拓扑物态的量子模拟和物性研究

拓扑物态是指一类特殊的物质态，其性质（尤其是电学性质）不受物理条件扰 动的影响。拓扑物态的鲁棒性使其在拓扑量子计算和自旋电子学器件领域具有广 泛的潜在应用。因此，拓扑物态的测量与表征的研究已成为物理学领域的研究热 点。本论文旨在研究拓扑物态，主要包括具有长程纠缠的拓扑态（即拓扑序）和受 对称性保护的短程纠缠拓扑态。我们对这两种拓扑物态进行了深入研究，探讨了 如何区分不同拓扑序的方法，提出并实现了完整确定拓扑序的实验测量方案；同 时也研究了对称保护拓扑态受维度变化的影响，包括一维拓扑态向二维拓扑态的 推广；以及三维拓扑态在量子限制下的输运性质变化。

本文的第二、三章讨论了长程纠缠的拓扑序的量子模拟、分类区分和完备测 量的问题。在第二章中，我们考虑在环面上的弦网模型，其中包括阿贝尔和非阿 贝尔的拓扑序模型，用量子散射电路测量了模数据信息（𝑆 矩阵和 𝑇 矩阵），成功 区分了阿贝尔和非阿尔贝的拓扑序。由于模数据信息并不能完备地区分所有的拓 扑序，因此在第三章中，我们研究了完备描述拓扑序信息的测量方案。基于体边 对应关系和任意子凝聚性质，我们设计了测量 𝐹 矩阵和 𝑅 矩阵的测量方案，并通 过核磁共振系统模拟了 toric code 模型且成功地实现了两个矩阵的测量，并证明了 这两个矩阵可以完备地描述拓扑序。

本文的第四、五章研究了维度变化对于对称保护拓扑态的影响。在第四章中， 我们对最简单的拓扑模型 Su-Schrieffer-Heeger（SSH）模型进行了维度的扩展，详 细地计算和分析了二维 SSH 模型的电子结构和拓扑性质，得到了拓扑边缘态的平 带特征。通过分析这些边缘态的空间分布，我们解释了边缘平带的起源，并表明 键合方块在平带的形成中起着至关重要的作用。在第五章中，我们通过对三维拓 扑节点环超导体施加量子维度限制，研究了拓扑节点环超导体的多个马约拉纳表 面态，探讨了（1）金属/势垒/拓扑节点环超导体结和（2）拓扑节点环超导体上接 两个薄膜这两种构型下马约拉纳表面态辅助的安德烈夫反射的特性。这两种构型 分别可以测量动量空间和实空间中的多个马约拉纳表面态分布。利用垂直于结方 向的马约拉纳表面态的非局域性，这种空间移动的安德烈夫反射以及干涉图案可 能提供另一种途径来确认量子限制系统中的马约拉纳束缚态。

在本论文中，我们取得了以下研究成果。首先，我们针对具有长程纠缠的拓扑 序进行了量子模拟、分类区分和完备测量，特别是编织操作相关的 𝑅 矩阵的测量，为拓扑序在拓扑量子计算中的应用奠定了基础。另一方面，我们从对称性保护拓 扑态的性质研究入手，通过改变维度来观察拓扑物态的物性变化。发现了拓扑边 缘态的平带特征以及多个马约拉纳表面态的存在，这些新奇的拓扑物态性质为自 旋电子学和拓扑量子计算的应用提供了新的可能。我们相信，本论文的研究成果 为进一步研究拓扑物态的理论研究和实验实现产生了积极的影响。

[1] FREEDMAN M, KITAEV A, LARSEN M, et al. Topological quantum computation[J]. J. Am. Math. Soc., 2003, 40(1): 31-38.
[2] KITAEV A. Fault-tolerant Quantum Computation by Anyons[J]. Ann. Phys., 2003, 303(1): 2 - 30.
[3] NAYAK C, SIMON S H, STERN A, et al. Non-abelian Anyons and Topological Quantum Computation[J]. Rev. Mod. Phys., 2008, 80: 1083-1159.
[4] QI X L, HUGHES T L, ZHANG S C. Topological field theory of time-reversal invariant insulators[J]. Phys. Rev. B, 2008, 78: 195424.
[5] CHEN Y L, ANALYTIS J G, CHU J H, et al. Experimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3[J]. Science, 2009, 325(5937): 178-181.
[6] HASAN M Z, KANE C L. Colloquium: Topological insulators[J]. Rev. Mod. Phys., 2010, 82: 3045-3067.
[7] QI X L, ZHANG S C. Topological insulators and superconductors[J]. Rev. Mod. Phys., 2011, 83: 1057-1110.
[8] WEN X G. Colloquium: Zoo of quantum-topological phases of matter[J]. Rev. Mod. Phys., 2017, 89: 041004.
[9] TAO R, WU Y S. Gauge Invariance and Fractional Quantum Hall Effect[J]. Phys. Rev. B, 1984, 30: 1097-1098.
[10] KALMEYER V, LAUGHLIN R B. Equivalence of the resonating-valence-bond and fractional quantum Hall states[J]. Phys. Rev. Lett., 1987, 59: 2095-2098.
[11] BLOK B, WEN X G. Effective Theories of the Fractional Quantum Hall Effect at Generic Filling Fractions[J]. Phys. Rev. B, 1990, 42: 8133-8144.
[12] READ N. Excitation Structure of the Hierarchy Scheme in the Fractional Quantum Hall Effect [J]. Phys. Rev. Lett., 1990, 65: 1502-1505.
[13] TSUI D C, STORMER H L, GOSSARD A C. Two-Dimensional Magnetotransport in the Extreme Quantum Limit[J]. Phys. Rev. Lett., 1982, 48: 1559-1562.
[14] WITTEN E. Quantum field theory and the Jones polynomial[J]. Commun. Math. Phys., 1989, 121(3): 351-399.
[15] WEN X G. Vacuum degeneracy of chiral spin states in compactified space[J]. Phys. Rev. B, 1989, 40(10): 7387.
[16] 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(6): 494.
[17] LAUGHLIN R B. Quantized Hall conductivity in two dimensions[J]. Phys. Rev. B, 1981, 23 (10): 5632.
[18] HALPERIN B I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential[J]. Phys. Rev. B, 1982, 25(4): 2185.
[19] QI X L, WU Y S, ZHANG S C. Topological quantization of the spin Hall effect in twodimensional paramagnetic semiconductors[J]. Phys. Rev. B, 2006, 74(8): 085308.
[20] BERNEVIG B A, HUGHES T L, ZHANG S C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells[J]. Science, 2006, 314(5806): 1757-1761.
[21] BERNEVIG B A, ZHANG S C. Quantum spin Hall effect[J]. Phys. Rev. Lett., 2006, 96(10): 106802.
[22] 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 Phys., 2009, 5(6): 438-442.
[23] WEN X G. Topological orders in rigid states[J]. Int. J. Mod. Phys. B, 1990, 4(02): 239-271.
[24] WEN X G, NIU Q. Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces[J]. Phys. Rev. B, 1990, 41: 9377-9396.
[25] CHEN X, GU Z C, WEN X G. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order[J]. Phys. Rev. B, 2010, 82: 155138.
[26] LEVIN M, WEN X G. Colloquium: Photons and electrons as emergent phenomena[J]. Rev. Mod. Phys., 2005, 77: 871-879.
[27] READ N, SACHDEV S. Large-N expansion for frustrated quantum antiferromagnets[J]. Phys. Rev. Lett., 1991, 66: 1773-1776.
