拓扑体系的表征及物性研究

近四十年来，拓扑能带理论得到了充分的发展并扩展到各种体系，如拓扑绝缘体、超导体和Weyl 半金属。在这些体系中，由于非平庸的体拓扑，产生了各种受拓扑保护的边界态。对于这些体系的电子与拓扑性质的研究，仍然是目前科学研究的前沿领域。本文研究了两种拓扑体系的电子与拓扑性质，并使用不同的方案对它们进行表征，主要内容如下：

我们通过构造不同的构型来进一步扩展二维 Su-Schrieffer-Heeger 模型，包括所有 4 格点原胞的可能构型，以及具有更多格点原胞的复杂构型。我们对这些构型的电子和拓扑结构进行了详细的表征、计算和分析，发现了一 些丰富而新颖的性质，如拓扑保护的边界态，不同形状的金链，金属系统中的体边分离和拓扑边界态的平带特征。我们的研究可以推广到更多的构型和更高的维度，为进一步的理论和实验探索提供了基础。

我们使用动力学分类方案来表征外尔半金属的拓扑性质。通过淬灭操作，可以使用低维拓扑不变量来表征外尔半金属。更进一步，我们进行连续的淬灭与降维操作，可以在最低零维的表面(点)上表征外尔半金属的拓扑属性。为了方便实验上的操作，我们放宽了淬灭动力学的条件，使用两种不同的方案，表征和分析了外尔半金属的拓扑性质。我们的研究拓展了动力学分类方法的适用范围，证明了动力学分类方法的合理性与便利性。

[1] KLITZING K V, DORDA G, PEPPER M. New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance[J]. Phys. Rev. Lett., 1980, 45:494-497.
[2] KÖNIG M, WIEDMANN S, BRÜNE C, et al. Quantum Spin Hall Insulator State in HgTe Quantum Wells[J]. Science, 2007, 318:766-770.
[3] CHANG C Z, ZHANG J, FENG X, et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator[J]. 2013, 340:167-170.
[4] BAYM G, PETHICK C J. Ground-State Properties of Magnetically Trapped Bose-Condensed Rubidium Gas[J]. Phys. Rev. Lett, 1996, 76:6-9.
[5] OH S. The Complete Quantum Hall Trio[J].Science, 2013, 340:153-154.
[6] FULGA I C, VAN HECK B, EDGE J M, et al. Statistical topological insulators[J]. Phys. Rev. B, 2014, 89:155424.
[7] WANG Z, QI X L, ZHANG S C. Topological Order Parameters for Interacting Topological Insulators[J]. Phys. Rev. Lett., 2010, 105:256803.
[8] HASAN M Z, KANE C L. Colloquium: Topological insulators[J]. Rev. Mod. Phys., 2010, 82:3045-3067.
[9] YI C R, ZHANG L, ZHANG L, et al. Observing Topological Charges and Dynamical Bulk-Surface Correspondence with Ultracold Atoms[Jl. Phys. Rev. Lett., 2019, 123:190603.
[10] LIN S, JIN L, SONG Z. Symmetry protected topological phases characterized by isolated exceptional points[J]. Phys. Rev. B, 2019, 99:165148.
[11] COOPER N R, DALIBARD J, SPIELMAN I B. Topological bands for ultracold atoms[J]. Rev. Mod. Phys, 2019, 91:015005.
[12] SU W P, SCHRIEFFER J R, HEEGER A J. Solitons in Polyacetylene[J].Phys. Rev. Lett, 1979, 42:1698-1701.
[13] KITAEV A Y. Unpaired Majorana fermions in quantum wires[J]. Physics-Uspekhi, 2001, 44(10S):131.
[14] HALDANE F D M. Model for a Quantum Hall Effect without Landan Levels: Condensed-Matter Realization of the “Parity Anomaly”[J]. Phys. Rev. Lett., 1988, 61:2015-2018.
[15] QI X L, WU Y S, ZHANG S C. Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors[J]. Phys. Rev. B, 2006, 74:085308.
[16] QI X L, ZHANG S C. Topological Insulators and Superconductors[J]. Rev. Mod. Phys., 2011, 83:1057–110.
[17] KING-SMITH RD, VANDERBILT D. Theory of Polarization of Crystalline Solids[J]. Phys. Rev. B, 1993, 47:1651–4
[18] SHEN S Q. Topological Insulators: Dirac Equation in Condensed Matter[M]. 2nd ed. Singapore: Springer. 2017.
[19] BERNEVIG B A, HUGHES T L. Topological Insulators and Topological Superconductors[M]. Princeton, New Jersey: Princeton University Press, 2013.
[20] LIU F, WAKABAYASHI K. Novel Topological Phase with a Zero berry Curvature[J]. Phys. Rev. Lett., 2017, 118:076803.
[21] OBANA D, LIU F, WAKABAYASHI K. Topological edge states in the Su-Schrieffer-Heeger model[J]. Phys. Rev. B, 2019, 100:075437.
[22] LIU F, DENG HAI-YAO, WAKABAYASHI K. Topological photonic crystals with zero berry curvature[J]. Phys. Rev. B, 2018, 97:035442.
