多$Q$磁序的磁场-温度phase diagrams：长程相互作用自旋系统的精确最陡下降法
Magnetic field-temperature phase diagrams for multiple-$Q$ magnetic orderings: Exact steepest descent approach to long-range interacting spin systems
摘要：
多重- $Q$磁有序表示由多重自旋密度波或自旋螺旋的叠加组成的磁织构，如由Skyrmion晶体和刺猬晶格表示的。近年来，在各种磁性材料中观察到了这种磁有序性，特别是从拓扑学和非共面磁性结构产生的电磁场的角度，引起了人们极大的关注。虽然它们经常在改变温度和外部磁场的同时表现出不同的多$Q$状态之间的连续相变，但要阐明相图并不简单，这主要是由于缺乏简明的理论工具以及适当的微观模型。在这里，我们提供了一个理论框架，一类有效的自旋模型与长程磁相互作用介导的传导电子在磁性金属。我们的框架是基于最速下降法与一组自洽方程，导致在热力学极限的精确解，并有许多优势，现有的方法，如偏变分计算和数值蒙特卡罗模拟。将该框架应用于三重和六重$Q$磁序不稳定性模型，我们发现有趣的重入相变，其中多个$Q$相只出现在有限温度和/或非零磁场下。此外，我们发现，多个$Q$态可以拓扑非平凡堆叠skyrmion晶体或刺猬晶格，表现出大的净自旋标量手性与非零skyrmion数。结果表明，我们的框架可以是一个通用的工具，用于研究磁性和拓扑相变和相关的量子现象，在实际的磁性金属托管多个- $Q$磁有序。
英文摘要：
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.
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