[28] WEN X G. Mean-field theory of spin-liquid states with finite energy gap and topological orders [J]. Phys. Rev. B, 1991, 44: 2664-2672.
[29] LAUGHLIN R B. Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations[J]. Phys. Rev. Lett., 1983, 50: 1395-1398.
[30] MOORE J E. Not trivial to realize[J]. Nature Phys., 2015, 11(11): 897-898.
[31] LEVIN M A, WEN X G. String-net condensation: A physical mechanism for topological phases [J]. Phys. Rev. B, 2005, 71: 045110.
[32] DENNIS E, KITAEV A, LANDAHL A, et al. Topological quantum memory[J]. J. Math. Phys., 2002, 43(9): 4452-4505.
[33] WEN X G. Gapless boundary excitations in the quantum Hall states and in the chiral spin states [J]. Phys. Rev. B, 1991, 43: 11025-11036.
[34] KANE C L, MELE E J. Quantum Spin Hall Effect in Graphene[J]. Phys. Rev. Lett., 2005, 95: 226801.
[35] MURAKAMI S, NAGAOSA N, ZHANG S C. Spin-Hall Insulator[J]. Phys. Rev. Lett., 2004, 93: 156804.
[36] THOULESS D J, KOHMOTO M, NIGHTINGALE M P, et al. Quantized Hall Conductance in a Two-Dimensional Periodic Potential[J]. Phys. Rev. Lett., 1982, 49: 405-408.
[37] WILCZEK F. Quantum Mechanics of Fractional-Spin Particles[J]. Phys. Rev. Lett., 1982, 49: 957-959.
[38] MOORE G, READ N. Nonabelions in the fractional quantum Hall effect[J]. Nucl. Phys. B, 1991, 360(2-3): 362-396.
[39] KITAEV A. Anyons in an Exactly Solved Model and Beyond[J]. Ann. Phys., 2006, 321(1): 2 - 111.
[40] LEVIN M, WEN X G. Detecting Topological Order in a Ground State Wave Function[J]. Phys. Rev. Lett., 2006, 96: 110405.
[41] KITAEV A, PRESKILL J. Topological Entanglement Entropy[J]. Phys. Rev. Lett., 2006, 96: 110404.
[42] KITAEV A, KONG L. Models for Gapped Boundaries and Domain Walls[J]. Commun. Math. Phys., 2012, 313(2): 351-373.
[43] KONG L. Anyon Condensation and Tensor Categories[J]. Nucl. Phys. B, 2014, 886: 436 - 482.
[44] KONG L, ZHENG H. Categories of quantum liquids I[J]. J. High Energy Phys., 2022, 2022 (8): 1-44.
[45] KOHMOTO M. Topological invariant and the quantization of the Hall conductance[J]. Annals of Physics, 1985, 160(2): 343-354.
[46] KONIG M, WIEDMANN S, BRUNE C, et al. Quantum spin Hall insulator state in HgTe quantum wells[J]. Science, 2007, 318(5851): 766-770.
[47] WENG H, FANG C, FANG Z, et al. Weyl semimetal phase in noncentrosymmetric transitionmetal monophosphides[J]. Phys. Rev. X, 2015, 5(1): 011029.
[48] SOLUYANOV A A, GRESCH D, WANG Z, et al. Type-ii weyl semimetals[J]. Nature, 2015, 527(7579): 495-498.
[49] KITAEV A Y. Unpaired Majorana fermions in quantum wires[J]. Physics-uspekhi, 2001, 44 (10S): 131.
[50] HALDANE F D M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the” parity anomaly”[J]. Phys. Rev. Lett., 1988, 61(18): 2015.
[51] FU L, KANE C L, MELE E J. Topological insulators in three dimensions[J]. Phys. Rev. Lett., 2007, 98(10): 106803.
[52] HSIEH D, XIA Y, WRAY L, et al. Observation of unconventional quantum spin textures in topological insulators[J]. Science, 2009, 323(5916): 919-922.
[53] BURKOV A, HOOK M, BALENTS L. Topological nodal semimetals[J]. Phys. Rev. B, 2011, 84(23): 235126.
[54] WANG Z, SUN Y, CHEN X Q, et al. Dirac semimetal and topological phase transitions in A 3 Bi (A= Na, K, Rb)[J]. Phys. Rev. B, 2012, 85(19): 195320.
[55] LV B, WENG H, FU B, et al. Experimental discovery of Weyl semimetal TaAs[J]. Phys. Rev. X, 2015, 5(3): 031013.
[56] ZHU Z, WINKLER G W, WU Q, et al. Triple point topological metals[J]. Phys. Rev. X, 2016, 6(3): 031003.
[57] SCHINDLER F, COOK A M, VERGNIORY M G, et al. Higher-order topological insulators [J]. Sci. Adv., 2018, 4(6): eaat0346.
[58] READ N, GREEN D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect[J]. Phys. Rev. B, 2000, 61: 10267-10297.
[59] FU L, KANE C L. Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator[J]. Phys. Rev. Lett., 2008, 100: 096407.
[60] CHUNG S B, QI X L, MACIEJKO J, et al. Conductance and noise signatures of Majorana backscattering[J]. Phys. Rev. B, 2011, 83(10): 100512.
[61] MOURIK V, ZUO K, FROLOV S M, et al. Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[J]. Science, 2012, 336(6084): 1003-1007.
[62] GE J F, LIU Z L, LIU C, et al. Superconductivity above 100 K in single-layer FeSe films on doped SrTiO 3[J]. Nature Mater., 2015, 14(3): 285-289.
[63] LIAO M, ZANG Y, GUAN Z, et al. Superconductivity in few-layer stanene[J]. Nature Phys., 2018, 14(4): 344-348.
[64] ZHANG H, LIU C X, GAZIBEGOVIC S, et al. Quantized majorana conductance[J]. Nature (London), 2018, 556(7699): 74-79.
[65] ZHANG H, LIU C X, GAZIBEGOVIC S, et al. Retraction note: Quantized majorana conductance[J]. Nature (London), 2021, 591(7851): E30-E30.
[66] RIEFFEL E G, POLAK W H. Quantum computing: A gentle introduction[M]. MIT Press, 2011.
[67] DIVINCENZO D P. The physical implementation of quantum computation[J]. Fortschritte der Phys.: Rep. Prog. Phys., 2000, 48(9-11): 771-783.
[68] NIELSEN M A, CHUANG I. Quantum computation and quantum information[M]. American Association of Physics Teachers, 2002.
[69] KAYE P, LAFLAMME R, MOSCA M. An introduction to quantum computing[M]. OUP Oxford, 2006.
[70] BULUTA I, NORI F. Quantum simulators[J]. Science, 2009, 326(5949): 108-111.
[71] GEORGESCU I M, ASHHAB S, NORI F. Quantum simulation[J]. Rev. Mod. Phys., 2014, 86: 153-185.
[72] LLOYD S. Universal quantum simulators[J]. Science, 1996, 273(5278): 1073-1078.
[73] WIESNER S. Simulations of many-body quantum systems by a quantum computer[A]. 1996.
[74] ABRAMS D S, LLOYD S. Simulation of many-body Fermi systems on a universal quantum computer[J]. Phys. Rev. Lett., 1997, 79(13): 2586.
[75] LIDAR D A, BIHAM O. Simulating Ising spin glasses on a quantum computer[J]. Phys. Rev. E, 1997, 56(3): 3661.
[76] ZALKA C. Simulating quantum systems on a quantum computer[J]. Proc. Math. Phys. Eng.es, 1998, 454(1969): 313-322.
