{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T22:28:58Z","timestamp":1784068138057,"version":"3.55.0"},"reference-count":193,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T00:00:00Z","timestamp":1693008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12204356"],"award-info":[{"award-number":["12204356"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2022QA024"],"award-info":[{"award-number":["ZR2022QA024"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2022QA085"],"award-info":[{"award-number":["ZR2022QA085"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["12204356"],"award-info":[{"award-number":["12204356"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2022QA024"],"award-info":[{"award-number":["ZR2022QA024"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2022QA085"],"award-info":[{"award-number":["ZR2022QA085"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Weifang University","award":["12204356"],"award-info":[{"award-number":["12204356"]}]},{"name":"Weifang University","award":["ZR2022QA024"],"award-info":[{"award-number":["ZR2022QA024"]}]},{"name":"Weifang University","award":["ZR2022QA085"],"award-info":[{"award-number":["ZR2022QA085"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The exploration of topologically nontrivial states in condensed matter systems, along with their novel transport properties, has garnered significant research interest. This review aims to provide a comprehensive overview of representative topological phases, starting from the initial proposal of the quantum Hall insulator. We begin with a concise introduction, followed by a detailed examination of first-order topological quantum phases, including gapped and gapless systems, encompassing relevant materials and associated phenomena in experiment. Subsequently, we delve into the realm of exotic higher-order topological quantum phases, examining both theoretical propositions and experimental findings. Moreover, we discuss the mechanisms underlying the emergence of higher-order topology, as well as the challenges involved in experimentally verifying materials exhibiting such properties. Finally, we outline future research directions. This review not only systematically surveys various types of topological quantum states, spanning from first-order to higher-order, but also proposes potential approaches for realizing higher-order topological phases, thereby offering guidance for the detection of related quantum phenomena in experiments.<\/jats:p>","DOI":"10.3390\/sym15091651","type":"journal-article","created":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T01:56:57Z","timestamp":1693187817000},"page":"1651","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Research Progress of Topological Quantum Materials: From First-Order to Higher-Order"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3668-3531","authenticated-orcid":false,"given":"Bing","family":"Liu","sequence":"first","affiliation":[{"name":"School of Physics and Electronic Information, Weifang University, Weifang 261061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenjun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Physics and Electronic Information, Weifang University, Weifang 261061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21004","DOI":"10.1103\/RevModPhys.88.021004","article-title":"Colloquium: Topological band theory","volume":"88","author":"Bansil","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.1103\/RevModPhys.82.3045","article-title":"Colloquium: Topological insulators","volume":"82","author":"Hasan","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1103\/RevModPhys.83.1057","article-title":"Topological insulators and superconductors","volume":"83","author":"Qi","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15001","DOI":"10.1103\/RevModPhys.90.015001","article-title":"Weyl and Dirac semimetals in three-dimensional solids","volume":"90","author":"Armitage","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1038\/nphoton.2014.248","article-title":"Topological photonics","volume":"8","author":"Lu","year":"2014","journal-title":"Nat. Photonics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1038\/s42005-018-0094-4","article-title":"Topological sound","volume":"1","author":"Zhang","year":"2018","journal-title":"Commun. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"114301","DOI":"10.1103\/PhysRevLett.114.114301","article-title":"Topological acoustics","volume":"114","author":"Yang","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1038\/nphys3867","article-title":"Acoustic topological insulator and robust one-way sound transport","volume":"12","author":"He","year":"2016","journal-title":"Nat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1038\/nphys3458","article-title":"Synthetic gauge flux and Weyl points in acoustic systems","volume":"11","author":"Xiao","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"104302","DOI":"10.1103\/PhysRevLett.122.104302","article-title":"Experimental realization of type-II Weyl points and fermi arcs in phononic crystal","volume":"122","author":"Xie","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"44086","DOI":"10.1103\/PhysRevApplied.11.044086","article-title":"Acoustic topological transport and refraction in a Kekul\u00e9 Lattice","volume":"11","author":"Xie","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1038\/nphys3801","article-title":"Topological mechanics","volume":"12","author":"Huber","year":"2016","journal-title":"Nat. