The IM-GCM framework — multireference in-medium similarity renormalization group (MR-IMSRG) evolution of chiral two- plus three-nucleon Hamiltonians followed by quantum-number projected generator coordinate method (PGCM) diagonalization — reproduces shape coexistence and tracks the emergence of the N=20N=20 island of inversion in neutron-rich Mg isotopes, but at the MR-IMSRG(2)/normal-ordered-two-body (NO2B) level with zero-cranking configurations it systematically overestimates excitation energies (by roughly 50% for the first 2+2^+ state of 32^{32}Mg) and leaves parts of the 33^{33}Mg level scheme in tension with experiment. The open problem is to bring this description under controlled many-body truncation: incorporate flowing three-body operators (IMSRG(3)) or controlled approximations to them in the multireference IMSRG evolution, and cranked or quasiparticle-excited configurations in the PGCM mixing; and determine, with quantified uncertainties, whether the excitation-energy bias is removed, whether the intruder-emergence mechanism survives the improved treatment, and whether the 33^{33}Mg discrepancies are resolved. As of August 2026, single-reference/valence-space IMSRG(3) and factorized approximations exist and the automated derivation of the multireference three-body flow equations has been completed, but no production MR-IMSRG(3)+PGCM calculation has been published, and the newest systematic study of the region still carries the excitation-energy bias. The problem remains open.

Background

Certain neutron numbers — the 'magic numbers', such as N=20N=20 — close a shell of neutron orbitals, so nuclei near them were long expected to be spherical and relatively inert. From the 1970s onward, however, neutron-rich nuclei near 31^{31}Na proved more bound and more collective than the spherical shell model predicts. This region is now called the 'island of inversion' (IoI): the usual (normal) configurations built from the sdsd shell are energetically inverted with so-called intruder configurations, in which neutrons are excited across the N=20N=20 gap into the fpfp shell. The intruder configurations are deformed, so IoI nuclei display shape coexistence — low-lying nuclear states of different intrinsic deformation (for example weakly and strongly deformed) appearing at comparable excitation energy.

'Ab initio' nuclear theory computes nuclear structure directly from internucleon forces derived from chiral effective field theory, a systematic low-energy expansion of quantum chromodynamics in terms of two- and three-nucleon (NN+3N) interactions. The in-medium similarity renormalization group (IMSRG) is a main ab initio tool for medium-mass nuclei: a continuous unitary transformation decouples a chosen reference state from particle-hole excitations, evolving the Hamiltonian into an effective one. In practice all operators are truncated at the normal-ordered two-body (NO2B) level — the Hamiltonian is normal-ordered with respect to a reference state and the residual three-body terms are discarded — which also drops the 'flowing' three-body operators generated during the evolution; retaining them defines IMSRG(3). For open-shell, deformed nuclei the reference is a symmetry-breaking Hartree–Fock–Bogoliubov state — the mean-field product wavefunction, which breaks particle-number and rotational symmetries so that pairing and intrinsic deformation can be described — giving the multireference (MR) IMSRG. The collective correlations that remain — pairing and shape mixing — are then added by the quantum-number projected generator coordinate method (PGCM): the evolved Hamiltonian is diagonalized in a basis of reference states labeled by deformation coordinates, after projecting each onto good particle number and angular momentum. The combination is the IM-GCM. 'Cranking' means adding a term −ωJx-\omega J_x to the mean-field Hamiltonian so the intrinsic state carries a finite angular-momentum projection along the xx axis; mixing such cranked configurations is known, from symmetry-restored GCM studies in heavier nuclei, to compress excitation energies by factors of order 1.3–1.6.

With IM-GCM and the EM1.8/2.0 chiral NN+3N interaction (the widely used Entem–Machleidt 1.8/2.0 parameterization), Zhou et al. (2025) reproduced the coexistence of weakly and strongly deformed low-lying states in 30–34^{30–34}Mg, tracked how the deformed intruder ground states emerge continuously during the IMSRG evolution of the chiral Hamiltonian, and predicted for 33^{33}Mg a strongly deformed 3/2−3/2^- ground state together with a lowest 7/2−7/2^- state that is a shape isomer — a relatively long-lived excited state of a different intrinsic shape. They also showed that normal ordering with respect to the evolved reference (the ENO scheme) outperforms normal ordering with respect to the original unevolved reference (the GNO scheme) for shape-coexistence nuclei. However, at the MR-IMSRG(2)/NO2B level with only zero-cranking configurations, the method systematically overestimates excitation energies — by about 50% for the first 2+2^+ state of 32^{32}Mg, denoted Ex(21+)E_x(2_1^+) — and the detailed level ordering in 33^{33}Mg remains in partial tension with the knockout-reaction data. The remedies the authors themselves name — triple corrections beyond NO2B in the IMSRG evolution, and cranked or quasiparticle-excited configurations in the PGCM space — were left unimplemented, and the uncertainty assessment (flow-parameter and basis-size extrapolations) was incomplete.

Problem Statement

Bring the IM-GCM (multireference IMSRG + quantum-number projected generator coordinate method) description of shape coexistence and the N=20N=20 island of inversion under controlled many-body truncation, as one coherent quantitative-spectroscopy objective: incorporate the correlations that the present MR-IMSRG(2)/NO2B, zero-cranking implementation omits — flowing three-body operators (IMSRG(3)) or controlled approximations such as factorized triples in the multireference IMSRG evolution, together with cranked-frequency and/or quasiparticle-excited configurations in the PGCM mixing — and determine, with quantified uncertainties:

(a) whether the systematic overestimation of excitation energies (approximately 50% for Ex(21+)E_x(2_1^+) in 32^{32}Mg at the NO2B, zero-cranking level) is removed;

(b) whether the microscopic mechanism by which deformed intruder ground states emerge during the continuous IMSRG evolution, and the shape-coexistence and island-of-inversion picture established for the neutron-rich Mg isotopes 30–34^{30–34}Mg, survive the improved treatment;

(c) whether the remaining discrepancies with experiment — notably the disputed level ordering and decay properties of 33^{33}Mg, including the predicted 7/2−7/2^- shape isomer — are resolved.

The problem window covers whatever method development, benchmarking against existing single-reference/valence-space IMSRG(3) results, and systematic calculations across the neutron-rich Mg isotopes this objective jointly requires; components (a)–(c) are judged together as one ab initio accuracy claim about the same framework and region, not as independent deliverables.

References

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