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arXiv:cond-mat/9712036 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Chaos thresholds in finite Fermi systems

P. G. Silvestrov

Published Nov 1, 1998• 35 citations
arXivHTMLPDF

Abstract

The development of quantum chaos in finite interacting Fermi systems is considered. At sufficiently high excitation energy the direct two-particle interaction may mix into an eigenstate the exponentially large number of simple Slater determinant states. Nevertheless, the transition from Poisson to Wigner-Dyson statistics of energy levels is governed by the effective high-order interaction between states very distant in the Fock space. The concrete form of the transition depends on the way one chooses to work out the problem of factorial divergence of the number of Feynman diagrams. In the proposed scheme the change of statistics has a form of narrow phase transition and may happen even below the direct interaction threshold.

DOI
10.1103/physreve.58.5629

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The development of quantum chaos in finite interacting Fermi systems is considered. At sufficiently high excitation energy the direct two-particle interaction may mix into an eigenstate the exponentially large number of simple Slater determinant states. Nevertheless, the transition from Poisson to Wigner-Dyson statistics of energy levels is governed by the effective high-order interaction between states very distant in the Fock space. The concrete form of the transition depends on the way one chooses to work out the problem of factorial divergence of the number of Feynman diagrams. In the proposed scheme the change of statistics has a form of narrow phase transition and may happen even below the direct interaction threshold.
The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution P(s)P(s)P(s) as a function of interaction strength U and the excitation energy $\ensuremath{\epsilon}$ above the Fermi level. As U increases, P(s)P(s)P(s) undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by $Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},$ where u0{u}_{0}u0​ is a constant. While the exponent $\ensuremath{\alpha},$ which determines the global change of the chaos border, is indecisive within a broad range of $0.9\ensuremath{\sim}2.0,$ the finiteness of $\ensuremath{\nu},$ which comes from the increase of the Fock space size with $\ensuremath{\epsilon},$ suggests that the transition becomes sharp as $\ensuremath{\epsilon}$ increases.
We study the level spacing statistics P(s)P(s)P(s) in many-body Fermi systems and determine a critical two-body interaction strength Uc{U}_{c}Uc​ at which a crossover from Poisson to Wigner-Dyson statistics takes place. Near the Fermi level, the results allow one to find a critical temperature Tch{T}_{\mathrm{ch}}Tch​ above which quantum chaos and thermalization set in.
For many-body Fermi systems we determine the dependence of the Breit-Wigner width $\ensuremath{\gamma}$ and inverse participation ratio $\ensuremath{\xi}$ on interaction strength $U\ensuremath{\ge}{U}_{c}$ and energy excitation $\ensuremath{\delta}E\ensuremath{\ge}\ensuremath{\delta}{E}_{\mathrm{ch}}$ when a crossover from Poisson to Wigner-Dyson P(s)P(s)P(s) statistics takes place. At $U\ensuremath{\ge}{U}_{c}$ the eigenstates are composed of a large number of noninteracting states and even for U<UcU<{U}_{c}U<Uc​ there is a regime where P(s)P(s)P(s) is close to the Poisson distribution but $\ensuremath{\xi}\ensuremath{\gg}1$.
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.
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.
We have studied the appearance of chaos in the many-body spectrum of interacting Fermions. The coupling of a single state to the Fermi sea is considered. This state is coupled to a hierarchy of states corresponding to one or several particle-hole excitations. We have considered various couplings between two successive generations of this hierarchy and determined under which conditions this coupling can lead to Wigner-Dyson correlations. We have found that the cross-over from a Poisson to a Wigner distribution is characterized not only by the ratio V = Δc, but also by the ratio Δc = δ. V is the typical interaction matrix element, δ is the energy distance between many-body states and Δc is the distance between many-body states coupled by the interaction.