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14201","DOI":"10.1038\/ncomms14201","article-title":"Transformable topological mechanical metamaterials","volume":"8","author":"Rocklin","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1103\/PhysRevLett.45.494","article-title":"New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance","volume":"45","author":"Klitzing","year":"1980","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","unstructured":"Cage, M.E., Klitzing, K., Chang, A.M., Duncan, F., Haldane, M., Laughlin, R.B., Pruisken, A., and Thouless, D.J. (2012). The Quantum Hall Effect, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5632","DOI":"10.1103\/PhysRevB.23.5632","article-title":"Quantized Hall conductivity in two dimensions","volume":"23","author":"Laughlin","year":"1981","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRevLett.49.405","article-title":"Quantized Hall conductance in a two-dimensional periodic potential","volume":"49","author":"Thouless","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.1103\/PhysRevLett.61.2015","article-title":"Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the \u201cparity anomaly\u201d","volume":"61","author":"Haldane","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3697","DOI":"10.1103\/PhysRevLett.71.3697","article-title":"Chern number and edge states in the integer quantum Hall effect","volume":"71","author":"Hatsugai","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"195424","DOI":"10.1103\/PhysRevB.78.195424","article-title":"Topological field theory of time-reversal invariant insulators","volume":"78","author":"Qi","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"40502","DOI":"10.1103\/RevModPhys.89.040502","article-title":"Nobel lecture: Topological quantum matter","volume":"89","author":"Haldane","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"146802","DOI":"10.1103\/PhysRevLett.95.146802","article-title":"Z2 topological order and the quantum spin Hall effect","volume":"95","author":"Kane","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1126\/science.1133734","article-title":"Quantum spin Hall effect and topological phase transition in HgTe quantum wells","volume":"314","author":"Bernevig","year":"2006","journal-title":"Science"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1126\/science.1148047","article-title":"Quantum spin Hall insulator state in HgTe quantum wells","volume":"318","author":"Wiedmann","year":"2007","journal-title":"Science"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1126\/science.1187485","article-title":"Quantized anomalous Hall effect in magnetic topological insulators","volume":"329","author":"Yu","year":"2010","journal-title":"Science"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1126\/science.1234414","article-title":"Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator","volume":"340","author":"Chang","year":"2013","journal-title":"Science"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1126\/science.1237215","article-title":"The complete quantum Hall trio","volume":"340","author":"Oh","year":"2013","journal-title":"Science"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1080\/00018732.2015.1068524","article-title":"Quantum anomalous Hall effect and related topological electronic states","volume":"64","author":"Weng","year":"2015","journal-title":"Adv. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"106803","DOI":"10.1103\/PhysRevLett.98.106803","article-title":"Topological insulators in three dimensions","volume":"98","author":"Fu","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"45302","DOI":"10.1103\/PhysRevB.76.045302","article-title":"Topological insulators with inversion symmetry","volume":"76","author":"Fu","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1038\/nature06843","article-title":"A topological Dirac insulator in a quantum spin Hall phase","volume":"452","author":"Hsieh","year":"2008","journal-title":"Nature"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3841","DOI":"10.1038\/ncomms4841","article-title":"Observation of quantum-tunnelling-modulated spin texture in ultrathin topological insulator Bi2Se3 films","volume":"5","author":"Neupane","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1038\/nphys1274","article-title":"Observation of a large-gap topological-insulator class with a single Dirac cone on the surface","volume":"5","author":"Xia","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1126\/science.1173034","article-title":"Experimental realization of a three-dimensional topological insulator, Bi2Te3","volume":"325","author":"Chen","year":"2009","journal-title":"Science"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"146401","DOI":"10.1103\/PhysRevLett.103.146401","article-title":"Observation of time-reversal-protected single-Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3","volume":"103","author":"Hsieh","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1038\/nphys1270","article-title":"Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface","volume":"5","author":"Zhang","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1038\/nature08308","article-title":"Topological surface states protected from backscattering by chiral spin texture","volume":"460","author":"Roushan","year":"2009","journal-title":"Nature"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"081407","DOI":"10.1103\/PhysRevB.88.081407","article-title":"Emerging weak localization effects on a topological insulator\u2013insulating ferromagnet (Bi2Se3-EuS) interface","volume":"88","author":"Yang","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"235315","DOI":"10.1103\/PhysRevB.103.235315","article-title":"Current-induced dynamics of isolated antiferromagnetic antiskyrmion and antiskyrmionium","volume":"103","author":"Arimoto","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"140405","DOI":"10.1103\/PhysRevLett.108.140405","article-title":"Dirac semimetal in three dimensions","volume":"108","author":"Young","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"195320","DOI":"10.1103\/PhysRevB.85.195320","article-title":"Dirac semimetal and topological phase transitions in A3Bi (A=Na, K, Rb)","volume":"85","author":"Wang","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"205101","DOI":"10.1103\/PhysRevB.83.205101","article-title":"Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates","volume":"83","author":"Wan","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_43","first-page":"11029","article-title":"Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides","volume":"5","author":"Weng","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_44","first-page":"31013","article-title":"Experimental discovery of Weyl semimetal TaAs","volume":"5","author":"Lv","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1126\/science.aav2873","article-title":"Magnetic Weyl semimetal phase in a kagom\u00e9 crystal","volume":"365","author":"Liu","year":"2019","journal-title":"Science"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"186402","DOI":"10.1103\/PhysRevLett.116.186402","article-title":"Double Dirac semimetals in three dimensions","volume":"116","author":"Wieder","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"115423","DOI":"10.1103\/PhysRevB.107.115423","article-title":"Symmetry-engineered nodal lines and hourglass fermions in patterned two-dimensional electron gas","volume":"107","author":"Liu","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"36806","DOI":"10.1103\/PhysRevLett.115.036806","article-title":"Dirac line nodes in inversion-symmetric crystals","volume":"115","author":"Kim","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"36807","DOI":"10.1103\/PhysRevLett.115.036807","article-title":"Topological node-line semimetal and Dirac semimetal state in antiperovskite Cu3PdN","volume":"115","author":"Yu","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"117106","DOI":"10.1088\/1674-1056\/25\/11\/117106","article-title":"Topological nodal line semimetals","volume":"25","author":"Fang","year":"2016","journal-title":"Chin. Phys. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"155138","DOI":"10.1103\/PhysRevB.97.155138","article-title":"Pressure-induced organic topological nodal-line semimetal in the three-dimensional molecular crystal Pd(dddt)2","volume":"97","author":"Liu","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1038\/s41567-017-0032-5","article-title":"Threes company","volume":"14","author":"Wieder","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1126\/science.aaf5037","article-title":"Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals","volume":"353","author":"Bradlyn","year":"2016","journal-title":"Science"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1038\/nmat4023","article-title":"Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2","volume":"13","author":"Jeon","year":"2014","journal-title":"Nat. Mater."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"081306","DOI":"10.1103\/PhysRevB.92.081306","article-title":"Large linear magnetoresistance in Dirac semimetal Cd3As2 with fermi surfaces close to the Dirac points","volume":"92","author":"Feng","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"10735","DOI":"10.1038\/ncomms10735","article-title":"Signatures of the Adler\u2013Bell\u2013Jackiw chiral anomaly in a Weyl fermion semimetal","volume":"7","author":"Zhang","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_57","first-page":"31023","article-title":"Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs","volume":"5","author":"Huang","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1606202","DOI":"10.1002\/adma.201606202","article-title":"Weyl semimetals as hydrogen evolution catalysts","volume":"29","author":"Rajamathi","year":"2017","journal-title":"Adv. Mater."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s40843-017-9178-4","article-title":"Topological quantum catalyst: Dirac nodal line states and a potential electrocatalyst of hydrogen evolution in the TiSi family","volume":"61","author":"Li","year":"2018","journal-title":"Sci. China Mater."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1103\/RevModPhys.80.1083","article-title":"Non-Abelian anyons and topological quantum computation","volume":"80","author":"Nayak","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1640003","DOI":"10.1142\/S2010324716400038","article-title":"Dirac and Weyl materials: Fundamental aspects and some spintronics applications","volume":"6","author":"Yang","year":"2016","journal-title":"Spin"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1126\/science.1102896","article-title":"Electric field effect in atomically thin carbon films","volume":"306","author":"Novoselov","year":"2004","journal-title":"Science"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1038\/nature04235","article-title":"Experimental observation of the quantum Hall effect and Berry\u2019s phase in graphene","volume":"438","author":"Zhang","year":"2005","journal-title":"Nature"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1038\/s41586-020-2260-6","article-title":"Tunable correlated states and spin-polarized phases in twisted bilayer-bilayer graphene","volume":"583","author":"Cao","year":"2020","journal-title":"Nature"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1038\/s41586-020-2255-3","article-title":"Mapping the twist-angle disorder and Landau levels in magic-angle graphene","volume":"581","author":"Uri","year":"2020","journal-title":"Nature"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1126\/science.1245085","article-title":"Discovery of a three-dimensional topological Dirac semimetal, Na3Bi","volume":"343","author":"Liu","year":"2014","journal-title":"Science"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1126\/science.1256742","article-title":"Observation of fermi arc surface states in a topological metal","volume":"347","author":"Xu","year":"2015","journal-title":"Science"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1126\/science.aac6089","article-title":"Evidence for the chiral anomaly in the Dirac semimetal Na3Bi","volume":"350","author":"Xiong","year":"2015","journal-title":"Science"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"27002","DOI":"10.1209\/0295-5075\/114\/27002","article-title":"Anomalous conductivity tensor in the Dirac semimetal Na3Bi","volume":"114","author":"Xiong","year":"2016","journal-title":"Europhys. Lett."