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arXiv:cond-mat/9607035 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Towards a statistical theory of finite Fermi systems and compound states: Random two-body interaction approach

V. V. Flambaum, Felix M. Izrailev, Giulio Casati

Published Aug 1, 1996• 86 citations
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Abstract

The model of Fermi particles with random two-body interaction is investigated. This model allows one to study the origin and accuracy of statistical laws in few-body systems, the role of interaction and chaos in thermalization, the Fermi-Dirac distribution for quasiparticles with spreading widths, matrix elements of external fields, and enhancement of weak perturbation in chaotic compound states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.54.2136

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The model of Fermi particles with random two-body interaction is investigated. This model allows one to study the origin and accuracy of statistical laws in few-body systems, the role of interaction and chaos in thermalization, the Fermi-Dirac distribution for quasiparticles with spreading widths, matrix elements of external fields, and enhancement of weak perturbation in chaotic compound states. \textcopyright{} 1996 The American Physical Society.
In this review the problem of statistical description of isolated quantum systems of interacting particles is discussed. Main attention is paid to a recently developed approach which is based on chaotic properties of compound states in the basis of non-interacting particles. In order to demonstrate the most important aspects of this approach, the matrix model of two-body random interaction between Fermi-particles has been used. Different problems have been considered such as the onset of chaos and statistical equilibrium, the relation between the structure of eigenstates and distribution of occupation numbers, the emergence of the Fermi-Dirac distribution in isolated systems of finite number of particles and many others. The application of the approach to dynamical systems with the classical limit is discussed as well.
A theory of strongly interacting Fermi systems of a few particles is developed. At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual two-body interaction. This regime can be described as many-body quantum chaos. Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy. Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g. rare earths) and multicharged ions, molecules, clusters and quantum dots in solids. The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis. A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix. We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
A theory of strongly interacting Fermi systems of a few particles is developed.At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual twobody interaction.This regime can be described as many-body quantum chaos.Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy.Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g.rare earths) and multicharged ions, molecules, clusters and quantum dots in solids.The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis.A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix.We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
It is shown that the two-body character of the interaction in a many-body system gives rise to specific correlations between the components of compound states, even if this interaction is completely random. Surprisingly, these correlations increase with the increase of the number of active (valence) particles. Statistical theory of transition amplitudes between compound states, which takes into account these correlations is developed and tested within the framework of the two-body random interaction model. It is demonstrated that a feature, which can be called ``correlation resonance,'' appears in the distribution of the transition matrix amplitudes, since the correlations strongly reduce the transition amplitudes at the tails and increase them near the maximum of the distribution. \textcopyright{} 1996 The American Physical Society.
It is shown that statistical mechanics is applicable to quantum systems with finite numbers of particles, such as complex atoms, atomic clusters, etc., where the residual two-body interaction is sufficiently strong. This interaction mixes the unperturbed shell-model basis states and produces ``chaotic'' many-body eigenstates. As a result, an interaction-induced equilibrium emerges in the system, and temperature can be introduced. However, the interaction between the particles and their finite number can lead to prominent deviations of the equilibrium occupation numbers distribution from the Fermi-Dirac shape. For example, this takes place in the cerium atom with four valence electrons, which was used to compare the theory with realistic numerical calculations.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.