Thouless, D. J. et al. Quantized hall conductance in a two-dimensional periodic potential.Phys. Rev. Lett.49, 405–408 (1982)..

Klitzing, K. V., Dorda, G.&Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance.Phys. Rev. Lett.45, 494–497 (1980)..

Landau, L. D.&Lifshitz, E. M.Course of Theoretical PhysicsVol 5. (USSR Academy of Sciences, Moscow, 1980).

Hasan, M. Z.&Kane, C. L. Colloquium: topological insulators.Rev. Mod. Phys.82, 3045–3067 (2010)..

Liu, Y. Z., Chen, X. B.&Xu, Y. Topological phononics: from fundamental models to real materials.Adv. Funct. Mater.30, 1904784 (2020)..

Huber, S. D. Topological mechanics.Nat. Phys.12, 621–623 (2016)..

Cooper, N. R., Dalibard, J.&Spielman, I. B. Topological bands for ultracold atoms.Rev. Mod. Phys.91, 015005 (2019)..

Khanikaev, A. B. et al. Photonic topological insulators.Nat. Mater.12, 233–239 (2013)..

Bliokh, K. Y., Smirnova, D.&Nori, F. Quantum spin Hall effect of light.Science348, 1448–1451 (2015)..

Lu, L., Joannopoulos, J. D.&Soljačić, M. Topological photonics.Nat. Photonics8, 821–829 (2014)..

Xie, B. Y. et al. Photonics meets topology.Opt. Express26, 24531–24550 (2018)..

Khanikaev, A. B.&Shvets, G. Two-dimensional topological photonics.Nat. Photonics11, 763–773 (2017)..

Ozawa, T. et al. Topological photonics.Rev. Mod. Phys.91, 015006 (2019)..

Yuan, L. Q. et al. Synthetic dimension in photonics.Optica5, 1396–1405 (2018)..

Rider, M. S. et al. A perspective on topological nanophotonics: current status and future challenges.J. Appl. Phys.125, 120901 (2019)..

Sun, X. C. et al. Two-dimensional topological photonic systems.Prog. Quantum Electron.55, 52–73 (2017)..

Ota, Y. et al. Active topological photonics.Nanophotonics9, 547–567 (2020)..

Smirnova, D. et al. Nonlinear topological photonics.Appl. Phys. Rev.7, 021306 (2020)..

Liu, C. X., Zhang, S. C.&Qi, X. L. The quantum anomalous hall effect: theory and experiment.Annu. Rev. Condens. Matter Phys.7, 301–321 (2016)..

Weng, H. M. et al. Quantum anomalous Hall effect and related topological electronic states.Adv. Phys.64, 227–282 (2015)..

Berestetskii, V. B., Lifshitz, E. M.&Pitaevskii, L. P.Quantum Electrodynamics. 2nd edn. (Butterworth-Heinemann, Oxford, 1982).

Martin, G. E.Transformation Geometry: An Introduction to Symmetry. (New York: Springer, 1982).

Heaviside, O. X. I. On the forces, stresses, and fluxes of energy in the electromagnetic field.Philos. Trans. R. Soc. A183, 423–480 (1892)..

Bouchaud, J. P.&Zérah, P. G. Spontaneous resonances and universal behavior in ferrimagnets: effective-medium theory.Phys. Rev. Lett.63, 1000–1003 (1989)..

Asadchy, V., Díaz-Rubio, A.&Tretyakov, S. A. Bianisotropic metasurfaces: physics and applications.Nanophotonics7, 1069–1094 (2018)..

Lekner, J. Optical properties of isotropic chiral media.Pure Appl. Opt.5, 417–443 (1996)..

Poddubny, A. et al. Hyperbolic metamaterials.Nat. Photonics7, 948–957 (2013)..

Simovski, C. R.&Belov, P. A. Low-frequency spatial dispersion in wire media.Phys. Rev. E70, 046616 (2004)..

Rayleigh, L. XXVI. On the remarkable phenomenon of crystalline reflexion described by Prof. Stokes.Lond. Edinb. Dublin Philos. Mag. J. Sci.26, 256–265 (1888)..

