  Topological defects, called magnetic hedgehogs, realize emergent magnetic
monopoles, which are not allowed in the ordinary electromagnetism described by
Maxwell's equations. Such monopoles were experimentally discovered in magnets
in two different forms: tetrahedral $4Q$ and cubic $3Q$ hedgehog lattices. The
spin textures are modulated by the chemical composition, an applied magnetic
field, and temperature, leading to quantum transport and optical phenomena
through movement and pair annihilation of magnetic monopoles, but the
theoretical understanding remains elusive, especially in the regions where
different types of hedgehog lattices are competing. Here we propose a
theoretical model that can stabilize both tetrahedral and cubic hedgehog
lattices, and perform a thorough investigation of the phase diagram while
changing the interaction parameters, magnetic field, and temperature, by using
a recently developed method that delivers exact solutions in the thermodynamic
limit. We find that the model exhibits various types of topological transitions
with changes of the density of monopoles and antimonopoles, some of which are
accompanied by anomalies in the thermodynamic quantities, while the others are
hidden with less or no anomaly. We also find another hidden topological
transition with pair annihilation of two-dimensional vortices in the
three-dimensional system. These results not only provide useful information for
understanding the existing experimental data, but also challenge the
identification of hidden topological transitions and the exploration of
emergent electromagnetism in magnetic monopole lattices.