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"125427","DOI":"10.1103\/PhysRevB.88.125427","article-title":"Three-dimensional Dirac semimetal and quantum transport in Cd3As2","volume":"88","author":"Wang","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"27603","DOI":"10.1103\/PhysRevLett.113.027603","article-title":"Experimental realization of a three-dimensional Dirac semimetal","volume":"113","author":"Borisenko","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1038\/nmat3990","article-title":"A stable three-dimensional topological Dirac semimetal Cd3As2","volume":"13","author":"Liu","year":"2014","journal-title":"Nat. Mater."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"3786","DOI":"10.1038\/ncomms4786","article-title":"Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2","volume":"5","author":"Neupane","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1038\/nmat4143","article-title":"Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2","volume":"14","author":"Liang","year":"2015","journal-title":"Nat. Mater."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"10137","DOI":"10.1038\/ncomms10137","article-title":"Giant negative magnetoresistance induced by the chiral anomaly in individual Cd3As2 nanowires","volume":"6","author":"Li","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"10301","DOI":"10.1038\/ncomms10301","article-title":"Negative magnetoresistance in Dirac semimetal Cd3As2","volume":"7","author":"Li","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"136806","DOI":"10.1103\/PhysRevLett.119.136806","article-title":"3D quantum Hall effect of fermi arcs in topological semimetals","volume":"119","author":"Wang","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1038\/s41467-017-01438-y","article-title":"Evolution of Weyl orbit and quantum Hall effect in Dirac semimetal Cd3As2","volume":"8","author":"Zhang","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"7373","DOI":"10.1038\/ncomms8373","article-title":"A Weyl fermion semimetal with surface fermi arcs in the transition metal monopnictide TaAs class","volume":"6","author":"Huang","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1038\/nphys3426","article-title":"Observation of Weyl nodes in TaAs","volume":"11","author":"Lv","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"e1501092","DOI":"10.1126\/sciadv.1501092","article-title":"Experimental discovery of a topological Weyl semimetal state in TaP","volume":"1","author":"Xu","year":"2015","journal-title":"Sci. Adv."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1126\/science.aaa9297","article-title":"Discovery of a Weyl fermion semimetal and topological fermi arcs","volume":"349","author":"Xu","year":"2015","journal-title":"Science"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1038\/nphys3437","article-title":"Discovery of a Weyl fermion state with fermi arcs in niobium arsenide","volume":"11","author":"Xu","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1038\/nmat4457","article-title":"Evolution of the fermi surface of Weyl semimetals in the transition metal pnictide family","volume":"15","author":"Liu","year":"2016","journal-title":"Nat. Mater."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"11006","DOI":"10.1038\/ncomms11006","article-title":"Observation of Weyl nodes and fermi arcs in tantalum phosphide","volume":"7","author":"Xu","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"11615","DOI":"10.1038\/ncomms11615","article-title":"Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP","volume":"7","author":"Arnold","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1038\/nature15768","article-title":"Type-II Weyl semimetals","volume":"527","author":"Soluyanov","year":"2015","journal-title":"Nature"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"56805","DOI":"10.1103\/PhysRevLett.117.056805","article-title":"MoTe2: A type-II Weyl topological metal","volume":"117","author":"Wang","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"13973","DOI":"10.1038\/ncomms13973","article-title":"Signature of type-II Weyl semimetal phase in MoTe2","volume":"8","author":"Jiang","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_90","first-page":"31021","article-title":"Fermi arcs and their topological character in the candidate type-II Weyl semimetal MoTe2","volume":"6","author":"Tamai","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"161107","DOI":"10.1103\/PhysRevB.92.161107","article-title":"Prediction of Weyl semimetal in orthorhombic MoTe2","volume":"92","author":"Sun","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1038\/nphys3871","article-title":"Experimental observation of topological fermi arcs in type-II Weyl semimetal MoTe2","volume":"12","author":"Deng","year":"2016","journal-title":"Nat. Phys."