[23] ZHENG L Y, ACHILLEOS V, RICHOUX O, et al. Observation of Edge Waves in a Two-Dimensional Su-Schrieffer-Heeger Acoustic Network[J]. Phys. Rev. Appl., 2019, 12:034014.
[24] LIU S, GAO W, ZHANG Q, et al. Topologically Protected Edge State in Two-DimensionalSu–Schrieffer–Heeger Circuit[J]. Research, 2019, 5:8609875.
[25] BENALCAZAR W A, BERNEVIG B A, HUGHES T L. Quantized Electric Multipole Insulators[J]. Science, 2017, 357:61-6.
[26] BENALCAZAR W A, BERNEVIG B A, HUGHES T L. Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators[J]. Phys. Rev. B, 2017, 96:245115.
[27] VOLPEZ Y, LOSS D, KLINOVAJAL J. Second-Order Topological Superconductivity \mathrm{\pi}-Junction Rashba Layers[J]. Phys. Rev. Lett. 2019, 122:126402.
[28] SHENG X L, CHEN C, LIU H, et al. Two-Dimensional Second-Order Topological Insulator in Graphdiyne[J]. Phys. Rev. Lett., 2019, 123:256402.
[29] PENG Y, REFAEL G. Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries[J]. Phys. Rev. Lett., 2019, 123:016806.
[30] HUA C B, CHEN R, ZHOU B, et al. Higher-order Topological Insulator in a Dodecagonal Quasicrystal[J]. Phys. Rev. B, 2020, 102:241102.
[31] CHEN R, CHEN C Z, GAO J H, et al. Higher-Order Topological Insulators in Quasicrystals. Phys. Rev. Lett., 2020, 124:036803.
[32] LI C A, ZHANG S B, LI J, TRAUZETTE B. Higher-Order Fabry-Pérot Interferometer from Topological Hinge States. Phys. Rev. Lett., 2021, 127:026803.
[33] ZHANG R X, CPLE W S, WU X, et al. Higher-Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures[J]. Phys. Rev. Lett., 2019, 123:167001.
[34] ZHANG S B, TRAUZETTEL B. Detection of Second-Order Topological Superconductors by Josephson Junctions[J]. Phys. Rev. Res., 2020, 2:012018.
[35] WU Y J, HOU J, LI Y M, et al. In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an S -Wave Superconductor Heterostructure[J]. Phys. Rev. Lett., 2020, 124:227001.
[36] ZHANG S B, RUI W B, CALZONA A, et al. Topological and Holonomic Quantum Computation Based on Second-Order Topological Superconductors[J]. Phys. Rev. Res., 2020 2:043025.
[37] WANG Z, WIEDER B J, LI J, et al. Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe2 ( X=Mo,W )[J]. Phys. Rev. Lett., 2019 123:186401.
[38] WANG H X, LIN Z K, JIANG B, et al. Higher-Order Weyl Semimetals[J]. Phys. Rev. Lett., 2020, 125:146401.
[39] GHORASHI S A A, LI T, HUGHES T L. Higher-Order Weyl Semimetals[J]. Phys. Rev. Lett., 2020, 125:266804.
[40] MURAKAMI S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase[J]. J. Phys., 2007, 9:356.
[41] B G, HALÁSZ, BALENTS L. Time-reversal invariant realization of the Weyl semimetal phase[J]. Phys. Rev. B, 2012, 85:035103
[42] ZYUZIN A A, WU S, BURKOV A A. Weyl semimetal with broken time reversal and inversion symmetries[J]. Phys. Rev. B, 2012, 85:165110.
[43] LIU C X, YE P, QI X L. Chiral gauge field and axial anomaly in a Weyl semimeta[J]. Phys. Rev. B, 2013, 87:235306.
[44] WAN X, TURNER A M, VISHWANATH A, et al. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[J]. Phys. Rev. B, 2011, 83:205101.
[45] XU G, WENG H, WANG Z, et al. Chern Semimetal and the Quantized Anomalous Hall Effect in HgCr2Se4[J]. Phys. Rev. Lett., 2011, 107:186806.
[46] POTTER A C, KIMCHI I, VISHWANATH A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals[J]. Nat. Commun., 2014, 5:5161.
[47] PARAMESWARAN S A, GROVER T, ABANIN D A, et al. Probing the Chiral Anomaly with Nonlocal Transport in Three-Dimensional Topological Semimetals[J]. Phys. Rev. X, 2014, 4:031035.
[48] WENG H, FANG C, FANG Z, et al. Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[J]. Phys. Rev. X, 2015, 5:011029.
[49] Xu S Y, BELOPOLSKI I, ALIDOUST N, et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs[J]. Science, 2015, 349:613-617.
[50] ARNOLD F, SHEKHAR C, WU S C, et al. Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP[J]. Nat. Commun., 2016, 7:11615.
[51] SON D T, SPIVAK B Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals[J]. Phys. Rev. B, 2013, 88:104412.
[52] CIUDAD D. Massless yet real[J]. Nat. Mater, 2015, 14:863.
