Juan Diego Urbina and Klaus Richter

The understanding of the interplay between electron-electron interactions and chaos has evolved to a prominent field in mesoscopic physics. While interaction effects in disordered systems (a particulary kind of chaotic systems have been intensively studied in the recent past, accounts on the inter-relation between many-body effects and chaotic dynamics are still rare. The latter systems can be modelled by quantum billiards which have served as prototypes to investigate quantum signatures of integrable and chaotic single-particle dynamics. Hence generalization of quantum chaos concepts to interacting particles in billiards appears natural (see Fig. 1).

Semiclassical methods provide a direct link between classical and quantum mechanics, for instance by means of Gutwiller's trace formula. While these mehods have proved successful for the treatment of quantum chaotic single-particle problems or systems with few interacting particles, their generalization to interacting many-particle systems (corresponding to high-dimensional phase space) is an open and challenging problem. The situation is similar for supersymmetrical methods, which have proven powerful for single-particle problems, but its extension to interacting systems is still lacking.

One possible way to understand the interplay between chaotic single-particle behaviour and interactions is to make explicit their appearance in the full many-body problem. This program goes in two steps: First, the effect of the billiard's shape ist taken into account on the level of the single-particle wave functions. These eigenfunctions will be affected by the regular, mixed, or chaotic classical phase space which in turn depends on the shape of the billiard. Second, the full interacting hamiltonian is represented in the basis of the non-interacting problem controlling in this way the strength and functional form of the interaction.

An approach inspired by the success of the supersymmetric technique and Random Matrix Theory for disordered systems could start with the study of statistical features of interaction matrix elements in the single-particle basis. Interesting questions, currently under investigation, are:

  • What is the statistical distribution of these matrix entries?
  • Is there any way to find such a distribution using just one-particle techniques?
  • Do the main features of the distributions depend in universal way on, for example, the energy?

Extensive numerical simulations allow us to reach the universal chaotic regime for the single-particle problem and to calculate explicitly the probability distribution of the interaction matrix elements. Typical examples for the diagonal and non-diagonal matrix elements between different single-particle states are shown in Fig. 2.

Our results suggest that, in contrast to usual Random Matrix Theory assumptions, these distributions are not Gaussian. Such deviations and their implications are under current investigation.