  We study thermodynamic properties of an antiferromagnetic Ising model on the
inverse perovskite lattice by using Monte Carlo simulations. The lattice
structure is composed of corner-sharing octahedra and contains
three-dimensional geometrical frustration in terms of magnetic interactions.
The system with the nearest-neighbor interactions alone does not exhibit any
phase transition, leading to a degenerate ground state with large residual
entropy. The degeneracy is lifted by an external magnetic field or by an
anisotropy in the interactions. Depending on the anisotropy, they stabilize
either a 3D ferrimagnetic state or a partially-disordered antiferromagnetic
(PDAF) state with a dimensionality reduction to 2D. By the degeneracy-lifting
perturbations, all the transition temperatures of these different ordered
states continuously grow from zero, leaving an unusual zero-temperature
critical point at the unperturbed point. Such a zero-temperature
multicriticality is not observed in other frustrated structures such as
face-centered cubic and pyrochlore. The transition to the PDAF state is
represented by either the first- or second-order boundaries separated by
tricritical lines, whereas the PDAF phase shows 1/3 magnetization plateaus.