[53] VOLOVIK G, ZUBKOV M. Emergent Weyl spinors in multi-fermion systems[J]. Nuclear Phys. B, 2014,881:514.
[54] XU Y, ZHANG F, ZHANG C. Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids[J]. Phys. Rev. Lett., 2015, 115:265304.
[55] SOLUYANOV A A, GRESCH D, WANG Z, et al. Type-II Weyl semimetals[J]. Nature, 2015, 527:495-498.
[56] ZHANG D, SHI M J, ZHU T S, et al. Topological Axion States in the Magnetic Insulator MnBi2Te4 with the Quantized Magnetoelectric Effect[J]. Phys. Rev. Lett., 2019,122:206401.
[57] JIAN S K, JIANG Y F, YAO H. Emergent Spacetime Supersymmetry in 3D Weyl Semimetals and 2D Dirac Semimetals[J]. Phys. Rev. Lett., 2015, 114:237001.
[58] LIU D F, LIANG A J, LIU E K, et al. Magnetic Weyl semimetal phase in a Kagomé crystal[J]. Science, 2019, 365:1282–1285.
[59] MORALI N, BATABYAL R, NAG P K, et al. Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2[J]. Science, 2019, 365:1286–1291.
[60] Ma J Z, NIE S M, YI C J, et al. Spin fluctuation induced Weyl semimetal state in the paramagnetic phase of EuCd2As2[J]. Sci. Adv., 2019, 5:eaaw4718.
[61] JO N H, KUTHANAZHI B, WU Y, et al. Manipulating magnetism in the topological semimetal EuCd2As2[J]. Phys. Rev. B, 2020, 101:140402.
[62] HOFSTADTER, DOUGLAS R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields[J]. Phys. Rev. B, 1976, 14:2239–2249.
[63] AIDELSBURGER M, ATALA M, LOHSE M, et al. Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices[J]. Phys. Rev. Lett., 2013:111:185301.
[64] AIDELSBURGER M, LOHSE M, SCHWEIZER C, et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms[J]. Nature Phys., 2015, 11:162–166.
[65] HIROKAZU M, G A SIVILOGLOU, KENNEDY C J, et al. Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices[J]. Phys. Rev. Lett. 2013, 111:185301.
[66] F D M HALDANE. Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly” [J]. Phys. Rev. Lett., 1988, 61:2015-2018.
[67] JOTZU G, MESSER M, DESBUQUOIS R, et al. Experimental realization of the topological Haldane model with ultracold fermions[J]. Nature, 2014, 515:237–240.
[68] WU Z, ZHANG L, SUN W, et al. Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates[J]. Science, 2016, 354:83-88.
[69] WANG C, ZHANG P F, CHEN X, et al. Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics[J]. Phys. Rev. Lett. 2017, 118:185701.
[70] ZHANG L, ZHANG L, NIU S, et al. Dynamical classification of topological quantum phases[J]. Science Bulletin. 2018(63): 1385-1391.
[71] WANG K K, QIU X, XIAO L, et al. Observation of emergent momentum–time skyrmions in parity–time-symmetric non-unitary quench dynamics[J]. Nat Commun., 2019, 10:2293.
[72] HU H P, ZHAO E. Topological Invariants for Quantum Quench Dynamics from Unitary Evolution[J]. Phys. Rev. Lett., 2020, 124:160402.
[73] LEYKAM D, ANDREANOV A, FLACH S. Artificial flat band systems: from lattice models to experiments[J]. Adv. Phys.: X, 2018, 3:1473052.
[74] MAIMAITI W, ANDREANOV A, FLACH S. Flat-band generator in two dimensions. Phys. Rev. B, 2021, 103:165116.
[75] BISTRITZER R, MACDONALD A H. Moiré bands in twisted double-layer graphene[J]. Proc. Natl. Acad. Sci. USA, 2011, 108:12233.
[76] PAL H K, SPITZ S, KINDERMANN M. Emergent Geometric Frustration and Flat Band in Moiré Bilayer Graphene[J]. Phys. Rev. Lett., 2019, 123:186402.
[77] SHASHKIN A A, DOLGOPOLOV V T, CLARK J W, et al. Merging of Landau Levels in a Strongly Interacting Two-Dimensional Electron System in Silicon[J]. Phys. Rev. Lett., 2014, 112:186402.
[78] DANIELI C, ANDREANOV A, FLACH S. Many-body flat band localization[J]. Phys. Rev. B, 2020, 102:041116(R).
[79] TASAKI H. Hubbard model and the origin of ferromagnetism[J]. Eur. Phys. J. B, 64, 2008, 365.
[80] MONDAINI R, BATROUNI G G, GRÉMAUD B. Pairing and superconductivity in the flat band: Creutz lattice[J]. Phys. Rev. B, 2018, 98:155142.
[81] VOLOVIK G E. Graphite, Graphene, and the Flat Band Superconductivity. JETP Lett., 2018, 107:516.
[82] MIELKE A, Pair Formation of Hard Core Bosons in Flat Band Systems[J]. J. Stat. Phys., 2018, 171:679.