Bernal Stacking and Symmetry-Inequivalent Antiferromagnetism in MSi$_2$N$_4$ Heterobilayers

Layered MA$_2$Z$_4$ compounds, structural relatives of MoS$_2$ discovered in 2020, exhibit rich magnetic behavior arising from reduced dimensionality, noncentrosymmetric lattice symmetries, and stacking-dependent exchange interactions. Here, we investigate Bernal-like stackings in H-phase MA$_2$Z$_4$ (M = Mn and Fe; A = Si; Z = N) monolayers and bilayers by combining first-principles spin-dependent relaxation energies with a localized-spin Heisenberg description. From density-functional calculations, we extract the dominant intralayer exchange couplings up to third-nearest neighbors and the leading interlayer exchanges up to second-nearest neighbors, enabling construction of an effective bilayer spin Hamiltonian. We first analyze interface-driven proximity effects within a ferromagnetic reference configuration, demonstrating how recovery of AB-type stacking and spin alignment--while varying only the transition-metal species--provides a route for selectively tuning magnetic order and symmetry breaking within the P$\bar{6}$m2 space group. Building on this microscopic understanding of the bonding environment, we then examine antiferromagnetic ordering tendencies in the coupled layers. Exact diagonalization of the resulting bilayer Hamiltonian reveals the magnetic ground state and low-lying excitation spectrum, showing that the interlayer exchange is not merely perturbative but competes directly with intralayer interactions in stabilizing the observed spin configurations. These results establish Bernal-stacked MA$_2$Z$_4$ bilayers as a platform in which stacking geometry and exchange hierarchy jointly govern magnetic reconstruction, offering a controlled pathway toward domain selection and spin-texture engineering in low-dimensional van der Waals materials.