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arXiv:2411.01439 · Physical Review C

Shannon entropy of optimized proton-neutron pair condensates

Shu-Yuan Liang, Yi Lu, Yang Lei, Calvin W. Johnson, G. J. Fu, Jiajie Shen

Published Feb 13, 2025• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevc.111.024310

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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.
The Short-Range Correlations between nucleons in nuclei is regarded as a complex system. We investigate the relationship between the orbital entanglement entropy of SRCs Sij in nuclear structures and Tan contact cij, and find that the orbital entanglement entropies and Tan contacts corresponding to proton-proton SRC pairs and neutron-proton SRC pairs in nuclei demonstrate a scaling relation. More specifically, the proportionality of entanglement entropy between proton-proton pairs and neutron-proton pairs is directly related to the ratio of nuclear contacts within the atomic nucleus, demonstrating an approximate ratio of 2.0. Our research suggests that this scaling relationship should hold true for all symmetric nuclei, furthermore, we offer a possible explanation for this phenomenon.
Quantum entanglement provides a unique perspective for probing nuclear structure. In this work, we employ quantum entanglement measures, including proton-neutron entanglement entropy, mutual information, and quantum relative entropy, to investigate the evolution of entanglement patterns as we approach neutron-rich nuclei. The study is carried out in the vicinity of the N=20N=20N=20 island of inversion region consisting of even-AAA Ne, Mg, and Si isotopes, and also for isotones corresponding to N=20N=20N=20. The state-of-the-art \textit{ab initio} valence space in-medium similarity renormalization group method has been used for this purpose. We have highlighted the role of proton-neutron entanglement entropy in the formation of the island of inversion region. Mutual information provides insight into the strength of correlations between proton-proton, neutron-neutron, and proton-neutron single-particle states. While these correlations are relatively weak between protons and neutrons in the ground states, they become comparable to like-particle correlations in excited states. The quantum relative entropy is also studied between 0+0^+0+ and 2+2^+2+ states of the Ne, Mg, and Si isotopes, as well as N=20N=20N=20 isotones, using the Kullback-Leibler divergence and Jensen-Shannon divergence. We have performed these calculations by expressing the nuclear wavefunctions in a Slater-determinant basis and analyzing them through complementary partitions, including proton-neutron and mode-resolved factorizations.
The entanglement of pure states of the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-shell nucleon pairs has been studied. The von Neumann entropy of the partial density matrix has been employed to quantify the entanglement of the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-shell nucleon pairs. The Slater decomposition theorem has been used to verify if any pure state of a nucleon pair is an entangled state. Results of calculations have evidenced that a dominant part of the isospin [Formula: see text] proton–neutron states of the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-shell nucleon pairs, respectively, are strongly entangled. It is shown that the calculated data are a source of valuable information on the spin quantum numbers of the entangled protons from two-proton decay.
How do protons and neutrons bind to form nuclei?This is the central question of ab initio nuclear structure theory.While the answer may seem as simple as the fact that nuclear forces are attractive, the full story is more complex and interesting.In this work we present numerical evidence from ab initio lattice simulations showing that nature is near a quantum phase transition, a zero-temperature transition driven by quantum fluctuations.Using lattice effective field theory, we perform Monte Carlo simulations for systems with up to twenty nucleons.For even and equal numbers of protons and neutrons, we discover a first-order transition at zero temperature from a Bose-condensed gas of alpha particles ( 4 He nuclei) to a nuclear liquid.Whether one has an alpha-particle gas or nuclear liquid is determined by the strength of the alpha-alpha interactions, and we show that the alpha-alpha interactions depend on the strength and locality of the nucleon-nucleon interactions.This insight should be useful in improving calculations of nuclear structure and important astrophysical reactions involving alpha capture on nuclei.Our findings also provide a tool to probe the structure of alpha cluster states such as the Hoyle state responsible for the production of carbon in red giant stars and point to a connection between nuclear states and the universal physics of bosons at large scattering length.
Abstract We compute the proton-neutron entanglement entropy in the interacting nuclear shell model for a variety of nuclides and interactions. Some results make intuitive sense, for example, that the shell structure, as governed by single-particle and monopole energies, strongly affects the energetically available space and thus the entanglement entropy. We also find a surprising result: that the entanglement entropy at low excitation energy tends to decrease for nuclides when N ≠ Z. While we provide evidence this arises from the physical nuclear force by contrasting with random two-body interactions which shows no such decrease, the exact mechanism is unclear. Nonetheless, the low entanglement suggests that in models of neutron-rich nuclides, the coupling between protons and neutrons may be less computationally demanding than one might otherwise expect.
How do protons and neutrons bind to form nuclei? This is the central question of ab initio nuclear structure theory. While the answer may seem as simple as the fact that nuclear forces are attractive, the full story is more complex and interesting. In this work we present numerical evidence from ab initio lattice simulations showing that nature is near a quantum phase transition, a zero-temperature transition driven by quantum fluctuations. Using lattice effective field theory, we perform Monte Carlo simulations for systems with up to twenty nucleons. For even and equal numbers of protons and neutrons, we discover a first-order transition at zero temperature from a Bose-condensed gas of alpha particles (^{4}He nuclei) to a nuclear liquid. Whether one has an alpha-particle gas or nuclear liquid is determined by the strength of the alpha-alpha interactions, and we show that the alpha-alpha interactions depend on the strength and locality of the nucleon-nucleon interactions. This insight should be useful in improving calculations of nuclear structure and important astrophysical reactions involving alpha capture on nuclei. Our findings also provide a tool to probe the structure of alpha cluster states such as the Hoyle state responsible for the production of carbon in red giant stars and point to a connection between nuclear states and the universal physics of bosons at large scattering length.