Challenges for first-principles nuclear structure: $^{11}$Li and $^{29}$F

Ab initio calculations of atomic nuclei have had many successes in recent years. Nonetheless, important challenges that resist even brute-force calculation remain. As archetypal examples of these challenges, we consider $^{11}$Li and $^{29}$F, well known halo nuclides situated on islands of inversion. The deformed intruder levels, which are primarily two-particle, two-hole neutron excitations with respect to naive spherical shell model configurations, are slow, with respect to increases of the model space, to take their rightful place among, and potentially mix with, the lowest levels. We suggest these systems prototype the challenges for other important intruder states, and can serve as useful testbeds for potential approaches.

💡 Research Summary

The paper “Challenges for first‑principles nuclear structure: ¹¹Li and ²⁹F” investigates why ab initio nuclear‑structure calculations, even when performed with the powerful no‑core shell model (NCSM), still encounter severe difficulties for certain nuclei. The authors focus on two archetypal cases—¹¹Li, a classic two‑neutron halo nucleus, and ²⁹F, a member of the “island of inversion” where intruder configurations dominate the ground state. Both nuclei exhibit strong two‑particle–two‑hole (2p‑2h) neutron excitations that are low in energy, contrary to the naive expectation from a spherical shell‑model picture.

In the traditional spherical shell model, the Aufbau principle orders single‑particle orbitals and defines a valence space (P) separated from an excluded space (Q). Intruder states are those that, despite being dominated by Q‑space components, appear low in the spectrum. Historically, such states caused a “perturbative crisis” because perturbation theory assumes a large energy gap between P and Q. The introduction of the in‑medium similarity renormalization group (IMSRG) and, later, the NCSM alleviated the crisis by treating all excitations non‑perturbatively. However, the authors point out a new problem: in the NCSM the intruder levels converge far more slowly with the model‑space truncation parameter N_max than the normal (0 ℏω) states. Consequently, for modest N_max the calculated ground state is often a normal configuration, while the true ground state is an intruder or a strong mixture of both.

Two realistic nucleon‑nucleon interactions are employed. The first is the Entem‑Machleidt N³LO chiral interaction softened by a similarity‑renormalization‑group (SRG) evolution to λ = 2.0 fm⁻¹, without three‑body forces. The second is Daejeon16, which is an even softer interaction (SRG to λ = 1.5 fm⁻¹ followed by phase‑shift‑equivalent adjustments) that reproduces binding energies up to A = 16 without induced three‑body terms. The softer Daejeon16 interaction leads to faster convergence, allowing the authors to illustrate the intrinsic difficulty more clearly.

For ¹¹Li the NCSM spectra as a function of N_max are shown for both interactions. At low N_max the lowest‑lying 3/2⁻ and 1/2⁻ states are normal (dominantly 0 ℏω) configurations, with the dominant component being a filled neutron 0p shell and a valence proton in the p₃/₂ or p₁/₂ orbital. Intruder states appear at higher excitation energy, characterized by negligible 0 ℏω content and a dominant 2 ℏω component (i.e., 2p‑2h neutron excitations). As N_max increases, the intruder levels drop rapidly in energy, especially for Daejeon16, where by N_max ≈ 8 they are already close to the normal levels. The authors observe an avoided crossing between the normal 1/2⁻ and intruder 1/2⁻ states, indicating strong mixing. Decomposition of the wave functions into N_ex (the number of oscillator quanta) and SU(3) irreducible representations confirms that normal states are essentially spherical (SU(3) (0,0)), whereas intruder states carry large deformation (e.g., SU(3) (8,0)).

To quantify deformation, the dimensionless ratio Q/r² (electric quadrupole moment divided by the mean‑square radius) is examined for both protons and neutrons. The proton Q_p/r_p² for the ground state is negative, as expected for a single particle outside a closed shell, and the calculated values approach the experimental measurement as N_max grows, though full convergence is not reached. The neutron ratio Q_n/r_n² is substantially larger, reflecting the pronounced deformation of the neutron halo.

A parallel analysis for ²⁹F shows the same pattern. In a naïve spherical picture the valence neutrons would occupy the 0s‑0p‑1s0d shells, but experimental data and the NCSM calculations indicate that a 2p‑2h intruder configuration (neutrons promoted across the N = 20 gap) dominates. The intruder states again start high in energy at small N_max and descend slowly, with Daejeon16 providing a more rapid approach to the true ordering. The SU(3) analysis reveals strong prolate deformation for the intruder, consistent with the large measured B(E2) values in this region.

The authors conclude that the slow convergence of intruder configurations constitutes a fundamental bottleneck for current ab initio approaches. Even with powerful non‑perturbative methods, achieving the N_max → ∞ limit for nuclei where intruders dominate would require computational resources far beyond present capabilities. They suggest that ¹¹Li and ²⁹F serve as benchmark “testbeds” for developing new strategies—such as importance‑truncated bases, adaptive SRG evolutions, or machine‑learning‑guided model‑space reductions—that can accelerate the convergence of intruder‑dominated states. The paper thus highlights both a specific technical challenge and a broader direction for future research in first‑principles nuclear theory.