suggest that the inverse chi-squared superstatistics could be used for studying the whole symmetry-breaking process.     

Transition from Poisson to circular unitary ensemble

Transitions to universality classes of random matrix ensembles have been useful in the study of weakly-broken symmetries in quantum chaotic systems. Transitions involving Poisson as the initial ensemble have been particularly interesting. The exact two-point correlation function was derived by one of the present authors for the Poisson to circular unitary ensemble (CUE) transition with uniform initial density. This is given in terms of a rescaled symmetry breaking parameter Λ. The same result was obtained for Poisson to Gaussian unitary ensemble (GUE) transition by Kunz and Shapiro, using the contour-integral method of Brezin and Hikami. We show that their method is applicable to Poisson to CUE transition with arbitrary initial density. Their method is also applicable to the more general ℓCUE to CUE transition where ℓCUE refers to the superposition of ℓ independent CUE spectra in arbitrary ratio.     

Statistical theory of energy levels and random matrices in physics

In this paper the physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed. After a preliminary introduction we summarize the symmetry properties of physical systems. Different kinds of ensembles are then discussed. This includes the Gaussian, orthogonal, and unitary ensembles. The problem of eigenvalue-eigenvector distributions of the Gaussian ensemble is then discussed, followed by a discussion on the distribution of the widths. In the appendices we discuss the symplectic group and quaternions, and the Gaussian ensemble in detail.     

Extreme statistics of complex random and quantum chaotic states

Complex random states have the statistical properties of the Gaussian and circular unitary ensemble eigenstates of random matrix theory. Even though their components are correlated by the normalization constraint, it is nevertheless possible to derive compact analytic formulas for their extreme values' statistical properties for all dimensionalities. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-N formulas.     

Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of the Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.