Power-spectrum characterization of the continuous Gaussian ensemble.

The continuous Gaussian ensemble, also known as the nu -Gaussian or nu -Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real (nu=1) , complex (nu=2) , or quaternion (nu=4) matrices, where nu is allowed to take any positive value. From a physical point of view, this ensemble may be useful to describe transitions between different symmetries or to describe the terrace-width distributions of vicinal surfaces. Moreover, its simple form allows one to speed up and increase the efficiency of numerical simulations dealing with large matrix dimensions. We analyze the long-range spectral correlations of this ensemble by means of the delta(n) statistic. We derive an analytical expression for the average power spectrum of this statistic, P(k)(delta)[over ] , based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of delta(n) evolves from P(k)(delta)[over ] proportional, variant1/k at nu=1 to P(k)(delta)[over ] proportional, variant1/k(2) at nu=0 . Relevantly, the transition is not homogeneous with a 1/f(alpha) noise at all scales, but heterogeneous with coexisting 1/f and 1/f(2) noises. There exists a critical frequency k(c) proportional, variant nu that separates both behaviors: below k(c) , P(k)(delta)[over ] follows a 1/f power law, while beyond k(c) , it transits abruptly to a 1/f(2) power law. For nu >1 the 1/f noise dominates through the whole frequency range, unveiling that the 1/f correlation structure remains constant as we increase the level repulsion and reduce to zero the amplitude of the spectral fluctuations. All these results are confirmed by stringent numerical calculations involving matrices with dimensions up to 10(5) .

1/f noise and very high spectral rigidity.

It was recently pointed out that the spectral fluctuations of quantum systems are formally analogous to discrete time series, and therefore their structure can be characterized by the power spectrum of the signal. Moreover, it is found that the power spectrum of chaotic spectra displays a 1/f behavior, while that of regular systems follows a 1/f2 law. This analogy provides a link between the concepts of spectral rigidity and antipersistence. Trying to get a deeper understanding of this relationship, we have studied the correlation structure of spectra with high spectral rigidity. Using an appropriate family of random Hamiltonians, we increase the spectral rigidity up to hindering completely the spectral fluctuations. Analyzing the long range correlation structure a neat power law 1/f has been found for all the spectra, along the whole process. Therefore, 1/f noise is the characteristic fingerprint of a transition that, preserving the scale-free correlation structure, hinders completely the fluctuations of the spectrum.

Spectral statistics in noninteracting many-particle systems.

It is widely accepted that the statistical properties of energy level spectra provide an essential characterization of quantum chaos. Indeed, the spectral fluctuations of many different systems like quantum billiards, atoms, or atomic nuclei have been studied. However, noninteracting many-body systems have received little attention, since it is assumed that they must exhibit Poisson-like fluctuations. Apart from a heuristic argument of Bloch, there are neither systematic numerical calculations nor a rigorous derivation of this fact. Here we present a rigorous study of the spectral fluctuations of noninteracting identical particles moving freely in a mean field emphasizing the evolution with the number of particles N as well as with the energy. Our results are conclusive. For N > or =2 the spectra of these systems exhibit Poisson fluctuations provided that we consider sufficiently high excitation energies. Nevertheless, when the mean field is chaotic there exists a critical energy scale L(c); beyond this scale, the fluctuations deviate from the Poisson statistics as a reminiscence of the statistical properties of the mean field.

1/f alpha noise in spectral fluctuations of quantum systems.

The power law 1/f(alpha) in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law 1/f(alpha) is found at all the transition stages, and it is shown that the exponent alpha is related to the chaotic component of the classical phase space of the quantum system.

Correlation structure of the deltan statistic for chaotic quantum systems.

The existence of a formal analogy between quantum energy spectra and discrete time series has been recently pointed out. When the energy level fluctuations are described by means of the statistic, it is found that chaotic quantum systems are characterized by noise, while regular systems are characterized by . In order to investigate the correlation structure of the statistic, we study the -order height-height correlation function , which measures the momentum of order , i.e., the average power of the signal change after a time delay . It is shown that this function has a logarithmic behavior for the spectra of chaotic quantum systems, modeled by means of random matrix theory. On the other hand, since the power spectrum of chaotic energy spectra considered as time series exhibit noise, we investigate whether the -order height-height correlation function of other time series with noise exhibits the same properties. A time series of this kind can be generated as a linear combination of cosine functions with arbitrary phases. We find that the logarithmic behavior arises with great accuracy for time series generated with random phases.

Stringent numerical test of the Poisson distribution for finite quantum integrable Hamiltonians.

Using a class of exactly solvable models based on the pairing interaction, we show that it is possible to construct integrable Hamiltonians with a Wigner distribution of nearest-neighbor level spacings. However, these Hamiltonians involve many-body interactions and the addition of a small integrable perturbation very quickly leads the system to a Poisson distribution. Besides this exceptional case, we show that the accumulated distribution of an ensemble of random integrable two-body pairing Hamiltonians is in perfect agreement with the Poisson limit. These numerical results for quantum integrable Hamiltonians provide a further empirical confirmation of the work of Berry and Tabor in the semiclassical limit.

1/f noise in the two-body random ensemble.

We show that the spectral fluctuations of the two-body random ensemble exhibit 1/f noise. This result supports a recent conjecture stating that chaotic quantum systems are characterized by 1/f noise in their energy level fluctuations. After suitable individual averaging, we also study the distribution of the exponent alpha in the 1/ f(alpha) noise for the individual members of the ensemble. Almost all the exponents lie inside a narrow interval around alpha=1, suggesting that also individual members exhibit 1/f noise, provided they are individually unfolded.

Theoretical derivation of 1/f noise in quantum chaos.

It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/f (1/f(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.

Quantum chaos and 1/f noise.

It is shown that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series. The power spectrum behavior of such a signal for different systems suggests the following conjecture: The energy spectra of chaotic quantum systems are characterized by 1/f noise.

Misleading signatures of quantum chaos.

The main signature of chaos in a quantum system is provided by spectral statistical analysis of the nearest-neighbor spacing distribution P(s) and the spectral rigidity given by the Delta(3)(L) statistic. It is shown that some standard unfolding procedures, such as local unfolding and Gaussian broadening, lead to a spurious saturation of Delta(3)(L) that spoils the relationship of this statistic with the regular or chaotic motion of the system. This effect can also be misinterpreted as Berry's saturation.