Brody Distribution Research Articles

Two mass-imbalanced atoms in a hard-wall trap: Deep learning integrability of many-body systems.

The study of integrable systems has led to significant advancements in our understanding of many-body physics. We design a series of numerical experiments to analyze the integrability of a mass-imbalanced two-body system through energy-level statistics and deep learning of wave functions. The level spacing distributions are fitted by a Brody distribution and the fitting parameter ω is found to separate the integrable and nonintegrable mass ratios by a critical line ω=0. The convolutional neural network built from the probability density images could identify the transition points between integrable and nonintegrable systems with high accuracy, yet in a much shorter computation time. A brilliant example of the network's ability is to identify a new integrable mass ratio 1/3 by learning from the known integrable case of equal mass, with a remarkable network confidence of 98.22%. The robustness of our neural networks is further enhanced by adversarial learning, where samples are generated by standard and quantum perturbations mixed in the probability density images and the wave functions, respectively.

Missing levels in intermediate spectra

We derive an expression for the nearest-neighbor spacing distribution P(s) of the energy levels of quantum systems with intermediate dynamics between regularity and chaos and missing levels due to random experimental errors. The expression is based on the Brody distribution, the most widely used for fitting mixed spectra as a function of one parameter. By using Monte Carlo simulations of intermediate spectra based on the β-Hermite ensemble of random matrix theory (RMT), we evaluate the quality of the formula and its suitability for fitting purposes. Estimations of the Brody parameter and the fraction of missing levels can be obtained by a least-square two-parameter fitting of the experimental P(s). The results should be important to distinguish the origins of deviations from RMT in experimental spectra.

Ground-state energy distribution of disordered many-body quantum systems.

Extreme-value distributions are studied in the context of a broad range of problems, from the equilibrium properties of low-temperature disordered systems to the occurrence of natural disasters. Our focus here is on the ground-state energy distribution of disordered many-body quantum systems. We derive an analytical expression that, upon tuning a parameter, reproduces with high accuracy the ground-state energy distribution of the systems that we consider. For some models, it agrees with the Tracy-Widom distribution obtained from Gaussian random matrices. They include transverse Ising models, the Sachdev-Ye model, and a randomized version of the PXP model. For other systems, such as Bose-Hubbard models with random couplings and the disordered spin-1/2 Heisenberg chain used to investigate many-body localization, the shapes are at odds with the Tracy-Widom distribution. Our analytical expression captures all of these distributions, thus playing a role to the lowest energy level similar to that played by the Brody distribution to the bulk of the spectrum.

Complex collisions of ultracold molecules: A toy model

We introduce a model to study the collisions of two ultracold diatomic molecules in one dimension interacting via pairwise potentials. We present results for this system and argue that it offers lessons for real molecular collisions in three dimensions. We analyze the distribution of the adiabatic potentials in the hyperspherical coordinate representation as well as the distribution of near-threshold four-body bound states, systematically studying the effects of molecular properties, such as interaction strength, interaction range, and atomic mass. It is found that the adiabatic potential's nearest-neighbor energy level distribution transitions from significant level repulsion characteristic of chaos (Brody distribution) to nonchaotic (Poisson distribution) as the two molecules are separated. For the near-threshold four-atom bound states, the case where all atoms have equal masses shows a Poissonian spacing distribution, while the unequal-mass system exhibits significant level repulsion characterized by a nonzero Brody parameter. We derive a semiclassical formula for the density of states and extract from it simple scaling laws with potential depth and range. We find good agreement between the semiclassical predictions for the density of states and the full quantum mechanical calculations.

Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals

The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.

A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories

The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse χ 2 distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.

Spectral statistics of light in one-dimensional graphene-based photonic crystals.

