Topological wave insulators: a review

Farzad Zangeneh-Nejad; Andrea Alù; Romain Fleury

doi:10.5802/crphys.3

Topological wave insulators: a review
[Isolants topologiques pour les ondes : un état de l’art]

Farzad Zangeneh-Nejad ¹ ; Andrea Alù ² ; Romain Fleury ¹

¹ Laboratory of Wave Engineering, School of Electrical Engineering, EPFL, Station 11, 1015 Lausanne, Switzerland
² Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY 10031, USA

Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 467-499.

Résumé
Abstract

Découverte à l’origine en matière condensée, la notion d’isolant topologique (IT) a été étendue à divers domaines de la physique des ondes classiques, notamment la photonique, la phononique, l’acoustique, la mécanique et les micro-ondes. Dans leur volume, comme tout autre isolant, les IT électroniques présentent une résistance excessivement élevée à l’écoulement des charges, interdisant la conduction métallique. Sur leur surface, cependant, ils présentent des états conducteurs unidirectionnels avec une protection inhérente contre certains types de défauts, au-delà de ce que pouvait laisser présager la physique du transport électronique en présence d’impuretés. Transposés aux ondes classiques, les IT ouvrent une multitude d’applications passionnantes en ingénierie, comme le routage, les lasers, le traitement du signal, les commutations, etc. avec une robustesse sans précédent face à différentes classes de défauts. Dans cet article, nous passons d’abord en revue le concept de base des IT appliqué aux ondes classiques, à partir de l’exemple simple et monodimensionnel du modèle Su–Schrieffer–Heeger (SSH). Nous passons ensuite aux IT à ondes bidimensionnelles, en discutant des analogues pour les ondes classiques des IT de Chern, d’effet Hall quantique, de spin-Hall, de Valley-Hall, et de Floquet. Enfin, nous passons en revue les développements les plus récents dans le domaine, y compris les semi-métaux de Weyl et nodaux, les isolants topologiques d’ordre supérieur et les états topologiques non linéaires auto-induits.

Originally discovered in condensed matter systems, topological insulators (TIs) have been ubiquitously extended to various fields of classical wave physics including photonics, phononics, acoustics, mechanics, and microwaves. In the bulk, like any other insulator, electronic TIs exhibit an excessively high resistance to the flow of mobile charges, prohibiting metallic conduction. On their surface, however, they support one-way conductive states with inherent protection against certain types of disorder and defects, defying the common physical wisdom of electronic transport in presence of impurities. When transposed to classical waves, TIs open a wealth of exciting engineering-oriented applications, such as robust routing, lasing, signal processing, switching, etc., with unprecedented robustness against various classes of defects. In this article, we first review the basic concept of topological order applied to classical waves, starting from the simple one-dimensional example of the Su–Schrieffer–Heeger (SSH) model. We then move on to two-dimensional wave TIs, discussing classical wave analogues of Chern, quantum Hall, spin-Hall, Valley-Hall, and Floquet TIs. Finally, we review the most recent developments in the field, including Weyl and nodal semimetals, higher-order topological insulators, and self-induced non-linear topological states.

Première publication : 2020-11-03
Publié le : 2020-12-16

DOI : 10.5802/crphys.3

Keywords: Condensed matter, Photonics, Phononics, Acoustics, Mechanics, Microwaves
Mot clés : Matière condensée, Photonique, Phononique, Acoustique, Mécanique, Micro-ondes

Affiliations des auteurs :

Farzad Zangeneh-Nejad ¹ ; Andrea Alù ² ; Romain Fleury ¹

¹ Laboratory of Wave Engineering, School of Electrical Engineering, EPFL, Station 11, 1015 Lausanne, Switzerland
² Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY 10031, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRPHYS_2020__21_4-5_467_0,
     author = {Farzad Zangeneh-Nejad and Andrea Al\`u and Romain Fleury},
     title = {Topological wave insulators: a review},
     journal = {Comptes Rendus. Physique},
     pages = {467--499},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {21},
     number = {4-5},
     year = {2020},
     doi = {10.5802/crphys.3},
     language = {en},
}

TY  - JOUR
AU  - Farzad Zangeneh-Nejad
AU  - Andrea Alù
AU  - Romain Fleury
TI  - Topological wave insulators: a review
JO  - Comptes Rendus. Physique
PY  - 2020
SP  - 467
EP  - 499
VL  - 21
IS  - 4-5
PB  - Académie des sciences, Paris
DO  - 10.5802/crphys.3
LA  - en
ID  - CRPHYS_2020__21_4-5_467_0
ER  -

%0 Journal Article
%A Farzad Zangeneh-Nejad
%A Andrea Alù
%A Romain Fleury
%T Topological wave insulators: a review
%J Comptes Rendus. Physique
%D 2020
%P 467-499
%V 21
%N 4-5
%I Académie des sciences, Paris
%R 10.5802/crphys.3
%G en
%F CRPHYS_2020__21_4-5_467_0

Farzad Zangeneh-Nejad; Andrea Alù; Romain Fleury. Topological wave insulators: a review. Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 467-499. doi : 10.5802/crphys.3. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.3/

Bibliographie
Cité par

[1] L. D. Landau On the theory of phase transitions, Ukr. J. Phys., Volume 11 (1937), pp. 19-32

[2] F. D. M. Haldane Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., Volume 61 (1988) no. 18, p. 2015 | DOI

[3] X.-G. Wen Topological orders in rigid states, Intl J. Modern Phys. B, Volume 4 (1990) no. 02, pp. 239-271 | DOI | MR

[4] M. Z. Hasan; C. L Kane Colloquium: topological insulators, Rev. Mod. Phys., Volume 82 (2010) no. 4, p. 3045 | DOI

[5] X.-L. Qi; S.-C. Zhang Topological insulators and superconductors, Rev. Mod. Phys., Volume 83 (2011) no. 4, p. 1057

[6] B. A. Bernevig; T. L. Hughes Topological Insulators and Topological Superconductors, Princeton University Press, 2013 | DOI | Zbl

[7] Y. Hatsugai Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett., Volume 71 (1993) no. 22, p. 3697 | DOI | MR | Zbl

[8] C. Wang; A. C. Potter; T. Senthil Classification of interacting electronic topological insulators in three dimensions, Science, Volume 343 (2014) no. 6171, pp. 629-631 | DOI

[9] C. Nayak et al. Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys., Volume 80 (2008) no. 3, p. 1083 | DOI | MR | Zbl

[10] J. D. Sau et al. Generic new platform for topological quantum computation using semiconductor heterostructures, Phys. Rev. Lett., Volume 104 (2010) no. 4, 040502

[11] M. Freedman et al. Topological quantum computation, Bull. Am. Math. Soc., Volume 40 (2003) no. 1, pp. 31-38 | DOI | MR

[12] D. Pesin; A. H. MacDonald Spintronics and pseudospintronics in graphene and topological insulators, Nat. Mater., Volume 11 (2012) no. 5, p. 409 | DOI

[13] L. Šmejkal et al. Topological antiferromagnetic spintronics, Nat. Phys. (2018), p. 1

[14] M. V. Berry Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A. Math. Phys. Sci., Volume 392 (1984) no. 1802, pp. 45-57 | MR | Zbl

[15] F. D. M. Haldane; S. Raghu Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett., Volume 100 (2008) no. 1, 013904

[16] L. Lu; J. D Joannopoulos; M. Soljačić Topological photonics, Nat. Photonics, Volume 8 (2014) no. 11, p. 821 | DOI

[17] A. B. Khanikaev; G. Shvets Two-dimensional topological photonics, Nat. Photonics, Volume 11 (2017) no. 12, p. 763 | DOI

[18] T. Ozawa et al. Topological photonics, Rev. Mod. Phys., Volume 91 (2019) no. 1, 015006 | MR

[19] C. Liu et al. Disorder-induced topological state transition in photonic metamaterials, Phys. Rev. Lett., Volume 119 (2017) no. 18, 183901

[20] S. Barik et al. A topological quantum optics interface, Science, Volume 359 (2018) no. 6376, pp. 666-668 | DOI | MR | Zbl

[21] W. Gao et al. Topological photonic phase in chiral hyperbolic metamaterials, Phys. Rev. Lett., Volume 114 (2015) no. 3, 037402

[22] S. Kruk et al. Edge states and topological phase transitions in chains of dielectric nanoparticles, Small, Volume 13 (2017) no. 11, 1603190 | DOI

[23] C. Yin et al. Realizing topological edge states in a silicon nitride microring-based photonic integrated circuit, Opt. Lett., Volume 41 (2016) no. 20, pp. 4791-4794 | DOI

[24] J. Noh et al. Topological protection of photonic mid-gap defect modes, Nat. Photonics, Volume 12 (2018) no. 7, p. 408 | DOI

[25] S. R. Pocock et al. Topological plasmonic chain with retardation and radiative effects, Acs Photonics, Volume 5 (2018) no. 6, pp. 2271-2279 | DOI

[26] A. V. Poshakinskiy; A. N. Poddubny Optomechanical Kerker effect, Phys. Rev. X, Volume 9 (2019) no. 1, 011008

[27] J. Perczel et al. Topological quantum optics in two-dimensional atomic arrays, Phys. Rev. Lett., Volume 119 (2017) no. 2, 023603 | DOI

[28] B. Yang et al. Direct observation of topological surface-state arcs in photonic metamaterials, Nat. Commun., Volume 8 (2017) no. 1, p. 97

[29] W. Tan et al. Photonic simulation of topological excitations in metamaterials, Sci. Rep., Volume 4 (2014), p. 3842

[30] X.-D. Chen; X.-T. He; J.-W. Dong All-dielectric layered photonic topological insulators, Laser Photonics Rev., Volume 13 (2019) no. 8, 1900091

[31] J. Yuen-Zhou et al. Topologically protected excitons in porphyrin thin films, Nat. Mater., Volume 13 (2014) no. 11, p. 1026 | DOI

[32] T. Ma; G. Shvets Scattering-free edge states between heterogeneous photonic topological insulators, Phys. Rev. B, Volume 95 (2017) no. 16, 165102

[33] D. A. Jacobs et al. Photonic topological Chern insulators based on Tellegen metacrystals, New J. Phys., Volume 17 (2015) no. 12, 125015

[34] R. E. Christiansen; F. Wang; O. Sigmund Topological insulators by topology optimization, Phys. Rev. Lett., Volume 122 (2019) no. 23, 234502 | DOI

[35] F.-F. Li et al. Topological light-trapping on a dislocation, Nat. Commun., Volume 9 (2018) no. 1, p. 2462

[36] T. Karzig et al. Topological polaritons, Phys. Rev. X, Volume 5 (2015) no. 3, 031001

[37] M. C. Rechtsman et al. Topological creation and destruction of edge states in photonic graphene, Phys. Rev. Lett., Volume 111 (2013) no. 10, 103901 | DOI