John, S. Strong localization of photons in certain disordered dielectric superlattices.Phys. Rev. Lett.58, 2486–2489 (1987)..

Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics.Phys. Rev. Lett.58, 2059–2062 (1987)..

Joannopoulos, J. D., Meade, R. D.&Winn, J. N.Photonic Crystals: Molding the Flow of Light. (Princeton University Press, Princeton, NJ, 1995).

Kane, C. L. Topological band theory and the$$ {\Bbb{Z_{2}}}$$invariant.Contemp. Concepts Condens. Matter Sci.6, 3–34 (2013)..

Jackiw, R.&Rebbi, C. Solitons with fermion number 1/2.Phys. Rev. D13, 3398–3409 (1976)..

Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect.Phys. Rev. Lett.71, 3697–3700 (1993)..

Kohmoto, M. Topological invariant and the quantization of the Hall conductance.Ann. Phys.160, 343–354 (1985)..

Berry, M. V. Quantal phase factors accompanying adiabatic changes.Proc. R. Soc. A392, 45–57 (1984)..

Haldane, F. D. M.&Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry.Phys. Rev. Lett.100, 013904 (2008)..

Raghu, S.&Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals.Phys. Rev. A78, 033834 (2008)..

Wang, Z. et al. Reflection-free one-way edge modes in a gyromagnetic photonic crystal.Phys. Rev. Lett.100, 013905 (2008)..

Wang, Z. et al. Observation of unidirectional backscattering-immune topological electromagnetic states.Nature461, 772–775 (2009)..

Rechtsman, M. C. et al. Photonic Floquet topological insulators.Nature496, 196–200 (2013)..

Fang, K. J., Yu, Z. F.&Fan, S. H. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation.Nat. Photonics6, 782–787 (2012)..

Jin, D. F. et al. Topological magnetoplasmon.Nat. Commun.7, 13486 (2016)..

Jin, D. F. et al. Infrared topological plasmons in graphene.Phys. Rev. Lett.118, 245301 (2017)..

Kramers, H. A. Théorie générale de la rotation paramagnétique dans les cristaux.Proc. R. Neth. Acad. Arts Sci.33, 959–972 (1930)..

Van Mechelen, T.&Jacob, Z. Universal spin-momentum locking of evanescent waves.Optica3, 118–126 (2016)..

Kalhor, F., Thundat, T.&Jacob, Z. Universal spin-momentum locked optical forces.Appl. Phys. Lett.108, 061102 (2016)..

Kane, C. L.&Mele, E. J.Z2topological order and the quantum spin hall effect.Phys. Rev. Lett.95, 146802 (2005)..

Roy, R.Z2classification of quantum spin Hall systems: an approach using time-reversal invariance.Phys. Rev. B79, 195321 (2009)..

Van Mechelen, T.&Jacob, Z. Photonic Dirac monopoles and skyrmions: spin-1 quantization [Invited].Opt. Mater. Express9, 95–111 (2019)..

Piao, X. J., Yu, S.&Park, N. Design of transverse spinning of light with globally unique handedness.Phys. Rev. Lett.120, 203901 (2018)..

Wu, L. H.&Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material.Phys. Rev. Lett.114, 223901 (2015)..

Hafezi, M. et al. Imaging topological edge states in silicon photonics.Nat. Photonics7, 1001–1005 (2013)..

Mittal, S. et al. Photonic anomalous quantum hall effect.Phys. Rev. Lett.123, 043201 (2019)..

Barik, S. etal. A topological quantum optics interface.Science359, 666–668 (2018)..

Jalali Mehrabad, M. et al. A semiconductor topological photonic ring resonator.Appl. Phys. Lett.116, 061102 (2020)..

Saba, M., Wong, S., Elman, M., Oh, S. S.&Hess, O. Nature of topological protection in photonic spin and valley Hall insulators.Phys. Rev. B101, 054307 (2020)..

Ma, T.&Shvets, G. All-Si valley-Hall photonic topological insulator.New J. Phys.18, 025012 (2016)..

Wu, X. X. et al. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals.Nat. Commun.8, 1304 (2017)..

Gao, Z. et al. Valley surface-wave photonic crystal and its bulk/edge transport.Phys. Rev. B96, 201402 (2017)..

