Energy Centroids of Spin I States by Random Two-body Interactions

In this paper we study the behavior of energy centroids (denoted as 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 ’s discussed in terms of the socalled 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 the geometric chaoticity. It is shown that the inclusion of isospin and parity does not “break” the regularities of EI ’s. PACS number: 05.30.Fk, 05.45.-a, 21.60Cs, 24.60.Lz Typeset using REVTEX 1 Low-lying spectra of many-body systems with an even number of particles were examined by Johnson, Bertsch and Dean in Ref. [1] by using random two-body interactions (TBRE). Their results showed a preponderance of spin = 0 ground states. Many efforts have been made to understand this very interesting observation and to study other regularities of many-body systems in the presence of random interactions. For instance, studies of oddeven staggering of binding energies, generic collectivity, behavior of energy centroids for spin I states, correlations, have attracted much attention in recent years. See Ref. [2] for a recent review. Among many works along the context of regularities under the TBRE Hamiltonian, regularities of energy centroids (denoted as EI ’s) of spin I states are very interesting. We denote by P(I) the probability that EI is the lowest energy among all EI′ ’s. It was shown in Refs. [3,4] that P(I) is large only when I ≃ Imin or I ≃ Imax. One thus divides the TBRE into two subsets, one subset has EI≃Imin as the lowest energy, and the other has EI≃Imax as the lowest energy. We define 〈EI〉min (〈EI〉max) as the value obtained by averaging EI over the subset where EI≃Imin (EI≃Imax) is the lowest energy. It was shown in Ref. [4] that 〈EI〉min ≃ CI(I + 1) and 〈EI〉max ≃ C [Imax(Imax + 1)− I(I + 1)], where C is a constant depending on the occupied single-particle orbits and the choice of the ensemble. These features were discussed by using the quasi-randomness of two-body coefficients of fractional parentage (cfp’s) in Ref. [4], and by using other statistical views in Refs. [5,6]. The purpose of this Brief Report is to revisit regularities of EI . We shall show that the above regularities of P(I)’s and 〈EI〉min’s hold even for very simple systems in which one cannot assume that two-body cfp’s are random. Previous studies of EI under random twobody interactions have been restricted to identical fermions in one-j shell or two-j shells. Here we shall extend the study of EI under random interactions to systems of many-j shells with the inclusion of parity and/or isospin. In this paper we use the general shell model Hamiltonians defined in Ref. [2], and take the TBRE of Ref. [1] for two-body matrix elements. EI and P(I) for “± ” parity states are denoted by EI± and P(I±), respectively. The number of particles is denoted by n. Proton (neutron) degree of freedom is denoted by “p” (“n”). Our statistics are based on 1000 sets of the TBRE Hamiltonian. We begin with simple systems, i.e., fermions in a small single-j shell (j ≤ 7 2 ) and bosons with a small spin l. First we study fermions in a j = 5/2 or 7/2 shell. EI for three fermions were given in Eqs. (2.1) and (D1) of Ref. [7]. EI for four fermions in a j = 7/2 shell can be obtained based on Eq. (5) of Ref. [8]. P(I)’s obtained by using the TBRE Hamiltonian and those by applying the empirical rule of Ref. [7] are plotted and compared in Fig. 1(a-c), where a reasonable agreement is achieved. One sees that P(I)’s are large for I ≃ Imin or I ≃ Imax, except that this pattern is not very striking for Fig. 1(a) where there are only three EI ’s given by three two-body matrix elements. For j = 7 2 , P(I)’s are small for “medium” I. Let us look at 〈EI〉min’s, which are obtained by averaging EI over the subset with EI≃Imin being the lowest energy. We plot 〈EI〉min’s for n = 3 with j = 5/2, n = 3 with j = 7/2, and