【作者】 唐峰；

【导师】 万贤纲；

【作者基本信息】 南京大学 ， 凝聚态物理， 2019， 博士

【摘要】 凝聚态物理学家在研究二维电子气的量子霍尔效应时引入拓扑这一数学家用来描述几何形状不受连续变化改变的性质的概念。人们发现在量子霍尔效应中描述系统的拓扑不变量是第一类Chern数(可取任意整数),刻画了鲁棒的霍尔电导。而拓扑物理兴起于2005年人们提出的时间反演对称保护的量子自旋霍尔效应(即2维拓扑绝缘体),其拓扑不变量可取0或1,即冗2拓扑不变量。此后人们推广得到3维拓扑绝缘体。此外考虑到空间对称人们也提出了多种拓扑晶体绝缘体相:镜面Chern绝缘体、实现Hourglass型表面态的拓扑晶体绝缘体、高阶拓扑绝缘体等等。除了绝缘体相,人们也发现金属相同样可以具有拓扑性质,相关的拓扑材料包括:Weyl半金属、Dirac半金属、Nodal-line半金属、Hopf-link半金属等等。其中Weyl半金属不需要任何对称性保护,而其余半金属中的能带交点需要一些晶格对称的保护。对称性不仅保护了某种拓扑相,它也可以用来快速计算其他对称操作所保护的拓扑相的拓扑不变量。其中一个著名的例子是Fu和Kane提出的中心反演对称系统的拓扑绝缘体的Fu-Kane宇称判据,基于此人们可以对于具有中心反演对称的体系利用中心反演对称本征值去很快计算出刻画拓扑绝缘体与否的Z2拓扑不变量。此外,和Fu-Kane理论类似,人们也可以根据一些空间群操作的本征值去计算不同拓扑不变量,这种计算前提是系统是绝缘体或者布里渊区(BZ)内存在连续有限的直接带隙。最近通过分析凝聚态体系实空间与倒空间的匹配性人们提出了对称性指标理论,从而对整个230个空间群可能的拓扑行为进行了分类。基于对称性指标理论,我们发展出来通过构建原子绝缘体基组,以高效判断拓扑材料的全新理论方案。我们的理论方案不仅可以判定拓扑的绝缘相,也能判断出具有能带交点的拓扑金属相。我们根据该方案做了系统的晶体库拓扑材料搜索,因为不需要通过预设任何拓扑相,而且只需要看展开系数,所以能够高效地找寻出晶体库中所打(可通过对称性判断的)拓扑材料。本论文主要研究利用对称性指标进行高效拓扑材料的判断和搜索,具体论文内容包括:第一,在简介了对称性指标理论后,我们着重介绍我们发展的基于原子绝缘体基组的拓扑材料判断方案。第二,该理论方案由于其高效性和全面性,适合大规模搜索,因此我们对晶体库中所有合适的材料(这里“合适”指没有原子位置分数占据,不含磁性元素和放射性元素)进行了拓扑分类,得到数千种拓扑材料,这说明拓扑材料在自然界中并不罕见。我们构造的拓扑材料数据库为寻找有实用价值的拓扑体系提供了基础。第三,在强Z4对称性指标群的体系中,通过进一步的分析,我们发现,11号空间群的MoTe2可以实现螺旋对称保护的1维铰链态,12号空间群的BfiBr可以实现旋转反常,这些材料预言也被后续的研究验证。第四,对于具有强Z2对称性指标群的体系,它们没有中心反演对称,而我们的方案很快发现,216号空间群的AgNaO为强拓扑绝缘体。第五,对于具有强Z12、强Z8的大指数对称性指标群的空间群,我们不但发现,很多被我们熟知的材料,比如超导体MgB2,石墨等都是拓扑晶体绝缘体,我们也发现了许多属于这些空间群的其他拓扑晶体绝缘体,并选取了石墨,Pt3Ge,PbPt3,Au4Ti和Ti2Sn进行了具体分析。综上所述,我们发展了一个全新的拓扑材料判断算法,它和传统的方案有很大的区别,不仅在计算速度上有优势,也不会出现传统方案漏判的情况。我们的算法,可以很容易推广到二维材料,磁性材料甚至玻色子体系,有望推动拓扑材料早日得到实际应用。更多还原

