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Frontiers of Physics

ISSN 2095-0462

ISSN 2095-0470(Online)

CN 11-5994/O4

邮发代号 80-965

2019 Impact Factor: 2.502

Frontiers of Physics  2023, Vol. 18 Issue (1): 13308   https://doi.org/10.1007/s11467-022-1221-y
  本期目录
Topological hinge modes in Dirac semimetals
Xu-Tao Zeng1, Ziyu Chen1, Cong Chen1,2, Bin-Bin Liu1, Xian-Lei Sheng1,3(), Shengyuan A. Yang4,5
1. School of Physics, Beihang University, Beijing 100191, China
2. Department of Physics, The University of Hong Kong, Hong Kong, China
3. Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
4. Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
5. Center for Quantum Transport and Thermal Energy Science, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China
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Abstract

Dirac semimetals (DSMs) are an important class of topological states of matter. Here, focusing on DSMs of band inversion type, we investigate their boundary modes from the effective model perspective. We show that in order to properly capture the boundary modes, k-cubic terms must be included in the effective model, which would drive an evolution of surface degeneracy manifold from a nodal line to a nodal point. Sizable k-cubic terms are also needed for better exposing the topological hinge modes in the spectrum. Using first-principles calculations, we demonstrate that this feature and the topological hinge modes can be clearly exhibited in β-CuI. We extend the discussion to magnetic DSMs and show that the time-reversal symmetry breaking can gap out the surface bands and hence is beneficial for the experimental detection of hinge modes. Furthermore, we show that magnetic DSMs serve as a parent state for realizing multiple other higher-order topological phases, including higher-order Weyl-point/nodal-line semimetals and higher-order topological insulators.