[38] Q. Huang et al. Observation of a topological edge state in the X-ray band, Laser Photonics Rev., Volume 13 (2019) no. 6, 1800339

[39] F. Liu; H.-Y. Deng; K. Wakabayashi Topological photonic crystals with zero Berry curvature, Phys. Rev. B, Volume 97 (2018) no. 3, 035442

[40] J. Han; C. Gneiting; D. Leykam Helical transport in coupled resonator waveguides, Phys. Rev. B, Volume 99 (2019) no. 22, 224201

[41] F. Lindel et al. Inducing and controlling rotation on small objects using photonic topological materials, Phys. Rev. B, Volume 98 (2018) no. 14, 144101 | DOI

[42] L. Xu et al. Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals, Opt. Express, Volume 24 (2016) no. 16, pp. 18059-18071 | DOI

[43] B.-Y. Xie et al. Photonics meets topology, Opt. Express, Volume 26 (2018) no. 19, pp. 24531-24550 | DOI

[44] M. Bello et al. Unconventional quantum optics in topological waveguide QED, Sci. Adv., Volume 5 (2019) no. 7, eaaw0297 | DOI

[45] Y. Wang et al. Topological protection of two-photon quantum correlation on a photonic chip, Optica, Volume 6 (2019) no. 8, pp. 955-960 | DOI

[46] T. Dubček et al. The Harper–Hofstadter Hamiltonian and conical diffraction in photonic lattices with grating assisted tunneling, New J. Phys., Volume 17 (2015) no. 12, 125002

[47] Z. A. Kudyshev et al. Photonic topological phase transition on demand, Nanophotonics, Volume 8 (2019), pp. 1349-1356 | DOI

[48] S. A. Sato et al. Microscopic theory for the light-induced anomalous Hall effect in graphene, Phys. Rev. B, Volume 99 (2019) no. 21, 214302

[49] Y. Li; J. Mei Double Dirac cones in two-dimensional dielectric photonic crystals, Opt. Express, Volume 23 (2015) no. 9, pp. 12089-12099 | DOI

[50] X.-C. Sun et al. Two-dimensional topological photonic systems, Prog. Quantum Electron., Volume 55 (2017), pp. 52-73 | DOI

[51] L. Wang et al. Subwavelength topological edge states based on localized spoof surface plasmonic metaparticle arrays, Opt. Express, Volume 27 (2019) no. 10, pp. 14407-14422 | DOI

[52] J. W. McIver et al. Control over topological insulator photocurrents with light polarization, Nat. Nanotechnol., Volume 7 (2012) no. 2, p. 96 | DOI

[53] Y. Long et al. Inverse design of photonic topological state via machine learning, Appl. Phys. Lett., Volume 114 (2019) no. 18, 181105 | DOI

[54] P. Di Pietro et al. Observation of Dirac plasmons in a topological insulator, Nat. Nanotechnol., Volume 8 (2013) no. 8, p. 556 | DOI

[55] F. Gao et al. Probing topological protection using a designer surface plasmon structure, Nat. Commun., Volume 7 (2016), 11619

[56] Y. Ke et al. Topological phase transitions and thouless pumping of light in photonic waveguide arrays, Laser Photonics Rev., Volume 10 (2016) no. 6, pp. 995-1001 | DOI

[57] M. S. Rider et al. A perspective on topological nanophotonics: current status and future challenges, J. Appl. Phys., Volume 125 (2019) no. 12, 120901 | DOI

[58] M. I. Shalaev; W. Walasik; N. M. Litchinitser Optically tunable topological photonic crystal, Optica, Volume 6 (2019) no. 7, pp. 839-844 | DOI

[59] S. Klembt et al. Exciton-polariton topological insulator, Nature, Volume 562 (2018) no. 7728, p. 552 | DOI

[60] X. Yao; M. Tokman; A. Belyanin Efficient nonlinear generation of THz plasmons in graphene and topological insulators, Phys. Rev. Lett., Volume 112 (2014) no. 5, 055501

[61] F. Cardano et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons, Nat. Commun., Volume 8 (2017), 15516

[62] T. Kitagawa et al. Observation of topologically protected bound states in photonic quantum walks, Nat. Commun., Volume 3 (2012), p. 882

[63] N. P. Mitchell et al. Amorphous topological insulators constructed from random point sets, Nat. Phys., Volume 14 (2018) no. 4, p. 380 | DOI

[64] E. Lustig et al. Curved-space topological phases in photonic lattices, Phys. Rev. A, Volume 96 (2017) no. 4, 041804 | DOI | MR

[65] P. N. Dyachenko et al. Controlling thermal emission with refractory epsilon-near-zero metamaterials via topological transitions, Nat. Commun., Volume 7 (2016), 11809

[66] T. Ma; G. Shvets All-Si valley-Hall photonic topological insulator, New J. Phys., Volume 18 (2016) no. 2, 025012

[67] B. Yang et al. Topological states in amorphous magnetic photonic lattices, Phys. Rev. B, Volume 99 (2019) no. 4, 045307 | DOI

[68] A. Girschik; F. Libisch; S. Rotter Percolating states in the topological Anderson insulator, Phys. Rev. B, Volume 91 (2015) no. 21, 214204 | DOI

[69] J.-P. Xia et al. Programmable coding acoustic topological insulator, Adv. Mater., Volume 30 (2018) no. 46, 1805002

[70] C. Brendel et al. Snowflake phononic topological insulator at the nanoscale, Phys. Rev. B, Volume 97 (2018) no. 2, 020102 | DOI

[71] Z. Yang et al. Strain-induced gauge field and Landau levels in acoustic structures, Phys. Rev. Lett., Volume 118 (2017) no. 19, 194301 | DOI

[72] Y. Meng et al. Designing topological interface states in phononic crystals based on the full phase diagrams, New J. Phys., Volume 20 (2018) no. 7, 073032

[73] J. Chen et al. Self-ordering induces multiple topological transitions for in-plane bulk waves in solid phononic crystals, Phys. Rev. B, Volume 98 (2018) no. 1, 014302 | DOI

[74] F. Zangeneh-Nejad; R. Fleury Active times for acoustic metamaterials, Rev. Phys. (2019), 100031 | DOI

[75] X. Ni et al. Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow, New J. Phys., Volume 17 (2015) no. 5, 053016

[76] X. Wen et al. Acoustic Landau quantization and quantum-Hall-like edge states, Nat. Phys., Volume 15 (2019), pp. 352-356 | DOI

[77] D. Zhao et al. Topological interface modes in local resonant acoustic systems, Phys. Rev. B, Volume 98 (2018) no. 1, 014110 | DOI

[78] M.-J. Tuo et al. Twist-projected two-dimensional acoustic topological insulators, Phys. Rev. B, Volume 99 (2019) no. 20, 205432

[79] L.-Y. Zheng et al. Observation of edge waves in a two-dimensional Su–Schrieffer–Heeger acoustic network, Phys. Rev. Appl., Volume 12 (2019) no. 3, 034014

[80] G. Arregui et al. Coherent generation and detection of acoustic phonons in topological nanocavities, APL Photonics, Volume 4 (2019) no. 3, 030805 | DOI

[81] G. Baardink et al. Localizing softness and stress along loops in 3D topological metamaterials, Proc. Natl Acad. Sci. USA, Volume 115 (2018) no. 3, pp. 489-494 | DOI | MR | Zbl

[82] J. Ma et al. Optically Controlled Topologically Protected Acoustic Wave Amplification, IEEE J. Selected Topics Quantum Electron., Volume 26 (2019), pp. 1-10 | DOI

[83] X. Zhang et al. Topological sound, Commun. Phys., Volume 1 (2018) no. 1, p. 97 | DOI

[84] J. Yin et al. Band transition and topological interface modes in 1D elastic phononic crystals, Sci. Rep., Volume 8 (2018) no. 1, p. 6806

[85] V. Peano et al. Topological phases of sound and light, Phys. Rev. X, Volume 5 (2015) no. 3, 031011

[86] S. Shankar; M. J. Bowick; M. C. Marchetti Topological sound and flocking on curved surfaces, Phys. Rev. X, Volume 7 (2017) no. 3, 031039

[87] B. Xie et al. Acoustic topological transport and refraction in a Kekulé lattice, Phys. Rev. Appl., Volume 11 (2019) no. 4, 044086

[88] Y. Liu et al. Pseudospins and topological effects of phonons in a Kekulé lattice, Phys. Rev. Lett., Volume 119 (2017) no. 25, 255901

[89] H. Abbaszadeh et al. Sonic Landau levels and synthetic gauge fields in mechanical metamaterials, Phys. Rev. Lett., Volume 119 (2017) no. 19, 195502 | DOI

[90] Z.-G. Chen; Y. Wu Tunable topological phononic crystals, Phys. Rev. Appl., Volume 5 (2016) no. 5, 054021

[91] A. Darabi; M. J. Leamy Reconfigurable topological insulator for elastic waves, J. Acoust. Soc. Am., Volume 146 (2019) no. 1, pp. 773-781 | DOI

[92] Z.-Y. Ong; C. H. Lee Transport and localization in a topological phononic lattice with correlated disorder, Phys. Rev. B, Volume 94 (2016) no. 13, 134203

[93] G. Gupta et al. Role of acoustic phonons in Bi $_{2}$ Se $_{3}$ topological insulator slabs: a quantum transport investigation, Phys. Rev. B, Volume 89 (2014) no. 24, 245419 | DOI

[94] Z. Yang; F. Gao; B. Zhang Topological water wave states in a one-dimensional structure, Sci. Rep., Volume 6 (2016), 29202

[95] S.-Y. Huo; J.-J. Chen; H.-B. Huang Topologically protected edge states for out-of-plane and in-plane bulk elastic waves, J. Phys.: Condens. Matter, Volume 30 (2018) no. 14, 145403

[96] G. Ma; M. Xiao; C. T. Chan Topological phases in acoustic and mechanical systems, Nat. Rev. Phys., Volume 1 (2019), pp. 281-294 | DOI

[97] K. Saha; I. Garate Phonon-induced topological insulation, Phys. Rev. B, Volume 89 (2014) no. 20, 205103 | DOI

[98] T. Lee; H. Iizuka Bragg scattering based acoustic topological transition controlled by local resonance, Phys. Rev. B, Volume 99 (2019) no. 6, 064305

[99] Y. Liu et al. Model for topological phononics and phonon diode, Phys. Rev. B, Volume 96 (2017) no. 6, 064106

[100] Z. Yu; Z. Ren; J. Lee Phononic topological insulators based on six-petal holey silicon structures, Sci. Rep., Volume 9 (2019) no. 1, p. 1805

[101] R. Süsstrunk; S. D. Huber Observation of phononic helical edge states in a mechanical topological insulator, Science, Volume 349 (2015) no. 6243, pp. 47-50 | DOI

[102] S. D. Huber Topological mechanics, Nat. Phys., Volume 12 (2016) no. 7, p. 621 | DOI

