黄侯迪,常治文,刘鑫

• 1:北京工业大学理学部物理系理论物理研究所

摘要(Abstract)：

本专题旨在为感兴趣且学有余力的本科生和低年级研究生提供一条从大学物理到当前二维、三维拓扑新材料研究前沿的学习路径。专题包含两篇，本文是第二篇。基于第一篇回顾过的各种量子霍尔效应及其中的拓扑不变量，本篇介绍多种新材料，包括石墨烯、拓扑绝缘体和拓扑半金属。重点将放在晶格结构的紧束缚模型处理，以蜂窝六角晶格和方晶格为例推导出描述量子反常霍尔效应的两带模型，介绍能带结构与拓扑不变量(陈数、拓扑映射度等)的联系及其所导致的体-边对应关系(bulk-boundary correspondence)。其中自旋轨道耦合(spin-orbitcoupling)的写法、哈密顿量的傅里叶(Fourier)变换、能带结构的求解、拓扑不变量及边缘态的计算都有助于夯实初学者的理论基础。另外本篇还将简述不同种类的拓扑半金属的能带结构及其表面态。

关键词(KeyWords)： 石墨烯;拓扑绝缘体;两带模型;拓扑半金属

Abstract:

Keywords:

基金项目(Foundation): 北京市自然科学基金重点项目(Z180007);; 国家自然科学基金(11572005)

