奇点、燕尾和狄拉克点。哈密顿量族的分析及其在线网络特别是陀螺中的应用
Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid
摘要：
受双陀螺纳米线网络的启发，我们开发了检测狄拉克点和分类水平交叉点的方法。在一个家庭的哈密顿谱奇点。我们使用的方法是奇点理论。使用这种语言，我们得到了狄拉克点的一个特征，也表明，分支行为的水平交叉给出了一个展开的$A_n$型奇点。哪种类型的奇异性发生可以从$A_k$奇异性的微小展开内的特征区域读出。然后，我们将这些方法应用于设置家庭的图形哈密顿，如有线网络。在双陀螺仪的特殊情况下，我们对其奇异性进行了解析分类，并表明它具有狄拉克点。这表明这种类型的纳米线系统应该具有非常特殊的物理性质。
英文摘要：
Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of $A_n$ type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an $A_k$ singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.
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