[103] H. Chen; H. Nassar; G. Huang “Topological mechanics of edge waves in Kagome lattices”, preprint, arXiv:1802.04404 (2018)

[104] B. Liu et al. Topological kinematics of origami metamaterials, Nat. Phys., Volume 14 (2018) no. 8, p. 811 | DOI

[105] A. S. Meeussen; J. Paulose; V. Vitelli Geared topological metamaterials with tunable mechanical stability, Phys. Rev. X, Volume 6 (2016) no. 4, 041029

[106] T. Tian et al. Observation of dynamical phase transitions in a topological nanomechanical system, Phys. Rev. B, Volume 100 (2019) no. 2, 024310 | DOI

[107] J. Cha; K. W. Kim; C. Daraio Experimental realization of on-chip topological nanoelectromechanical metamaterials, Nature, Volume 564 (2018) no. 7735, p. 229 | DOI

[108] Y.-W. Tsai et al. Topological phase transition in a one-dimensional elastic string system, Crystals, Volume 9 (2019) no. 6, p. 313 | DOI

[109] Y. Zhou et al. Kink-antikink asymmetry and impurity interactions in topological mechanical chains, Phys. Rev. E, Volume 95 (2017) no. 2, 022202

[110] E. Prodan et al. Dynamical Majorana edge modes in a broad class of topological mechanical systems, Nat. Commun., Volume 8 (2017), 14587

[111] J. Paulose; B. G.-g. Chen; V. Vitelli Topological modes bound to dislocations in mechanical metamaterials, Nat. Phys., Volume 11 (2015) no. 2, p. 153 | DOI

[112] J. Köpfler et al. Topologically protected twist edge states for a resonant mechanical laser-beam scanner, Phys. Rev. Appl., Volume 11 (2019) no. 3, 034059

[113] P. Deymier; K. Runge One-dimensional mass-spring chains supporting elastic waves with non-conventional topology, Crystals, Volume 6 (2016) no. 4, p. 44 | DOI

[114] R. Chaunsali; F. Li; J. Yang Stress wave isolation by purely mechanical topological phononic crystals, Sci. Rep., Volume 6 (2016), 30662

[115] M. Brandenbourger et al. Non-reciprocal robotic metamaterials, Nat. Commun., Volume 10 (2019) no. 1, pp. 1-8 | DOI

[116] S. M. Young et al. Theoretical investigation of the evolution of the topological phase of Bi $_{2}$ Se $_{3}$ under mechanical strain, Phys. Rev. B, Volume 84 (2011) no. 8, 085106 | DOI

[117] Y.-T. Wang; P.-G. Luan; S. Zhang Coriolis force induced topological order for classical mechanical vibrations, New J. Phys., Volume 17 (2015) no. 7, 073031

[118] D. Zeb. Rocklin et al. Transformable topological mechanical metamaterials, Nat. Commun., Volume 8 (2017), 14201

[119] G. Wang; H. Xu; Y.-C. Lai Mechanical topological semimetals with massless quasiparticles and a finite Berry curvature, Phys. Rev. B, Volume 95 (2017) no. 23, 235159 | DOI

[120] J. Attig et al. Topological mechanics from supersymmetry, Phys. Rev. Res., Volume 1 (2019) no. 3, 032047 | DOI

[121] Z. Xiong et al. Topological node lines in mechanical metacrystals, Phys. Rev. B, Volume 97 (2018) no. 18, 180101 | DOI

[122] H. Chen; H. Nassar; G. L. Huang A study of topological effects in 1D and 2D mechanical lattices, J. Mech. Phys. Solids, Volume 117 (2018), pp. 22-36 | DOI | MR

[123] M. Fruchart; D. Carpentier An introduction to topological insulators, C. R. Phys., Volume 14 (2013) no. 9–10, pp. 779-815 | DOI

[124] J. K. Asbóth; L. Oroszlány; A. Pályi A short course on topological insulators, Lecture Notes in Physics, Volume 919 (2016), p. 166 | MR | Zbl

[125] P. A. Kalozoumis et al. Finite-size effects on topological interface states in one-dimensional scattering systems, Phys. Rev. A, Volume 98 (2018) no. 2, 023838 | DOI

[126] C. E. Whittaker et al. Effect of photonic spin-orbit coupling on the topological edge modes of a Su–Schrieffer–Heeger chain, Phys. Rev. B, Volume 99 (2019) no. 8, 081402 | DOI

[127] L. Ge et al. Topological phase transition and interface states in hybrid plasmonic-photonic systems, J. Opt., Volume 19 (2017) no. 6, p. 06LT02

[128] C. L. Kane; T. C. Lubensky Topological boundary modes in isostatic lattices, Nat. Phys., Volume 10 (2014) no. 1, p. 39 | DOI

[129] Y. Hadad; V. Vitelli; A. Alu Solitons and propagating domain walls in topological resonator arrays, ACS Photonics, Volume 4 (2017) no. 8, pp. 1974-1979 | DOI

[130] B. Midya; L. Feng Topological multiband photonic superlattices, Phys. Rev. A, Volume 98 (2018) no. 4, 043838 | DOI

[131] Q. Cheng et al. Topologically protected interface mode in plasmonic waveguide arrays, Laser Photonics Rev., Volume 9 (2015) no. 4, pp. 392-398 | DOI

[132] C. W. Ling et al. Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles, Opt. Express, Volume 23 (2015) no. 3, pp. 2021-2031 | DOI

[133] F. Bleckmann et al. Spectral imaging of topological edge states in plasmonic waveguide arrays, Phys. Rev. B, Volume 96 (2017) no. 4, 045417 | DOI

[134] Z. Zhang et al. Experimental realization of multiple topological edge states in a 1D photonic lattice, Laser Photonics Rev., Volume 13 (2019) no. 2, 1800202 | DOI

[135] R. K. Pal; J. Vila; M. Ruzzene Topologically protected edge states in mechanical metamaterials, Adv. Crystals Elastic Metamaterials, Volume 52 (2019), p. 147 | DOI

[136] A. Altland; M. R. Zirnbauer Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B, Volume 55 (1997) no. 2, p. 1142 | DOI

[137] P. St-Jean et al. Lasing in topological edge states of a one-dimensional lattice, Nat. Photonics, Volume 11 (2017) no. 10, p. 651 | DOI

[138] M. Xiao et al. Geometric phase and band inversion in periodic acoustic systems, Nat. Phys., Volume 11 (2015) no. 3, p. 240 | DOI

[139] M. Parto et al. Edge-mode lasing in 1D topological active arrays, Phys. Rev. Lett., Volume 120 (2018) no. 11, 113901 | DOI

[140] D. Woolard; J. L. Jensen Spectral Sensing Research for Water Monitoring Applications and Frontier Science and Technology for Chemical, Biological and Radiological Defense, Volume 48, World Scientific, 2008

[141] Y. Zhang et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature, Volume 438 (2005) no. 7065, p. 201 | DOI

[142] Z. Wang et al. Observation of unidirectional backscattering-immune topological electromagnetic states, Nature, Volume 461 (2009) no. 7265, p. 772 | DOI

[143] W. Śmigaj et al. Magneto-optical circulator designed for operation in a uniform external magnetic field, Opt. Lett., Volume 35 (2010) no. 4, pp. 568-570 | DOI

[144] R. Fleury et al. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator, Science, Volume 343 (2014) no. 6170, pp. 516-519 | DOI

[145] A. B. Khanikaev et al. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice, Nat. Commun., Volume 6 (2015), p. 8260

[146] Z. Yang et al. Topological acoustics, Phys. Rev. Lett., Volume 114 (2015) no. 11, 114301 | DOI

[147] A. Souslov et al. Topological waves in fluids with odd viscosity, Phys. Rev. Lett., Volume 122 (2019) no. 12, 128001 | DOI | MR

[148] A. Souslov et al. Topological sound in active-liquid metamaterials, Nat. Phys., Volume 13 (2017) no. 11, p. 1091 | DOI

[149] Y. Ding et al. Experimental demonstration of acoustic Chern insulators, Phys. Rev. Lett., Volume 122 (2019) no. 1, 014302 | DOI

[150] L. M. Nash et al. Topological mechanics of gyroscopic metamaterials, Proc. Natl Acad. Sci. USA, Volume 112 (2015) no. 47, pp. 14495-14500 | DOI

[151] C. L. Kane; E. J. Mele Quantum spin Hall effect in graphene, Phys. Rev. Lett., Volume 95 (2005) no. 22, 226801

[152] B. A. Bernevig; T. L. Hughes; S.-C. Zhang Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science, Volume 314 (2006) no. 5806, pp. 1757-1761 | DOI

[153] R. K. Pal; M. Schaeffer; M. Ruzzene Helical edge states and topological phase transitions in phononic systems using bi-layered lattices, J. Appl. Phys., Volume 119 (2016) no. 8, 084305

[154] X.-Y. Zhu et al. Z $_{2}$ topological edge state in honeycomb lattice of coupled resonant optical waveguides with a flat band, Opt. Express, Volume 26 (2018) no. 19, pp. 24307-24317 | DOI

[155] M. L. N. Chen et al. Pseudospin-polarized topological line defects in dielectric photonic crystals, IEEE Trans. Antennas Propagation, Volume 68 (2019), pp. 609-613 | DOI

[156] H. Zhong et al. Topological insulator properties of photonic kagome helical waveguide arrays, Results Phys., Volume 12 (2019), pp. 996-1001 | DOI

[157] H. Xiong et al. Polarization-resolved edge states in terahertz topological photonic crystal, Opt. Express, Volume 27 (2019) no. 16, pp. 22819-22826 | DOI

[158] O. Gröning et al. Engineering of robust topological quantum phases in graphene nanoribbons, Nature, Volume 560 (2018) no. 7717, p. 209 | DOI

[159] A. P. Slobozhanyuk et al. Enhanced photonic spin Hall effect with subwavelength topological edge states, Laser Photonics Rev., Volume 10 (2016) no. 4, pp. 656-664 | DOI

[160] X.-C. Sun et al. Photonic topological states in a two-dimensional gyrotropic photonic crystal, Crystals, Volume 9 (2019) no. 3, p. 137 | DOI

[161] V. K. Kozin et al. Topological metamaterials based on polariton rings, Phys. Rev. B, Volume 98 (2018) no. 12, 125115 | DOI

[162] R. E. Christiansen et al. Designing photonic topological insulators with quantum-spin-Hall edge states using topology optimization, Nanophotonics, Volume 8 (2019), pp. 1363-1369 | DOI

[163] K. Y. Bliokh; D. Smirnova; F. Nori Quantum spin Hall effect of light, Science, Volume 348 (2015) no. 6242, pp. 1448-1451 | DOI | MR | Zbl

[164] C. He et al. Acoustic topological insulator and robust one-way sound transport, Nat. Phys., Volume 12 (2016) no. 12, p. 1124 | DOI

