标题： 量子系统的多重简并点和矩阵流形的奇异性 显示英文标题

标题： Multifold degeneracy points of quantum systems and singularities of matrix varieties

摘要： 参数依赖的量子系统通常表现出能量简并点，其全面描述自然导致奇点理论方法的应用。 一个典型的例子是电子能带结构，在动量空间的一个点上两个能量水平重合。 可能会发生这种情况，这也是我们工作的重点，即三个或更多水平在参数点上重合，称为多倍简并。 在一般的扰动下，这种多倍简并点会分解为一组外尔点，即一般的二重简并点。 在本工作中，我们提供了从多倍简并点产生的外尔点数量的上限。 为了计算这个上限，我们描述了复矩阵空间中的几何简并流形。 我们计算了对应于多倍简并的某些奇异点处的流形的重数，以及相对于该流形的全纯映射芽的重数。 我们的工作详细涵盖了物理和数学方面，并试图连接这两个学科和社区。 为了自洽性，我们综述了量子系统和凝聚态物理中的多倍简并示例，以及我们用来确定上限的局部代数几何的已建立工具。 显示英文摘要

摘要： Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally lead to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic two-fold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail, and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multi-fold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.