[77] ZALKA C. Efficient simulation of quantum systems by quantum computers[J]. Fortschritte der Phys.: Rep. Prog. Phys., 1998, 46(6-8): 877-879.
[78] TERHAL B M, DIVINCENZO D P. Problem of equilibration and the computation of correlationfunctions on a quantum computer[J]. Phys. Rev. A, 2000, 61(2): 022301.
[79] MARZUOLI A, RASETTI M. Spin network quantum simulator[J]. Phys. Lett. A, 2002, 306 (2-3): 79-87.
[80] VERSTRAETE F, CIRAC J I, LATORRE J I. Quantum circuits for strongly correlated quantum systems[J]. Phys. Rev. A, 2009, 79(3): 032316.
[81] RAEISI S, WIEBE N, SANDERS B C. Quantum-circuit design for efficient simulations of many-body quantum dynamics[J]. New. J. Phys., 2012, 14(10): 103017.
[82] LI K, WAN Y, HUNG L Y, et al. Experimental Identification of Non-Abelian Topological Orders on a Quantum Simulator[J]. Phys. Rev. Lett., 2017, 118: 080502.
[83] ZHANG Z, LONG X, ZHAO X, et al. Identifying Abelian and non-Abelian topological orders in the string-net model using a quantum scattering circuit[J]. Phys. Rev. A, 2022, 105(3): L030402.
[84] HAI Y J, ZHANG Z, ZHENG H, et al. Uniquely identifying topological order based on boundary-bulk duality and anyon condensation[J]. Natl. Sci. Rev., 2023, 10(3): nwac264.
[85] DASKIN A, KAIS S. Decomposition of unitary matrices for finding quantum circuits: application to molecular Hamiltonians[J]. J. Chem. Phys., 2011, 134(14): 144112.
[86] ABRAMS D S, LLOYD S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors[J]. Phys. Rev. Lett., 1999, 83(24): 5162.
[87] ASPURU-GUZIK A, DUTOI A D, LOVE P J, et al. Simulated quantum computation of molecular energies[J]. Science, 2005, 309(5741): 1704-1707.
[88] MEYER D A. Quantum computing classical physics[J]. Philos. Trans. Royal Soc. A, 2002, 360 (1792): 395-405.
[89] SINHA S, RUSSER P. Quantum computing algorithm for electromagnetic field simulation[J]. Quantum Inf. Process., 2010, 9: 385-404.
[90] YUNG M H, NAGAJ D, WHITFIELD J D, et al. Simulation of classical thermal states on a quantum computer: A transfer-matrix approach[J]. Phys. Rev. A, 2010, 82(6): 060302.
[91] MANOUSAKIS E. A quantum-dot array as model for copper-oxide superconductors: A dedicated quantum simulator for the many-fermion problem[J]. J. Low Temp. Phys, 2002, 126: 1501-1513.
[92] FISCHER U R, SCHÜTZHOLD R. Quantum simulation of cosmic inflation in two-component Bose-Einstein condensates[J]. Phys. Rev. A, 2004, 70(6): 063615.
[93] PORRAS D, CIRAC J I. Effective quantum spin systems with trapped ions[J]. Phys. Rev. Lett., 2004, 92(20): 207901.
[94] ZAGOSKIN A M, SAVEL’EV S, NORI F. Modeling an adiabatic quantum computer via an exact map to a gas of particles[J]. Phys. Rev. Lett., 2007, 98(12): 120503.
[95] LAMATA L, LEÓN J, SCHÄTZ T, et al. Dirac equation and quantum relativistic effects in a single trapped ion[J]. Phys. Rev. Lett., 2007, 98(25): 253005.
[96] GERRITSMA R, KIRCHMAIR G, ZÄHRINGER F, et al. Quantum simulation of the Dirac equation[J]. Nature, 2010, 463(7277): 68-71.
[97] AGUADO M, BRENNEN G K, VERSTRAETE F, et al. Creation, Manipulation, and Detectionof Abelian and Non-Abelian Anyons in Optical Lattices[J]. Phys. Rev. Lett., 2008, 101: 260501.
[98] YOU J Q, SHI X F, HU X, et al. Quantum emulation of a spin system with topologicallyprotected ground states using superconducting quantum circuits[J]. Phys. Rev. B, 2010, 81:014505.
[99] GIORGI G L, PAGANELLI S, GALVE F. Ion-trap simulation of the quantum phase transitionin an exactly solvable model of spins coupled to bosons[J]. Phys. Rev. A, 2010, 81: 052118.
[100] BYRNES T, KIM N Y, KUSUDO K, et al. Quantum simulation of Fermi-Hubbard models insemiconductor quantum-dot arrays[J]. Phys. Rev. B, 2008, 78: 075320.
[101] PENG X, DU J, SUTER D. Quantum phase transition of ground-state entanglement in a Heisenberg spin chain simulated in an NMR quantum computer[J]. Phys. Rev. A, 2005, 71: 012307.
[102] HAN Y J, RAUSSENDORF R, DUAN L M. Scheme for demonstration of fractional statisticsof anyons in an exactly solvable model[J]. Phys. Rev. Lett., 2007, 98(15): 150404.
[103] LU C Y, GAO W B, GÜHNE O, et al. Demonstrating anyonic fractional statistics with a sixqubit quantum simulator[J]. Phys. Rev. Lett., 2009, 102(3): 030502.
[104] FENG G, LONG G, LAFLAMME R. Experimental simulation of anyonic fractional statisticswith an NMR quantum-information processor[J]. Phys. Rev. A, 2013, 88: 022305.
[105] PARK A J, MCKAY E, LU D, et al. Simulation of anyonic statistics and its topological pathindependence using a seven-qubit quantum simulator[J]. New J. Phys., 2016, 18(4): 043043.
[106] ZHONG Y P, XU D, WANG P, et al. Emulating Anyonic Fractional Statistical Behavior in aSuperconducting Quantum Circuit[J]. Phys. Rev. Lett., 2016, 117: 110501.
[107] DAI H N, YANG B, REINGRUBER A, et al. Four-body ring-exchange interactions and anyonicstatistics within a minimal toric-code Hamiltonian[J]. Nat. Phys., 2017, 13(12): 1195-1200.
[108] LUO Z, LI J, LI Z, et al. Experimentally Probing Topological Order and Its Breakdown throughModular Matrices[J]. Nat. Phys., 2018, 14(2): 160-165.
[109] SONG C, XU D, ZHANG P, et al. Demonstration of topological robustness of anyonic braidingstatistics with a superconducting quantum circuit[J]. Phys. Rev. Lett., 2018, 121(3): 030502.
[110] SONG C, XU D, ZHANG P, et al. Demonstration of Topological Robustness of Anyonic Braiding Statistics with a Superconducting Quantum Circuit[J]. Phys. Rev. Lett., 2018, 121: 030502.
[111] ANDERSEN C K, REMM A, LAZAR S, et al. Repeated quantum error detection in a surfacecode[J]. Nature Phys., 2020, 16(8): 875-880.
[112] ERHARD A, NAUTRUP H P, METH M, et al. Entangling logical qubits with lattice surgery[J]. Nature (London), 2021, 589(7841): 220-224.
[113] SATZINGER K, LIU Y J, SMITH A, et al. Realizing topologically ordered states on a quantumprocessor[J]. Science, 2021, 374(6572): 1237-1241.
[114] FAN Y A, LI Y, HU Y, et al. Experimental realization of a topologically protected Hadamardgate via braiding Fibonacci anyons[A]. 2022.
[115] MIGNARD M, SCHAUENBURG P. Modular categories are not determined by their modulardata[A]. 2017.