[165] S. S. Nanthakumar et al. Inverse design of quantum spin hall-based phononic topological insulators, J. Mech. Phys. Solids, Volume 125 (2019), pp. 550-571 | DOI | MR

[166] D. Jia et al. Pseudospin-dependent acoustic topological insulator by airborne sonic crystals with a triangular lattice, Appl. Phys. Express, Volume 12 (2019) no. 4, 044003

[167] H. Chen et al. Elastic quantum spin Hall effect in kagome lattices, Phys. Rev. B, Volume 98 (2018) no. 9, 094302 | DOI

[168] F. Ju; Y. Cheng; X. Liu Acoustic spin Hall-like effect in hyperbolic metamaterials controlled by the helical wave, Sci. Rep., Volume 8 (2018) no. 1, 11113

[169] B.-Z. Xia et al. Topological phononic insulator with robust pseudospin-dependent transport, Phys. Rev. B, Volume 96 (2017) no. 9, 094106

[170] Y. Liu; Y. Xu; W. Duan “Phononic topological insulators with tunable pseudospin physics”, preprint, arXiv:1809.05721 (2018)

[171] X.-F. Zhu et al. “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices”, preprint, arXiv:1508.06243 (2015)

[172] S. Wang; G. Ma; C. T. Chan Topological transport of sound mediated by spin-redirection geometric phase, Sci. Adv., Volume 4 (2018) no. 2, eaaq1475 | DOI

[173] A. Slobozhanyuk et al. Three-dimensional all-dielectric photonic topological insulator, Nat. Photonics, Volume 11 (2017) no. 2, p. 130 | DOI

[174] C. He et al. Photonic topological insulator with broken time-reversal symmetry, Proc. Natl Acad. Sci. USA, Volume 113 (2016) no. 18, pp. 4924-4928 | DOI

[175] L.-y. Feng et al. Reconfigurable topological phononic crystal slabs, Phys. Lett. A, Volume 382 (2018) no. 39, pp. 2880-2885 | DOI

[176] Zhen Gao et al. Flexible photonic topological insulator, Adv. Opt. Mater., Volume 6 (2018) no. 17, 1800532

[177] S. H. Mousavi; A. B Khanikaev; Z. Wang Topologically protected elastic waves in phononic metamaterials, Nat. Commun., Volume 6 (2015), p. 8682

[178] Y. Yang; Z. H. Hang Topological whispering gallery modes in two-dimensional photonic crystal cavities, Opt. Express, Volume 26 (2018) no. 16, pp. 21235-21241 | DOI

[179] Y. Li et al. Topological LC-circuits based on microstrips and observation of electromagnetic modes with orbital angular momentum, Nat. Commun., Volume 9 (2018) no. 1, p. 4598

[180] J. Mei; Z. Chen; Y. Wu Pseudo-time-reversal symmetry and topological edge states in two-dimensional acoustic crystals, Sci. Rep., Volume 6 (2016), 32752

[181] Y. Deng et al. Observation of zone folding induced acoustic topological insulators and the role of spin-mixing defects, Phys. Rev. B, Volume 96 (2017) no. 18, 184305 | DOI

[182] C. He et al. Topological phononic states of underwater sound based on coupled ring resonators, Appl. Phys. Lett., Volume 108 (2016) no. 3, 031904

[183] H. Dai et al. Subwavelength acoustic topological edge states realized by zone folding and the role of boundaries selection, J. Appl. Phys., Volume 124 (2018) no. 17, 175107

[184] S. Li et al. Observation of elastic topological states in soft materials, Nat. Commun., Volume 9 (2018) no. 1, p. 1370

[185] X.-D. Chen et al. Accidental double dirac cones and robust edge states in topological anisotropic photonic crystals, Laser Photonics Rev., Volume 12 (2018) no. 11, 1800073

[186] Y. Zhou; P. R. Bandaru; D. F. Sievenpiper Quantum-spin-Hall topological insulator in a spring-mass system, New J. Phys., Volume 20 (2018) no. 12, 123011

[187] A. B. Khanikaev et al. Photonic topological insulators, Nat. Mater., Volume 12 (2013) no. 3, p. 233 | DOI

[188] M. Miniaci et al. Experimental observation of topologically protected helical edge modes in patterned elastic plates, Phys. Rev. X, Volume 8 (2018) no. 3, 031074

[189] E. Martini; M. G. Silveirinha; S. Maci Exact solution for the protected TEM edge mode in a PTD-symmetric parallel-plate waveguide, IEEE Trans. Antennas Propagation, Volume 67 (2018) no. 2, pp. 1035-1044 | DOI

[190] M. G. Silveirinha P. T. D symmetry-protected scattering anomaly in optics, Phys. Rev. B, Volume 95 (2017) no. 3, 035153 | DOI

[191] L.-H. Wu; X. Hu Scheme for achieving a topological photonic crystal by using dielectric material, Phys. Rev. Lett., Volume 114 (2015) no. 22, 223901

[192] S. Yves et al. Crystalline metamaterials for topological properties at subwavelength scales, Nat. Commun., Volume 8 (2017), 16023

[193] L. Wang et al. The existence of topological edge states in honeycomb plasmonic lattices, New J. Phys., Volume 18 (2016) no. 10, 103029

[194] S. Barik et al. Two-dimensionally confined topological edge states in photonic crystals, New J. Phys., Volume 18 (2016) no. 11, 113013

[195] Y. Yang et al. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials, Phys. Rev. Lett., Volume 120 (2018) no. 21, 217401 | DOI

[196] M. I. Shalaev et al. Robust topologically protected transport in photonic crystals at telecommunication wavelengths, Nat. Nanotechnol., Volume 14 (2019) no. 1, p. 31 | DOI

[197] M. Honari-Latifpour; Y. Leila Topological plasmonic edge states in a planar array of metallic nanoparticles, Nanophotonics, Volume 8 (2019), pp. 799-806 | DOI

[198] S. Wu; Y. Wu; J. Mei Topological helical edge states in water waves over a topographical bottom, New J. Phys., Volume 20 (2018) no. 2, 023051

[199] R. Chaunsali; C.-W. Chen; J. Yang Experimental demonstration of topological waveguiding in elastic plates with local resonators, New J. Phys., Volume 20 (2018) no. 11, 113036

[200] Y. Chen; X. Liu; G. Hu Topological phase transition in mechanical honeycomb lattice, J. Mech. Phys. Solids, Volume 122 (2019), pp. 54-68 | DOI | MR

[201] S.-Y. Yu et al. Elastic pseudospin transport for integratable topological phononic circuits, Nat. Commun., Volume 9 (2018) no. 1, p. 3072

[202] Z. Zhang et al. Topological creation of acoustic pseudospin multipoles in a flow-free symmetry-broken metamaterial lattice, Phys. Rev. Lett., Volume 118 (2017) no. 8, 084303 | DOI

[203] Z. Zhang et al. Experimental verification of acoustic pseudospin multipoles in a symmetry-broken snowflakelike topological insulator, Phys. Rev. B, Volume 96 (2017) no. 24, 241306 | DOI

[204] Z.-G. Geng et al. Topologically protected edge transport of sound in coupled cavities of a modified honeycomb lattice, J. Phys.: Condens. Matter, Volume 30 (2018) no. 34, 345401

[205] S. Yves et al. Topological acoustic polaritons: robust sound manipulation at the subwavelength scale, New J. Phys., Volume 19 (2017) no. 7, 075003

[206] B. Bradlyn et al. Topological quantum chemistry, Nature, Volume 547 (2017) no. 7663, p. 298 | DOI

[207] B. Orazbayev; R. Fleury Quantitative robustness analysis of topological edge modes in C6 and Valley-Hall metamaterial waveguides, Nanophotonics, Volume 8 (2019), pp. 1433-1441 | DOI

[208] X.-T. He et al. A silicon-on-insulator slab for topological valley transport, Nat. Commun., Volume 10 (2019) no. 1, p. 872

[209] Z. Zhu et al. Negative refraction and partition in acoustic valley materials of a square lattice, Phys. Rev. Appl., Volume 12 (2019) no. 2, 024007

[210] L. Ye et al. Observation of valley-selective microwave transport in photonic crystals, Appl. Phys. Lett., Volume 111 (2017) no. 25, 251107

[211] X. Han et al. Experimental demonstration of acoustic valley hall topological insulators with the robust selection of C 3v-symmetric scatterers, Phys. Rev. Appl., Volume 12 (2019) no. 1, 014046

[212] X. Wu et al. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals, Nat. Commun., Volume 8 (2017) no. 1, p. 1304

[213] Y. Deng; Y. Jing A comparison study between topological insulators based on valley Hall and quantum spin Hall effects, J. Acoust. Soc. Am., Volume 145 (2019) no. 3, p. 1762-1762 | DOI

[214] D. Song et al. Valley vortex states and degeneracy lifting via photonic higher-band excitation, Phys. Rev. Lett., Volume 122 (2019) no. 12, 123903 | DOI

[215] Z. Gao et al. Valley surface-wave photonic crystal and its bulk/edge transport, Phys. Rev. B, Volume 96 (2017) no. 20, 201402

[216] Qiaolu Chen et al. Valley-Hall photonic topological insulators with dual-band kink states, Adv. Optical Mater. (2019), 1900036

[217] R. K. Pal; M. Ruzzene Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect, New J. Phys., Volume 19 (2017) no. 2, 025001

[218] C. He et al. Three-dimensional topological acoustic crystals with pseudospin-valley coupled saddle surface states, Nat. Commun., Volume 9 (2018) no. 1, p. 4555

[219] T.-W. Liu; F. Semperlotti Experimental evidence of robust acoustic valley Hall edge states in a nonresonant topological elastic waveguide, Phys. Rev. Appl., Volume 11 (2019) no. 1, 014040

[220] T.-W. Liu; F. Semperlotti Tunable acoustic valley-Hall edge states in reconfigurable phononic elastic waveguides, Phys. Rev. Appl., Volume 9 (2018) no. 1, 014001

[221] X. Wu et al. Interlayer topological transport and devices based on layer pseudospins in photonic valley-Hall phases, Adv. Opt. Mater., Volume 7 (2019), 1900872

[222] J. Lu et al. Valley topological phases in bilayer sonic crystals, Phys. Rev. Lett., Volume 120 (2018) no. 11, 116802

[223] J.-J. Chen et al. Topological valley transport of plate-mode waves in a homogenous thin plate with periodic stubbed surface, AIP Adv., Volume 7 (2017) no. 11, 115215

[224] Y. Shen et al. Valley-projected edge modes observed in underwater sonic crystals, Appl. Phys. Lett., Volume 114 (2019) no. 2, 023501 | DOI

[225] H. Dai; B. Xia; D. Yu Temperature-controlled tunable underwater acoustic topological insulators, J. Appl. Phys., Volume 125 (2019) no. 23, 235105

