Predictions Of Random Matrix Theory Research Articles

Statistical properties at the spectrum edge of the QCD Dirac operator

The statistical properties of the spectrum of the staggered Dirac operator in an SU(2) lattice gauge theory are analyzed both in the bulk of the spectrum and at the spectrum edge. Two commonly used statistics, the number variance and the spectral rigidity, are investigated. While the spectral fluctuations at the edge are suppressed to the same extent as in the bulk, the spectra are more rigid at the edge. To study this effect, we introduce a microscopic unfolding procedure to separate the variation of the microscopic spectral density from the fluctuations. For the unfolded data, the number variance shows oscillations of the same kind as before unfolding, and the average spectral rigidity becomes larger than the one in the bulk. In addition, the short-range statistics at the origin is studied. The lattice data are compared to predictions of chiral random-matrix theory, and agreement with the chiral Gaussian Symplectic Ensemble is found.

Wave Function Intensity Statistics from Unstable Periodic Orbits

We examine the effect of short unstable periodic orbits on wave function statistics in a classically chaotic system, and find that the tail of the wave function intensity distribution in phase space is dominated by scarring associated with the least unstable periodic orbits. In an ensemble average over systems with classical orbits of different instabilities, a power-law tail is found, in sharp contrast to the exponential prediction of random matrix theory. The calculations are compared with numerical data, and quantitative agreement is obtained.

On the convergence to ergodic behaviour of quantum wavefunctions

We study the decrease of fluctuations of diagonal matrix elements of observables and of Husimi densities of quantum mechanical wave functions around their mean value upon approaching the semi-classical regime ($\hbar \rightarrow 0$). The model studied is a spin (SU(2)) one in a classically strongly chaotic regime. We show that the fluctuations are Gaussian distributed, with a width $\sigma^2$ decreasing as the square root of Planck's constant. This is consistent with Random Matrix Theory (RMT) predictions, and previous studies on these fluctuations. We further study the width of the probability distribution of $\hbar$-dependent fluctuations and compare it to the Gaussian Orthogonal Ensemble (GOE) of RMT.

Quantum dot self-consistent electronic structure and the Coulomb blockade.

We employ density functional theory to calculate the self consistent electronic structure, free energy and linear source-drain conductance of a lateral semiconductor quantum dot patterned via surface gates on the 2DEG formed at the interface of a $GaAs-AlGaAs$ heterostructure. The Schr\"{o}dinger equation is reduced from 3D to multi-component 2D and solved via an eigenfunction expansion in the dot. This permits the solution of the electronic structure for dot electron number $N \sim 100$. We present details of our derivation of the total dot-lead-gates interacting free energy in terms of the electronic structure results, which is free of capacitance parameters. Statistical properties of the dot level spacings and connection coefficients to the leads are computed in the presence of varying degrees of order in the donor layer. Based on the self-consistently computed free energy as a function of gate voltages, $V_i$, and N, we modify the semi-classical expression for the tunneling conductance as a function of gate voltage through the dot in the linear source-drain, Coulomb blockade regime. Among the many results presented, we demonstrate the existence of a shell structure in the dot levels which (a) results in envelope modulation of Coulomb oscillation peak heights, (b) which influences the dot capacitances and should be observable in terms of variations in the activation energy for conductance in a Coulomb oscillation minimum, and (c) which possibly contributes to departure of recent experimental results from the predictions of random matrix theory.

Theoretical study of the unimolecular dissociation HO2→H+O2. II. Calculation of resonant states, dissociation rates, and O2 product state distributions

Three-dimensional quantum mechanical calculations have been carried out, using a modification of the log-derivative version of Kohn’s variational principle, to study the dissociation of HO2 into H and O2. In a previous paper, over 360 bound states were found for each parity, and these are shown to extend into the continuum, forming many resonant states. Analysis of the bound states close to the dissociation threshold have revealed that HO2 is a mainly irregular system and in this paper it is demonstrated how this irregularity persists in the continuum. At low energies above the threshold, these resonances are isolated and have widths that fluctuate strongly over more than two orders of magnitude. At higher energies, the resonances begin to overlap, while the fluctuations in the widths decrease. The fluctuations in the lifetimes and the intensities in an absorption-type spectrum are compared to the predictions of random matrix theory, and are found to be in fair agreement. The Rampsberger–Rice–Kassel–Marcus (RRKM) rates, calculated using variational transition state theory, compare well to the average of the quantum mechanical rates. The vibrational/rotational state distributions of O2 show strong fluctuations in the same way as the dissociation rates. However, their averages do not agree well with the predictions of statistical models, neither phase space theory (PST) nor the statistical adiabatic channel model (SACM), as these are dependent on the dynamical features of the exit channel. The results of classical trajectory calculations agree well on average with those of the quantum calculations.

