  Multiple-$Q$ magnetic orderings represent magnetic textures composed of
superpositions of multiple spin density waves or spin spirals, as represented
by skyrmion crystals and hedgehog lattices. Such magnetic orderings have been
observed in various magnetic materials in recent years, and attracted enormous
attention, especially from the viewpoint of topology and emergent
electromagnetic fields originating from noncoplanar magnetic structures.
Although they often exhibit successive phase transitions among different
multiple-$Q$ states while changing temperature and an external magnetic field,
it is not straightforward to elucidate the phase diagrams, mainly due to the
lack of concise theoretical tools as well as appropriate microscopic models.
Here, we provide a theoretical framework for a class of effective spin models
with long-range magnetic interactions mediated by conduction electrons in
magnetic metals. Our framework is based on the steepest descent method with a
set of self-consistent equations that leads to exact solutions in the
thermodynamic limit, and has many advantages over existing methods such as
biased variational calculations and numerical Monte Carlo simulations. Applying
the framework to the models with instabilities toward triple- and hextuple-$Q$
magnetic orderings, we find that interesting reentrant phase transitions where
the multiple-$Q$ phases appear only at finite temperature and/or nonzero
magnetic field. Furthermore, we show that the multiple-$Q$ states can be
topologically-nontrivial stacked skyrmion crystals or hedgehog lattices, which
exhibit large net spin scalar chirality associated with nonzero skyrmion
number. The results demonstrate that our framework could be a versatile tool
for studying magnetic and topological phase transitions and related quantum
phenomena in actual magnetic metals hosting multiple-$Q$ magnetic orderings.