[116] LIU S, GAO W, ZHANG Q, et al. Topologically protected edge state in two-dimensional Su–Schrieffer–Heeger circuit[J]. Res., 2019, 2019.
[117] XIE B Y, SU G X, WANG H F, et al. Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional Dielectric Photonic Crystals[J]. Phys. Rev. Lett., 2019, 122:233903.
[118] LIU F, WAKABAYASHI K. Novel Topological Phase with a Zero Berry Curvature[J]. Phys.Rev. Lett., 2017, 118: 076803.
[119] XIE B Y, WANG H F, WANG H X, et al. Second-order photonic topological insulator withcorner states[J]. Phys. Rev. B, 2018, 98: 205147.
[120] OBANA D, LIU F, WAKABAYASHI K. Topological edge states in the Su-Schrieffer-Heegermodel[J]. Phys. Rev. B, 2019, 100: 075437.
[121] ZHENG L Y, ACHILLEOS V, RICHOUX O, et al. Observation of Edge Waves in a TwoDimensional Su-Schrieffer-Heeger Acoustic Network[J]. Phys. Rev. Appl., 2019, 12: 034014.
[122] BANSIL A, LIN H, DAS T. Colloquium: Topological band theory[J]. Rev. Mod. Phys., 2016,88: 021004.
[123] ARMITAGE N, MELE E, VISHWANATH A. Weyl and Dirac semimetals in three-dimensionalsolids[J]. Rev. Mod. Phys., 2018, 90(1): 015001.
[124] HUANG C, ZHOU B T, ZHANG H, et al. Proximity-induced surface superconductivity in Diracsemimetal Cd3As2[J]. Nat. Commun., 2019, 10(1): 2217.
[125] LI Y, WU Z, ZHOU J, et al. Enhanced anisotropic superconductivity in the topological nodalline semimetal In x TaS 2[J]. Phys. Rev. B, 2020, 102(22): 224503.
[126] LI Y, WU Y, XU C, et al. Anisotropic gapping of topological Weyl rings in the charge-densitywave superconductor InxTaSe2[J]. Science Bulletin, 2021, 66(3): 243-249.
[127] BIAN G, CHANG T R, ZHENG H, et al. Drumhead surface states and topological nodal-linefermions in TlTaSe 2[J]. Phys. Rev. B, 2016, 93(12): 121113.
[128] ZHANG C L, YUAN Z, BIAN G, et al. Superconducting properties in single crystals of thetopological nodal semimetal PbTaSe 2[J]. Phys. Rev. B, 2016, 93(5): 054520.
[129] GUAN S Y, CHEN P J, CHU M W, et al. Superconducting topological surface states in thenoncentrosymmetric bulk superconductor PbTaSe2[J]. Sci. Adv., 2016, 2(11): e1600894.
[130] DU Y, BO X, WANG D, et al. Emergence of topological nodal lines and type-II Weyl nodesin the strong spin-orbit coupling system InNb X 2 (X= S, Se)[J]. Phys. Rev. B, 2017, 96(23):235152.
[131] SCHNYDER A P, BRYDON P, TIMM C. Types of topological surface states in nodal noncentrosymmetric superconductors[J]. Phys. Rev. B, 2012, 85(2): 024522.
[132] SCHNYDER A P, BRYDON P M. Topological surface states in nodal superconductors[J]. J.Phys. Condens., 2015, 27(24): 243201.
[133] MENG T, BALENTS L. Weyl superconductors[J]. Phys. Rev. B, 2012, 86(5): 054504.
[134] BEDNIK G, ZYUZIN A, BURKOV A. Superconductivity in Weyl metals[J]. Phys. Rev. B,2015, 92(3): 035153.
[135] KOBAYASHI S, SATO M. Topological superconductivity in Dirac semimetals[J]. Phys. Rev.Lett., 2015, 115(18): 187001.
[136] QIN S, HU L, LE C, et al. Quasi-1D topological nodal vortex line phase in doped superconducting 3D Dirac semimetals[J]. Phys. Rev. Lett., 2019, 123(2): 027003.
[137] NANDKISHORE R. Weyl and Dirac loop superconductors[J]. Phys. Rev. B, 2016, 93(2):020506.
[138] WANG Y, NANDKISHORE R M. Topological surface superconductivity in doped Weyl loopmaterials[J]. Phys. Rev. B, 2017, 95(6): 060506.
[139] FU P H, LIU J F, WU J. Transport properties of Majorana drumhead surface states in topologicalnodal-ring superconductors[J]. Phys. Rev. B, 2020, 102(7): 075430.
[140] TSUI D C, STORMER H L, GOSSARD A C. Two-dimensional Magnetotransport in the Extreme Quantum Limit[J]. Phys. Rev. Lett., 1982, 48: 1559-1562.
[141] WEN X G. Topological Orders in Rigid States[J]. Int. J. Mod. Phys. B, 1990, 04(02): 239-271.
[142] WEN X G, NIU Q. Ground-state Degeneracy of the Fractional Quantum Hall States in thePresence of a Random Potential and on High-genus Riemann Surfaces[J]. Phys. Rev. B, 1990,41: 9377-9396.
[143] WEN X, ZEE A. Quantum statistics and superconductivity in two spatial dimensions[J]. Nucl.Phys. B, Proc. Suppl., 1990, 15: 135-156.
[144] WEN X G. Topological Orders and Chern-Simons Theory in Strongly Correlated QuantumLiquid[J]. Int. J. Mod. Phys. B, 1991, 05(10): 1641-1648.
[145] CHEN X, GU Z C, LIU Z X, et al. Symmetry-Protected Topological Orders in InteractingBosonic Systems[J]. Science, 2012, 338(6114): 1604-1606.
[146] DAS SARMA S, FREEDMAN M, NAYAK C. Topologically Protected Qubits from a PossibleNon-Abelian Fractional Quantum Hall State[J]. Phys. Rev. Lett., 2005, 94: 166802.
[147] NAYAK C, SIMON S H, STERN A, et al. Non-Abelian anyons and topological quantum computation[J]. Rev. Mod. Phys., 2008, 80: 1083-1159.
[148] TERHAL B M. Quantum error correction for quantum memories[J]. Rev. Mod. Phys., 2015,87: 307-346.
[149] BRAVYI S B, KITAEV A Y. Quantum Codes on a Lattice with Boundary[A].
[150] WEN X G. Colloquium: Zoo of Quantum-topological Phases of Matter[J]. Rev. Mod. Phys.,2017, 89: 041004.
[151] HAN Y J, RAUSSENDORF R, DUAN L M. Scheme for Demonstration of Fractional Statisticsof Anyons in an Exactly Solvable Model[J]. Phys. Rev. Lett., 2007, 98: 150404.
[152] ROWELL E, STONG R, WANG Z. On Classification of Modular Tensor Categories[J]. Commun. Math. Phys., 2009, 292(2): 343-389.
[153] WEN X G. Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states[A].
[154] ZHANG Y, GROVER T, TURNER A, et al. Quasiparticle statistics and braiding from groundstate entanglement[J]. Phys. Rev. B, 2012, 85: 235151
[155] CINCIO L, VIDAL G. Characterizing Topological Order by Studying the Ground States on anInfinite Cylinder[J]. Phys. Rev. Lett., 2013, 110: 067208.
[156] LIU F, WANG Z, YOU Y Z, et al. Modular transformations and topological orders in twodimensions[A]. 2014. arXiv: 1303.0829.
[157] JIANG S, MESAROS A, RAN Y. Generalized Modular Transformations in (3 + 1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding[J]. Phys. Rev. X, 2014, 4:031048.
