Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
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DOI：
10.1103/physrevb.96.155105
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发表时间：
2017-10-04
影响因子：
3.7
通讯作者：
Sigrist, Manfred
中科院分区：
文献类型：
--
作者：
Bzdusek, Tomas;Sigrist, Manfred
三维空间中的外尔点的特征在于一个Z值的电荷-陈数-这使得它们对大范围的扰动是稳定的。一组外尔点只有在它们的净电荷为零时才能相互湮灭，这一性质我们称为鲁棒性。虽然节点环在这个意义上通常不是鲁棒的，但最近使用同伦论证表明，在AI对称类的中心对称扩展中，它们仍然会产生类似于陈数的Z（2）电荷。携带该Z（2）电荷的非平凡值的节环是鲁棒的，即，它们只能通过成对的湮灭而不是它们自己被隔开。由于这是一个额外的电荷，独立于沿着能带简并流动的Berry π相，因此这种节环实际上是双重带电的。在这份手稿中，我们推广同伦讨论的中心对称扩展的所有Atland-Zirnbauer类。我们开发了一个量身定制的数学框架，称为A-+I分类，并表明，在三个空间维度，这样的强大和多电荷节点出现在四个这样的中心对称扩展，即A-+I类CI和AI导致双电荷节点线，而D和BDI支持双电荷节点表面。我们注意到，没有进一步的晶体对称性，除了空间反转是必要的，他们的稳定性。我们提供了相应的拓扑电荷的描述，并开发简单的紧束缚模型的各种半金属和超导相，表现出这些节点。我们还表明如何强大的和多电荷节点的概念推广到其他空间维度。
Weyl points in three spatial dimensions are characterized by a Z-valued charge-the Chern number-which makes them stable against a wide range of perturbations. A set of Weyl points can mutually annihilate only if their net charge vanishes, a property we refer to as robustness. While nodal loops are usually not robust in this sense, it has recently been shown using homotopy arguments that in the centrosymmetric extension of the AI symmetry class they nevertheless develop a Z(2) charge analogous to the Chern number. Nodal loops carrying a nontrivial value of this Z(2) charge are robust, i.e., they can be gapped out only by a pairwise annihilation and not on their own. As this is an additional charge independent of the Berry pi-phase flowing along the band degeneracy, such nodal loops are, in fact, doubly charged. In this manuscript, we generalize the homotopy discussion to the centrosymmetric extensions of all Atland-Zirnbauer classes. We develop a tailored mathematical framework dubbed the A-+I classification and show that in three spatial dimensions such robust and multiply charged nodes appear in four of such centrosymmetric extensions, namely, A-+I classes CI and AI lead to doubly charged nodal lines, while D and BDI support doubly charged nodal surfaces. We remark that no further crystalline symmetries apart from the spatial inversion are necessary for their stability. We provide a description of the corresponding topological charges, and develop simple tight-binding models of various semimetallic and superconducting phases that exhibit these nodes. We also indicate how the concept of robust and multiply charged nodes generalizes to other spatial dimensions.