Wong, S. et al. Gapless unidirectional photonic transport using all-dielectric kagome lattices.Phys. Rev. Res.2, 012011 (2020)..

Noh, J. et al. Observation of photonic topological valley hall edge states.Phys. Rev. Lett.120, 063902 (2018)..

He, X. T. et al. A silicon-on-insulator slab for topological valley transport.Nat. Commun.10, 872 (2019)..

Shalaev, M. I. et al. Robust topologically protected transport in photonic crystals at telecommunication wavelengths.Nat. Nanotechnol.14, 31–34 (2019)..

Chen, Q. L. et al. Valley-hall photonic topological insulators with dual-band kink states.Adv. Opt. Mater.7, 1900036 (2019)..

Zeng, Y. Q. et al. Electrically pumped topological laser with valley edge modes.Nature578, 246–250 (2020)..

Dong, J. W. et al. Valley photonic crystals for control of spin and topology.Nat. Mater.16, 298–302 (2017)..

Xue, H. R. et al. Spin-valley-controlled photonic topological insulator. Preprint at:https://arxiv.org/abs/1811.00393https://arxiv.org/abs/1811.00393(2018).

Floquet, G. Sur les équations différentielles linéaires à coefficients périodiques.Annales scientifiques de. l'École Norm. Supérieure12, 47–88 (1883)..

Kittle, C.Introduction to Solid State Physics. 5th edn. (Wiley, New York, 1976).

Noether, E. Invariant variation problems.Transp. Theory Stat. Phys.1, 186–207 (1971)..

Kitagawa, T. et al. Topological characterization of periodically driven quantum systems.Phys. Rev. B82, 235114 (2010)..

Lindner, N. H., Refael, G.&Galitski, V. Floquet topological insulator in semiconductor quantum wells.Nat. Phys.7, 490–495 (2011)..

Gu, Z. H. et al. Floquet spectrum and transport through an irradiated graphene ribbon.Phys. Rev. Lett.107, 216601 (2011)..

Oka, T.&Aoki, H. Photovoltaic Hall effect in graphene.Phys. Rev. B79, 081406 (2009)..

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

Lin, Q. et al. A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension.Sci. Adv.4, eaat2774 (2018)..

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

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

Zak, J. Berry's phase for energy bands in solids.Phys. Rev. Lett.62, 2747–2750 (1989)..

Liu, F.&Wakabayashi, K. Novel topological phase with a zero berry curvature.Phys. Rev. Lett.118, 076803 (2017)..

Fu, L., Kane, C. L.&Mele, E. J. Topological insulators in three dimensions.Phys. Rev. Lett.98, 106803 (2007)..

Moore, J. E.&Balents, L. Topological invariants of time-reversal-invariant band structures.Phys. Rev. B75, 121306 (2007)..

Roy, R. Topological phases and the quantum spin Hall effect in three dimensions.Phys. Rev. B79, 195322 (2009)..

Yang, Y. H. et al. Realization of a three-dimensional photonic topological insulator.Nature565, 622–626 (2019)..

Nakahara, M.Geometry, Topology and Physics. 2nd edn. (CRC Press, Bristol, 2003).

Zhang, S. C.&Hu, J. P. A four-dimensional generalization of the quantum hall effect.Science294, 823–828 (2001)..

Halperin, B. I. Possible states for a three-dimensional electron gas in a strong magnetic field.Jpn J. Appl. Phys.26, 1913 (1987)..

Kohmoto, M., Halperin, B. I.&Wu, Y. S. Diophantine equation for the three-dimensional quantum Hall effect.Phys. Rev. B45, 13488–13493 (1992)..

De Blas, A. Z. E.&Axel, F. Diophantine equation for the 3D transport coefficients of Bloch electrons in a strong tilted magnetic field with quantum Hall effect.J. Phys. Condens. Matter16, 7673–7708 (2004)..

Störmer, H. L. et al. Quantization of the Hall effect in an anisotropic three-dimensional electronic system.Phys. Rev. Lett.56, 85–88 (1986)..

Tang, F. D. et al. Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5.Nature569, 537–541 (2019)..

