Correlation structure of the δ_(n) 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 δ_(n) statistic, it is found that chaotic quantum systems are characterized by 1/f noise, while regular systems are characterized by 1/f(2). In order to investigate the correlation structure of the δ_(n) statistic, we study the qth-order height-height correlation function C-q(tau), which measures the momentum of order q, i.e., the average qth power of the signal change after a time delay tau. 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 1/f noise, we investigate whether the qth-order height-height correlation function of other time series with 1/f 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

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

Power spectrum characterization of the continuous gaussian ensemble

The continuous Gaussian ensemble, also known as the ν-Gaussian or ν-Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real (ν=1), complex (ν=2), or quaternion (ν=4) matrices, where ν 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 δn statistic. We derive an analytical expression for the average power spectrum of this statistic, Pkδ̅ , based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of δn evolves from Pkδ̅ ∝1/k at ν=1 to Pkδ̅ ∝1/k2 at ν=0. Relevantly, the transition is not homogeneous with a 1/fα noise at all scales, but heterogeneous with coexisting 1/f and 1/f2 noises. There exists a critical frequency kc∝ν that separates both behaviors: below kc, Pkδ̅ follows a 1/f power law, while beyond kc, it transits abruptly to a 1/f2 power law. For ν>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

Dos estudios de estructura nuclear: la transición de forma en N [aproximadamente] 20 lejos de la estabilidad y la colectividad orbital magnética en núcleos ligeros

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de lectura: 15-10-199

Excited-state phase transition leading to symmetry-breaking steady states in the Dicke model

We study the phase diagram of the Dicke model in terms of the excitation energy and the radiation-matter coupling constant lambda. Below a certain critical value lambda(c), all the energy levels have a well-defined parity. For lambda > lambda(c) the energy spectrum exhibits two different phases separated by a critical energy E-c that proves to be independent of lambda. In the upper phase, the energy levels have also a well-defined parity, but below E-c the energy levels are doubly degenerated. We show that the long-time behavior of appropriate parity-breaking observables distinguishes between these two different phases of the energy spectrum. Steady states reached from symmetry-breaking initial conditions restore the symmetry only if their expected energies are above the critical. This fact makes it possible to experimentally explore the complete phase diagram of the excitation spectrum of the Dicke model

Quantum phase transitions and spontaneous symmetry-breaking in Dicke Model

A method to find the Excited-States Quantum Phase Transitions (ESQPT's) from parity-symmetry in the Dicke model is studied and presented. This method allows us to stablish a critical energy where ESQPT's take places, and divides the whole energy spectrum in two regions with different properties

Spectral-statistics properties of the experimental and theoretical light meson spectra

We present a robust analysis of the spectral fluctuations exhibited by the light meson spectrum. This analysis provides information about the degree of chaos in light mesons and may be useful to get some insight on the underlying interactions. Our analysis unveils that the statistical properties of the light meson spectrum are close, but not exactly equal, to those of chaotic systems. In addition, we have analyzed several theoretical spectra including the latest lattice QCD calculation. With a single exception, their statistical properties are close to those of a generic integrable system, and thus incompatible with the experimental spectrum

Chaos in hadrons

In the last decade quantum chaos has become a well established discipline with outreach to different fields, from condensed-matter to nuclear physics. The most important signature of quantum chaos is the statistical analysis of the energy spectrum, which distinguishes between systems with integrable and chaotic classical analogues. In recent years, spectral statistical techniques inherited from quantum chaos have been applied successfully to the baryon spectrum revealing its likely chaotic behaviour even at the lowest energies. However, the theoretical spectra present a behaviour closer to the statistics of integrable systems which makes theory and experiment statistically incompatible. The usual statement of missing resonances in the experimental spectrum when compared to the theoretical ones cannot account for the discrepancies. In this communication we report an improved analysis of the baryon spectrum, taking into account the low statistics and the error bars associated with each resonance. Our findings give a major support to the previous conclusions. Besides, analogue analyses are performed in the experimental meson spectrum, with comparison to theoretical models

Thermalization in the two-body random ensemble

Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit