Skyrmion lattices in chiral ferromagnets

铁磁体是由分子组成的材料，每个分子的行为都像一个微小的磁铁，它们相互作用，使邻近的分子希望它们的磁铁对齐。这种材料的状态可以用一个单位长度的矢量场来表示，称为磁化矢量，它记录了磁体在空间中每个点的方向。如果底层分子缺乏反射对称性，则称铁磁体是手性的。在这种情况下，当暴露在外部磁场中时，铁磁体的最低能量配置可能不是磁化矢量与磁场对齐的明显状态，而是一种更神秘的状态，称为skyrmion。这是一种构型，在这种构型中，磁化矢量以一种类似于结的二维形式将自身绑在一起，当一个人在空间中移动时，它会覆盖所有可能的方向。这种打结性不能被系统的任何连续变形所消除——用数学语言来说，这是一个拓扑孤子。磁性存储器的一维阵列是下一代数据存储设备（所谓的赛道存储器）的基础。他们的数学研究几乎完全集中在单个孤立的天空中，或者小的孤立的天空中。但在真实的铁磁系统中，斯基子自发地出现在规则的二维阵列中——斯基子晶格。值得注意的是，目前还没有对这种格进行系统的数学研究。我们建议研究斯基子晶格的存在性、稳定性、通用性和几何形状，特别关注这些性质如何依赖于外加磁场的强度和方向。为了做到这一点，我们将采用最初在核物理和超导模型中发展的数学思想，在大规模计算机模拟中实现这些思想。

Ferromagnets are materials made up of molecules each of which behaves like a tiny magnet, which interact with one another so that neighbouring molecules want their magnets to align. The state of such a material can be represented by a unit length vector field called the magnetization vector, which records the direction of the magnet at each point in space. If the underlying molecules lack reflexion symmetry, the ferromagnet is said to be chiral. In this case, the lowest energy configuration of the ferromagnet, when exposed to an external magnetic field, may not be the obvious state in which the magnetization vector aligns with the field, but a rather more mysterious state called a skyrmion. This is a configuration in which the magnetization vector ties itself up in a two-dimensional analogue of a knot, ranging through all possible orientations as one moves in space. The knottedness cannot be undone by any continuous deformation of the system - in mathematical language, this is a topological soliton. One dimensional arrays of magnetic skyrmions are the basis of proposed next-generation data storage devices (so-called race track memory). Their mathematical study has focussed almost exclusively on single isolated skyrmions, or small isolated clusters of skyrmions. But in real ferromagnetic systems, skyrmions occur spontaneously in regular two dimensional arrays - skyrmion lattices. Remarkably, there has been no systematic mathematical study of such lattices. We propose to study the existence, stability, genericity, and geometry of skyrmion lattices, focussing in particular on how these properties depend on the strength and direction of the applied magnetic field. To do this, we will adapt mathematical ideas developed originally in models of nuclear physics and superconductivity, implementing these in large-scale computer simulations.