标题： 量子相图与陈数拓扑绝缘体的转变 显示英文标题

标题： Quantum Phase diagrams and transitions for Chern topological insulators

摘要： 诸如陈类这样的拓扑不变量现在已成为分类拓扑相的标准方法。 在这种系统中引入和改变参数会导致相图，在这种相图中，当穿越临界集时，陈类可能会跳跃。 当考虑高维系统的截面或者考虑具有参数的系统时，这些系统自然会出现。 由于陈类是拓扑不变量，它们只能在“拓扑结构崩溃”时发生变化。 我们给出了这一现象的精确数学表述，并证明通过覆盖（即绕数映射），可以合成设计并实现任意陈拓扑相的相图。 在这里，我们提供了实现任意陈跳跃的显式族。 这些映射的临界集由经典的玫瑰曲线描述。 这些实现了从局部电荷观点得出的必要Dirac点数量的下限。 我们研究了几个具体的模型，并展示了它们具有预测的通用行为。 特别是，我们关注不同类型的格子和紧束缚模型，并证明使用k个最近邻可以实现有效的绕数映射和较高的陈数。 我们给出了使用虚二次域扩展及其范数的一族二维格子的显式公式。 我们的研究包括正方形、三角形、蜂窝状和Kagome格子。 显示英文摘要

摘要： Topological invariants such as Chern classes are by now a standard way to classify topological phases. Introducing and varying parameters in such systems leads to phase diagrams, where the Chern classes may jump when crossing a critical locus. These systems appear naturally when considering slicing of higher dimensional systems or when considering systems with parameters. As the Chern classes are topological invariants, they can only change if the "topology breaks down". We give a precise mathematical formulation of this phenomenon and show that synthetically any phase diagram of Chern topological phases can be designed and realized by a physical system, using covering, aka. winding maps. Here we provide explicit families realizing arbitrary Chern jumps. The critical locus of these maps is described by the classical rose curves. These realize the lower bound on the number of Dirac points necessary obtained from viewing them as local charges. We treat several concrete models and show that they have the predicted generic behavior. In particular, we focus on different types of lattices and tight-binding models, and show that effective winding maps, and thus higher Chern numbers, can be achieved using k-th nearest neighbors. We give explicit formulas for a family of 2D lattices using imaginary quadratic field extensions and their norms. Our study includes the square, triangular, honeycomb and Kagome lattices.