Shannon entropy of optimized proton-neutron pair condensates

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 $T=0$ proton-neutron pair condensate phase in $N=Z$ nuclei. With an explicit formalism of general pair condensate with good particle numbers, we optimize proton-neutron pair condensates for all $N=Z$ nuclei between $^{16}\mathrm{O}$ and $^{100}\mathrm{Sn}$, 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 $T=1$ phase to an entangled $T=0$ phase, i.e., pairing phase transition driven by external parameters. In the $T=0$ limit, the proton-neutron pair condensate optimized for $^{24}\mathrm{Mg}$ turns out to be a purely $T=0$ P-pair condensate with large entanglement entropy, although such cases may occur in cold atom systems, unlikely in atomic nuclei.