The magnetism in Mn$_3$Si$_2$Te$_6$ has been investigated using thermodynamic measurements, first principles calculations, neutron diffraction and diffuse neutron scattering on single crystals. These data confirm that Mn$_3$Si$_2$Te$_6$ is a ferrimagnet below a Curie temperature of $T_C$ approximately 78K. The magnetism is anisotropic, with magnetization and neutron diffraction demonstrating that the moments lie within the basal plane of the trigonal structure. The saturation magnetization of approximately 1.6$mu_B$/Mn at 5K originates from the different multiplicities of the two antiferromagnetically-aligned Mn sites. First principles calculations reveal antiferromagnetic exchange for the three nearest Mn-Mn pairs, which leads to a competition between the ferrimagnetic ground state and three other magnetic configurations. The ferrimagnetic state results from the energy associated with the third-nearest neighbor interaction, and thus long-range interactions are essential for the observed behavior. Diffuse magnetic scattering is observed around the 002 Bragg reflection at 120K, which indicates the presence of strong spin correlations well above $T_C$. These are promoted by the competing ground states that result in a relative suppression of $T_C$, and may be associated with a small ferromagnetic component that produces anisotropic magnetism below $approx$330K.
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