Quantum chaos in many-body systems: what can we learn from the Ce atom?

Abstract

Results of an extensive study of a real quantum chaotic many-body system -the Ce atom -are presented. We discuss the origins of the quantum chaotic behaviour of the system, analyse statistical and dynamical properties of the multi-particle chaotic eigenstates and consider matrix elements or transition amplitudes between them. We show that based on the universal properties of the chaotic eigenstates a statistical theory of finite few-particle systems with strong interaction can be developed. We also discuss such important physical effects as enhancement of weak perturbations in many-body quantum chaotic systems, distribution of single-particle occupation numbers and its deviations from the standard Fermi-Dirac shape, and ways of introducing statistical temperature-based description in such systems.

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

“…It is well established that quantized versions of classically integrable and fully chaotic systems can be distinguished by their quantum energy level statistics [1,3,4,41,42,43]. Of particular interest is the distribution of energy level spacings, P (s), where s is the spacing between neighboring energy levels after the spectrum has been appropriately unfolded, so that the density of states is everywhere equal to one.…”

Prediction of quantum many-body chaos in the protactinium atom

“…In this regard Pa atom is very similar to highly chaotic Ce atom thoroughly investigated before [4,6,7]; therefore properties of its Hamiltonian can be treated statistically [5]. Approximate quantum numbers such as total electron orbital angular momentum L and spin S disappear due to the enhancement of the spin-orbit interaction and such classification of atomic energy levels, which is still present in the Tables, becomes meaningless [4][5][6][7]. Number of principal components participating in excited eigenstates of Pa is N ∼ 10 3 , an order of magnitude larger than for Ce and closer to that of compound nuclei (N ∼ 10 4 − 10 6 ).…”

“…RTBI model demonstrates some aspects of behavior close to pure random matrix theory, such as Wigner distribution of spacings between energy levels, but it differs, for instance, in the composition of its eigenfunctions [8,33,34]. In this regard Pa atom is very similar to highly chaotic Ce atom thoroughly investigated before [4,6,7]; therefore properties of its Hamiltonian can be treated statistically [5]. Approximate quantum numbers such as total electron orbital angular momentum L and spin S disappear due to the enhancement of the spin-orbit interaction and such classification of atomic energy levels, which is still present in the Tables, becomes meaningless [4][5][6][7].…”

Structure of wave functions in(1+2)-body random matrix ensembles

“…For a EGOE (2), with N >> m >> 2, the normalized state density ρ(E) = δ(H − E) takes Gaussian form and it is defined by its centroid ǫ = H and variance σ 2 = (H − ǫ) 2 . In order to explicitly state that the state density is generated by the hamiltonian H, sometimes ρ(E) is denoted as ρ H (E) and similarly ǫ as ǫ H and σ as σ H .…”

“…variate Gaussian form for EGOE (2) and it is defined by the centroids (ǫ i , ǫ f ) and widths (σ i , σ f ) of its two marginals and the bivariate correlation coefficient which is given by…”