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"66402","DOI":"10.1103\/PhysRevLett.117.066402","article-title":"Robust type-II Weyl semimetal phase in transition metal diphosphides XP2 (X=Mo, W)","volume":"117","author":"Gresch","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"201101","DOI":"10.1103\/PhysRevB.93.201101","article-title":"TaIrTe4: A ternary type-II Weyl semimetal","volume":"93","author":"Koepernik","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"121113","DOI":"10.1103\/PhysRevB.94.121113","article-title":"Observation of fermi arcs in the type-II Weyl semimetal candidate WTe2","volume":"94","author":"Wu","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1038\/nature13763","article-title":"Large, non-saturating magnetoresistance in WTe2","volume":"514","author":"Ali","year":"2014","journal-title":"Nature"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"2150","DOI":"10.1038\/s41467-017-02237-1","article-title":"Evidence for topological type-II Weyl semimetal WTe2","volume":"8","author":"Li","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"7804","DOI":"10.1038\/ncomms8804","article-title":"Superconductivity emerging from a suppressed large magnetoresistant state in tungsten ditelluride","volume":"6","author":"Kang","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"235126","DOI":"10.1103\/PhysRevB.84.235126","article-title":"Topological nodal semimetals","volume":"84","author":"Burkov","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"161101","DOI":"10.1103\/PhysRevB.95.161101","article-title":"Tunability of the topological nodal-line semimetal phase in ZrSiX-type materials (X=S, Se, Te)","volume":"95","author":"Hosen","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"201104","DOI":"10.1103\/PhysRevB.93.201104","article-title":"Observation of topological nodal fermion semimetal phase in ZrSiS","volume":"93","author":"Neupane","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"11696","DOI":"10.1038\/ncomms11696","article-title":"Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS","volume":"7","author":"Schoop","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"16602","DOI":"10.1103\/PhysRevLett.117.016602","article-title":"Evidence of topological nodal-line fermions in ZrSiSe and ZrSiTe","volume":"117","author":"Hu","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"121108","DOI":"10.1103\/PhysRevB.94.121108","article-title":"Dirac-node arc in the topological line-node semimetal HfSiS","volume":"94","author":"Takane","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"10556","DOI":"10.1038\/ncomms10556","article-title":"Topological nodal-line fermions in spin-orbit metal PbTaSe2","volume":"7","author":"Bian","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_106","first-page":"1414631","article-title":"Symmetry demanded topological nodal-line materials","volume":"3","author":"Yang","year":"2018","journal-title":"Adv. Phys-X"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1063\/1.3149495","article-title":"Periodic table for topological insulators and superconductors","volume":"1134","author":"Kitaev","year":"2009","journal-title":"AIP Conf. Proc."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"45001","DOI":"10.1088\/1751-8113\/44\/4\/045001","article-title":"Symmetries, dimensions and topological insulators: The mechanism behind the face of the Bott clock","volume":"44","author":"Stone","year":"2010","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"35005","DOI":"10.1103\/RevModPhys.88.035005","article-title":"Classification of topological quantum matter with symmetries","volume":"88","author":"Chiu","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"106802","DOI":"10.1103\/PhysRevLett.106.106802","article-title":"Topological crystalline insulators","volume":"106","author":"Fu","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"982","DOI":"10.1038\/ncomms1969","article-title":"Topological crystalline insulators in the SnTe material class","volume":"3","author":"Hsieh","year":"2012","journal-title":"Nat. Commun."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1038\/nphys2442","article-title":"Experimental realization of a topological crystalline insulator in SnTe","volume":"8","author":"Tanaka","year":"2012","journal-title":"Nat. Phys."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1038\/nmat3449","article-title":"Topological crystalline insulator states in Pb1\u2212xSnxSe","volume":"11","author":"Dziawa","year":"2012","journal-title":"Nat. Mater."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1038\/nature17410","article-title":"Hourglass fermions","volume":"532","author":"Wang","year":"2016","journal-title":"Nature"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"e1602415","DOI":"10.1126\/sciadv.1602415","article-title":"Experimental evidence of hourglass fermion in the candidate nonsymmorphic topological insulator KHgSb","volume":"3","author":"Ma","year":"2017","journal-title":"Sci. Adv."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"85304","DOI":"10.1103\/PhysRevB.90.085304","article-title":"Topological nonsymmorphic crystalline insulators","volume":"90","author":"Liu","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"161105","DOI":"10.1103\/PhysRevB.91.161105","article-title":"New classes of three-dimensional topological crystalline insulators: Nonsymmorphic and magnetic","volume":"91","author":"Fang","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"155120","DOI":"10.1103\/PhysRevB.91.155120","article-title":"Z2 topology in nonsymmorphic crystalline insulators: M\u00f6bius twist in surface states","volume":"91","author":"Shiozaki","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"eaat2374","DOI":"10.1126\/sciadv.aat2374","article-title":"New classes of topological crystalline insulators having surface rotation anomaly","volume":"5","author":"Fang","year":"2019","journal-title":"Sci. Adv."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"12001","DOI":"10.1103\/PhysRevResearch.1.012001","article-title":"Topological crystalline insulators with C2 rotation anomaly","volume":"1","author":"Zhang","year":"2019","journal-title":"Phys. Rev. Res."