Key wordstopological    hinge    Dirac    semimetals
收稿日期: 2022-08-09      出版日期: 2022-11-17
Corresponding Author(s): Xian-Lei Sheng   
 引用本文:   
. [J]. Frontiers of Physics, 2023, 18(1): 13308.
Xu-Tao Zeng, Ziyu Chen, Cong Chen, Bin-Bin Liu, Xian-Lei Sheng, Shengyuan A. Yang. Topological hinge modes in Dirac semimetals. Front. Phys. , 2023, 18(1): 13308.
 链接本文:  
https://academic.hep.com.cn/fop/CN/10.1007/s11467-022-1221-y
https://academic.hep.com.cn/fop/CN/Y2023/V18/I1/13308
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1 Z. Hasan M., L. Kane C.. Topological insulators. Rev. Mod. Phys., 2010, 82(4): 3045
https://doi.org/10.1103/RevModPhys.82.3045
2 L. Qi X., C. Zhang S.. Topological insulators and superconductors. Rev. Mod. Phys., 2011, 83(4): 1057
https://doi.org/10.1103/RevModPhys.83.1057
3 Q. Shen S., Topological Insulators, Vol. 174, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012
4 A. Bernevig B.L. Hughes T., Topological Insulators and Topological Superconductors, Princeton University Press, 2013
5 Bansil A., Lin H., Das T.. Topological band theory. Rev. Mod. Phys., 2016, 88(2): 021004
https://doi.org/10.1103/RevModPhys.88.021004
6 K. Chiu C., C. Y. Teo J., P. Schnyder A., Ryu S.. Classification of topological quantum matter with symmetries. Rev. Mod. Phys., 2016, 88(3): 035005
https://doi.org/10.1103/RevModPhys.88.035005
7 A. Yang S.. Dirac and Weyl materials: Fundamental aspects and some spintronics applications. Spin, 2016, 6(2): 1640003
https://doi.org/10.1142/S2010324716400038
8 Dai X.. Weyl fermions go into orbit. Nat. Phys., 2016, 12(8): 727
https://doi.org/10.1038/nphys3841
9 A. Burkov A.. Topological semimetals. Nat. Mater., 2016, 15(11): 1145
https://doi.org/10.1038/nmat4788
10 P. Armitage N., J. Mele E., Vishwanath A.. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys., 2018, 90(1): 015001
https://doi.org/10.1103/RevModPhys.90.015001
11 Qi J., Liu H., Jiang H., C. Xie X.. Dephasing effects in topological insulators. Front. Phys., 2019, 14(4): 43403
https://doi.org/10.1007/s11467-019-0907-2
12 D. M. Haldane F.. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett., 1988, 61(18): 2015
https://doi.org/10.1103/PhysRevLett.61.2015
13 Wan X., M. Turner A., Vishwanath A., Y. Savrasov S.. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B, 2011, 83(20): 205101
https://doi.org/10.1103/PhysRevB.83.205101
14 M. Young S., Zaheer S., C. Y. Teo J., L. Kane C., J. Mele E., M. Rappe A.. Dirac semimetal in three dimensions. Phys. Rev. Lett., 2012, 108(14): 140405
https://doi.org/10.1103/PhysRevLett.108.140405
15 Wang Z., Sun Y., Q. Chen X., Franchini C., Xu G., Weng H., Dai X., Fang Z.. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B, 2012, 85(19): 195320
https://doi.org/10.1103/PhysRevB.85.195320
16 Wang Z., Weng H., Wu Q., Dai X., Fang Z.. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. Phys. Rev. B, 2013, 88(12): 125427
https://doi.org/10.1103/PhysRevB.88.125427
17 Li S., M. Yu Z., Yao Y., A. Yang S.. Type-II topological metals. Front. Phys., 2020, 15(4): 43201
https://doi.org/10.1007/s11467-020-0963-7
18 A. Steinberg J., M. Young S., Zaheer S., L. Kane C., J. Mele E., M. Rappe A.. Bulk Dirac points in distorted spinels. Phys. Rev. Lett., 2014, 112(3): 036403
https://doi.org/10.1103/PhysRevLett.112.036403
19 K. Liu Z., Zhou B., Zhang Y., J. Wang Z., M. Weng H., Prabhakaran D., K. Mo S., X. Shen Z., Fang Z., Dai X., Hussain Z., L. Chen Y.. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science, 2014, 343(6173): 864
https://doi.org/10.1126/science.1245085
20 K. Liu Z., Jiang J., Zhou B., J. Wang Z., Zhang Y., M. Weng H., Prabhakaran D., K. Mo S., Peng H., Dudin P., Kim T., Hoesch M., Fang Z., Dai X., X. Shen Z., L. Feng D., Hussain Z., L. Chen Y.. A stable three-dimensional topological Dirac semimetal Cd3As2. Nat. Mater., 2014, 13(7): 677