[226] J. Lu et al. Valley vortex states in sonic crystals, Phys. Rev. Lett., Volume 116 (2016) no. 9, 093901

[227] J. Vila; R. K. Pal; M. Ruzzene Observation of topological valley modes in an elastic hexagonal lattice, Phys. Rev. B, Volume 96 (2017) no. 13, 134307 | DOI

[228] H. Zhu; T.-W. Liu; F. Semperlotti Design and experimental observation of valley-Hall edge states in diatomic-graphene-like elastic waveguides, Phys. Rev. B, Volume 97 (2018) no. 17, 174301

[229] M. Miniaci et al. Valley-based splitting of topologically protected helical waves in elastic plates, Phys. Rev. B, Volume 100 (2019) no. 2, 024304 | DOI

[230] Z. Yu; Z. Ren; J. Lee Phononic topological insulators based on six-petal holey silicon structures, Sci. Rep., Volume 9 (2019) no. 1, p. 1805

[231] Z.-G. Geng et al. Mirror-symmetry induced topological valley transport along programmable boundaries in a hexagonal sonic crystal, J. Phys.: Condens. Matter, Volume 31 (2019) no. 24, 245403

[232] B.-Z. Xia et al. Observation of valleylike edge states of sound at a momentum away from the high-symmetry points, Phys. Rev. B, Volume 97 (2018) no. 15, 155124

[233] C. Chen et al. Observation of topological locally resonate and Bragg edge modes in a two-dimensional slit-typed sonic crystal, Appl. Phys. Express, Volume 12 (2019) no. 9, 097001 | DOI

[234] M. Chen et al. Tunable Dirac cones in two-dimensional acoustic metamaterials with matryoshka structure, J. Acoust. Soc. Am., Volume 146 (2019) no. 1, pp. 767-772 | DOI

[235] Y.-F. Tang et al. Topological phononic crystals with tunable interface state based on local resonance, Appl. Phys. Express, Volume 12 (2019) no. 9, 094002

[236] Y. Yang; Z. Yang; B. Zhang Acoustic valley edge states in a graphene-like resonator system, J. Appl. Phys., Volume 123 (2018) no. 9, 091713 | DOI

[237] G. G. Gentili et al. Towards topological protection based millimeter wave devices, Phys. Rev. B, Volume 100 (2019) no. 12, 125108 | DOI

[238] M. Yan et al. On-chip valley topological materials for elastic wave manipulation, Nat. Mater., Volume 17 (2018) no. 11, p. 993 | DOI

[239] F. Gao et al. Topologically protected refraction of robust kink states in valley photonic crystals, Nat. Phys., Volume 14 (2018) no. 2, p. 140 | DOI

[240] Z. Wang et al. Guiding robust valley-dependent edge states by surface acoustic waves, J. Appl. Phys., Volume 125 (2019) no. 4, 044502 | DOI

[241] X.-D. Chen et al. Valley-contrasting physics in all-dielectric photonic crystals: orbital angular momentum and topological propagation, Phys. Rev. B, Volume 96 (2017) no. 2, 020202

[242] X. Liu; Q. Guo; J. Yang Tunable acoustic valley edge states in a flow-free resonator system, Appl. Phys. Lett., Volume 115 (2019) no. 7, 074102

[243] H. Jiang et al. Acoustic valley edge states in a graphene-like system with sub-wavelength resonator, J. Acoust. Soc. Am., Volume 146 (2019) no. 1, pp. 736-741 | DOI

[244] X. Wen et al. Acoustic Dirac degeneracy and topological phase transitions realized by rotating scatterers, J. Appl. Phys., Volume 123 (2018) no. 9, 091703

[245] D. Jia et al. Acoustic topological insulator by honeycomb sonic crystals with direct and indirect band gaps, New J. Phys., Volume 20 (2018) no. 9, 093027

[246] J. Noh et al. Observation of photonic topological valley Hall edge states, Phys. Rev. Lett., Volume 120 (2018) no. 6, 063902

[247] J. Lu et al. Observation of topological valley transport of sound in sonic crystals, Nat. Phys., Volume 13 (2017) no. 4, p. 369 | DOI

[248] N. Laforge et al. Observation of topological gravity-capillary waves in a water wave crystal, New J. Phys., Volume 21 (2019) no. 8, 083031 | DOI

[249] J. K. Asbóth; B. Tarasinski; P. Delplace Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems, Phys. Rev. B, Volume 90 (2014) no. 12, 125143 | DOI

[250] V. Dal Lago; M. Atala; L. E. F. Foa Torres Floquet topological transitions in a driven one-dimensional topological insulator, Phys. Rev. A, Volume 92 (2015) no. 2, 023624 | DOI

[251] M. Fruchart Complex classes of periodically driven topological lattice systems, Phys. Rev. B, Volume 93 (2016) no. 11, 115429 | DOI

[252] N. H. Lindner; G. Refael; V. Galitski Floquet topological insulator in semiconductor quantum wells, Nat. Phys., Volume 7 (2011) no. 6, p. 490 | DOI

[253] L. He et al. Floquet Chern insulators of light, Nat. Commun., Volume 10 (2019) no. 1, pp. 1-6

[254] L. J. Maczewsky et al. Observation of photonic anomalous Floquet topological insulators, Nat. Commun., Volume 8 (2017), 13756

[255] X.-L. Lü; H. Xie Topological phases and pumps in the Su–Schrieffer–Heeger model periodically modulated in time, J. Phys.: Condens. Matter, Volume 31 (2019) no. 49, 495401

[256] Q. Cheng et al. Observation of anomalous $π$ modes in photonic Floquet engineering, Phys. Rev. Lett., Volume 122 (2019) no. 17, 173901 | DOI

[257] X. Liu; Q. Guo; J. Yang Miniaturization of Floquet topological insulators for airborne acoustics by thermal control, Appl. Phys. Lett., Volume 114 (2019) no. 5, 054102

[258] L. He et al. “Floquet Chern Insulators of Light”, preprint, arXiv:1902.08560 (2019)

[259] C. M. Dai; W. Wang; X. X. Yi Photonic Floquet topological insulators with fluctuations and disorders, Phys. Rev. A, Volume 99 (2019) no. 3, 033844

[260] Y. Long; J. Ren Floquet topological acoustic resonators and acoustic Thouless pumping, J. Acoust. Soc. Am., Volume 146 (2019) no. 1, pp. 742-747 | DOI

[261] Y.-G. Peng; Z.-G. Geng; X.-F. Zhu Topologically protected bound states in one-dimensional Floquet acoustic waveguide systems, J. Appl. Phys., Volume 123 (2018) no. 9, 091716

[262] Y.-G. Peng et al. Low-loss and broadband anomalous Floquet topological insulator for airborne sound, Appl. Phys. Lett., Volume 110 (2017) no. 17, 173505

[263] W. Zhang; X. Chen; F. Ye Plasmonic topological insulators for topological nanophotonics, Opt. Lett., Volume 42 (2017) no. 20, pp. 4063-4066 | DOI

[264] S. Mukherjee et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice, Nat. Commun., Volume 8 (2017), 13918

[265] M. Oudich et al. Space-time phononic crystals with anomalous topological edge states, Phys. Rev. Res., Volume 1 (2019) no. 3, 033069 | DOI

[266] Y. Zhang et al. Photonic Floquet topological insulators in atomic ensembles, Laser Photonics Rev., Volume 9 (2015) no. 3, pp. 331-338 | DOI

[267] H. Chen et al. Mechanical quantum Hall effect in time-modulated elastic materials, Phys. Rev. Appl., Volume 11 (2019) no. 4, 044029

[268] M. C. Rechtsman et al. Photonic Floquet topological insulators, Nature, Volume 496 (2013) no. 7444, p. 196 | DOI

[269] R. Fleury; A. B. Khanikaev; A. Alu Floquet topological insulators for sound, Nat. Commun., Volume 7 (2016), 11744

[270] M. Pasek; Y. D. Chong Network models of photonic Floquet topological insulators, Phys. Rev. B, Volume 89 (2014) no. 7, 075113 | DOI

[271] P. Delplace; M. Fruchart; C. Tauber Phase rotation symmetry and the topology of oriented scattering networks, Phys. Rev. B, Volume 95 (2017) no. 20, 205413 | DOI

[272] M. S. Rudner et al. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X, Volume 3 (2013) no. 3, 031005

[273] W. Hu et al. Measurement of a topological edge invariant in a microwave network, Phys. Rev. X, Volume 5 (2015) no. 1, 011012

[274] Y.-G. Peng et al. Experimental demonstration of anomalous Floquet topological insulator for sound, Nat. Commun., Volume 7 (2016), 13368

[275] D. Leykam; M. C. Rechtsman; Y. D. Chong Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators, Phys. Rev. Lett., Volume 117 (2016) no. 1, 013902 | DOI

[276] M. Neupane et al. Observation of topological nodal fermion semimetal phase in ZrSiS, Phys. Rev. B, Volume 93 (2016) no. 20, 201104 | DOI

[277] L. Xia et al. Observation of hourglass nodal lines in photonics, Phys. Rev. Lett., Volume 122 (2019) no. 10, 103903

[278] M. Kim et al. Topologically nontrivial photonic nodal surface in a photonic metamaterial, Phys. Rev. B, Volume 99 (2019) no. 23, 235423

[279] W. Deng et al. Nodal rings and drumhead surface states in phononic crystals, Nat. Commun., Volume 10 (2019) no. 1, p. 1769

[280] H. Weng et al. Topological node-line semimetal in three-dimensional graphene networks, Phys. Rev. B, Volume 92 (2015) no. 4, 045108 | DOI

[281] W. Gao et al. Experimental observation of photonic nodal line degeneracies in metacrystals, Nat. Commun., Volume 9 (2018) no. 1, p. 950

[282] H. C. Po; Y. Bahri; A. Vishwanath Phonon analog of topological nodal semimetals, Phys. Rev. B, Volume 93 (2016) no. 20, 205158

[283] L. Lu et al. Experimental observation of Weyl points, Science, Volume 349 (2015) no. 6248, pp. 622-624 | DOI | MR | Zbl

[284] S.-Y. Xu et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science, Volume 349 (2015) no. 6248, pp. 613-617

[285] Y. Liu; Y. Xu; W. Duan Three-dimensional topological states of phonons with tunable pseudospin physics, Research, Volume 2019 (2019), 5173580

[286] M. Kim et al. Extremely broadband topological surface states in a photonic topological metamaterial, Adv. Opt. Mater., Volume 7 (2019), 1900900

[287] L. Wang; S.-K. Jian; H. Yao Topological photonic crystal with equifrequency Weyl points, Phys. Rev. A, Volume 93 (2016) no. 6, 061801 | DOI

[288] W. Ye et al. Photonic Hall effect and helical Zitterbewegung in a synthetic Weyl system, Light: Sci. Appl., Volume 8 (2019) no. 1, p. 49 | DOI

