The nature of order in low-temperature phases of some materials is not directly seen by experiment. Such hidden orders (HO) may inspire decades of research to identify the mechanism underlying those exotic states of matter. In insulators, HO phases originate in degenerate many-electron states on localized f or d shells that may harbor high-rank multipole moments. Coupled by inter-site exchange, those moments form a vast space of competing order parameters. Here, we show how the ground state order and magnetic excitations of a prototypical HO system, neptunium dioxide NpO$_2$, can be fully described by a low-energy Hamiltonian derived by a many-body ab initio force-theorem method. Superexchange interactions between the lowest crystal-field quadruplet of Np$^{4+}$ ions induce a primary non-collinear order of time-odd rank-5 (triakontadipolar) moments with a secondary quadrupole order preserving the cubic symmetry of NpO$_2$. Our study also reveals an unconventional multipolar exchange-striction mechanism behind the anomalous volume contraction of the NpO$_2$ HO phase.
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