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1038\/nature23268","article-title":"Topological quantum chemistry","volume":"547","author":"Bradlyn","year":"2017","journal-title":"Nature"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1038\/s41467-017-00133-2","article-title":"Symmetry-based indicators of band topology in the 230 space groups","volume":"8","author":"Po","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1038\/s41586-019-0944-6","article-title":"Catalogue of topological electronic materials","volume":"566","author":"Zhang","year":"2019","journal-title":"Nature"},{"key":"ref_124","first-page":"11037","article-title":"Proof of the bulk-edge correspondence through a link between topological photonics and fluctuation-electrodynamics","volume":"9","author":"Silveirinha","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1103\/Physics.10.132","article-title":"Topological insulators turn a corner","volume":"10","author":"Parameswaransa","year":"2017","journal-title":"Physics"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"eaat0346","DOI":"10.1126\/sciadv.aat0346","article-title":"Higher-order topological insulators","volume":"4","author":"Schindler","year":"2018","journal-title":"Sci. Adv."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1038\/nature25156","article-title":"Observation of a phononic quadrupole topological insulator","volume":"555","author":"Peri","year":"2018","journal-title":"Nature"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1038\/nature25777","article-title":"A quantized microwave quadrupole insulator with topologically protected corner states","volume":"555","author":"Peterson","year":"2018","journal-title":"Nature"},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1038\/s41567-018-0246-1","article-title":"Topolectrical-circuit realization of topological corner modes","volume":"14","author":"Imhof","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1038\/s41563-018-0251-x","article-title":"Acoustic higher-order topological insulator on a kagome lattice","volume":"18","author":"Xue","year":"2019","journal-title":"Nat. Mater."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1038\/s41563-018-0252-9","article-title":"Observation of higher-order topological acoustic states protected by generalized chiral symmetry","volume":"18","author":"Ni","year":"2019","journal-title":"Nat. Mater."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1038\/s41566-018-0179-3","article-title":"Topological protection of photonic mid-gap defect modes","volume":"12","author":"Noh","year":"2018","journal-title":"Nat. Photonics"},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"204301","DOI":"10.1103\/PhysRevLett.122.204301","article-title":"Elastic higher-order topological insulator with topologically protected corner states","volume":"122","author":"Fan","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1126\/science.aah6442","article-title":"Quantized electric multipole insulators","volume":"357","author":"Benalcazar","year":"2017","journal-title":"Science"},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"245115","DOI":"10.1103\/PhysRevB.96.245115","article-title":"Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators","volume":"96","author":"Benalcazar","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_136","doi-asserted-by":"crossref","unstructured":"Raab, R.E., and De Lange, O.L. (2004). Multipole Theory in Electromagnetism: Classical, Quantum, and Symmetry Aspects, with Applications, OUP.","DOI":"10.1093\/acprof:oso\/9780198567271.001.0001"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1038\/s41566-019-0452-0","article-title":"Photonic quadrupole topological phases","volume":"13","author":"Mittal","year":"2019","journal-title":"Nat. Photonics"},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"3119","DOI":"10.1038\/s41467-020-16916-z","article-title":"Quadrupole topological photonic crystals","volume":"11","author":"He","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"2000010","DOI":"10.1002\/lpor.202000010","article-title":"Twisted quadrupole topological photonic crystals","volume":"14","author":"Zhou","year":"2020","journal-title":"Laser Photon. Rev."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"1698","DOI":"10.1103\/PhysRevLett.42.1698","article-title":"Solitons in polyacetylene","volume":"42","author":"Su","year":"1979","journal-title":"Phys. Rev. Lett."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"76803","DOI":"10.1103\/PhysRevLett.118.076803","article-title":"Novel topological phase with a zero berry curvature","volume":"118","author":"Liu","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"026801","DOI":"10.1103\/PhysRevLett.120.026801","article-title":"Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices","volume":"120","author":"Ezawa","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"205147","DOI":"10.1103\/PhysRevB.98.205147","article-title":"Second-order photonic topological insulator with corner states","volume":"98","author":"Xie","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"233902","DOI":"10.1103\/PhysRevLett.122.233902","article-title":"Direct observation of corner states in second-order topological photonic crystal slabs","volume":"122","author":"Chen","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"233903","DOI":"10.1103\/PhysRevLett.122.233903","article-title":"Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals","volume":"122","author":"Xie","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"245151","DOI":"10.1103\/PhysRevB.99.245151","article-title":"Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators","volume":"99","author":"Benalcazar","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1038\/s41566-019-0561-9","article-title":"Higher-order topological states in photonic kagome crystals with longrange interactions","volume":"14","author":"Li","year":"2020","journal-title":"Nat. Photonics"},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1038\/s41566-019-0519-y","article-title":"Corner states of light in photonic waveguides","volume":"13","author":"Kunst","year":"2019","journal-title":"Nat. Photonics"},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"246401","DOI":"10.1103\/PhysRevLett.119.246401","article-title":"Reflection-symmetric second-order