Gaussian Unitary Ensemble Statistics Research Articles

Experimental test of the Rosenzweig-Porter model for the transition from Poisson to Gaussian unitary ensemble statistics

Experimental test of the Rosenzweig-Porter model for the transition from Poisson to Gaussian unitary ensemble statistics

On the number variance of zeta zeros and a conjecture of Berry

AbstractAssuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non‐zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non‐universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).

Onset of universality in the dynamical mixing of a pure state

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the GOE to the Gaussian unitary ensemble (GUE) for short and large times respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the Hilbert space dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace.

Spacing ratio characterization of the spectra of directed random networks.

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdős-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Exciton-phonon interaction breaking all antiunitary symmetries in external magnetic fields

Recent experimental investigations by M. A{\ss}mann et al. [Nature Mater. 15, 741 (2016)] on the spectrum of magnetoexcitons in cuprous oxide revealed the statistics of a Gaussian unitary ensemble (GUE). The model of F. Schweiner et al. [Phys. Rev. Lett. 118, 046401 (2017)], which includes the complete cubic valence band structure of the solid, can explain the appearance of GUE statistics if the magnetic field is not oriented in one of the symmetry planes of the cubic lattice. However, it cannot explain the experimental observation of GUE statistics for all orientations of the field. In this paper we investigate the effect of quasi-particle interactions or especially the exciton-phonon interaction on the level statistics of magnetoexcitons and show that the motional Stark field induced by the exciton-phonon interaction leads to the occurrence of GUE statistics for arbitrary orientations of the magnetic field in agreement with experimental observations. Importantly, the breaking of all antiunitary symmetries can be explained only by considering both the exciton-phonon interaction and the cubic crystal lattice.

GOE-GUE-Poisson transitions in the nearest-neighbor spacing distribution of magnetoexcitons.

Recent investigations on the Hamiltonian of excitons by Schweiner et al.. [Phys. Rev. Lett. 118, 046401 (2017)]PRLTAO0031-900710.1103/PhysRevLett.118.046401 revealed that the combined presence of a cubic band structure and external fields breaks all antiunitary symmetries. The nearest-neighbor spacing distribution of magnetoexcitons can exhibit Poissonian statistics, the statistics of a Gaussian orthogonal ensemble (GOE), or a Gaussian unitary ensemble (GUE) depending on the system parameters. Hence, magnetoexcitons are an ideal system to investigate the transitions between these statistics. Here we investigate the transitions between GOE and GUE statistics and between Poissonian and GUE statistics by changing the angle of the magnetic field with respect to the crystal lattice and by changing the scaled energy known from the hydrogen atom in external fields. Comparing our results with analytical formulas for these transitions derived with random matrix theory, we obtain a very good agreement and thus confirm the Wigner surmise for the exciton system.

Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

Random geometry and the Kardar–Parisi–Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on , which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as , with a fluctuation exponent , while the lateral spread of the minimizing geodesic between two points at a distance L grows as , with wandering exponent value . Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar–Parisi–Zhang universality class of surface kinetic roughening, with ξ and χ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy–Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW–GUE statistics with good accuracy in arrival times.

Semiclassical approach to discrete symmetries in quantum chaos

We use semiclassical methods to evaluate the spectral two-point correlation function of quantum chaotic systems with discrete geometrical symmetries. The energy spectra of these systems can be divided into subspectra that are associated with irreducible representations of the corresponding symmetry group. We show that for (spinless) time-reversal invariant systems, the statistics inside these subspectra depends on the type of irreducible representation. For real representations the spectral statistics agrees with those of the Gaussian orthogonal ensemble of random matrix theory (RMT), whereas complex representations correspond to the Gaussian unitary ensemble (GUE). For systems without time-reversal invariance, all subspectra show GUE statistics. There are no correlations between non-degenerate subspectra. Our techniques generalize recent developments in the semiclassical approach to quantum chaos allowing one to obtain full agreement with the two-point correlation function predicted by RMT, including oscillatory contributions.

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.

Quantum chaos in nanoelectromechanical systems

We present a theoretical study of the electron-phonon coupling in suspended nanoelectromechanical systems (NEMS) and investigate the resulting quantum chaotic behavior. The phonons are associated with the vibrational modes of a suspended rectangular dielectric plate, with free or clamped boundary conditions, whereas the electrons are confined to a large quantum dot (QD) on the plate's surface. The deformation potential and piezoelectric interactions are considered. By performing standard energy-level statistics we demonstrate that the spectral fluctuations exhibit the same distributions as those of the Gaussian Orthogonal Ensemble (GOE) or the Gaussian Unitary Ensemble (GUE), therefore evidencing the emergence of quantum chaos. That is verified for a large range of material and geometry parameters. In particular, the GUE statistics occurs only in the case of a circular QD. It represents an anomalous phenomenon, previously reported for just a small number of systems, since the problem is time-reversal invariant. The obtained results are explained through a detailed analysis of the Hamiltonian matrix structure.

Test of a numerical approach to the quantization of billiards

A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from C(3v) to C3 . Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.

Phase shift experiments identifying Kramers doublets in a chaotic superconducting microwave billiard of threefold symmetry.

The spectral properties of a two-dimensional microwave billiard showing threefold symmetry have been studied with a new experimental technique. This method is based on the behavior of the eigenmodes under variation of a phase shift between two input channels, which strongly depends on the symmetries of the eigenfunctions. Thereby a complete set of 108 Kramers doublets has been identified by a simple and purely experimental method. This set clearly shows Gaussian unitary ensemble statistics, although the system is time-reversal invariant.

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display, in certain irreducible subspaces, spectral statistics corresponding to the Gaussian-unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers-type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss.

Gaussian unitary ensemble statistics in a time-reversal invariant microwave triangular billiard

The spectrum of a chaotic two-dimensional quantum billiard with threefold symmetry has been studied in an experiment with a superconducting microwave cavity. In total 622 eigenvalues were identified experimentally and compared with numerical calculations. The statistical analysis of the data shows that Gaussian unitary ensemble statistics can be observed for a spectrum of a time-reversal invariant system.

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.

Wave Chaos Experiments with and without Time Reversal Symmetry: GUE and GOE Statistics.

It has been predicted that in the semiclassical regime the level statistics of a classically chaotic system correspond to that of the Gaussian unitary ensemble (GUE) of random matrices when time reversal symmetry is broken. This Letter presents the first experimental test of this prediction. The system employed is a microwave cavity containing a thin ferrite strip adjacent to one of the walls. When a sufficiently large magnetic field is applied to the ferrite (thus breaking the time reversal symmetry) good agreement with GUE statistics is obtained. The transition from Gaussian orthogonal ensemble (GOE) (which applies in the absence of the applied field) to GUE is also investigated.