Quantum spin models of commensurate $p$-wave magnets

The $p$-wave magnet has emerged as a new type of magnetism exhibiting odd-parity, time-reversal-symmetric spin splitting in momentum space, and has attracted considerable interest as a promising platform for spintronic applications. However, the theoretical understanding of the fundamental mechanism responsible for stabilizing this phase remains limited. In this work, we identify a microscopic interacting model that realizes the $p$-wave magnet as its ground state. We first introduce a Hubbard model and derive the corresponding low-energy spin Hamiltonian. At the classical level, we find that the $p$-wave magnet is stabilized but remains energetically degenerate with competing noncoplanar states. Quantum fluctuations lift this degeneracy, selecting the $p$-wave magnet as the unique ground state. The resulting electronic structure exhibits finite spin accumulation via the Edelstein effect, highlighting the potential of $p$-wave magnetism for spintronic applications. We further discuss the relevance of our theory to quasi-two-dimensional honeycomb magnets such as Ni$_2$Mo$_3$O$_8$. Our findings establish the possibility of spontaneous $p$-wave magnetism.