https://doi.org/10.1038/nmat3990
21 Neupane M., Y. Xu S., Sankar R., Alidoust N., Bian G., Liu C., Belopolski I., R. Chang T., T. Jeng H., Lin H., Bansil A., Chou F., Z. Hasan M.. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2. Nat. Commun., 2014, 5(1): 3786
https://doi.org/10.1038/ncomms4786
22 Jeon S., B. Zhou B., Gyenis A., E. Feldman B., Kimchi I., C. Potter A., D. Gibson Q., J. Cava R., Vishwanath A., Yazdani A.. Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2. Nat. Mater., 2014, 13(9): 851
https://doi.org/10.1038/nmat4023
23 Borisenko S., Gibson Q., Evtushinsky D., Zabolotnyy V., Büchner B., J. Cava R.. Experimental realization of a three-dimensional Dirac semimetal. Phys. Rev. Lett., 2014, 113(2): 027603
https://doi.org/10.1103/PhysRevLett.113.027603
24 Liang T., Gibson Q., N. Ali M., Liu M., J. Cava R., P. Ong N.. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2. Nat. Mater., 2015, 14(3): 280
https://doi.org/10.1038/nmat4143
25 Y. Xu S., Liu C., K. Kushwaha S., Sankar R., W. Krizan J., Belopolski I., Neupane M., Bian G., Alidoust N., R. Chang T., T. Jeng H., Y. Huang C., F. Tsai W., Lin H., P. Shibayev P., C. Chou F., J. Cava R., Z. Hasan M.. Observation of Fermi arc surface states in a topological metal. Science, 2015, 347(6219): 294
https://doi.org/10.1126/science.1256742
26 Xiong J., K. Kushwaha S., Liang T., W. Krizan J., Hirschberger M., Wang W., J. Cava R., P. Ong N.. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science, 2015, 350(6259): 413
https://doi.org/10.1126/science.aac6089
27 Kargarian M., Randeria M., M. Lu Y.. Are the surface Fermi arcs in Dirac semimetals topologically protected. Proc. Natl. Acad. Sci. USA, 2016, 113(31): 8648
https://doi.org/10.1073/pnas.1524787113
28 Zhang F., L. Kane C., J. Mele E.. Surface state magnetization and chiral edge states on topological insulators. Phys. Rev. Lett., 2013, 110(4): 046404
https://doi.org/10.1103/PhysRevLett.110.046404
29 A. Benalcazar W., A. Bernevig B., L. Hughes T.. Quantized electric multipole insulators. Science, 2017, 357(6346): 61
https://doi.org/10.1126/science.aah6442
30 Langbehn J., Peng Y., Trifunovic L., von Oppen F., W. Brouwer P.. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett., 2017, 119(24): 246401
https://doi.org/10.1103/PhysRevLett.119.246401
31 Song Z., Fang Z., Fang C.. (d−2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett., 2017, 119(24): 246402
https://doi.org/10.1103/PhysRevLett.119.246402
32 Schindler F., M. Cook A., G. Vergniory M., Wang Z., S. P. Parkin S., A. Bernevig B., Neupert T., Andrei Bernevig B., Neupert T.. Higher-order topological insulators. Sci. Adv., 2018, 4(6): eaat0346
https://doi.org/10.1126/sciadv.aat0346
33 Schindler F., Wang Z., G. Vergniory M., M. Cook A., Murani A., Sengupta S., Y. Kasumov A., Deblock R., Jeon S., Drozdov I., Bouchiat H., Guéron S., Yazdani A., A. Bernevig B., Neupert T.. Higher-order topology in bismuth. Nat. Phys., 2018, 14(9): 918
https://doi.org/10.1038/s41567-018-0224-7
34 L. Sheng X., Chen C., Liu H., Chen Z., M. Yu Z., X. Zhao Y., A. Yang S.. Two-dimensional second-order topological insulator in graphdiyne. Phys. Rev. Lett., 2019, 123(25): 256402
https://doi.org/10.1103/PhysRevLett.123.256402
35 X. Wang H., K. Lin Z., Jiang B., Y. Guo G., H. Jiang J.. Higher-order Weyl semimetals. Phys. Rev. Lett., 2020, 125(14): 146401
https://doi.org/10.1103/PhysRevLett.125.146401
36 A. A. Ghorashi S., Li T., L. Hughes T.. Higher-order Weyl semimetals. Phys. Rev. Lett., 2020, 125(26): 266804
https://doi.org/10.1103/PhysRevLett.125.266804
37 Qiu H., Xiao M., Zhang F., Qiu C.. Higher-order Dirac sonic crystals. Phys. Rev. Lett., 2021, 127(14): 146601
https://doi.org/10.1103/PhysRevLett.127.146601
38 Chen C., T. Zeng X., Chen Z., X. Zhao Y., L. Sheng X., A. Yang S.. Second-order real nodal-line semimetal in three-dimensional graphdiyne. Phys. Rev. Lett., 2022, 128(2): 026405