[158] MEI J W, CHEN J Y, HE H, et al. Gapped spin liquid with ℤ2topological order for the kagomeHeisenberg model[J]. Phys. Rev. B, 2017, 95: 235107.
[159] HU Y, STIRLING S D, WU Y S. Ground-state degeneracy in the Levin-Wen model for topological phases[J]. Phys. Rev. B, 2012, 85: 075107.
[160] LU C Y, GAO W B, GÜHNE O, et al. Demonstrating Anyonic Fractional Statistics with aSix-Qubit Quantum Simulator[J]. Phys. Rev. Lett., 2009, 102: 030502.
[161] ZHONG Y P, XU D, WANG P, et al. Emulating Anyonic Fractional Statistical Behavior in aSuperconducting Quantum Circuit[J]. Phys. Rev. Lett., 2016, 117: 110501.
[162] ANDERSEN C K, REMM A, LAZAR S, et al. Repeated quantum error detection in a surfacecode[J]. Nat. Phys., 2020, 16(8): 875-880.
[163] SATZINGER K J, LIU Y J, SMITH A, et al. Realizing topologically ordered states on a quantumprocessor[J]. Science, 2021, 374(6572): 1237-1241.
[164] CINCIO L, VIDAL G. Characterizing Topological Order by Studying the Ground States on anInfinite Cylinder[J]. Phys. Rev. Lett., 2013, 110: 067208.
[165] KNILL E, LAFLAMME R. Power of One Bit of Quantum Information[J]. Phys. Rev. Lett.,1998, 81: 5672-5675.
[166] MIQUEL C, PAZ J P, SARACENO M, et al. Interpretation of tomography and spectroscopy asdual forms of quantum computation[J]. Nature (London), 2002, 418(6893): 59-62.
[167] HUNG L Y, WAN Y. String-net models with 𝑍𝑁 fusion algebra[J]. Phys. Rev. B, 2012, 86:235132.
[168] ETINGOF P, GELAKI S, NIKSHYCH D, et al. Tensor Categories: volume 205[M]. Am. Math.Soc., 2016.
[169] CONG I, CHENG M, WANG Z. Universal Quantum Computation with Gapped Boundaries[J].Phys. Rev. Lett., 2017, 119: 170504.
[170] CLEVE R, EKERT A, MACCHIAVELLO C, et al. Quantum algorithms revisited[J]. Proc.Math. Phys. Eng., 1998, 454(1969): 339-354.
[171] BARENCO A, BENNETT C H, CLEVE R, et al. Elementary gates for quantum computation[J]. Phys. Rev. A, 1995, 52: 3457-3467.
[172] ABRAMS D S, LLOYD S. Quantum Algorithm Providing Exponential Speed Increase forFinding Eigenvalues and Eigenvectors[J]. Phys. Rev. Lett., 1999, 83: 5162-5165.
[173] KNILL E, LAFLAMME R. Power of One Bit of Quantum Information[J]. Phys. Rev. Lett.,1998, 81: 5672-5675.
[174] PASSANTE G, MOUSSA O, RYAN C A, et al. Experimental Approximation of the JonesPolynomial with One Quantum Bit[J]. Phys. Rev. Lett., 2009, 103: 250501.
[175] KHOURY A Z, SOUZA A M, OXMAN L E, et al. Intrinsic bounds of a two-qudit randomevolution[J]. Phys. Rev. A, 2018, 97: 042343.
[176] VIND F A, FOERSTER A, OLIVEIRA I S, et al. Experimental realization of the Yang-BaxterEquation via NMR interferometry[J]. Sci. Rep., 2016, 6(1): 1-8.
[177] NIE X, ZHANG Z, ZHAO X, et al. Detecting scrambling via statistical correlations betweenrandomized measurements on an NMR quantum simulator[A]. 2019.
[178] NIE X, WEI B B, CHEN X, et al. Experimental Observation of Equilibrium and DynamicalQuantum Phase Transitions via Out-of-Time-Ordered Correlators[J]. Phys. Rev. Lett., 2020,124: 250601.
[179] NIE X, ZHU X, XI C, et al. Experimental Realization of a Quantum Refrigerator Driven byIndefinite Causal Orders[A]. 2020.
[180] RYAN C A, NEGREVERGNE C, LAFOREST M, et al. Liquid-state nuclear magnetic resonance as a testbed for developing quantum control methods[J]. Phys. Rev. A, 2008, 78: 012328.
[181] KHANEJA N, REISS T, KEHLET C, et al. Optimal control of coupled spin dynamics: designof NMR pulse sequences by gradient ascent algorithms[J]. J. Magn. Reson., 2005, 172(2):296-305.
[182] ZHANG S C, HANSSON T H, KIVELSON S. Effective-field-theory Model for the FractionalQuantum Hall Effect[J]. Phys. Rev. Lett., 1989, 62: 82-85.
[183] DENNIS E, KITAEV A, LANDAHL A, et al. Topological Quantum Memory[J]. J. Math. Phys.,2002, 43(9): 4452-4505.
[184] BRAVYI S B, KITAEV A Y. Quantum Codes on a Lattice with Boundary[A]. 1998.
[185] FELDMAN D E, HALPERIN B I. Fractional charge and fractional statistics in the quantumHall effects[J]. Reports on Progress in Physics, 2021, 84(7): 076501.
[186] BONDERSON P, SHTENGEL K, SLINGERLAND J K. Probing Non-Abelian Statistics withQuasiparticle Interferometry[J]. Phys. Rev. Lett., 2006, 97: 016401.
[187] MCCLURE D T, CHANG W, MARCUS C M, et al. Fabry-Perot Interferometry with FractionalCharges[J]. Phys. Rev. Lett., 2012, 108: 256804.
[188] YANG G. Probing the 𝜈 =52quantum Hall state with electronic Mach-Zehnder interferometry[J]. Phys. Rev. B, 2015, 91: 115109.
[189] SIVAN I, BHATTACHARYYA R, CHOI H K, et al. Interaction-induced interference in theinteger quantum Hall effect[J]. Phys. Rev. B, 2018, 97: 125405.
[190] BHATTACHARYYA R, BANERJEE M, HEIBLUM M, et al. Melting of Interference in theFractional Quantum Hall Effect: Appearance of Neutral Modes[J]. Phys. Rev. Lett., 2019, 122:246801.
[191] KANE C L, FISHER M P A. Quantized thermal transport in the fractional quantum Hall effect[J]. Phys. Rev. B, 1997, 55: 15832-15837.
[192] JEZOUIN S, PARMENTIER F, ANTHORE A, et al. Quantum limit of heat flow across a singleelectronic channel[J]. Science, 2013, 342(6158): 601-604.
[193] SIVRE E, ANTHORE A, PARMENTIER F, et al. Heat Coulomb blockade of one ballisticchannel[J]. Nature Phys., 2018, 14(2): 145-148.
[194] SIMON S H. Interpretation of thermal conductance of the = 5/2 edge[J]. Phys. Rev. B, 2018,97: 121406.
[195] MA K K W, FELDMAN D E. Thermal Equilibration on the Edges of Topological Liquids[J].Phys. Rev. Lett., 2020, 125: 016801.
[196] GEORGESCU I M, ASHHAB S, NORI F. Quantum simulation[J]. Rev. Mod. Phys., 2014, 86:153-185.
[197] ZHONG Y P, XU D, WANG P, et al. Emulating Anyonic Fractional Statistical Behavior in aSuperconducting Quantum Circuit[J]. Phys. Rev. Lett., 2016, 117: 110501.
[198] KITAEV A, PRESKILL J. Topological Entanglement Entropy[J]. Phys. Rev. Lett., 2006, 96:110404.
