Colloquium: Random matrices and chaos in nuclear spectra

Abstract

Chaos occurs in quantum systems if the statistical properties of the eigenvalue spectrum coincide with predictions of random-matrix theory. Chaos is a typical feature of atomic nuclei and other self-bound Fermi systems. How can the existence of chaos be reconciled with the known dynamical features of spherical nuclei? Such nuclei are described by the shell model ͑a mean-field theory͒ plus a residual interaction. The question is answered using a statistical approach ͑the two-body random ensemble͒: The matrix elements of the residual interaction are taken to be random variables. Chaos is shown to be a generic feature of the ensemble and some of its properties are displayed, emphasizing those which differ from standard random-matrix theory. In particular, the existence of correlations among spectra carrying different quantum numbers is demonstrated. These are subject to experimental verification.

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

“…The second equality O(t)O(t)H P (k) m = O(t)H P (k)O(t) m follows from the cyclic invariance of the m-particle average. Equation (20) gives the moments, (11) and (13). Thus, the non-trivial moments M P Q for P + Q ≤ 4 are M 11 , M 13 = M 31 and M 22 .…”

“…However, in the context of isolated finite many-particle quantum systems, classical random matrix ensembles are too unspecific to account for important features of the physical system at hand. One refinement which retains the basic stochastic approach but allows for such features consists in the use of embedded random matrix ensembles [8][9][10][11][12][13].…”

Assortative and disassortative mixing investigated using the spectra of graphs

“…Other values of β indicates that the distribution lies intermediate to these two. The NNSD provides a correlation measure of subsequent eigenvalues, whereas the ∆ 3 (L) statistic measures how the eigenvalues which are L distance apart are correlated, and can be estimated using the least-square deviation of the spectral staircase function representing average integrated eigenvalue densityN (λ) from the best fitted straight line for a finite interval of length L of the spectrum given by [32]:…”

“…This paper presents a systematic analysis of impact of degree-degree correlations on the spectral properties of various networks under the random matrix theory (RMT) framework. Since its introduction in 1960s, in the context of nuclear spectra, the theory has been successfully applied to a wide range of complex systems ranging from the quantum chaos to galaxy [31,32]. Recently, with a spurt in the activities of network framework, the RMT got its extension in analysis of spectral properties of various model networks [33,34] as well as those arising from real world systems [35,36].…”

Eigenvalue correlations and the distribution of ground state angular momenta for random many-body quantum systems

“…The observation of a statistical preference of L = 0 ground states for ensembles of random two-body interactions has sparked a large number of investigations to further explore the properties of these random systems and to understand the mechanism for the emergence of regular ordered spectral features from random interactions [6][7][8][9]. The appearance of ordered spectra in systems with chaotic dynamics is a robust property that does not depend on the specific choice of the (two-body) ensemble of random interactions [2,[10][11][12], timereversal symmetry [10], and the restriction of the Hamiltonian to one-and two-body interactions [13], nor is it limited to yrast states with small angular momentum L = 0, 2, 4 [14] as used in the original studies [2,3].…”