<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dipole strength from projected generator coordinate method calculations in the <mml:math xmlns:mml="http://www.w3.org/1

Background: The low-energy enhancement observed in the deexcitation $\ensuremath{\gamma}$-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 $\ensuremath{\gamma}$ 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 $B(M1)$ 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.