Lu, L. et al. Symmetry-protected topological photonic crystal in three dimensions.Nat. Phys.12, 337–340 (2016)..

Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface.Nat. Phys.5, 398–402 (2009)..

Zhang, H. J. et al. Topological insulators in Bi2Se3, Bi2Te3and Sb2Te3with a single Dirac cone on the surface.Nat. Phys.5, 438–442 (2009)..

Yannopapas, V. Gapless surface states in a lattice of coupled cavities: a photonic analog of topological crystalline insulators.Phys. Rev. B84, 195126 (2011)..

Fu, L. Topological crystalline insulators.Phys. Rev. Lett.106, 106802 (2011)..

Slobozhanyuk, A. et al. Three-dimensional all-dielectric photonic topological insulator.Nat. Photonics11, 130–136 (2017)..

Ochiai, T. Gapless surface states originating from accidentally degenerate quadratic band touching in a three-dimensional tetragonal photonic crystal.Phys. Rev. A96, 043842 (2017)..

Wan, X. G. et al. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates.Phys. Rev. B83, 205101 (2011)..

Burkov, A. A.&Balents, L. Weyl semimetal in a topological insulator multilayer.Phys. Rev. Lett.107, 127205 (2011)..

Xu, S. Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs.Science349, 613–617 (2015)..

Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs.Phys. Rev. X5, 031013 (2015)..

Yang, K. Y., Lu, Y. M.&Ran, Y. Quantum Hall effects in a Weyl semimetal: possible application in pyrochlore iridates.Phys. Rev. B84, 075129 (2011)..

Soluyanov, A. A. et al. Type-II Weyl semimetals.Nature527, 495–498 (2015)..

Weyl, H. Gravitation and the electron,Proc. Natl Acad. Sci. USA15, 323 (1929)..

Lu, L. et al. Weyl points and line nodes in gyroid photonic crystals.Nat. Photonics7, 294–299 (2013)..

Yang, B. et al. Direct observation of topological surface-state arcs in photonic metamaterials.Nat. Commun.8, 97 (2017)..

Wang, D. Y. et al. Photonic Weyl points due to broken time-reversal symmetry in magnetized semiconductor.Nat. Phys.15, 1150–1155 (2019)..

Gao, W. L. et al. Photonic Weyl degeneracies in magnetized plasma.Nat. Commun.7, 12435 (2016)..

Yang, Z. J. et al. Weyl points in a magnetic tetrahedral photonic crystal.Opt. Express25, 15772–15777 (2017)..

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

Lu, L. et al. Experimental observation of Weyl points.Science349, 622–624 (2015)..

Yang, B. et al. Ideal Weyl points and helicoid surface states in artificial photonic crystal structures.Science359, 1013–1016 (2018)..

Noh, J. et al. Experimental observation of optical Weyl points and Fermi arc-like surface states.Nat. Phys.13, 611–617 (2017)..

Xiao, M., Lin, Q.&Fan, S. H. Hyperbolic Weyl point in reciprocal chiral metamaterials.Phys. Rev. Lett.117, 057401 (2016)..

Jia, H. W. et al. Observation of chiral zero mode in inhomogeneous three-dimensional Weyl metamaterials.Science363, 148–151 (2019)..

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

Potter, A. C., Kimchi, I.&Vishwanath, A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals.Nat. Commun.5, 5161 (2014)..

Nielsen, H. B.&Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal.Phys. Lett. B130, 389–396 (1983)..

Mansha, S.&Chong, Y. D. Robust edge states in amorphous gyromagnetic photonic lattices.Phys. Rev. B96, 121405 (2017)..

Skirlo, S. A., Lu, L.&Soljačić, M. Multimode one-way waveguides of large chern numbers.Phys. Rev. Lett.113, 113904 (2014)..

Liu, G. G. et al. Observation of an unpaired photonic Dirac point.Nat. Commun.11, 1873 (2020)..

Fu, J. X., Liu, R. J.&Li, Z. Y. Experimental demonstration of tunable gyromagnetic photonic crystals controlled by dc magnetic fields.EPL89, 64003 (2010)..

Poo, Y. et al. Experimental realization of self-guiding unidirectional electromagnetic edge states.Phys. Rev. Lett.106, 093903 (2011)..

