Composite Superconducting Orders and Magnetism in CeRh$_2$As$_2$
Locally noncentrosymmetric materials are attracting significant attention due to the unique phenomena associated with sublattice degrees of freedom. The recently discovered heavy-fermion superconductor CeRh$_2$As$_2$ has emerged as a compelling example of this class, garnering widespread interest for its remarkable temperature-magnetic-field phase diagram, which features a field-induced first-order superconductor-to-superconductor phase transition with nontrivial dependence on the field direction and high critical fields, as well as antiferromagnetic and potentially higher multipole orders. To investigate the complex interplay of the ordered phases in CeRh$_2$As$_2$, we develop a theoretical framework based on symmetry analysis applied to a Bogoliubov--de Gennes Hamiltonian and Landau methods. This approach allows us to propose probable symmetries of the superconducting states and elucidate their close relationship with magnetism. Among other results, we find that the near degeneracy of two pairing symmetries is naturally explained if and only if intralayer spin-orbit coupling is large compared to interlayer hopping. Intriguingly, we find that the first-order transition can be interpreted as a transition between coexistence phases of the same superconducting order parameters, albeit with distinct admixtures. This line may end in a critical end point below the superconducting critical temperature. Our approach accurately reproduces current experimental phase diagrams for varying temperature as well as out-of-plane and in-plane magnetic field, both if the transition to a magnetic phase occurs below the superconducting critical temperature and if it occurs above. Furthermore, we calculate the magnetic susceptibility and the specific heat and compare these quantities to recent experimental results.