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arXiv:nucl-th/0412083 · Physical Review C

Energy centroids of spin I states by random two-body interactions

Y. M. Zhao, A. Arima, K. Ogawa

Published Jan 31, 2005• 6 citations
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Abstract

In this paper we study the behavior of energy centroids (denoted as $\overline{{E}_{I}}$) of spin I states in the presence of random two-body interactions, for systems ranging from very simple systems (e.g., single-j shell for very small j) to very complicated systems (e.g., many-j shells with different parities and with isospin degree of freedom). Regularities of $\overline{{E}_{I}}$'s discussed in terms of the so-called geometric chaoticity (or quasi-randomness of two-body coefficients of fractional parentage) in earlier works are found to hold even for very simple systems in which one cannot assume geometric chaoticity. It is shown that the inclusion of isospin and parity does not ``break'' the regularities of $\overline{{E}_{I}}$'s.

DOI
10.1103/physrevc.71.017304

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In this paper we study the behavior of energy centroids (denoted as EI‾\overline{{E}_{I}}EI​​) of spin I states in the presence of random two-body interactions, for systems ranging from very simple systems (e.g., single-j shell for very small j) to very complicated systems (e.g., many-j shells with different parities and with isospin degree of freedom). Regularities of EI‾\overline{{E}_{I}}EI​​'s discussed in terms of the so-called geometric chaoticity (or quasi-randomness of two-body coefficients of fractional parentage) in earlier works are found to hold even for very simple systems in which one cannot assume geometric chaoticity. It is shown that the inclusion of isospin and parity does not ``break'' the regularities of EI‾\overline{{E}_{I}}EI​​'s.
In this paper, we discuss the regularities of average energies with a fixed angular momentum I (denoted as EI′s){E}_{I}'\mathrm{s})EI′​s) in many-body systems interacting via a two-body random ensemble. It is found that EI′s{E}_{I}'\mathrm{s}EI′​s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ (minimum of I)I)I) or $I\ensuremath{\sim}{I}_{\mathrm{max}}$ (maximum of I)I)I) have large probabilities [denoted as P(I)]\mathcal{P}(I)]P(I)] to be the smallest in energy, and P(I)\mathcal{P}(I)P(I) is close to zero elsewhere. A simple argument assuming the randomness of the two-particle coefficients of fractional parentage is given to explain these observations. A compact trajectory of the energy EI{E}_{I}EI​ vs I(I+1)I(I+1)I(I+1) is found to be robust. Other regularities, such that there are two or three sizable P(I)′s\mathcal{P}(I)'\mathrm{s}P(I)′s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ but $\mathcal{P}(I)\ensuremath{\ll}\mathcal{P}{(I}_{\mathrm{max}})'\mathrm{s}$ with $I\ensuremath{\sim}{I}_{\mathrm{max}},$ and that the coefficients C defined by 〈EI〉min=CI(I+1)〈{E}_{I}{〉}_{\mathrm{min}}=CI(I+1)〈EI​〉min​=CI(I+1) are sensitive to the orbits and not sensitive to particle number, etc., are discovered and studied for the first time.
In this talk I shall discuss some regularities of many‐body systems in the presence of random interactions and regularities of a single‐j shell for the J pairing interaction which works only when two particles are coupled to spin J. I shall first explain an empirical rule to predict the spin I ground state probability. Then I shall present some interesting results of a single‐j shell under the J pairing interaction. Last I shall discuss some preliminary results of binding energies in the presence of random two‐body interactions.
Regular structures generated by random interactions in energy centroids defined over irreducible representations (irreps) of some of the group symmetries of the interacting boson models sdIBM, sdgIBM, sdIBM-T, and sdIBM-ST are studied by deriving trace propagation equations for the centroids. It is found that, with random interactions, the lowest and highest group irreps in general carry most of the probability for the corresponding centroids to be lowest in energy. This generalizes the result known earlier, via numerical diagonalization, for the more complicated fixed spin J centroids where simple trace propagation is not possible.