Background: The low-energy enhancement observed in the deexcitation $\gamma$-ray strength functions, attributed to magnetic dipole ($M1$) radiations, has spurred theoretical efforts to improve on its description. Among the most widely used approaches are the quasiparticle random-phase approximation (QRPA) and its extensions. However, these methods often struggle to reproduce the correct behavior of the $M1$ strength at the lowest $\gamma$ energies. An alternative framework, the projected generator coordinate method (PGCM), offers significant advantages over QRPA by restoring broken symmetries and incorporating both vibrational and rotational dynamics within a unified description. Due to these features, PGCM has been proposed as a promising tool to study the low-energy $M1$ strength function in atomic nuclei. However, comprehensive investigations employing this method are lacking.

Purpose: The PGCM is developed and tested in its description of $1^{+}$ excited states in even-even nuclei, along with the $M1$ transitions connecting them to the $0^{+}$ ground state.

Methods: The PGCM is presently used within the frame of $sd$-shell valence space calculations based on the USDB shell-model interaction to benchmark its performance against the solutions obtained via exact diagonalization, i.e., results from the latter are considered as “exact” reference results in the present study. The reliability of two different sets of generator coordinates in the PGCM calculations is gauged using ${}^{24}\mathrm{Mg}$ as a test case.

Results: Energies and $M1$ transition strengths of the lowest excited $1^{+}$ states extracted from the PGCM calculation reproduce well the exact results for both sets of collective coordinates. Eventually, PGCM accurately models the exact level density and cumulative $B\left(M1\right)$ strength up to about 20-MeV excitation energy in all considered nuclei.

Conclusions: The ability of the PGCM to reproduce results from exact diagonalization in the $sd$ valence space is demonstrated for $1^{+}$ states and $M1$ transitions. Future work will need to assess whether the proposed method can be applied systematically and extended to large-scale calculations while maintaining a reasonable computational cost.

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• Electromagnetic transitions
• Energy levels
• Gamma-ray strength functions
• Nuclear many-body theory
• Nuclear transition rates
• Shell model
• 20 ≤ A ≤ 38