Skirlo, S. A. et al. Experimental observation of large chern numbers in photonic crystals.Phys. Rev. Lett.115, 253901 (2015)..

Fu, J. X., Liu, R. J.&Li, Z. Y. Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces.Appl. Phys. Lett.97, 041112 (2010)..

Gangaraj, S. A. H.&Hanson, G. W. Topologically protected unidirectional surface states in biased ferrites: duality and application to directional couplers.IEEE Antennas Wirel. Propag. Lett.16, 449–452 (2017)..

Tan, W. et al. Photonic simulation of topological superconductor edge state and zero-energy mode at a vortex.Sci. Rep.4, 7381 (2015)..

Gooth, J. et al. Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP.Nature547, 324–327 (2017)..

Kharzeev, D. E. et al. Giant photocurrent in asymmetric Weyl semimetals from the helical magnetic effect.Phys. Rev. B98, 014305 (2018)..

O'Brien, T. E., Diez, M.&Beenakker, C. W. J. Magnetic breakdown and klein tunneling in a type-II Weyl semimetal.Phys. Rev. Lett.116, 236401 (2016)..

Ochiai, T. Floquet–Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice.J. Phys.28, 425501 (2016)..

Wang, H. L., Zhou, L. W.&Chong, Y. D. Floquet Weyl phases in a three-dimensional network model.Phys. Rev. B93, 144114 (2016)..

Zhang, Y.&Zhu, Y. Y. Generation of Weyl points in coupled optical microdisk-resonator arrays via external modulation.Phys. Rev. A96, 013811 (2017)..

Bravo-Abad, J. et al. Weyl points in photonic-crystal superlattices.2D Mater.2, 034013 (2015)..

Wang, L. Y., Jian, S. K.&Yao, H. Topological photonic crystal with equifrequency Weyl points.Phys. Rev. A93, 061801 (2016)..

Goi, E. et al. Observation of type I photonic Weyl points in optical frequencies.Laser Photonics Rev.12, 1700271 (2018)..

Chang, M. L. et al. Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals.Phys. Rev. B95, 125136 (2017)..

Wang, H. X. et al. Three-dimensional photonic Dirac points stabilized by point group symmetry.Phys. Rev. B93, 235155 (2016)..

Wang, H. X. et al. Type-II Dirac photons.npj Quantum Mater.2, 54 (2017)..

Sakoda, K. Dirac cone in two- and three-dimensional metamaterials.Opt. Express20, 3898–3917 (2012)..

Armitage, N. P., Mele, E. J.&Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018)..

Fang, C. et al. Topological semimetals with helicoid surface states.Nat. Phys.12, 936–941 (2016)..

Slobozhanyuk, A. P. et al. Experimental demonstration of topological effects in bianisotropic metamaterials.Sci. Rep.6, 22270 (2016)..

Ma, T. et al. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides.Phys. Rev. Lett.114, 127401 (2015)..

Xiao, B. et al. Exciting reflectionless unidirectional edge modes in a reciprocal photonic topological insulator medium.Phys. Rev. B94, 195427 (2016)..

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

Cheng, X. J. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator.Nat. Mater.15, 542–548 (2016)..

Slobozhanyuk, A. et al. Near-field imaging of spin-locked edge states in all-dielectric topological metasurfaces.Appl. Phys. Lett.114, 031103 (2019)..

Gao, W. L. et al. Topological photonic phase in chiral hyperbolic metamaterials.Phys. Rev. Lett.114, 037402 (2015)..

Kim, M. et al. Topologically nontrivial photonic nodal surface in a photonic metamaterial.Phys. Rev. B99, 235423 (2019)..

Chern, R. L.&Yu, Y. Z. Chiral surface waves on hyperbolic-gyromagnetic metamaterials.Opt. Express25, 11801–11812 (2017)..

Shiu, R. C. et al. Photonic Chern insulators made of gyromagnetic hyperbolic metamaterials.Phys. Rev. Mater.4, 065202 (2020)..

Guo, Q. H. et al. Three dimensional photonic dirac points in metamaterials.Phys. Rev. Lett.119, 213901 (2017)..

