Single-particle and collective motion in unbound deformed $ ^{39}\text{Mg} $

Background: Deformed neutron-rich magnesium isotopes constitute a fascinating territory where the interplay between collective rotation and single-particle motion is strongly affected by the neutron continuum. The unbound $fp$-shell nucleus $^{39}\text{Mg}$ is an ideal candidate to study this interplay. Purpose: In this work, we predict the properties of low-lying resonant states of $^{39}\text{Mg}$, using a suite of realistic theoretical approaches rooted in the open quantum system framework. Method: To describe the spectrum and decay modes of $^{39}\text{Mg}$ we use the conventional Shell Model, Gamow Shell Model, Resonating Group Method, Density Matrix Renormalization Group method, and the non-adiabatic Particle-Plus-Rotor model formulated in the Berggren basis. Results: The unbound ground state of $^{39}\text{Mg}$ is predicted to be either a $J^{\pi} = {7/2}^-$ state or a ${3/2}^-$ state. A narrow $J^{\pi} = {7/2}^-$ ground-state candidate exhibits a resonant structure reminiscent of that of its one-neutron halo neighbor $^{37}\text{Mg}$, which is dominated by the $f_{7/2}$ partial wave at short distances and a $p_{3/2}$ component at large distances. A $J^{\pi}={3/2}^-$ ground-state candidate is favored by the large deformation of the system. It can be associated with the ${1/2}^- [321]$ Nilsson orbital dominated by the $\ell=1$ wave; hence its predicted width is large. The excited $J^{\pi} = {1/2}^-$ and $5/2^-$ states are expected to be broad resonances, while the $J^{\pi} = {9/2}^-$ and ${11/2}^-$ members of the ground-state rotational band are predicted to have very small neutron decay widths. Conclusion: We demonstrate that the subtle interplay between deformation, shell structure, and continuum coupling can result in a variety of excitations in an unbound nucleus just outside the neutron drip line.