topological insulators and superconductors","volume":"119","author":"Langbehn","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1038\/s41567-018-0224-7","article-title":"Higher-order topology in bismuth","volume":"14","author":"Schindler","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"186401","DOI":"10.1103\/PhysRevLett.123.186401","article-title":"Higher-order topology, monopole nodal lines, and the origin of large fermi arcs in transition metal dichalcogenides XTe2 (X= Mo, W)","volume":"123","author":"Wang","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1038\/s41567-019-0457-0","article-title":"Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator Bi2\u2013xSmxSe3","volume":"15","author":"Yue","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"256402","DOI":"10.1103\/PhysRevLett.122.256402","article-title":"Higher-order topology of the axion insulator EuIn2As2","volume":"122","author":"Xu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"216803","DOI":"10.1103\/PhysRevLett.123.216803","article-title":"Higher-order topological insulator in twisted bilayer graphene","volume":"123","author":"Park","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"66401","DOI":"10.1103\/PhysRevLett.126.066401","article-title":"Higher-order band topology in twisted Moir\u00e9 superlattice","volume":"126","author":"Liu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"256402","DOI":"10.1103\/PhysRevLett.123.256402","article-title":"Two-dimensional second-order topological insulator in graphdiyne","volume":"123","author":"Sheng","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41535-019-0206-8","article-title":"Two-dimensional higher-order topology in monolayer graphdiyne","volume":"5","author":"Lee","year":"2020","journal-title":"npj Quantum Mater."},{"key":"ref_158","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.1021\/acs.nanolett.1c04239","article-title":"Kekule lattice in graphdiyne: Coexistence of phononic and electronic second-order topological insulator","volume":"22","author":"Mu","year":"2022","journal-title":"Nano Lett."},{"key":"ref_159","doi-asserted-by":"crossref","first-page":"6492","DOI":"10.1021\/acs.nanolett.9b02719","article-title":"Two-dimensional quadrupole topological insulator in \u03b3-graphyne","volume":"19","author":"Liu","year":"2019","journal-title":"Nano Lett."},{"key":"ref_160","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1038\/s41524-022-00761-3","article-title":"Antiferromagnetic second-order topological insulator with fractional mass-kink","volume":"8","author":"Mu","year":"2022","journal-title":"npj Comput. Mater."},{"key":"ref_161","doi-asserted-by":"crossref","first-page":"10905","DOI":"10.1021\/acs.jpclett.2c02683","article-title":"Intrinsic second-order topological insulator in two-dimensional covalent organic frameworks","volume":"13","author":"Hu","year":"2022","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_162","doi-asserted-by":"crossref","first-page":"1900425","DOI":"10.1002\/lpor.201900425","article-title":"Cavity quantum electrodynamics with second-order topological corner state","volume":"14","author":"Xie","year":"2020","journal-title":"Laser Photon. Rev."},{"key":"ref_163","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1038\/s41377-020-00352-1","article-title":"Low-threshold topological nanolasers based on the second-order corner state","volume":"9","author":"Zhang","year":"2020","journal-title":"Light Sci Appl"},{"key":"ref_164","doi-asserted-by":"crossref","first-page":"1902724","DOI":"10.1002\/advs.201902724","article-title":"Higher-order topological states in surface-wave photonic crystals","volume":"7","author":"Zhang","year":"2020","journal-title":"Adv. Sci."},{"key":"ref_165","doi-asserted-by":"crossref","first-page":"73601","DOI":"10.1103\/PhysRevLett.123.073601","article-title":"Higher-order topological corner states induced by gain and loss","volume":"123","author":"Luo","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_166","doi-asserted-by":"crossref","first-page":"76801","DOI":"10.1103\/PhysRevLett.122.076801","article-title":"Second-order topological phases in non-Hermitian systems","volume":"122","author":"Liu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_167","doi-asserted-by":"crossref","first-page":"155305","DOI":"10.1103\/PhysRevB.97.155305","article-title":"Magnetic second-order topological insulators and semimetals","volume":"97","author":"Ezawa","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_168","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1038\/s41563-021-00933-4","article-title":"Higher-order topological semimetal in acoustic crystals","volume":"20","author":"Wei","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_169","doi-asserted-by":"crossref","first-page":"146401","DOI":"10.1103\/PhysRevLett.125.146401","article-title":"Higher-order Weyl semimetals","volume":"125","author":"Wang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_170","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1038\/s41563-021-00985-6","article-title":"Observation of a phononic higher-order Weyl semimetal","volume":"20","author":"Luo","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_171","doi-asserted-by":"crossref","first-page":"L060101","DOI":"10.1103\/PhysRevB.105.L060101","article-title":"Higher-order Dirac semimetal in a photonic crystal","volume":"105","author":"Wang","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_172","doi-asserted-by":"crossref","first-page":"241103","DOI":"10.1103\/PhysRevB.98.241103","article-title":"Topological quadrupolar semimetals","volume":"98","author":"Lin","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_173","doi-asserted-by":"crossref","first-page":"266804","DOI":"10.1103\/PhysRevLett.125.266804","article-title":"Higher-order Weyl