Caos cuántico en sistemas esquemáticos de partículas idénticas

Zusammenfassung

The work is divided into two parts. The main subject is the analysis of the statistics of the spectral fluctuations in different systems in the context of Quantum Chaos. In Part One non-interacting systems are studied. The analysis is carried out by means of the usual statistics in Quantum Chaos, leading to the conclusion that these kind of spectra are integrable, regardless of the type of fluctuation properties of the single-particle spectrum. A rapid evolution is observed with the energy as well as with the number of particles, although a reminiscence of the fluctuations of the single-particle spectrum, when it is chaotic, is always present and it can be observed in the long range correlations. In Part Two a two-body random interaction is introduced by a control parameter which allows a gradual transition from the non-interacting system to the system with the two-body interaction (the matrix ensembles used are the so-called embedded ensembles). In this part we study the hamiltonian matrices in a tridiagonal form; this can be very advantageous both analytically and numerically and there are antecedents well established in previous works. Then we analyze the relation between the ensembles in tridiagonal form (characterized by smooth part, fluctuations and correlations between the elements) and the spectral statistics, and the main purpose is to construct a model of tridiagonal ensembles which reproduce the transition between the non-interacting system (integrable) and the system with a two-body interaction (chaotic). A detailed analysis of the tridiagonal form of the hamiltonian matrices along the transition leads to a very complete characterization of them, that allows us to establish some relations between the smooth part and fluctuations in the elements of the tridiagonal matrix and the level density and spectral statistics, respectively. However, correlations between the matrix elements are much more difficult to model and for the moment, at the end of the presentation of this work, we do not have a sufficiently detailed description of the correlation structure as to construct a model capable of succesfully reproducing the transition. Anyway this work can be useful as a base to continue with the proposed proyect, which is of great interest due to the important advantage that would be to have such a simple model for these ensembles, as they are so complicated to manage in their normal form.