https://doi.org/10.1103/PhysRevLett.128.026405
39 D. Scammell H., Ingham J., Geier M., Li T.. Intrinsic first- and higher-order topological superconductivity in a doped topological insulator. Phys. Rev. B, 2022, 105(19): 195149
https://doi.org/10.1103/PhysRevB.105.195149
40 J. Wieder B., Wang Z., Cano J., Dai X., M. Schoop L., Bradlyn B., A. Bernevig B.. Strong and fragile topological Dirac semimetals with higher-order Fermi arcs. Nat. Commun., 2020, 11(1): 627
https://doi.org/10.1038/s41467-020-14443-5
41 Fang Y., Cano J.. Classification of Dirac points with higher-order Fermi arcs. Phys. Rev. B, 2021, 104(24): 245101
https://doi.org/10.1103/PhysRevB.104.245101
42 A. Bernevig B., L. Hughes T., C. Zhang S.. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 2006, 314(5806): 1757
https://doi.org/10.1126/science.1133734
43 Le C., Wu X., Qin S., Li Y., Thomale R., C. Zhang F., Hu J.. Dirac semimetal in β-CuI without surface Fermi arcs. Proc. Natl. Acad. Sci. USA, 2018, 115(33): 8311
https://doi.org/10.1073/pnas.1803599115
44 Shan Y., Li G., Tian G., Han J., Wang C., Liu S., Du H., Yang Y.. Description of the phase transitions of cuprous iodide. J. Alloys Compd., 2009, 477(1−2): 403
https://doi.org/10.1016/j.jallcom.2008.10.026
45 Tang P., Zhou Q., Xu G., C. Zhang S.. Dirac fermions in an antiferromagnetic semimetal. Nat. Phys., 2016, 12(12): 1100
https://doi.org/10.1038/nphys3839
46 Hua G., Nie S., Song Z., Yu R., Xu G., Yao K.. Dirac semimetal in type-IV magnetic space groups. Phys. Rev. B, 2018, 98: 201116(R)
https://doi.org/10.1103/PhysRevB.98.201116
47 Wang K., X. Dai J., B. Shao L., A. Yang S., X. Zhao Y.. Boundary Criticality of PT-invariant topology and second-order nodal-line semimetals. Phys. Rev. Lett., 2020, 125(12): 126403
https://doi.org/10.1103/PhysRevLett.125.126403
48 Nie S.Chen J.Yue C.Le C.Yuan D. Zhang W.Weng H., Tunable Dirac semimetals with higher-order Fermi arcs in Kagome lattices Pd3Pb2X2 (X = S, Se), arXiv: 2203.03162 (2022)
49 Kresse G., Hafner J.. Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys. Rev. B, 1994, 49(20): 14251
https://doi.org/10.1103/PhysRevB.49.14251
50 Kresse G., Furthmüller J.. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 1996, 54(16): 11169
https://doi.org/10.1103/PhysRevB.54.11169
51 E. Blöchl P.. Projector augmented-wave method. Phys. Rev. B, 1994, 50(24): 17953
https://doi.org/10.1103/PhysRevB.50.17953
52 P. Perdew J., Burke K., Ernzerhof M.. Generalized gradient approximation made simple. Phys. Rev. Lett., 1996, 77(18): 3865
https://doi.org/10.1103/PhysRevLett.77.3865
53 J. Monkhorst H., D. Pack J.. Special points for Brillouin-zone integrations. Phys. Rev. B, 1976, 13(12): 5188
https://doi.org/10.1103/PhysRevB.13.5188
54 Marzari N., Vanderbilt D.. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B, 1997, 56(20): 12847
https://doi.org/10.1103/PhysRevB.56.12847
55 Souza I., Marzari N., Vanderbilt D.. Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B, 2001, 65(3): 035109
https://doi.org/10.1103/PhysRevB.65.035109
56 P. L. Sancho M., M. L. Sancho J., Rubio J.. Quick iterative scheme for the calculation of transfer matrices: application to Mo(100). J. Phys. F Met. Phys., 1984, 14(5): 1205
https://doi.org/10.1088/0305-4608/14/5/016
57 P. L. Sancho M., M. L. Sancho J., M. L. Sancho J., Rubio J.. Highly convergent schemes for the calculation of bulk and surface Green functions. J. Phys. F Met. Phys., 1985, 15(4): 851
https://doi.org/10.1088/0305-4608/15/4/009
58 Wu Q., Zhang S., F. Song H., Troyer M., A. Soluyanov A.. WannierTools: An open-source software package for novel topological materials. Comput. Phys. Commun., 2018, 224: 405
https://doi.org/10.1016/j.cpc.2017.09.033
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