[199] JIANG H C, WANG Z, BALENTS L. Identifying Topological Order by Entanglement Entropy[J]. Nat. Phys., 2012, 8(12): 902-905.
[200] SATZINGER K, LIU Y J, SMITH A, et al. Realizing topologically ordered states on a quantumprocessor[J]. Science, 2021, 374(6572): 1237-1241.
[201] SEMEGHINI G, LEVINE H, KEESLING A, et al. Probing topological spin liquids on a programmable quantum simulator[J]. Science, 2021, 374(6572): 1242-1247.
[202] KAWAGOE K, LEVIN M. Microscopic definitions of anyon data[J]. Phys. Rev. B, 2020, 101:115113.
[203] MOORE G, SEIBERG N. Classical and Quantum Conformal Field Theory[J]. Commun. Math.Phys., 1989, 123(2): 177-254.
[204] KONG L, WEN X G, ZHENG H. Boundary-bulk Relation in Topological Orders[J]. Nucl.Phys. B, 2017, 922: 62 - 76.
[205] KONG L. Anyon Condensation and Tensor Categories[J]. Nucl. Phys. B, 2014, 886: 436 - 482.
[206] LEVIN M. Protected Edge Modes without Symmetry[J]. Phys. Rev. X, 2013, 3(2): 021009.
[207] BARKESHLI M, JIAN C M, QI X L. Classification of Topological Defects in Abelian Topological States[J]. Phys. Rev. B, 2013, 88(24): 241103(R).
[208] ETINGOF P, GELAKI S, NIKSHYCH D, et al. Tensor Categories: volume 205[M]. AmericanMathematical Soc., 2016.
[209] NEGREVERGNE C, SOMMA R, ORTIZ G, et al. Liquid-state NMR simulations of quantummany-body problems[J]. Phys. Rev. A, 2005, 71: 032344.
[210] CORY D G, FAHMY A F, HAVEL T F. Ensemble quantum computing by NMR spectroscopy[J]. Proc. Nat. Acad. Sci., 1997, 94(5): 1634-1639.
[211] XIN T, PEDERNALES J S, SOLANO E, et al. Entanglement measures in embedding quantumsimulators with nuclear spins[J]. Phys. Rev. A, 2018, 97(2): 022322.
[212] AHARONOV Y, BOHM D. Significance of Electromagnetic Potentials in the Quantum Theory[J]. Phys. Rev., 1959, 115: 485-491.
[213] WEN X G. Quantum Orders in an Exact Soluble Model[J]. Phys. Rev. Lett., 2003, 90: 016803.
[214] KONG L, ZHENG H. A mathematical theory of gapless edges of 2d topological orders. Part I[J]. J. High Energy Phys., 2020, 2020(2): 1-62.
[215] KONG L, ZHENG H. A mathematical theory of gapless edges of 2d topological orders. Part II[J]. Nucl. Phys. B, 2021, 966: 115384.
[216] FOWLER A G, MARIANTONI M, MARTINIS J M, et al. Surface Codes: Towards PracticalLarge-scale Quantum Computation[J]. Phys. Rev. A, 2012, 86: 032324.
[217] CONG I, CHENG M, WANG Z. Universal Quantum Computation with Gapped Boundaries[J].Phys. Rev. Lett., 2017, 119: 170504.
[218] CONG I, CHENG M, WANG Z. Topological quantum computation with gapped boundaries[A]. 2016.
[219] FANG C, GILBERT M J, BERNEVIG B A. Bulk topological invariants in noninteracting pointgroup symmetric insulators[J]. Phys. Rev. B, 2012, 86: 115112.
[220] FU L. Topological Crystalline Insulators[J]. Phys. Rev. Lett., 2011, 106: 106802.
[221] ANDO Y, FU L. Topological Crystalline Insulators and Topological Superconductors: FromConcepts to Materials[J]. Annual Review of Condensed Matter Physics, 2015, 6(1): 361-381.
[222] SHENG D N, WENG Z Y, SHENG L, et al. Quantum Spin-Hall Effect and TopologicallyInvariant Chern Numbers[J]. Phys. Rev. Lett., 2006, 97: 036808.
[223] SHIOZAKI K, SATO M. Topology of crystalline insulators and superconductors[J]. Phys. Rev.B, 2014, 90: 165114.
[224] DONG X Y, LIU C X. Classification of topological crystalline insulators based on representationtheory[J]. Phys. Rev. B, 2016, 93: 045429.
[225] YU X L, JIANG L, QUAN Y M, et al. Topological phase transitions, Majorana modes, andquantum simulation of the Su–Schrieffer–Heeger model with nearest-neighbor interactions[J].Phys. Rev. B, 2020, 101: 045422.
[226] SU W P, SCHRIEFFER J R, HEEGER A J. Soliton excitations in polyacetylene[J]. Phys. Rev.B, 1980, 22: 2099-2111.
[227] HIRSCH J E. Effect of Coulomb Interactions on the Peierls Instability[J]. Phys. Rev. Lett.,1983, 51: 296-299.
[228] VOIT J. Superconductivity in models of conducting polymers[J]. Phys. Rev. Lett., 1990, 64:323-325.
[229] Mean-field approach to extended Su–Schrieffer–Heeger models[J]. Chem. Phys. Lett., 2003,377(3): 455-461.
[230] MIYAO T. Ground state properties of the SSH model[J]. Journal of statistical physics, 2012,149: 519-550.
[231] STICLET D, SEABRA L, POLLMANN F, et al. From fractionally charged solitons to Majoranabound states in a one-dimensional interacting model[J]. Phys. Rev. B, 2014, 89: 115430.
[232] SIRKER J, MAITI M, KONSTANTINIDIS N, et al. Boundary fidelity and entanglement in thesymmetry protected topological phase of the SSH model[J]. J. Stat. Mech. Theory Exp., 2014,2014(10): P10032.
[233] WEBER M, ASSAAD F F, HOHENADLER M. Excitation spectra and correlation functionsof quantum Su-Schrieffer-Heeger models[J]. Phys. Rev. B, 2015, 91: 245147.
[234] MEIER E J, AN F A, GADWAY B. Observation of the topological soliton state in the Su–Schrieffer–Heeger model[J]. Nat. Commun., 2016, 7(1): 13986.
[235] LEE C H, IMHOF S, BERGER C, et al. Topolectrical circuits[J]. Commun. Phys., 2018, 1(1):39.
[236] CAI W, HAN J, MEI F, et al. Observation of Topological Magnon Insulator States in a Superconducting Circuit[J]. Phys. Rev. Lett., 2019, 123: 080501.
[237] EZAWA M. Exact solutions and topological phase diagram in interacting dimerized Kitaevtopological superconductors[J]. Phys. Rev. B, 2017, 96: 121105.
[238] BARBIERO L, SANTOS L, GOLDMAN N. Quenched dynamics and spin-charge separationin an interacting topological lattice[J]. Phys. Rev. B, 2018, 97: 201115.
[239] WANG Y, MIAO J J, JIN H K, et al. Characterization of topological phases of dimerized Kitaevchain via edge correlation functions[J]. Phys. Rev. B, 2017, 96: 205428.
[240] DE LéSéLEUC S, LIENHARD V, SCHOLL P, et al. Observation of a symmetry-protectedtopological phase of interacting bosons with Rydberg atoms[J]. Science, 2019, 365(6455):775-780.
[241] ATALA M, AIDELSBURGER M, BARREIRO J T, et al. Direct measurement of the Zak phasein topological Bloch bands[J]. Nature Phys., 2013, 9(12): 795-800.