作者(Author): 黄侯迪,常治文,刘鑫

参考文献(References)：

• [1] WALLANCE P R.The Band Theory of Graphite[J].Physical Review,1947,71:622-634.
• [2] NOVOSELOVA K S,GEIMS A K,MOROZOV S V,et al.Electric field effect in atomically thin carbon films[J].Science,2004,306:666-669.
• [3] CASTRO N A H,GUINEA F,PERES N M R,et al.The electronic properties of graphene[J].Review of Modern Physics,2009,81:109-162.
• [4] 徐小志，余佳晨，张智宏，等.石墨烯打开带隙研究进展[J].科学通报，2017,62:2220-2232.XU X Z,YU J C,ZHANG Z H,et al.Bandgap opening in graphene[J].Chinese Science Bulletin,2017,62:2220-2232.(in Chinese)
• [5] LIU X,ZHANG R.Topological vortices in chiral gauge theory of graphene[J].Annals of Physics,2010,325:384-391.
• [6] HALDANE F D M.Model for a quantum Hall effect without Landau levels:Condensed matter realization of the “parity anomaly”[J].Physical Review Letters,1988,61:2015-2018.
• [7] BERNEVIG B A,HUGES T L.Topological insulators and topological superconductors.Princeton University Press,2013.
• [8] YAKOVENKO V M.Chern-Simons terms and n field in Haldane's model for the quantum Hall effect without landau levels[J].Physical Review Letters,1990,65:251-254.
• [9] DU Y,ZHANG J,LI Z,et al.Quasi-freestanding epitaxial silicene on Ag (111) by oxygen intercalation[J].Science Advances,2016,2:1600067.
• [10] LI Z,CHEN L,WANG L,et al.Realization of flat band with possible nontrivial topology in electronic Kagome lattice[J].Science Advances,2018,4:4511.
• [11] KANE C L,MELE E J.Z2 topological order and the quantum spin Hall effect[J].Physical Review Letters,2005,95:146802.
• [12] KANE C L,MELE E J.Quantum spin Hall effect in graphene[J].Physical Review Letters,2005,95:226801.
• [13] BERNEVIG B A,HIGES T A,ZHANG S C.Quantum spin Hall effect and topological phase transition in HgTe quantum wells[J].Science,2006,314:1757-1761.
• [14] CHANG C Z,ZHANG J,FENG X,et al.Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator[J].Science,2013,340:167-170.
• [15] QI X L,HUGES T L,ZHANG S C.Topological field theory of time-reversal invariant insulators[J].Physical Review B,2008,78:195424.
• [16] 常治文，刘鑫.拓扑绝缘体中两带模型的规范理论[J].物理与工程，2020,30:64-71.CHANG Z W,LIU X.A gauge theory in two-band model of topological insulator[J].Physics and Engineering,2020,30:64-71.(in Chinese)
• [17] CHANG Z W,HAO W C,LIU X.A gauge theory for two-band model of Chern insulators and induced topological defects[J].Communications in Theoretical Physics,2022,74,015701.
• [18] CHANF C Z,ZHAO W,KIM D Y,et al.Zero-field dissipationless chiral edge transport and the nature of dissipation in the quantum anomalous Hall state[J].Physical Review Letters,2015,115:057206.
• [19] ZHAO Y-F,ZHANG R,MEI R,et al.Tuning the Chern number in quantum anomalous Hall insulators[J].Nature,2020,588:419-423.
• [20] GE J,LIU Y,LI J,et al.High-Chern-number and high-temperature quantum Hall effect without Landau levels[J].National Science Review,2020,7:1280-1287.
• [21] FU L,KANE C L,MELE E J.Topological insulators in three dimensions[J].Physical Review Letters,2007,98:106803.
• [22] MOORE J E,BALENTS L.Topological invariants of time-reversal-invariant band structures[J].Physical Review B,2007,75:121306(R).
• [23] ROY R.Topological phases and quantum spin Hall effect in three dimensions[J].Physical Review B,2009,79:195322.
• [24] HEIEH D,QIAN D,WRAY L,et al.A topological Dirac insulator in a quantum spin Hall phase[J].Nature,2008,452:970-974.
• [25] HASAN M Z,LIN H,BANSIL A.Spin-textures,Berry's phase and quasiparticle interference in Bi2Te3:A topological insulator with warped surface states[J].Physics,2009,2:108.
• [26] XIA Y,QIAN D,HSIEH D,et al.Observation of a large-gap topological-insulator class with a single Dirac cone on the surface[J].Nature Physics,2009,2:398-402.
• [27] ZHANG H,LIU C X,QI X L,et al.Topological insulators in Bi2Se3,Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface[J].Nature Physics,2009,5,438-442.
• [28] 金国钧.凝聚体中的拓扑量子态[J].物理学进展，2019,39:187-241.JIN G J.Topological quantum states in condensed matter[J].Progress in Physics,2019,39:187-241.(in Chinese)
• [29] 段一士.量子场论[M].北京：高等教育出版社，2015.
• [30] BERRY M V.Quantal phase factors accompanying adiabatic changes[J].Proceedings of the Royal Society A,1984,392:45-57.
• [31] WAN X,TURNER A M,VISHWANATH A,et al.Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[J].Physical Review B,2011,83:205101.
• [32] XU G,WENG H,WANG Z,et al.Chern semimetal and quantized anomalous Hall effect in HgCr2Se4[J].Physical Review Letters,2011,107:186806.
• [33] YANG K Y,LU Y M,RAN Y.Quantum Hall effects in a Weyl semimetal:Possible application in pyrochlore iridates[J].2011,84:075129.
• [34] BURKOV A A,HOOK M D,BALENTS L.Topological nodal semimetals[J].Physical Review B,2011,84:235126.
• [35] FENG B,FUT,KASAMATSUL S.Experimental realization of two-dimensional Dirac nodal line fermions in monolayer Cu2Si[J].Nature Communications,2017,8:1007.
• [36] LIU L L,QANG C Z,LI J X,et al.Two-dimensional topological semimetal states in monolayer Cu2Ge,Fe2Ge,and Fe2Sn[J].Physical Review B,2020,101:165403.
• [37] YAN Z,BI R,SHEN H,et al.Nodal-link semimetals[J].Physical Review B,2017,96:041103(R).
• (1)原胞只需从图1中红色和蓝色三角形中任选其一。不失一般性,取红色三角形为原胞,s1和s2为基矢,可生成完整晶格。
• (2)在非相对论量子力学中测不准原理只限制到“不能同时测量电子的位置和动量”;但在量子力学中甚至完全不能测量电子的位置,任何精度都不行。因为一旦测量(某种意义设置了一个势垒)就将从真空中制造出若干电子-正电子对。这会导致无法分辨何者为电子,何者为新产生的电子。
• (3)“抖动”的产生是由于相对论性粒子的波函数不可能被限制在一个小于其量子电动力学特征尺度即康普顿波长的空间里。为解释这一点,狄拉克引入了负能态的概念(负能态后来被解释为反粒子)。在相对论速度下运动的电子能够孕育出它自己的反粒子,而它和自己反粒子之间的相互作用使得电子的运动路径发生抖动。
• (4)这里需要解释:物理规律反映现实世界,用不同数学工具的表达之下呈现不同的样貌。用分析工具处理,就是微分方程;用代数工具,就是某种代数关系(比如某种对易关系)。在代数语境下求某种代数的表示,就相当于在分析的语境下求解微分方程。这就不难理解,对γ-矩阵取不同的表示,意味着讨论不同类型解的性质。
• (5)一般物理上取时空为赝欧的闵可夫斯基(Minkowski)空间,那么狄拉克方程的符号差(signature)是■,群对称性是■。为看清本质,可进一步简化问题,考虑■。从群论角度看■有直和分解■,其中■是两个■子空间。这一事实和外尔的分解思想相容。