semimetals","volume":"125","author":"Ghorashi","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_174","doi-asserted-by":"crossref","first-page":"26405","DOI":"10.1103\/PhysRevLett.128.026405","article-title":"Second-order real nodal-line semimetal in three-dimensional graphdiyne","volume":"128","author":"Chen","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_175","doi-asserted-by":"crossref","first-page":"184101","DOI":"10.1103\/PhysRevB.106.184101","article-title":"Design of a higher-order nodal-line semimetal in a spring-shaped acoustic topological crystal","volume":"106","author":"Zhang","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_176","doi-asserted-by":"crossref","first-page":"126403","DOI":"10.1103\/PhysRevLett.125.126403","article-title":"Boundary criticality of PT-invariant topology and second-order nodal-line semimetals","volume":"125","author":"Wang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_177","doi-asserted-by":"crossref","first-page":"56401","DOI":"10.1103\/PhysRevLett.118.056401","article-title":"PT-symmetric real Dirac fermions and semimetals","volume":"118","author":"Zhao","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_178","doi-asserted-by":"crossref","first-page":"117101","DOI":"10.1088\/1674-1056\/ab4d3b","article-title":"Stiefel\u2013Whitney classes and topological phases in band theory","volume":"28","author":"Ahn","year":"2019","journal-title":"Chin. Phys. B"},{"key":"ref_179","doi-asserted-by":"crossref","first-page":"106403","DOI":"10.1103\/PhysRevLett.121.106403","article-title":"Band topology and linking structure of nodal line semimetals with Z2 monopole charges","volume":"121","author":"Ahn","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_180","doi-asserted-by":"crossref","first-page":"081201","DOI":"10.1103\/PhysRevB.92.081201","article-title":"Topological nodal line semimetals with and without spin-orbital coupling","volume":"92","author":"Fang","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_181","doi-asserted-by":"crossref","first-page":"86804","DOI":"10.1103\/PhysRevLett.122.086804","article-title":"Helical topological edge states in a quadrupole phase","volume":"122","author":"Liu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_182","doi-asserted-by":"crossref","first-page":"13255","DOI":"10.1073\/pnas.1900527116","article-title":"Topology on a new facet of bismuth","volume":"116","author":"Hsu","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_183","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1038\/s41563-020-0721-9","article-title":"Evidence of higher-order topology in multilayer WTe2 from Josephson coupling through anisotropic hinge states","volume":"19","author":"Choi","year":"2020","journal-title":"Nat. Mater."},{"key":"ref_184","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1038\/s41563-020-00871-7","article-title":"Evidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth-halide chains","volume":"20","author":"Noguchi","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_185","doi-asserted-by":"crossref","first-page":"116801","DOI":"10.1103\/PhysRevLett.121.116801","article-title":"Topological switch between second-order topological insulators and topological crystalline insulators","volume":"121","author":"Ezawa","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_186","doi-asserted-by":"crossref","first-page":"166804","DOI":"10.1103\/PhysRevLett.124.166804","article-title":"Engineering corner states from two-dimensional topological insulators","volume":"124","author":"Ren","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_187","doi-asserted-by":"crossref","first-page":"56402","DOI":"10.1103\/PhysRevLett.125.056402","article-title":"Universal approach to magnetic second-order topological insulator","volume":"125","author":"Chen","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_188","doi-asserted-by":"crossref","first-page":"136407","DOI":"10.1103\/PhysRevLett.124.136407","article-title":"M\u00f6bius insulator and higher-order topology in MnBi2nTe3n+1","volume":"124","author":"Zhang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_189","doi-asserted-by":"crossref","first-page":"nwab170","DOI":"10.1093\/nsr\/nwab170","article-title":"Structural buckling induced higher-order topology","volume":"9","author":"Huang","year":"2021","journal-title":"Natl. Sci. Rev."},{"key":"ref_190","doi-asserted-by":"crossref","first-page":"241402","DOI":"10.1103\/PhysRevB.97.241402","article-title":"Strong and weak second-order topological insulators with hexagonal symmetry and Z3 index","volume":"97","author":"Ezawa","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_191","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1038\/nmat3828","article-title":"Spin-filtered edge states with an electrically tunable gap in a two-dimensional topological crystalline insulator","volume":"13","author":"Liu","year":"2014","journal-title":"Nat. Mater."},{"key":"ref_192","doi-asserted-by":"crossref","first-page":"183103","DOI":"10.1063\/5.0047539","article-title":"Effect of out-of-plane strain on the phonon structures and anharmonicity of twisted multilayer graphene","volume":"118","author":"Zeng","year":"2021","journal-title":"Appl. Phys. Lett."},{"key":"ref_193","doi-asserted-by":"crossref","first-page":"82511","DOI":"10.1063\/1.5040009","article-title":"Strain tunable phononic topological bandgaps in two-dimensional hexagonal boron nitride","volume":"125","author":"Jiang","year":"2019","journal-title":"J. Appl. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1651\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:39:33Z","timestamp":1760128773000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1651"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,26]]},"references-count":193,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091651"],"URL":"https:\/\/doi.org\/10.3390\/sym15091651","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,26]]}}}