标题： D类实空间拓扑不变量的维度交叉 显示英文标题

标题： Dimensional crossover of class D real-space topological invariants

摘要： 材料的拓扑性质取决于其对称性、参数和空间维度。由于参数和对称性的变化而导致这些性质的变化，可以通过计算相应的拓扑不变量来理解。由于拓扑不变量通常定义在一个固定的空问维度上，因此不存在现有的框架来通过不变量来理解改变空间维度的影响。在这里，我们引入了一个框架，用于研究当系统维度发生变化时的拓扑相变，使用实空间拓扑标记。具体来说，我们考虑了Shiba晶格，这是一种由常规超导体表面的磁性原子形成的D类材料，并且表征了当初始的圆形吸附原子岛被变形为链时其拓扑结构的演变。我们还提供了相应抗无序性的度量。我们的框架可以推广到任意对称类和空间维度，可能通过识别，例如，维持三维拓扑性质所需的最小厚度来指导材料的设计。 显示英文摘要

摘要： The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological invariant. Since topological invariants are typically defined for a fixed spatial dimension, there is no existing framework to understand the effects of changing spatial dimensions via invariants. Here, we introduce a framework to study topological phase transitions as a system's dimensionality is altered using real-space topological markers. Specifically, we consider Shiba lattices, which are class D materials formed by magnetic atoms on the surface of a conventional superconductor, and characterize the evolution of their topology when an initial circular adatom island is deformed into a chain. We also provide a measure of the corresponding protection against disorder. Our framework is generalizable to any symmetry class and spatial dimension, potentially guiding the design of materials by identifying, for example, the minimum thickness of a slab required to maintain three-dimensional topological properties.