Microscopic modeling of the 3D quantum magnet Cu$_2$OSeO$_3$

Unlike most undoped cuprates, the $S=1/2$ Heisenberg magnet Cu$_2$OSeO$_3$ exhibits a ferrimagnetic ground state and sizable magnetoelectric coupling. Recent experiments reported magnetic-field-induced emergence of skyrmions in this material. Based on extensive DFT band structure calculations we evaluate the microscopic magnetic model, including isotropic (Heisenberg) and anisotropic (Dzyaloshinskii-Moriya) terms. We extract five relevant couplings that form a complex, but non-frustrated spin model which can be described as a pyrochlore lattice of magnetic tetrahedra. A peculiar feature of this lattice is the alternation of ``strong'' (the constituent spins are strongly coupled) and ``weak'' tetrahedra. Profiting from a separation of the energy scales we develop an effective model, treating strong tetrahedra either as a classical $S=1$ object or as a coherent quantum superposition of classical states. For the latter case, we find an excellent agreement with the quantum Monte Carlo simulations of the full model and the experimental magnetization and neutron diffraction data. Quite surprising for a 3D model we find distinct manifestations of quantum fluctuations. The developed effective model can be further used to model the field-induced behavior including the formation of skyrmions.