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arXiv:1607.06593 · Physical Review C

Properties of isoscalar-pair condensates

P. Van Isacker, Augusto O. Macchiavelli, P. Fallon, S. Zerguine

Published Aug 17, 2016• 9 citations
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Abstract

It is pointed out that the ground state of $n$ neutrons and $n$ protons in a single-$j$ shell, interacting through an isoscalar ($T=0$) pairing force, is not paired, $J=0$, but rather spin aligned, $J=n$. This observation is explained in the context of a model of isoscalar $P$ ($J=1$) pairs, which is mapped onto a system of $p$ bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in $jj$ coupling.

DOI
10.1103/physrevc.94.024324

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It is pointed out that the ground state of nnn neutrons and nnn protons in a single-jjj shell, interacting through an isoscalar (T=0T=0T=0) pairing force, is not paired, J=0J=0J=0, but rather spin aligned, J=nJ=nJ=n. This observation is explained in the context of a model of isoscalar PPP (J=1J=1J=1) pairs, which is mapped onto a system of ppp bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in jjjjjj coupling.
Previous studies have shown that the ground state of systems of nucleons composed by an equal number of protons and neutrons interacting via proton-neutron pairing forces can be described accurately by a condensate of α-like quartets. Here we extend these studies to the low-lowing excited states of these systems and show that these states can be accurately described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet. This approach, which is analogous to the one-broken-pair approximation employed for like-particle pairing, is analyzed for various isovector and isovector-isoscalar pairing Hamiltonians which can be solved exactly by diagonalization.
Proton-neutron pairing and like-nucleon pairing are two different facets of atomic nuclear configurations. While like-nucleon pair condensates manifest their superfluidic nature in semimagic nuclei, it is not absolutely clear if there exists a T=0T=0T=0 proton-neutron pair condensate phase in N=ZN=ZN=Z nuclei. With an explicit formalism of general pair condensate with good particle numbers, we optimize proton-neutron pair condensates for all N=ZN=ZN=Z nuclei between 16O^{16}\mathrm{O}16O and 100Sn^{100}\mathrm{Sn}100Sn, given shell model effective interactions. As comparison, we also optimize like-nucleon pair condensates for their semimagic isotones. Shannon entanglement entropy is a measurement of mixing among pair configurations, and can signal intrinsic phase transition. It turns out the like-nucleon pair condensates for semimagic nuclei have large entropies signaling an entangled phase, but the proton-neutron pair condensates end up not far from a Hartree-Fock solution with small entropy. With artificial pairing interaction strengths, we show that the general proton-neutron pair condensate can transit from an entangled T=1T=1T=1 phase to an entangled T=0T=0T=0 phase, i.e., pairing phase transition driven by external parameters. In the T=0T=0T=0 limit, the proton-neutron pair condensate optimized for 24Mg^{24}\mathrm{Mg}24Mg turns out to be a purely T=0T=0T=0 P-pair condensate with large entanglement entropy, although such cases may occur in cold atom systems, unlikely in atomic nuclei.
A method is proposed for determining the ground state of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k}{\ensuremath{\epsilon}}_{k}{N}_{k}\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{\mathrm{ijkl}}{V}_{\mathrm{ijkl}}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}$when the residual interaction is such that $〈{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}〉={〈{N}_{i}{N}_{j}(1\ensuremath{-}{N}_{k})(1\ensuremath{-}{N}_{l})〉}^{\frac{1}{2}}{〈{N}_{k}{N}_{l}(1\ensuremath{-}{N}_{i})(1\ensuremath{-}{N}_{j})〉}^{\frac{1}{2}}$is a good approximation. The method is applied to pairing interactions involving one type of nucleon as well as to the generalized pairing problem containing neutron-neutron, neutron-proton, and proton-proton interactions.
Expressions for the ground and excited state energies of a system of nucleons interacting through pairing forces are given as a power series in inverse powers of the number of particles. The expressions are valid for systems with superfluid ground states and either J=0 or L=0 pairing. The first three terms in the expansion are given explicitly, and they exhibit excitations with both vibrational and rotational (in isospin space) character. Analytical and numerical results are given for a model system with a two-level single-particle spectrum.
Neutron-proton pairing correlations are studied within the context of two solvable models, one based on the algebra SO(5) and the other on the algebra SO(8). In both of these models, particles interact in L=0L=0L=0 pairs only. Boson-mapping techniques are applied to these models and shown to provide a convenient methodological tool both for solving such problems and for gaining useful insight into general features of pairing. We first focus on the SO(5) model, which involves generalized T=1T=1T=1 pairing. Neither boson mean-field methods nor fermion-pair approximations are able to describe in detail neutron-proton pairing in this model. The analysis suggests, however, that the boson Hamiltonian obtained from a mapping of the fermion Hamiltonian contains a pairing force between bosons, pointing to the importance of boson-boson (or equivalently four-fermion) correlations with isospin T=0T=0T=0 and spin S=0S=0S=0. These correlations are investigated by carrying out a second boson mapping. Closed forms for the fermion wave functions are given in terms of the fermion-pair operators. Similar techniques are applied---albeit in less detail---to the SO(8) model, involving a competition between T=1T=1T=1 and T=0T=0T=0 pairing. Conclusions similar to those of the SO(5) analysis are reached regarding the importance of four-particle correlations in systems involving neutron-proton pairing.