【Abstract】 As a mathematical concept topology is used for describing the invariant properties of an object undergoing continuous deformation.In the context of quantum Hall effect of 2D free electron gas under a strong magnetic field,condensed matter physicists introduced topology to physics by relating the robust Hall conductivity with the first-class Chern number(which can take any integer).Especially,topological physics has undergone flourishing developments since 2005 when people proposed time-reversal symmetry protected quantum spin Hall effect(or synonymously,2D topological insulator),whose topological index can take 0 or 1,i.e.a Z2 invariant.Soon after that,topological insulator was generalized to 3D.Considering fruitful spatial symmetries in crystals,various topological crystalline insulating phases were also proposed,such as mirror Chern insulator,Hourglass topological crystalline insulator,high-order topological insulator etc.Other than insulators,metals can also host nontrivial band topology,starting from Weyl semimetal,also including Dirac semimetal,nodal-line semimetal,Hopf-link semimetal etc.Weyl points in Weyl semimetal are not protected by any symmetry while the band crossings in other types of topological semimetals require protection of spatial symmetries.Symmetry not only can protect some topological phase,it can also give implication for some topological phase protected by other symmetry,Fu-Kane parity criterion for centrosymmetric topological insulator being a famous example:according to the symmetry information of inversion eigenvalues at inversion-invariant momenta,one can obtain the Z2 topological invariant(s)quickly.Similarly,one can also exploit other symmetry information to calculate some topological invariant quickly,applicable for insulators.Based on the mismatch between real and reciprocal spaces for all the 230 space groups,a Fu-Kane-like theory,called symmetry-indicator theory gives a complete classification of band topology for electrons based on the symmetry information.Based on the symmetry-indicator theory,we develop a new and efficient algorithm of discovering topological materials by first-principles calculations starting from the atomic insulator basis for each space group.This new algorithm need not pre-assume any targeted kind of topological phase and can uncover all the nontrivial topological materials from symmetry properties including topological(crystalline)insulator and semimetal in one sweep.We apply this algorithm to a database search with thousands of topological materials discovered.This thesis is mainly focused on using symmetry-indicator to diagnose band topology and search for topological material efficiently,including:1.After introducing the symmetry-indicator theory in brief,we demonstrate our algorithm of topological materials discovery using the atomic insulator basis as an anchor in detail2.Because of the high efficiency of our algorithm which could uncover all band topology indicated by symmetry,it is suitable for materials search in a large scale.We thus make a database search for topological materials(firstly we filter out those inappropriate materials,e.g.containing magnetic or radioactive elements,or nonstoichiometric),finding thousands of candidates,providing a fruitful flatform for further theoretical or experimental study in future.This result shows that the topological material are actually ubiquitous in nature.3.In systems with strong Z4symmetry-indicator group,through a further analysis,we find that MoTe2 crystallizing in space group 11,can realize screw-protected ID hinge states and BiBr within space group 12 can realize rotation anomaly.They are verified by later works.4.For systems with strong Z2 symmetry-indicator group,they don’t own an inversion center.Although Fu-Kane criterion is not applicable,our method,alternatively,predicts that AgNaO in space group 216,is a strong topological insulator.5.In systems with large-order strong symmetry indicator group including Z8 and Z12,we not only find that many well-known materials,such as superconductor MgB2,graphite etc.are topologically nontrivial discovered by our study.We also find many other topological crystalline insulators for this kind and choose graphite,Pt3Ge,PbPt3,Au4Ti and Ti2Sn to give a detailed analysis.All in all,this thesis provides a newly-developed topological materials discovery algorithm.Compared with conventional algorithm,it is not only highly-efficient,but can also uncover any possible band topology indicated by symmetry.Our algorithm can be generalized to 2D systems,magnetic materials,or even bosonic systems,and is expected to promote topological materials to be applied to devices soon.更多还原