Quantum chaotic dynamics and random polynomials

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of “quantum chaotic dynamics.” It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials.

On the duality between periodic orbit statistics and quantum level statistics

We discuss consequences of a recent observation that the sequence of periodic orbits in a chaotic billiard behaves like a poissonian stochastic process on small scales. This enables the semiclassical form factor $K_{sc}(\tau)$ to agree with predictions of random matrix theories for other than infinitesimal $\tau$ in the semiclassical limit.

Periodic-orbit theory of the number variance Σ2( L) of strongly chaotic systems

We discuss the number variance Σ 2( L) and the spectral form factor F( τ) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodic-orbit representations of Σ 2( L) and F( τ) are derived which explain the breakdown of universality, i.e., the deviations from the predictions of random-matrix theory. The relation of the exact spectral form factor F( τ) to the commonly used approximation K( τ) is clarified. As an illustration the periodic-orbit representations are tested in the case of a strongly chaotic system at low and high energies including very long-range correlations up to L = 700. Good agreement between “experimental” data and theory is obtained.

Gaussian Orthogonal Ensemble Statistics in a Microwave Stadium Billiard with Chaotic Dynamics: Porter-Thomas Distribution and Algebraic Decay of Time Correlations

The complete set of resonance parameters for 950 resonances of a superconducting microwave cavity connected to three antennas has been measured. This cavity simulates the quantum mechanics of a particle in a Bunimovich stadium. The partial widths are found to follow a Porter-Thomas distribution. The Fourier transforms of the $S$-matrix autocorrelation functions decay algebraically (nonexponentially) in time. These results agree perfectly with the predictions of random-matrix theory. They constitute one of the most stringent tests ever of this expected connection between chaotic dynamics and randommatrix theory.

Rydberg atoms in external fields as an example of open quantum systems with classical chaos

We examine the quantum spectra of hydrogen atoms in external magnetic and electric fields above the ionization threshold with respect to signatures of classical chaos characteristics of open systems. The spectra are obtained by calculating wavefunctions and photoionization cross sections in the continuum region with the aid of the complex-coordinate-rotation method. We find that the photoionization cross sections exhibit strong Ericson fluctuations, a quantum feature characteristic of classically chaotic scattering, in energy-field regions where classical trajectory calculations reveal a fractal dependence of the classical ionization time on the initial conditions. We also compare the nearest-neighbour-spacing distributions of complex resonance energies with predictions of random-matrix theories and and that our results are well reproduced by a Ginibre distribution.

A matrix ensemble with a preferential basis and its application to disordered metals and insulators

Trie standard ensembles of random matrices are invariant under change of basis. In disordered systems, ibis invariance is broken and deviations tram trie random matrix theory predictions occur, especially for strong disorder. We consider a generalization of trie standard ensembles which includes a preferential basis and which gives rise ta a screened pairwise interaction between energy levels. As a function of the conductance g, we qualitatively describe level statistics in the menai, insulator and at trie mobility edge. In trie unitary case, an exact solution of this mortel is provided by earlier works of Gaudin.

EXACTLY SOLVABLE SCALING THEORY OF CONDUCTION IN DISORDERED WIRES

Recent developments in the scaling theory of phase-coherent conduction through a disordered wire are reviewed. The Dorokhov–Mello–Pereyra–Kumar equation for the distribution of transmission eigenvalues has been solved exactly, in the absence of time-reversal symmetry. Comparison with the previous prediction of random-matrix theory shows that this prediction was highly accurate but not exact: the repulsion of the smallest eigenvalues was overestimated by a factor of two. This factor of two resolves several disturbing discrepancies between random-matrix theory and microscopic calculations, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.

Exact solution for the distribution of transmission eigenvalues in a disordered wire and comparison with random-matrix theory.

We consider the complete probability distribution P({${\mathit{T}}_{\mathit{n}}$}) of the transmission eigenvalues ${\mathit{T}}_{1}$,${\mathit{T}}_{2}$,...,${\mathit{T}}_{\mathit{N}}$ of a disordered quasi-one-dimensional conductor (length L much greater than width W and mean free path l). The Fokker-Planck equation which describes the evolution of P with increasing L is mapped onto a Schr\"odinger equation by a Sutherland-type transformation. In the absence of time-reversal symmetry (e.g., because of a magnetic field), the mapping is onto a free-fermion problem, which we solve exactly. The resulting distribution is compared with the predictions of random-matrix theory (RMT) in the metallic regime (L\ensuremath{\ll}Nl) and in the insulating regime (L\ensuremath{\gg}Nl). We find that the logarithmic eigenvalue repulsion of RMT is exact for ${\mathit{T}}_{\mathit{n}}$'s close to unity, but overestimates the repulsion for weakly transmitting channels. The nonlogarithmic repulsion resolves several long-standing discrepancies between RMT and microscopic theory, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.