[289] Y. Lu et al. Probing the Berry curvature and Fermi arcs of a Weyl circuit, Phys. Rev. B, Volume 99 (2019) no. 2, 020302

[290] H. Zhou et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points, Science, Volume 359 (2018) no. 6379, pp. 1009-1012 | DOI | MR | Zbl

[291] A. A. Zyuzin; V. A. Zyuzin Chiral electromagnetic waves in Weyl semimetals, Phys. Rev. B, Volume 92 (2015) no. 11, 115310 | DOI

[292] E. Goi et al. Observation of type I photonic Weyl points in optical frequencies, Laser Photonics Rev., Volume 12 (2018) no. 2, 1700271

[293] Z. Yang; B. Zhang Acoustic type-II Weyl nodes from stacking dimerized chains, Phys. Rev. Lett., Volume 117 (2016) no. 22, 224301 | DOI

[294] B. Xie et al. Experimental realization of type-II Weyl points and Fermi arcs in phononic crystal, Phys. Rev. Lett., Volume 122 (2019) no. 10, 104302

[295] X. Shi et al. Elastic Weyl points and surface arc states in three-dimensional structures, Phys. Rev. Appl., Volume 12 (2019) no. 2, 024058

[296] Z. Song; X. Dai Hear the sound of Weyl fermions, Phys. Rev. X, Volume 9 (2019) no. 2, 021053

[297] Z. Yin et al. Tunable THz generalized Weyl points, Opt. Express, Volume 27 (2019) no. 2, pp. 512-522 | DOI

[298] H. Ge et al. Experimental observation of acoustic weyl points and topological surface states, Phys. Rev. Appl., Volume 10 (2018) no. 1, 014017

[299] M. Fruchart et al. Soft self-assembly of Weyl materials for light and sound, Proc. Natl Acad. Sci. USA, Volume 115 (2018) no. 16, p. E3655-E3664 | DOI

[300] D. Liu; J. Shi Circular phonon dichroism in Weyl semimetals, Phys. Rev. Lett., Volume 119 (2017) no. 7, 075301

[301] T. Zhang et al. Double-weyl phonons in transition-metal monosilicides, Phys. Rev. Lett., Volume 120 (2018) no. 1, 016401 | DOI

[302] W. Gao et al. Photonic Weyl degeneracies in magnetized plasma, Nat. Commun., Volume 7 (2016), 12435

[303] W.-J. Chen; M. Xiao; C. T. Chan Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states, Nat. Commun., Volume 7 (2016), 13038

[304] S. M. Young et al. Dirac semimetal in three dimensions, Phys. Rev. Lett., Volume 108 (2012) no. 14, 140405 | DOI

[305] L. Lu et al. Symmetry-protected topological photonic crystal in three dimensions, Nat. Phys., Volume 12 (2016) no. 4, p. 337 | DOI

[306] Y. Yang et al. Realization of a three-dimensional photonic topological insulator, Nature, Volume 565 (2019) no. 7741, p. 622 | DOI

[307] H.-X. Wang et al. Type-ii dirac photons, NPJ Quantum Mater., Volume 2 (2017) no. 1, p. 54 | DOI

[308] Q. Guo et al. Three dimensional photonic Dirac points in metamaterials, Phys. Rev. Lett., Volume 119 (2017) no. 21, 213901

[309] J. Y. Lin et al. Line nodes, Dirac points, and Lifshitz transition in two-dimensional nonsymmorphic photonic crystals, Phys. Rev. B, Volume 96 (2017) no. 7, 075438

[310] S. Borisenko et al. Experimental realization of a three-dimensional Dirac semimetal, Phys. Rev. Lett., Volume 113 (2014) no. 2, 027603 | DOI

[311] S. M. Young; C. L. Kane Dirac semimetals in two dimensions, Phys. Rev. Lett., Volume 115 (2015) no. 12, 126803 | DOI

[312] L. Lu et al. Weyl points and line nodes in gyroid photonic crystals, Nat. Photonics, Volume 7 (2013) no. 4, p. 294 | DOI

[313] M. Xiao et al. Synthetic gauge flux and Weyl points in acoustic systems, Nat. Phys., Volume 11 (2015) no. 11, p. 920 | DOI

[314] F. Li et al. Weyl points and Fermi arcs in a chiral phononic crystal, Nat. Phys., Volume 14 (2018) no. 1, p. 30 | DOI

[315] Q. Lin et al. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension, Nat. Commun., Volume 7 (2016), 13731

[316] M. G. Silveirinha Bulk-edge correspondence for topological photonic continua, Phys. Rev. B, Volume 94 (2016) no. 20, 205105 | DOI

[317] S. Afzal; V. Van Topological phases and the bulk-edge correspondence in 2D photonic microring resonator lattices, Opt. Express, Volume 26 (2018) no. 11, pp. 14567-14577 | DOI

[318] R.-J. Slager et al. Impurity-bound states and Green’s function zeros as local signatures of topology, Phys. Rev. B, Volume 92 (2015) no. 8, 085126

[319] A. E. Hassan et al. Corner states of light in photonic waveguides, Nat. Photon., Volume 13 (2019) no. 10, pp. 697-700 | DOI

[320] B.-Y. Xie et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals, Phys. Rev. Lett., Volume 122 (2019) no. 23, 233903

[321] F. Liu; H.-Y. Deng; K. Wakabayashi Helical topological edge states in a quadrupole phase, Phys. Rev. Lett., Volume 122 (2019) no. 8, 086804

[322] X. Zhang et al. “Acoustic hierarchical topological insulators”, preprint, arXiv:1811.05514 (2018)

[323] S.-y. Huo et al. “Edge states and corner modes in second-order topological phononic crystal plates”, preprint, arXiv:1905.09731 (2019)

[324] T. Mizoguchi; H. Araki; Y. Hatsugai Higher-order topological phase in a honeycomb-lattice model with anti-Kekulé distortion, J. Phys. Soc. Japan, Volume 88 (2019) no. 10, 104703 | DOI

[325] H. Fan et al. Elastic higher-order topological insulator with topologically protected corner states, Phys. Rev. Lett., Volume 122 (2019) no. 20, 204301

[326] W. A. Benalcazar; B. A. Bernevig; T. L. Hughes Quantized electric multipole insulators, Science, Volume 357 (2017) no. 6346, pp. 61-66 | DOI | MR | Zbl

[327] S. Imhof et al. Topolectrical-circuit realization of topological corner modes, Nat. Phys., Volume 14 (2018) no. 9, p. 925 | DOI

[328] C. W. Peterson et al. A quantized microwave quadrupole insulator with topologically protected corner states, Nature, Volume 555 (2018) no. 7696, p. 346 | DOI

[329] H. Xue et al. Acoustic higher-order topological insulator on a kagome lattice, Nat. Mater., Volume 18 (2019) no. 2, p. 108 | DOI

[330] X. Ni et al. Observation of higher-order topological acoustic states protected by generalized chiral symmetry, Nat. Mater., Volume 18 (2019) no. 2, p. 113 | DOI

[331] M. Serra-Garcia et al. Observation of a phononic quadrupole topological insulator, Nature, Volume 555 (2018) no. 7696, p. 342 | DOI

[332] X. Zhang et al. Second-order topology and multidimensional topological transitions in sonic crystals, Nat. Phys., Volume 15 (2019), pp. 582-588 | DOI

[333] S. A. A. Ghorashi et al. Second-order Dirac superconductors and magnetic field induced Majorana hinge modes, Phys. Rev. B, Volume 100 (2019) no. 2, 020509

[334] Y. Ota et al. Photonic crystal nanocavity based on a topological corner state, Optica, Volume 6 (2019) no. 6, pp. 786-789 | DOI

[335] Y. Chen; X. Lu; H. Chen Effect of truncation on photonic corner states in a Kagome lattice, Opt. Lett., Volume 44 (2019) no. 17, pp. 4251-4254 | DOI

[336] B. Liu et al. Two-dimensional quadrupole topological insulator in $γ$ -graphyne, Nano Lett., Volume 19 (2019) no. 9, pp. 6492-6497 | DOI

[337] L. Zhang et al. “Higher-order photonic topological states in surface-wave photonic crystals”, preprint, arXiv:1901.07154 (2019)

[338] S. N. Kempkes et al. Robust zero-energy modes in an electronic higher-order topological insulator, Nat. Mater., Volume 18 (2019), pp. 1292-1297 | DOI

[339] X.-D. Chen et al. Direct observation of corner states in second-order topological photonic crystal slabs, Phys. Rev. Lett., Volume 122 (2019) no. 23, 233902

[340] Y. Volpez; D. Loss; J. Klinovaja Second-order topological superconductivity in $π$ -junction rashba layers, Phys. Rev. Lett., Volume 122 (2019) no. 12, 126402 | DOI

[341] S.-B. Zhang; B. Trauzettel Detection of second-order topological superconductors by Josephson junctions, Phys. Rev. Res., Volume 2 (2020) no. 1, 012018

[342] X.-L. Sheng et al. “Two-dimensional second-order topological insulator in graphdiyne”, preprint, arXiv:1904.09985 (2019)

[343] A. Agarwala; V. Juricic; B. Roy “Higher Order Topological Insulators in Amorphous Solids”, preprint, arXiv:1902.00507 (2019)

[344] S. Mittal et al. Photonic quadrupole topological phases, Nat. Photonics, Volume 13 (2019), pp. 692-696 | DOI

[345] M. Weiner et al. “Demonstration of a 3rd order hierarchy of higher order topological states in a three-dimensional acoustic metamaterial”, preprint, arXiv:1903.00428 (2019)

[346] H. Xue et al. Realization of an acoustic third-order topological insulator, Phys. Rev. Lett., Volume 122 (2019) no. 24, 244301

[347] X. Zhou et al. Optical isolation with nonlinear topological photonics, New J. Phys., Volume 19 (2017) no. 9, 095002

[348] D. Leykam; Y. D. Chong Edge solitons in nonlinear-photonic topological insulators, Phys. Rev. Lett., Volume 117 (2016) no. 14, 143901 | DOI

[349] D. R. Gulevich et al. Exploring nonlinear topological states of matter with exciton-polaritons: edge solitons in kagome lattice, Sci. Rep., Volume 7 (2017) no. 1, p. 1780

[350] R. K. Pal et al. Amplitude-dependent topological edge states in nonlinear phononic lattices, Phys. Rev. E, Volume 97 (2018) no. 3, 032209

[351] B. G.-g. Chen; N. Upadhyaya; V. Vitelli Nonlinear conduction via solitons in a topological mechanical insulator, Proc. Natl Acad. Sci. USA, Volume 111 (2014) no. 36, pp. 13004-13009 | DOI | MR | Zbl

[352] D. D. J. M. Snee; Y.-P. Ma Edge solitons in a nonlinear mechanical topological insulator, Extreme Mech. Lett., Volume 76 (2019), 100487