The optical properties and spectral statistics of light in one-dimensional photonic crystals in the representative classes of (AB)^{N} (composed of dielectric layers) and (AGBG)^{N} (composed of periodic stacking of graphene-dielectric layers) have been investigated using the transfer matrix method and random matrix theory. The proposed method provides new predictions to determine the chaos and regularity of the optical systems. In this analysis, the chaoticity parameter with q=0 for Poisson distribution and q→1 for Wigner distribution is determined based on the random matrix theory. It has been shown that two kinds of chaos and regularity modes can be found with Brody distribution. Also, as a part of this work, we found out the regular pattern in both classes of (AB)^{N} and (AGBG)^{N} when results were fit to a Brody distribution. Moreover, the effects of different parameters such as the number of unit cells, incident angle, state of polarization, and chemical potential of the graphene nanolayers on the structures' regularity are discussed. It is found that the regular patterns are seen in the band gaps. The results show that the structure (AGBG)^{N} has an extra photonic band gap compared to (AB)^{N}, which is tunable by changing the chemical potential of the graphene nanolayers. Therefore, the possibility of external control of the regularity using a gate voltage in the graphene-based photonic crystals is obtained. Finally, comparing of TE and TM waves based on the random matrix theory, which interpolates between regular and chaotic systems, indicates that the Poisson statistics well describes the TE waves.

Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic in Phys. Today 46 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard (B = 0:5) with strong stickiness effects published in Phys. Rev. E 103 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases B = 0:42; 0:55; 0:6, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following universal properties of mixed-type systems without stickiness in the chaotic regions: (i) Using the Poincare-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent β, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a bimodal universal distribution, where the narrow peak at small A encompasses the regular eigenstates, theoretically well understood, while the peak at larger A comprises the chaotic eigenstates, and is well described by the beta distribution. (iii) Thus the BRB energy level spacing distribution captures two effects: the divided phase space dictated by the classical Berry-Robnik parameter ρ2 measuring the relative size of the largest chaotic region, in agreement with the Berry-Robnik picture, and the localization of chaotic PH functions characterized by the level repulsion (Brody) parameter β. (iv) Examination of the PH functions shows that they are supported either on the classical invariant tori in the regular islands or on the chaotic sea, where they are only weakly localized. With increasing energy the localization of chaotic states decreases, as the PH functions tend towards uniform spreading over the classical chaotic region, and correspondingly β tends to 1.

Quantum quench dynamics of the Jaynes-Cummings-Hubbard model with weak nearest-neighbor hopping

Motivated by the nonequilibrium dynamics experiment, we study the quantum quench dynamics of the JCH model with weak nearest-neighbor hopping strength by exact diagonalization. One of the crucial insights is to focus on the comparison between the time-averaged density matrix and the canonical ensemble in a wide range of values for the coupling strength between the cavity and the atom, and significant differences are found. Thus, this phenomenon can be used to infer that the system with weak nearest-neighbor hopping strength may be integrable. In order to verify our prediction, we perform the level spacing distribution of the system and find that they conform with the characteristics of the integrable system. Further, a nearly integrable system is presented by removing the degenerate energy level and fitting Brody distribution. Besides, the evolution of the momentum distribution functions of the photon and the equilibrium value predicted by the canonical ensemble are followed, which demonstrates that the system cannot be thermalized. Finally, the way how the system avoids thermalization is also discussed by the finite-size scaling of the fluctuations of the diagonal and the off-diagonal matrix elements.

Model for scattering with proliferating resonances: Many coupled square wells

We present a multichannel model for elastic interactions, comprised of an arbitrary number of coupled finite square-well potentials, and derive semi-analytic solutions for its scattering behavior. Despite the model's simplicity, it is flexible enough to include many coupled short-ranged resonances in the vicinity of the collision threshold, as is necessary to describe ongoing experiments in ultracold molecules and lanthanide atoms. We also introduce a simple, but physically realistic, statistical ensemble for parameters in this model. We compute the resulting probability distributions of nearest-neighbor resonance spacings and analyze them by fitting to the Brody distribution. We quantify the ability of alternative distribution functions, for resonance spacing and resonance number variance, to describe the crossover regime. The analysis demonstrates that the multichannel square-well model with the chosen ensemble of parameters naturally captures the crossover from integrable to chaotic scattering as a function of closed channel coupling strength.

