Monopole pairing correlations with random interactions

In this paper we study the monopole pairing correlation in $0^{+}$ states of semimagic nuclei under random interactions. We calculate overlaps between $S$-pair wave functions and exact shell-model wave functions. No dominance of the $S$-pair components is shown: In a single-$j$ shell $0^{+}$ states have random overlaps with the $S$-pair wave function; in a multi-$j$ space the $0_1^{+}$ state has a small overlap with the $S$-pair wave function, and this is particularly pronounced if the single-particle splitting is small and/or the spins $j$ of single-particle orbits are large. For the $0_1^{+}$ state in a multi-$j$ space, the generalized seniority scheme and the seniority truncation of the shell model provide similar results. We study the odd-even staggering of one-neutron separation energies under random interactions. The odd-even staggering phenomenon survives in random samples where all the even members have spin-zero ground states but disappears in random samples where all the even members have spin-nonzero ground states.

Figure 1

Distributions of overlaps between the $S$-pair wave function and the shell-model wave functions, ${\langle {\mathrm{\Psi }}_{\mathrm{GS}}|{\mathrm{\Psi }}_{\mathrm{SM}}\left(0_i^{+}\right)\rangle }^2$, with $i=1,2,...,d_{\mathrm{SM}}$, for four and six neutrons in a single-$j$ shell with the TBRE. The average value of the probability density is set to 1. $n$ is the neutron number, and $d_{\mathrm{SM}}$ is the dimension of the shell-model space for $0^{+}$ states. The curves in black are distributions of ${\langle {\mathrm{\Psi }}_{\mathrm{GS}}|{\mathrm{\Psi }}_{\mathrm{SM}}\left(0_i^{+}\right)\rangle }^2$ for the random ensemble. The dashed curve in green is the exact distribution of overlaps between a fixed unit vector and random unit vectors in a $d_{\mathrm{SM}}$-dimensional space.

Figure 2

Distribution of $0_1^{+}$-state shell-model wave function obtained under the TBRE in the $d_{\mathrm{SM}}$-dimensional Hilbert space. The average value of the probability density is set to 1 in panel (a) and $Y_{0,0}$ ($=1/\sqrt{4\pi }=0.282$) in panel (b). The coordinates of the $S$-pair wave function are also presented.

Figure 3

Distributions of the overlap between the seniority-truncated wave function and the shell-model wave function, ${\langle {\mathrm{\Psi }}_{\mathrm{ST}}\left(0_1^{+}\right)|{\mathrm{\Psi }}_{\mathrm{SM}}\left(0_1^{+}\right)\rangle }^2$, for semimagic nuclei in the schematic $g_{9/2}{i}_{11/2}$ shell under the TBRE-RSPE. Here $\varepsilon ={\varepsilon }_{i_{11/2}}-{\varepsilon }_{g_{9/2}}$. The average value of the probability density is set to 1.

Figure 4

Comparison of the generalized seniority scheme and the seniority truncation of the shell model for the $0_1^{+}$ state of semimagic nuclei in the $pf$ shell under the TBRE: the left two panels show ${\langle {\mathrm{\Psi }}_{\mathrm{GS}}|{\mathrm{\Psi }}_{\mathrm{SM}}\left(0_1^{+}\right)\rangle }^2$ versus ${\langle {\mathrm{\Psi }}_{\mathrm{ST}}\left(0_1^{+}\right)|{\mathrm{\Psi }}_{\mathrm{SM}}\left(0_1^{+}\right)\rangle }^2$, and the right two panels show $\delta E_{\mathrm{GS}}$ versus $\delta E_{\mathrm{ST}}$. The generalized seniority scheme and the seniority truncation of the shell model provide very similar results.

Figure 5

Distributions of ${\mathrm{\Delta }}_n^{\left(3\right)}$ for three to six neutrons in the single-$j=11/2$ shell under TBRE. The curve in black represents the random ensembles that both the $n=4$ and 6 systems have 0 g.s., and the curve in red represents those that both the $n=4$ and 6 systems have non-0 g.s. The average value of the probability density is set to 1.