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Magnetic field-temperature phase diagrams for multiple-$Q$ magnetic orderings: Exact steepest descent approach to long-range interacting spin systems

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.