[242] MURTA B, CATARINA G, FERNÁNDEZ-ROSSIER J. Berry phase estimation in gate-basedadiabatic quantum simulation[J]. Phys. Rev. A, 2020, 101: 020302.
[243] ST-JEAN P, GOBLOT V, GALOPIN E, et al. Lasing in topological edge states of a onedimensional lattice[J]. Nature Photon., 2017, 11(10): 651-656.
[244] CHAUNSALI R, KIM E, THAKKAR A, et al. Demonstrating an In Situ Topological BandTransition in Cylindrical Granular Chains[J]. Phys. Rev. Lett., 2017, 119: 024301.
[245] ZHANG Z, LONG H, LIU C, et al. Deep-subwavelength holey acoustic second-order topological insulators[J]. Advanced Materials, 2019, 31(49): 1904682.
[246] OLEKHNO N A, ROZENBLIT A D, KACHIN V I, et al. Experimental realization of topological corner states in long-range-coupled electrical circuits[J]. Phys. Rev. B, 2022, 105: L081107.
[247] CHEN X D, DENG W M, SHI F L, et al. Direct Observation of Corner States in Second-OrderTopological Photonic Crystal Slabs[J]. Phys. Rev. Lett., 2019, 122: 233902.
[248] ZAK J. Berry’s phase for energy bands in solids[J]. Phys. Rev. Lett., 1989, 62: 2747-2750.
[249] RICE M J, MELE E J. Elementary Excitations of a Linearly Conjugated Diatomic Polymer[J].Phys. Rev. Lett., 1982, 49: 1455-1459.
[250] LONGHI S. Zak phase of photons in optical waveguide lattices[J]. Optics letters, 2013, 38(19):3716-3719.
[251] TAN W, SUN Y, CHEN H, et al. Photonic simulation of topological excitations in metamaterials[J]. Sci. Rep., 2014, 4(1): 3842.
[252] CARDANO F, D’ERRICO A, DAUPHIN A, et al. Detection of Zak phases and topologicalinvariants in a chiral quantum walk of twisted photons[J]. Nat. Commun., 2017, 8(1): 15516.
[253] XIAO M, MA G, YANG Z, et al. Geometric phase and band inversion in periodic acousticsystems[J]. Nature Phys., 2015, 11(3): 240-244.
[254] YANG Z, GAO F, ZHANG B. Topological water wave states in a one-dimensional structure[J].Sci. Rep., 2016, 6(1): 29202.
[255] KING-SMITH R D, VANDERBILT D. Theory of polarization of crystalline solids[J]. Phys.Rev. B, 1993, 47: 1651-1654.
[256] RESTA R. Macroscopic polarization in crystalline dielectrics: the geometric phase approach[J]. Rev. Mod. Phys., 1994, 66: 899-915.
[257] DERZHKO O, RICHTER J, MAKSYMENKO M. Strongly correlated flat-band systems: Theroute from Heisenberg spins to Hubbard electrons[J]. Int. J. Mod. Phys. B, 2015, 29(12):1530007.
[258] LEYKAM D, ANDREANOV A, FLACH S. Artificial flat band systems: from lattice modelsto experiments[J]. Adv. Phys.: X, 2018, 3(1): 1473052.
[259] MAIMAITI W, ANDREANOV A, FLACH S. Flat-band generator in two dimensions[J]. Phys.Rev. B, 2021, 103: 165116.
[260] LIEB E H. Two theorems on the Hubbard model[J]. Phys. Rev. Lett., 1989, 62: 1201-1204.
[261] BISTRITZER R, MACDONALD A H. Moiré bands in twisted double-layer graphene[J]. Proc.Nat. Acad. Sci., 2011, 108(30): 12233-12237.
[262] HEIKKILÄ T T, VOLOVIK G E. Flat bands as a route to high-temperature superconductivityin graphite[J]. Basic Physics of Functionalized Graphite, 2016: 123-143.
[263] HEIKKILÄ T T, KOPNIN N B, VOLOVIK G E. Flat bands in topological media[J]. JETPLett., 2011, 94: 233-239.
[264] HYART T, OJAJÄRVI R, HEIKKILÄ T. Two topologically distinct Dirac-line semimetal phasesand topological phase transitions in rhombohedrally stacked honeycomb lattices[J]. J. LowTemp. Phys, 2018, 191: 35-48.
[265] PAL H K, SPITZ S, KINDERMANN M. Emergent geometric frustration and flat band in moirébilayer graphene[J]. Phys. Rev. Lett., 2019, 123(18): 186402.
[266] KHODEL V, SHAGINYAN V. Superfluidity in system with fermion condensate[J]. Jetp Lett,1990, 51(9): 553.
[267] VOLOVIK G. A new class of normal Fermi liquids[J]. JETP Lett, 1991, 53(4): 222.
[268] NOZIÈRES P. Properties of Fermi liquids with a finite range interaction[J]. Journal de PhysiqueI, 1992, 2(4): 443-458.
[269] YUDIN D, HIRSCHMEIER D, HAFERMANN H, et al. Fermi Condensation Near van HoveSingularities Within the Hubbard Model on the Triangular Lattice[J]. Phys. Rev. Lett., 2014,112: 070403.
[270] VOLOVIK G. The fermi condensate near the saddle point and in the vortex core[J]. JETP Lett.,1994, 59(11): 830.
[271] SHASHKIN A A, DOLGOPOLOV V T, CLARK J W, et al. Merging of Landau Levels in aStrongly Interacting Two-Dimensional Electron System in Silicon[J]. Phys. Rev. Lett., 2014,112: 186402.
[272] TOVMASYAN M, PEOTTA S, LIANG L, et al. Preformed pairs in flat Bloch bands[J]. Phys.Rev. B, 2018, 98: 134513.
[273] DANIELI C, ANDREANOV A, FLACH S. Many-body flatband localization[J]. Phys. Rev. B,2020, 102: 041116.
[274] KUNO Y, ORITO T, ICHINOSE I. Flat-band many-body localization and ergodicity breakingin the Creutz ladder[J]. New. J. Phys., 2020, 22(1): 013032.
[275] ORITO T, KUNO Y, ICHINOSE I. Exact projector Hamiltonian, local integrals of motion, andmany-body localization with symmetry-protected topological order[J]. Phys. Rev. B, 2020, 101:224308.
[276] MIELKE A. Ferromagnetism in the Hubbard model on line graphs and further considerations[J]. Journal of Physics A: Mathematical and General, 1991, 24(14): 3311.
[277] TASAKI H. Ferromagnetism in the Hubbard models with degenerate single-electron groundstates[J]. Phys. Rev. Lett., 1992, 69: 1608-1611.
[278] MIELKE A. Ferromagnetic ground states for the Hubbard model on line graphs[J]. Journal ofPhysics A: Mathematical and General, 1991, 24(2): L73.
[279] TASAKI H. Hubbard model and the origin of ferromagnetism[J]. The European PhysicalJournal B, 2008, 64: 365-372.
[280] TASAKI H. Stability of Ferromagnetism in the Hubbard Model[J]. Phys. Rev. Lett., 1994, 73:1158-1161.
[281] MAKSYMENKO M, HONECKER A, MOESSNER R, et al. Flat-Band Ferromagnetism as aPauli-Correlated Percolation Problem[J]. Phys. Rev. Lett., 2012, 109: 096404.
[282] PEOTTA S, TÖRMÄ P. Superfluidity in topologically nontrivial flat bands[J]. Nat. Commun.,2015, 6(1): 8944.
