标题： 量子相图和陈数拓扑绝缘体的相变 显示英文标题

标题： Quantum Phase diagrams and transitions for Chern topological insulators

摘要： 拓扑不变量如陈类现在是一种标准的分类拓扑相的方法。在一个族中改变系统会导致相图，在穿越临界轨迹时，陈类可能会跳跃。当考虑高维系统的切片或考虑具有参数的系统时，这些系统自然出现。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，即所谓的绕数映射，可以设计和实现任意的陈拓扑相的相图。这里我们提供了实现任意陈跳跃的具体族。这些映射的临界轨迹由经典的玫瑰曲线描述。这为一般情况下狄拉克点的数量提供了一个下界，对于二能级系统而言，这个界限是尖锐的。在这个过程中，我们处理了几个具体的模型。特别是，我们处理了格子和紧束缚模型，并展示了可以通过第$k$近邻来实现有效的绕数映射。我们给出了使用虚二次域扩张及其范数的一族二维格子的显式公式。这包括正方形、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. Varying systems in a family leads to phase diagrams, where the Chern classes may jump when crossingn 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 give a lower bond on the number of Dirac points in general that is sharp for 2-level systems. In the process, we treat several concrete models. In particular, we treat the lattices and tight--binding models, and show that effective winding maps 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. This includes the square, triangular, honeycomb and Kagome lattices