Statistical Analysis of the Distribution of 0+ State Energies in the Actinide Nuclei

The statistical analysis of the distributions of the 0+ state energies in actinide nuclei is carried out, by using the Brody distribution function. It reveals an intermediate character of the experimental 0+ spectra between order and chaos. This function is turned out to be different for the available experimental data and the calculated ones in the frame of the quasiparticle-phonon model.

A Random Matrix Model Whose Eigenvalue Spacings Are Closely Described by the Brody Distribution

A Random Matrix Model Whose Eigenvalue Spacings Are Closely Described by the Brody Distribution

Universality in the spectral and eigenfunction properties of random networks.

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

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.

Generalization of Brody distribution for statistical investigation

In this paper, Brody distribution is generalized to explore the Poisson, Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble limits of Random Matrix Theory in the nearest neighbor spacing statistic framework. Parameters of new distribution are extracted via Maximum Likelihood Estimation technique for different sequences. This general distribution suggests more exact results in comparison with the results of other estimation methods and distribution functions.

Statistical properties of spectra in harmonically trapped spin–orbit coupled systems

We compute single-particle energy spectra for a one-body Hamiltonian consisting of a two-dimensional deformed harmonic oscillator potential, the Rashba spin–orbit coupling and the Zeeman term. To investigate the statistical properties of the obtained spectra as functions of deformation, spin–orbit and Zeeman strengths we examine the distributions of the nearest neighbor spacings. We find that the shapes of these distributions depend strongly on the three potential parameters. We show that the obtained shapes in some cases can be well approximated with the standard Poisson, Brody and Wigner distributions. The Brody and Wigner distributions characterize irregular motion and help identify quantum chaotic systems. We present a special choice of deformation and spin–orbit strengths without the Zeeman term which provide a fair reproduction of the fourth-power repelling Wigner distribution. By adding the Zeeman field we can reproduce a Brody distribution, which is known to describe a transition between the Poisson and linear Wigner distributions.

GUE-type energy level statistics for spherical odd-mass nuclei

The spectral statistics of spherical nuclei are steadily considered with emphasis on nearest-neighbor spacing statistics. Sequences are constructed by using all the available empirical data for the considered nuclei in different mass regions and analyzed via the generalized Brody distribution and the maximum likelihood estimation technique. Our results indicate an approach to chaotic dynamics for the nuclei considered in this work. Also, spherical odd-mass nuclei in the 50A ≤ 100 mass region exhibit a slight deviation to the GUE spectral statistics rather than to the GOE.

Dynamical localization in chaotic systems: Spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems

We study the kicked rotator in the classically fully chaotic regime using Izrailev's N-dimensional model for various N≤4000, which in the limit N→∞ tends to the quantized kicked rotator. We do treat not only the case K=5, as studied previously, but also many different values of the classical kick parameter 5≤K≤35 and many different values of the quantum parameter kε[5,60]. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable Λ=l(∞)/N to the case of anomalous diffusion in the classical phase space by deriving the localization length l(∞) for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents β(BR). (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by β(loc) in the interval [0,1]. The level repulsion parameters β(BR) and β(loc) are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between β(loc) and the relative localization length Λ, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).

The intermediate level statistics in dynamically localized chaotic eigenstates

We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K = 7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincaré Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting at about 1000000 above the ground state). This separation confirms the Berry-Robnik picture, and is performed for the first time at such high energies.

Spins coupled to a spin bath: From integrability to chaos

Motivated by the hyperfine interaction of electron spins with surrounding nuclei, we investigate systems of central spins coupled to a bath of noninteracting spins in the framework of random matrix theory. With increasing number of central spins a transition from Poissonian statistics to the Gaussian orthogonal ensemble occurs which can be described by a generalized Brody distribution. These observations are unaltered upon applying an external magnetic field. In the transition region, the classical counterparts of the models studied have mixed phase space.