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arXiv:nucl-th/0009076 · Physical Review Letters

Two-Body Random Ensembles: From Nuclear Spectra to Random Polynomials

Dimitri Kusnezov

Published Oct 30, 2000• 37 citations
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Abstract

The two-body random ensemble for a many-body bosonic theory is mapped to a problem of random polynomials on the unit interval. In this way one can understand the predominance of 0(+) ground states, and analytic expressions can be derived for distributions of lowest eigenvalues, energy gaps, density of states, and so forth. Recently studied nuclear spectroscopic properties are addressed.

DOI
10.1103/physrevlett.85.3773

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The two-body random ensemble for a many-body bosonic theory is mapped to a problem of random polynomials on the unit interval. In this way one can understand the predominance of 0(+) ground states, and analytic expressions can be derived for distributions of lowest eigenvalues, energy gaps, density of states, and so forth. Recently studied nuclear spectroscopic properties are addressed.
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We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Our study based both on analytical as well as on numerical arguments shows that except for the most J-stretched states, the ground states in the higher J-sectors are more orderly and develop larger energy gaps than the ones in the J=0-sector. Due to different characteristic energy scales in different J-sectors the J=0 ground states may predominate only when all the states are taken together.
We investigate the low-lying spectra of many-body systems with random two-body interactions, specifying that the ensemble be invariant under particle-hole conjugation. Surprisingly we find patterns reminiscent of more orderly interactions, such as a predominance of J=0J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0J=0 ground states separated by a gap from the excited states, and evidence of phonon vibrations in the low-lying spectra.
Eigenstate structure in many-body bosonic systems: Analysis using random matrices and qqq-Hermite polynomials
We analyze the structure of eigenstates in many-body bosonic systems by modeling the Hamiltonian of these complex systems using Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) defined by a mean-field plus kkk-body random interactions. The quantities employed are the number of principal components (NPC), the localization length (lHl_HlH​) and the entropy production S(t)S(t)S(t). The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate qqq-Hermite polynomials for local density of states Fk(E∣q)F_k(E|q)Fk​(E∣q) and the bivariate qqq-Hermite polynomial form for bivariate eigenvalue density ρbiv:q(E,Ek)ρ_{biv:q}(E,E_k)ρbiv:q​(E,Ek​) that are valid in the strong interaction domain. We also compare transport efficiency in many-body bosonic systems using BEGOE in absence and presence of centrosymmetry. It is seen that the centrosymmetry enhances quantum efficiency.
We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Contrary to the common belief our study based both on analytical as well as on numerical arguments shows that these are the higher JJJ-sectors whose ground states are more orderly than the ones in the J=0-sector. A predominance of J=0 ground states turns out to be the result of putting on together states with different characteristic energy scales from different JJJ-sectors.