[353] Y. Hadad; A. B. Khanikaev; A. Alu Self-induced topological transitions and edge states supported by nonlinear staggered potentials, Phys. Rev. B, Volume 93 (2016) no. 15, 155112 | DOI

[354] Y. Hadad et al. Self-induced topological protection in nonlinear circuit arrays, Nat. Electron., Volume 1 (2018) no. 3, p. 178 | DOI

[355] D. A. Dobrykh et al. Nonlinear control of electromagnetic topological edge states, Phys. Rev. Lett., Volume 121 (2018) no. 16, 163901 | DOI

[356] R. Chaunsali; T. Georgios Self-induced topological transition in phononic crystals by nonlinearity management, Phys. Rev. B, Volume 100 (2019) no. 1, 014302 | DOI

[357] F. Zangeneh-Nejad; R. Fleury Nonlinear second-order topological insulators, Phys. Rev. Lett., Volume 123 (2019), 053902 | DOI

[358] A. Blanco-Redondo et al. Topological optical waveguiding in silicon and the transition between topological and trivial defect states, Phys. Rev. Lett., Volume 116 (2016) no. 16, 163901 | DOI

[359] L. Shen et al. Backscattering-immune one-way surface magnetoplasmons at terahertz frequencies, Opt. Express, Volume 23 (2015) no. 2, pp. 950-962 | DOI

[360] H. Xu et al. Topological energy transfer in an optomechanical system with exceptional points, Nature, Volume 537 (2016) no. 7618, p. 80 | DOI

[361] Y.-X. Shen et al. Observation of low-loss broadband supermode propagation in coupled acoustic waveguide complex, Sci. Rep., Volume 7 (2017), 45603

[362] Q. Wei et al. Experimental demonstration of topologically protected efficient sound propagation in an acoustic waveguide network, Phys. Rev. B, Volume 95 (2017) no. 9, 094305

[363] T. Jiang et al. Experimental demonstration of angular momentum-dependent topological transport using a transmission line network, Nat. Commun., Volume 10 (2019) no. 1, p. 434

[364] Y. Guo; T. Dekorsy; M. Hettich Topological guiding of elastic waves in phononic metamaterials based on 2D pentamode structures, Sci. Rep., Volume 7 (2017) no. 1, 18043

[365] O. Oltulu et al. Topological insulator based locally resonant phononic crystals: wave propagation and acoustic band gaps, Ferroelectrics, Volume 499 (2016) no. 1, pp. 123-129 | DOI

[366] R. Deshmukh et al. Long-range resonant energy transfer using optical topological transitions in metamaterials, ACS Photonics, Volume 5 (2018) no. 7, pp. 2737-2741 | DOI

[367] C.-C. Chien et al. Topological quantization of energy transport in micromechanical and nanomechanical lattices, Phys. Rev. B, Volume 97 (2018) no. 12, 125425

[368] V. Peano et al. Topological phase transitions and chiral inelastic transport induced by the squeezing of light, Nat. Commun., Volume 7 (2016), 10779

[369] S. A. H. Gangaraj; A. Nemilentsau; G. W. Hanson The effects of three-dimensional defects on one-way surface plasmon propagation for photonic topological insulators comprised of continuum media, Sci. Rep., Volume 6 (2016), 30055

[370] A. P. Slobozhanyuk et al. Subwavelength topological edge states in optically resonant dielectric structures, Phys. Rev. Lett., Volume 114 (2015) no. 12, 123901 | DOI

[371] C.-Y. Ji et al. Transport tuning of photonic topological edge states by optical cavities, Phys. Rev. A, Volume 99 (2019) no. 4, 043801

[372] W.-M. Deng et al. Vortex index identification and unidirectional propagation in Kagome photonic crystals, Nanophotonics, Volume 8 (2019) no. 5, pp. 833-840 | DOI

[373] M. He; L. Zhang; H. Wang Two-dimensional photonic crystal with ring degeneracy and its topological protected edge states, Sci. Rep., Volume 9 (2019) no. 1, p. 3815

[374] P. Wang; L. Lu; K. Bertoldi Topological phononic crystals with one-way elastic edge waves, Phys. Rev. Lett., Volume 115 (2015) no. 10, 104302 | DOI

[375] H. Dai et al. Observation of topological edge states of acoustic metamaterials at subwavelength scale, J. Phys. D: Appl. Phys., Volume 51 (2018) no. 17, 175302

[376] Y. Jin; D. Torrent; B. Djafari-Rouhani Robustness of conventional and topologically protected edge states in phononic crystal plates, Phys. Rev. B, Volume 98 (2018) no. 5, 054307

[377] I. Kim; S. Iwamoto; Y. Arakawa Topologically protected elastic waves in one-dimensional phononic crystals of continuous media, Appl. Phys. Express, Volume 11 (2017) no. 1, 017201

[378] H. Liu et al. Thermally tunable topological edge states for in-plane bulk waves in solid phononic crystals, Ultrasonics, Volume 94 (2019), pp. 227-234 | DOI

[379] B. Xie et al. Multiband asymmetric transmission of airborne sound by coded metasurfaces, Phys. Rev. Appl., Volume 7 (2017) no. 2, 024010

[380] Z.-G. Chen et al. Multiple topological phase transitions in a gyromagnetic photonic crystal, Phys. Rev. A, Volume 95 (2017) no. 4, 043827

[381] S. A. Mann; D. L. Sounas; A. Alù Broadband delay lines and nonreciprocal resonances in unidirectional waveguides, Phys. Rev. B, Volume 100 (2019), 020303

[382] Y. V. Kartashov; D. V. Skryabin Two-dimensional topological polariton laser, Phys. Rev. Lett., Volume 122 (2019) no. 8, 083902 | DOI

[383] B. Bahari et al. Nonreciprocal lasing in topological cavities of arbitrary geometries, Science, Volume 358 (2017) no. 6363, pp. 636-640 | DOI

[384] X.-C. Sun; X. Hu “Topological ring-cavity laser formed by honeycomb photonic crystals”, preprint, arXiv:1906.02464 (2019)

[385] C. Han et al. Lasing at topological edge states in a photonic crystal L3 nanocavity dimer array, Light: Sci. Appl., Volume 8 (2019) no. 1, p. 40 | DOI

[386] J.-L. Xu et al. Ultrasensitive nonlinear absorption response of large-size topological insulator and application in low-threshold bulk pulsed lasers, Sci. Rep., Volume 5 (2015), 14856

[387] H. Zhao et al. Topological hybrid silicon microlasers, Nat. Commun., Volume 9 (2018) no. 1, p. 981

[388] L. Pilozzi; C. Conti Topological cascade laser for frequency comb generation in PT-symmetric structures, Opt. Lett., Volume 42 (2017) no. 24, pp. 5174-5177 | DOI

[389] G. Harari et al. Topological insulator laser: theory, Science, Volume 359 (2018) no. 6381, eaar4003 | DOI

[390] M. Hafezi et al. Robust optical delay lines with topological protection, Nat. Phys., Volume 7 (2011) no. 11, p. 907 | DOI

[391] M. Hafezi et al. Imaging topological edge states in silicon photonics, Nat. Photon., Volume 7 (2013) no. 12, p. 1001 | DOI

[392] M. A. Bandres et al. Topological insulator laser: experiments, Science, Volume 359 (2018) no. 6381, eaar4005 | DOI

[393] F. Zangeneh-Nejad; R. Fleury Topological fano resonances, Phys. Rev. Lett., Volume 122 (2019) no. 1, 014301 | DOI

[394] A. Silva et al. Performing mathematical operations with metamaterials, Science, Volume 343 (2014) no. 6167, pp. 160-163 | DOI | MR | Zbl

[395] A. Youssefi et al. Analog computing by Brewster effect, Opt. Lett., Volume 41 (2016) no. 15, pp. 3467-3470 | DOI

[396] F. Zangeneh-Nejad; R. Fleury Performing mathematical operations using high-index acoustic metamaterials, New J. Phys., Volume 20 (2018) no. 7, 073001

[397] N. M. Estakhri; B. Edwards; N. Engheta Inverse-designed metastructures that solve equations, Science, Volume 363 (2019) no. 6433, pp. 1333-1338 | DOI | MR | Zbl

[398] F. Zangeneh-Nejad; A. Khavasi; B. Rejaei Analog optical computing by half-wavelength slabs, Opt. Commun., Volume 407 (2018), pp. 338-343 | DOI

[399] F. Zangeneh-Nejad; A. Khavasi Spatial integration by a dielectric slab and its planar graphene-based counterpart, Opt. Lett., Volume 42 (2017) no. 10, pp. 1954-1957 | DOI

[400] F. Zangeneh-Nejad; R. Fleury Topological analog signal processing, Nat. Commun., Volume 10 (2019) no. 1, p. 2058

[401] M. Ezawa Topological switch between second-order topological insulators and topological crystalline insulators, Phys. Rev. Lett., Volume 121 (2018) no. 11, 116801 | DOI

[402] R. Süsstrunk; P. Zimmermann; S. D. Huber Switchable topological phonon channels, New J. Phys., Volume 19 (2017) no. 1, 015013

[403] Y. Fan et al. Magnetization switching through giant spin-orbit torque in a magnetically doped topological insulator heterostructure, Nat. Mater., Volume 13 (2014) no. 7, p. 699 | DOI

[404] J. Han et al. Room-temperature spin-orbit torque switching induced by a topological insulator, Phys. Rev. Lett., Volume 119 (2017) no. 7, 077702

[405] N. H. D. Khang; Y. Ueda; P. N. Hai A conductive topological insulator with large spin Hall effect for ultralow power spin-orbit torque switching, Nat. Mater., Volume 17 (2018), pp. 808-813 | DOI

[406] J. Lee et al. All-fiberized, passively Q-switched 1.06 $μ$ m laser using a bulk-structured Bi $_{2}$ Te $_{3}$ topological insulator, J. Opt., Volume 16 (2014) no. 8, 085203

[407] Y. Chen et al. Large energy, wavelength widely tunable, topological insulator Q-switched erbium-doped fiber laser, IEEE J. Sel. Top. Quantum Electron., Volume 20 (2013) no. 5, pp. 315-322 | DOI

[408] H. Yu et al. Topological insulator as an optical modulator for pulsed solid-state lasers, Laser Photonics Rev., Volume 7 (2013) no. 6, p. L77-L83 | DOI

[409] X. B. Wang et al. Topological-insulator-based terahertz modulator, Sci. Rep., Volume 7 (2017) no. 1, 13486

[410] X. Xiao et al. All-electric spin modulator based on a two-dimensional topological insulator, Appl. Phys. Lett., Volume 108 (2016) no. 3, 032403 | DOI

[411] F. Hassler; A. R. Akhmerov; C. W. J. Beenakker Flat-lens focusing of electrons on the surface of a topological insulator, Phys. Rev. B, Volume 82 (2010) no. 12, 125423 | DOI