[283] JULKU A, PEOTTA S, VANHALA T I, et al. Geometric Origin of Superfluidity in the LiebLattice Flat Band[J]. Phys. Rev. Lett., 2016, 117: 045303.
[284] MONDAINI R, BATROUNI G G, GRÉMAUD B. Pairing and superconductivity in the flatband: Creutz lattice[J]. Phys. Rev. B, 2018, 98: 155142.
[285] VOLOVIK G E. Graphite, graphene, and the flat band superconductivity[J]. JETP Lett., 2018,107: 516-517.
[286] MIELKE A. Pair formation of hard core bosons in flat band systems[J]. Journal of StatisticalPhysics, 2018, 171: 679-695.
[287] CAPRIOTTI L, SORELLA S. Spontaneous Plaquette Dimerization in the J1 − −J2 HeisenbergModel[J]. Phys. Rev. Lett., 2000, 84: 3173-3176.
[288] MOSADEQ H, SHAHBAZI F, JAFARI S. Plaquette valence bond ordering in a J1–J2 Heisenberg antiferromagnet on a honeycomb lattice[J]. J. Phys. Condens., 2011, 23(22): 226006.
[289] HUGHES T L, PRODAN E, BERNEVIG B A. Inversion-symmetric topological insulators[J].Phys. Rev. B, 2011, 83: 245132.
[290] JIAO Z Q, LONGHI S, WANG X W, et al. Experimentally Detecting Quantized Zak Phaseswithout Chiral Symmetry in Photonic Lattices[J]. Phys. Rev. Lett., 2021, 127: 147401.
[291] PÉREZ-GONZÁLEZ B, BELLO M, GÓMEZ-LEÓN A, et al. Interplay between long-rangehopping and disorder in topological systems[J]. Phys. Rev. B, 2019, 99: 035146.
[292] ALICEA J. New directions in the pursuit of Majorana fermions in solid state systems[J]. Reportson progress in physics, 2012, 75(7): 076501.
[293] BEENAKKER C. Search for Majorana fermions in superconductors[J]. Annu. Rev. Condens.Matter Phys., 2013, 4(1): 113-136.
[294] STANESCU T D, TEWARI S. Majorana fermions in semiconductor nanowires: fundamentals,modeling, and experiment[J]. J. Phys. Condens., 2013, 25(23): 233201.
[295] ELLIOTT S R, FRANZ M. Colloquium: Majorana fermions in nuclear, particle, and solid-statephysics[J]. Rev. Mod. Phys., 2015, 87(1): 137.
[296] DENG M, VAITIEKĖNAS S, HANSEN E B, et al. Majorana bound state in a coupled quantumdot hybrid-nanowire system[J]. Science, 2016, 354(6319): 1557-1562.
[297] ZHANG H, GÜL Ö, CONESA-BOJ S, et al. Ballistic superconductivity in semiconductornanowires[J]. Nat. Commun., 2017, 8(1): 16025.
[298] GRIVNIN A, BOR E, HEIBLUM M, et al. Concomitant opening of a bulk-gap with an emergingpossible Majorana zero mode[J]. Nat. Commun., 2019, 10(1): 1940.
[299] DE MOOR M W, BOMMER J D, XU D, et al. Electric field tunable superconductorsemiconductor coupling in Majorana nanowires[J]. New. J. Phys., 2018, 20(10): 103049.
[300] BOMMER J D, ZHANG H, GÜL Ö, et al. Spin-orbit protection of induced superconductivityin Majorana nanowires[J]. Phys. Rev. Lett., 2019, 122(18): 187702.
[301] ZHANG H, LIU D E, WIMMER M, et al. Next steps of quantum transport in Majorana nanowiredevices[J]. Nat. Commun., 2019, 10(1): 5128.
[302] HE Q L, PAN L, STERN A L, et al. Chiral Majorana fermion modes in a quantum anomalousHall insulator–superconductor structure[J]. Science, 2017, 357(6348): 294-299.
[303] LUTCHYN R M, SAU J D, SARMA S D. Majorana fermions and a topological phase transitionin semiconductor-superconductor heterostructures[J]. Phys. Rev. Lett., 2010, 105(7): 077001.
[304] OREG Y, REFAEL G, VON OPPEN F. Helical liquids and Majorana bound states in quantumwires[J]. Phys. Rev. Lett., 2010, 105(17): 177002.
[305] QI X L, HUGHES T L, ZHANG S C. Chiral topological superconductor from the quantum Hall state[J]. Phys. Rev. B, 2010, 82(18): 184516.
[306] WANG J, ZHOU Q, LIAN B, et al. Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition[J]. Phys. Rev. B, 2015, 92(6): 064520.
[307] PRADA E, SAN-JOSE P, DE MOOR M W, et al. From Andreev to Majorana bound states inhybrid superconductor–semiconductor nanowires[J]. Nat. Rev. Phys., 2020, 2(10): 575-594.
[308] PAN H, SAU J D, SARMA S D. Three-terminal nonlocal conductance in Majorana nanowires:Distinguishing topological and trivial in realistic systems with disorder and inhomogeneouspotential[J]. Phys. Rev. B, 2021, 103(1): 014513.
[309] LAI Y H, SAU J D, SARMA S D. Presence versus absence of end-to-end nonlocal conductancecorrelations in Majorana nanowires: Majorana bound states versus Andreev bound states[J].Phys. Rev. B, 2019, 100(4): 045302.
[310] PAN H, SARMA S D. Physical mechanisms for zero-bias conductance peaks in Majorana nanowires[J]. Phys. Rev. Res., 2020, 2(1): 013377.
[311] LIU C X, SAU J D, SARMA S D. Distinguishing topological Majorana bound states from trivialAndreev bound states: Proposed tests through differential tunneling conductance spectroscopy[J]. Phys. Rev. B, 2018, 97(21): 214502.
[312] JI W, WEN X G. 1 2 (e 2/h) conductance plateau without 1d chiral majorana fermions[J]. Phys.Rev. Lett., 2018, 120(10): 107002.
[313] HUANG Y, SETIAWAN F, SAU J D. Disorder-induced half-integer quantized conductance plateau in quantum anomalous Hall insulator-superconductor structures[J]. Phys. Rev. B, 2018, 97(10): 100501.
[314] KAYYALHA M, XIAO D, ZHANG R, et al. Absence of evidence for chiral Majorana modesin quantum anomalous Hall-superconductor devices[J]. Science, 2020, 367(6473): 64-67.
[315] CHIU C K, TEO J C, SCHNYDER A P, et al. Classification of topological quantum matter with symmetries[J]. Rev. Mod. Phys., 2016, 88(3): 035005.
[316] HE J J, WU J, CHOY T P, et al. Correlated spin currents generated by resonant-crossed Andreev reflections in topological superconductors[J]. Nat. Commun., 2014, 5(1): 3232.
[317] DATTA S. Electronic transport in mesoscopic systems[M]. Cambridge university press, 1997.
[318] SANCHO M L, SANCHO J L, SANCHO J L, et al. Highly convergent schemes for the calculation of bulk and surface Green functions[J]. J. Phys. F, 1985, 15(4): 851.
[319] FISHER D S, LEE P A. Relation between conductivity and transmission matrix[J]. Phys. Rev. B, 1981, 23(12): 6851.
[320] LAW K T, LEE P A, NG T K. Majorana fermion induced resonant Andreev reflection[J]. Phys. Rev. Lett., 2009, 103(23): 237001.
[321] HECHT E. Optics[M]. Pearson Education India, 2012.
[322] SARMA S D, SAU J D, STANESCU T D. Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire[J]. Phys. Rev. B, 2012, 86(22): 220506