Coherence-length dependence of fluctuations in the conductance of normal-superconducting interfaces.

We compute sample-to-sample fluctuations in the electrical conductance ${\mathit{G}}_{\mathit{N}\mathit{S}}$ of mesoscopic normal-superconducting boundaries. If the superconductor is clean and the normal conductor diffusive, the rms deviation \ensuremath{\delta}${\mathit{G}}_{\mathit{N}\mathit{S}}$ is approximately a factor of 2 larger than the universal value obtained for normal disordered conductors. This is consistent with predictions of random matrix theories and recent numerical simulations. If the normal material is clean, but the superconductor diffusive, the boundary conductance fluctuation is greater than \ensuremath{\delta}${\mathit{G}}_{\mathit{N}\mathit{S}}$, by an amount which increases as the superconducting coherence length decreases. For a diffusive normal material in contact with a dirty superconductor, \ensuremath{\delta}${\mathit{G}}_{\mathit{N}\mathit{S}}$ is insensitive to weak disorder in the superconductor, but decreases in the extreme dirty limit, where quasiparticles within the superconductor become Anderson localized.

Random-matrix theory and eigenvector statistics of dissipative dynamical systems

The authors investigate the predictions of random-matrix theory for eigenvector statistics and compare them with numerical results obtained for the dissipative periodically kicked top. Different types of statistics are found. In contrast to conservative dynamics they are connected with the types of eigenvalue of the propagator rather than with the symmetries of Hamiltonian evolution.

Onset of chaos and its signature in the spectral autocorrelation function.

The signature of chaos in the spectral autocorrelation function and in its Fourier transform, the survival probability, is shown to be in good agreement with the predictions of random matrix theory. An expression is proposed for the survival probability of an experimentally prepared nonstationary state when the dynamics are intermediate between chaotic and regular. Its validity is tested through the study of a model Hamiltonian . Two parameters can be extracted from the above observable, one which characterizes the level statistics and one which characterizes the distribution of transition intensities.

Statistical properties of multichannel-quantum-defect-theory spectra.

We study the nearest-neighbor spacings (NNS) and spectral rigidities in quantum spectra generated by the equation of multichannel-quantum-defect theory for interacting Rydberg series. For weak coupling near the separable limit, the NNS distributions agree with a Poisson spectrum quite closely, but the spectral rigidities follow the Poisson expectations only for very short stretches of spectrum containing hardly more states than the number of channels involved. For intermediate and strong coupling, both the NNS distributions and the spectral rigidities agree well with the predictions of random matrix theory for the Gaussian orthogonal ensemble case.

Eigenmode statistics in large two‐dimensional systems

A finite difference model is used to numerically investigate the higher‐acoustic eigenmodes of membranes with random boundaries. Natural frequencies are found to distribute themselves with level repulsion and spectral rigidity like those of the “Gaussian orthogonal ensemble” in accord with the predictions of random matrix theory. These statistics have also been observed in recent experiments determining the higher eigenfrequencies of aluminum blocks. The present work goes further in that it also considers the statistics of the eigenmode shapes. In particular, values for the mean fourth power of modal amplitude (related to the “participation ratio” and to variances of power transfer functions in large systems) are reported, as are results on repulsion between mode shapes. [Work supported by NSF.]

Spin-orbit scattering for localized electrons: Absence of negative magnetoconductance

The distribution of hopping conductances in the strongly localized regime in the presence of both a magnetic field and spin-orbit (SO) scattering is calculated via an analytic independent-directed-path formalism, and a locator expansion which includes all correlations between paths. Both methods lead to a positive magnetoconductance for all strengths of SO scattering, contrary to recent random-matrix theory predictions. Extensive numerical simulations demonstrate that the crossover from negative to positive magnetoconductance occurs as the system size exceeds the localization length.

Quantum chaos in the hyperbola billiard

A few hundred energy levels of the hyperbola billiard, a strongly chaotic system, are computed using a boundary-element method. The level statistics is investigated and found to be consistent with the predictions of random-matrix theory for the Gaussian orthogonal ensemble. The energy spectrum is used for an application of Gutzwiller's periodic orbit theory, which nicely demonstrates that the quantal energies “know” the classical periodic orbits.