动量空间全局拓扑属性导致的无能隙和有能隙晶体拓扑相

The topological band theory has been blossoming since topological insulators were discovered. Recently, more and more attention has been paid to gapless topological phases, including Weyl and Dirac semimetals, nodal line semimetals and nodal superconductors, and it has been developed into an extremely rich field by considering crystalline topological phases protected by both symmorphic and nonsymmorphic spatial symmetries of crystals. In this project we plan to study crystalline gapless and gapped topological phases which are intrinsically dependent on the global topology of the first Brillouin zone under crystal symmetries, time-reversal symmetry and charge-conjugate symmetry, for which a complete or comprehensive understanding is still lacking. It is noteworthy that most works, if not all, on gapless systems to date merely concern local topological configurations in the Brillouin zone. The project consists of five interrelated topics. 1) We shall establish the concept of global topological property for gapless systems including topological semimetals and nodal topological superconductor. 2) We plan to develop theoretical techniques in the framework of K theory to topologically classify the global band structures of semimetals and insulators under crystal symmetries, time-reversal and charge-conjugate symmetry, in particular based on the real equivariant K theory, or the KRG theory for brevity. 3) We will develop the concept of topological charges for spatial symmetry operators, which should be distinguished from the topological invariants for systems, and look for the connections to topological properties of the crystal system. While it is impossible to exhaust all crystal symmetries, we will identify most elementary symmetry classes and focus on experimentally relevant ones. 4) New fermions are expected to be identified in the course of this project, which not only are topological charged but represent spatial symmetries of crystals, for instance Majorana fermions with topological charge preserving spatial symmetries as well. Note that in high energy physics, commonly only one Fermi point with the Poincaré symmetry is considered, but in a crystal, there are generically a set of topological Fermi points preserving the crystal symmetries, or a single Fermi point located at a high-symmetry point representing the corresponding little group. 5) We will study physical properties related to quantum anomalies in quantum field theory. For intance, in condensed matter, the whole system is always anomaly free, which requires anomaly cancellation leading to nontrivial topological low-energy effective theories.

拓扑绝缘体的发现导致了拓扑能带理论的充分发展。近来人们越来越多的关注无能隙拓扑相，包括外尔和狄拉克半金属，狄拉克线状半金属和无能隙的拓扑超导体，并且由简单和非简单晶体对称性保护的晶体拓扑相成了研究热点。目前对于无能隙的拓扑相，绝大多数工作研究的都是布里渊区局域的拓扑性质，比如外尔半金属涉及的只是外尔点的奇异性，而对于布里渊区全局拓扑性质，以及由此导致的物理结果的理解仍然很不足。在这个项目中，我们将重点研究对称性限制下的布里渊区全局的拓扑属性，以及受保护的无能隙和有能隙的拓扑相。本项目包括以下内容： 一、对无能隙的晶体系统发展具全局拓扑的新物理概念；二、在K理论特别是KRG理论的框架下，发展刻画全局拓扑的理论工具；三、探讨空间对称性算符的拓扑和系统拓扑的关系； 四、寻找新的兼具空间对称性和拓扑属性的费米子（Majorana费米子）；五、与量子反常相关的物理性质，包括输运性质和低能有效理论。

项目旨在具有晶体对称性的系统中探索新奇的具有全局属性的拓扑性质。项目按照既定方针顺利进行，各项目标基本达成。实施过程中产生了系列创新成果，发表SCI文章8篇，包括：PRL文章4篇，其中2篇为第一标注；PRB文章3篇，其中1篇为第一标注；NSR文章1篇。..代表性成果包括：.基于KO理论系统发展了PT对称性下的实数拓扑能带理论，提出了超越传统“一对一”假设的“一对多”体边对应，预言了实现新奇实数拓扑相的材料体系。 [第一标注、共同通讯PRL 125, 126403 (2020)].首次给出了拓扑费米子朗道能带指标定理的普适性证明，证实了拓扑荷与手性朗道能带数的关系，为强磁场探测拓扑荷的实验手段提供了理论基础。[第一标注、一作且单独通讯PRL 126, 046401 (2021)]..人才培养方面，课题组已毕业博士生3名，均受到了项目的资助，并在项目的实施过程中受到了有效训练。.此外，项目的实施促进了学术交流。在探索项目提出理论的材料实现过程中，课题组与德国马普所Schnyder研究员、香港大学汪子丹教授、新加坡科技设计大学杨声远教授、北京航空航天大学胜献雷教授等，建立了良好的合作基础。

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