[412] H. He et al. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal, Nature, Volume 560 (2018) no. 7716, p. 61 | DOI

[413] T. Fujita; M. B. A. Jalil; S. G. Tan Topological insulator cell for memory and magnetic sensor applications, Appl. Phys. Express, Volume 4 (2011) no. 9, 094201 | DOI

[414] L. Ye et al. Observation of acoustic valley vortex states and valley-chirality locked beam splitting, Phys. Rev. B, Volume 95 (2017) no. 17, 174106

[415] P. Qiu et al. Plasmonic valley chiral states in graphene based plasmonic crystals, J. Phys. D: Appl. Phys., Volume 52 (2018) no. 1, 015102

[416] M. P. Makwana; R. Craster; S. Guenneau “Novel topological beam-splitting in photonic crystals”, preprint, arXiv:1902.00072 (2019)

[417] D. G. Rothe; E. M. Hankiewicz Tunable polarization in a beam splitter based on two-dimensional topological insulators, Phys. Rev. B, Volume 89 (2014) no. 3, 035418 | DOI

[418] M. Makwana; R. Craster; S. Guenneau Topological beam-splitting in photonic crystals, Opt. Express, Volume 27 (2019) no. 11, pp. 16088-16102 | DOI

[419] J. Lee et al. Passively Q-Switched 1.89- $μ$ m fiber laser using a bulk-structured Bi $_{2}$ Te $_{3}$ topological insulator, IEEE J. Sel. Top. Quantum Electron., Volume 21 (2014) no. 1, pp. 31-36

[420] H. Liu et al. Femtosecond pulse generation from a topological insulator mode-locked fiber laser, Opt. Express, Volume 22 (2014) no. 6, pp. 6868-6873 | DOI

[421] Z.-C. Luo et al. 2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber, Opt. Lett., Volume 38 (2013) no. 24, pp. 5212-5215 | DOI

[422] M. Liu et al. Dual-wavelength harmonically mode-locked fiber laser with topological insulator saturable absorber, IEEE Photonics Technol. Lett., Volume 26 (2014) no. 10, pp. 983-986

[423] P. Yan; R. Lin; H. Chen; H. Zhang; A. Liu; H. Yang; S. Ruan Topological insulator solution filled in photonic crystal fiber for passive mode-locked fiber laser, IEEE Photonics Technol. Lett., Volume 27 (2014) no. 3, pp. 264-267

[424] F. Bernard et al. Towards mode-locked fiber laser using topological insulators, Nonlinear Photonics, Optical Society of America, 2012 | DOI

[425] Z.-G. Geng et al. Acoustic delay-line filters based on largely distorted topological insulators, Appl. Phys. Lett., Volume 113 (2018) no. 3, 033503

[426] Z. Zhang et al. Topological acoustic delay line, Phys. Rev. Appl., Volume 9 (2018) no. 3, 034032

[427] K. Lai et al. Experimental realization of a reflections-free compact delay line based on a photonic topological insulator, Sci. Rep., Volume 6 (2016), 28453

[428] Y. Wu et al. Applications of topological photonics in integrated photonic devices, Adv. Optical Mater., Volume 5 (2017) no. 18, 1700357

[429] Y. Yang et al. “Terahertz topological photonics for on-chip communication”, preprint, arXiv:1904.04213 (2019)

[430] Z.-G. Chen et al. Acoustic frequency filter based on anisotropic topological phononic crystals, Sci. Rep., Volume 7 (2017) no. 1, 15005

[431] F. Nathan; I. Martin; G. Refael Topological frequency conversion in a driven dissipative quantum cavity, Phys. Rev. B, Volume 99 (2019) no. 9, 094311 | DOI

[432] Y. Wang et al. Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial, Nat. Commun., Volume 10 (2019) no. 1, p. 1102

[433] C. Jürß; D. Bauer High-harmonic generation in Su–Schrieffer–Heeger chains, Phys. Rev. B, Volume 99 (2019) no. 19, 195428 | DOI

[434] R. Ilan; F. De Juan; J. E. Moore Spin-based Mach-Zehnder interferometry in topological insulator p–n junctions, Phys. Rev. Lett., Volume 115 (2015) no. 9, 096802 | DOI

[435] V. Peano et al. Topological quantum fluctuations and traveling wave amplifiers, Phys. Rev. X, Volume 6 (2016) no. 4, 041026

[436] D. Malz; J. Knolle; A. Nunnenkamp Topological magnon amplification, Nat. Commun., Volume 10 (2019) no. 1, pp. 1-7 | DOI

[437] D. Leykam et al. Edge modes, degeneracies, and topological numbers in non-Hermitian systems, Phys. Rev. Lett., Volume 118 (2017) no. 4, 040401 | DOI | MR

[438] K. Esaki et al. Edge states and topological phases in non-Hermitian systems, Phys. Rev. B, Volume 84 (2011) no. 20, 205128 | DOI

[439] H. Zhao et al. Non-Hermitian topological light steering, Science, Volume 365 (2019) no. 6458, pp. 1163-1166 | DOI

[440] M. S. Rudner; L. S. Levitov Topological transition in a non-hermitian quantum walk, Phys. Rev. Lett., Volume 102 (2009) no. 6, 065703 | DOI

[441] S. Longhi; D. Gatti; G. D. Valle Robust light transport in non-Hermitian photonic lattices, Sci. Rep., Volume 5 (2015), 13376

[442] L. Li; C. H. Lee; J. Gong Geometric characterization of non-Hermitian topological systems through the singularity ring in pseudospin vector space, Phys. Rev. B, Volume 100 (2019) no. 7, 075403

[443] B. Midya; H. Zhao; L. Feng Non-Hermitian photonics promises exceptional topology of light, Nat. Commun., Volume 9 (2018) no. 1, p. 2674

[444] K. Y. Bliokh et al. Topological non-Hermitian origin of surface Maxwell waves, Nat. Commun., Volume 10 (2019) no. 1, p. 580

[445] A. Cerjan et al. Experimental realization of a Weyl exceptional ring, Nat. Photon., Volume 13 (2019), pp. 623-628 | DOI

[446] S. Malzard; C. Poli; H. Schomerus Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry, Phys. Rev. Lett., Volume 115 (2015) no. 20, 200402 | DOI

[447] E. Edvardsson; F. K. Kunst; E. J. Bergholtz Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence, Phys. Rev. B, Volume 99 (2019) no. 8, 081302 | DOI

[448] R. Chen et al. Finite-size effects in non-Hermitian topological systems, Phys. Rev. B, Volume 99 (2019) no. 15, 155431 | DOI

[449] L. Xiao et al. Observation of topological edge states in parity-time-symmetric quantum walks, Nat. Phys., Volume 13 (2017) no. 11, p. 1117 | DOI

[450] C. Sheng et al. Definite photon deflections of topological defects in metasurfaces and symmetry-breaking phase transitions with material loss, Nat. Commun., Volume 9 (2018) no. 1, p. 4271

[451] Ş. K. Özdemir et al. Parity-time symmetry and exceptional points in photonics, Nat. Mater., Volume 18 (2019), pp. 783-798 | DOI

[452] H. Shen; B. Zhen; L. Fu Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett., Volume 120 (2018) no. 14, 146402 | DOI | MR

[453] Z. Gong et al. Topological phases of non-Hermitian systems, Phys. Rev. X, Volume 8 (2018) no. 3, 031079

[454] Y. Li et al. Waveguide metatronics: lumped circuitry based on structural dispersion, Sci. Adv., Volume 2 (2016) no. 6, p. e1501790 | DOI

[455] H. M. Price et al. Four-dimensional quantum Hall effect with ultracold atoms, Phys. Rev. Lett., Volume 115 (2015) no. 19, 195303 | DOI

[456] M. Fremling et al. “A Chern insulator in ln(8) $/$ ln(3) dimensions”, preprint, arXiv:1906.07387 (2019)

[457] Y. E. Kraus; Z. Ringel; O. Zilberberg Four-dimensional quantum Hall effect in a two-dimensional quasicrystal, Phys. Rev. Lett., Volume 111 (2013) no. 22, 226401 | DOI

[458] S.-C. Zhang; J. Hu A four-dimensional generalization of the quantum Hall effect, Science, Volume 294 (2001) no. 5543, pp. 823-828 | DOI

[459] L. Yuan et al. Synthetic dimension in photonics, Optica, Volume 5 (2018) no. 11, pp. 1396-1405 | DOI

[460] F. Mei et al. Topological insulator and particle pumping in a one-dimensional shaken optical lattice, Phys. Rev. A, Volume 90 (2014) no. 6, 063638

[461] E. Lustig et al. Photonic topological insulator in synthetic dimensions, Nature, Volume 567 (2019) no. 7748, p. 356 | DOI

[462] X.-W. Luo et al. Quantum simulation of 2D topological physics in a 1D array of optical cavities, Nat. Commun., Volume 6 (2015), p. 7704

[463] F. Mei et al. Simulating Z $_{2}$ topological insulators with cold atoms in a one-dimensional optical lattice, Phys. Rev. A, Volume 85 (2012) no. 1, 013638

[464] T. Ozawa; H. M. Price Topological quantum matter in synthetic dimensions, Nat. Rev. Phys., Volume 1 (2019), pp. 349-357 | DOI

[465] G. Salerno et al. Quantized Hall conductance of a single atomic wire: a proposal based on synthetic dimensions, Phys. Rev. X, Volume 9 (2019) no. 4, 041001

[466] C.-M. Jian; C. Xu Interacting topological insulators with synthetic dimensions, Phys. Rev. X, Volume 8 (2018) no. 4, 041030

[467] J. R. M. Silva et al. Phononic topological states in 1D quasicrystals, J. Phys.: Condens. Matter, Volume 31 (2019), 505405

[468] T. Ozawa et al. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics, Phys. Rev. A, Volume 93 (2016) no. 4, 043827 | DOI

[469] D. J. Apigo et al. Observation of topological edge modes in a quasiperiodic acoustic waveguide, Phys. Rev. Lett., Volume 122 (2019) no. 9, 095501 | DOI

[470] Y. E. Kraus et al. Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett., Volume 109 (2012) no. 10, 106402 | DOI

[471] M. Verbin et al. Observation of topological phase transitions in photonic quasicrystals, Phys. Rev. Lett., Volume 110 (2013) no. 7, 076403 | DOI

[472] Y. E. Kraus; O. Zilberberg Topological equivalence between the Fibonacci quasicrystal and the Harper model, Phys. Rev. Lett., Volume 109 (2012) no. 11, 116404 | DOI

[473] S. Ganeshan; K. Sun; S. D. Sarma Topological zero-energy modes in gapless commensurate Aubry–André–Harper models, Phys. Rev. Lett., Volume 110 (2013) no. 18, 180403 | DOI

Cité par Sources :

Commentaires - Politique