Guo, Q. H. et al. Observation of three-dimensional photonic dirac points and spin-polarized surface arcs.Phys. Rev. Lett.122, 203903 (2019)..

Yu, Y. Z.&Chern, R. L. Photonic topological phases in dispersive metamaterials.Sci. Rep.8, 17881 (2018)..

Liu, C. X. et al. Disorder-induced topological state transition in photonic metamaterials.Phys. Rev. Lett.119, 183901 (2017)..

Xu, X. D. et al. Spin and pseudospins in layered transition metal dichalcogenides.Nat. Phys.10, 343–350 (2014)..

Lu, J. Y. et al. Valley topological phases in bilayer sonic crystals.Phys. Rev. Lett.120, 116802 (2018)..

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

Chen, X. D., He, X. T.&Dong, J. W. All-dielectric layered photonic topological insulators.Laser Photonics Rev.13, 1900091 (2019)..

Bisharat, D. J.&Sievenpiper, D. F. Electromagnetic-dual metasurfaces for topological states along a 1D interface.Laser Photonics Rev.13, 1900126 (2019)..

Burkov, A. A., Hook, M. D.&Balents, L. Topological nodal semimetals.Phys. Rev. B84, 235126 (2011)..

Bi, R. et al. Nodal-knot semimetals.Phys. Rev. B96, 201305 (2017)..

Kawakami, T.&Hu, X. Symmetry-guaranteed nodal-line semimetals in an fcc lattice.Phys. Rev. B96, 235307 (2017)..

Lin, J. Y. et al. Line nodes, Dirac points, and Lifshitz transition in two-dimensional nonsymmorphic photonic crystals.Phys. Rev. B96, 075438 (2017)..

Gao, W. L. et al. Experimental observation of photonic nodal line degeneracies in metacrystals.Nat. Commun.9, 950 (2018)..

Yan, Q. H. et al. Experimental discovery of nodal chains.Nat. Phys.14, 461–464 (2018)..

Takahashi, R., Hirayama, M.&Murakami, S. Spinless hourglass nodal-line semimetals.Phys. Rev. B96, 155206 (2017)..

Fu, B. T. et al. Hourglasslike nodal net semimetal in Ag2BiO3.Phys. Rev. B98, 075146 (2018)..

Xia, L. B. et al. Observation of hourglass nodal lines in photonics.Phys. Rev. Lett.122, 103903 (2019)..

Yang, E. C. et al. Observation of non-Abelian nodal links in photonics. Preprint at:https://arxiv.org/abs/1910.01345https://arxiv.org/abs/1910.01345(2019).

Zhong, C. Y. et al. Towards three-dimensional Weyl-surface semimetals in graphene networks.Nanoscale8, 7232–7239 (2016)..

Bzdušek, T.&Sigrist, M. Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems.Phys. Rev. B96, 155105 (2017)..

Türker, O.&Moroz, S. Weyl nodal surfaces.Phys. Rev. B97, 075120 (2018)..

Xiao, M. et al. Experimental demonstration of acoustic semimetal with topologically charged nodal surface.Sci. Adv.6, eaav2360 (2020)..

Yang, Y. H. et al. Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal.Nat. Commun.10, 5185 (2019)..

König, A.&Mermin, N. D. Electronic level degeneracy in nonsymmorphic periodic or aperiodic crystals.Phys. Rev. B56, 13607–13610 (1997)..

Van Mechelen, T.&Jacob, Z. Nonlocal topological electromagnetic phases of matter.Phys. Rev. B99, 205146 (2019)..

Van Mechelen, T.&Jacob, Z. Unidirectional Maxwellian spin waves.Nanophotonics8, 1399–1416 (2019)..

Van Mechelen, T.&Jacob, Z. Quantum gyroelectric effect: photon spin-1 quantization in continuum topological bosonic phases.Phys. Rev. A98, 023842 (2018)..

Berdyugin, A. I. et al. Measuring Hall viscosity of graphene's electron fluid.Science364, 162–165 (2019)..

Barnett, S. M. Optical Dirac equation.New J. Phys.16, 093008 (2014)..

Bialynicki-Birula, I. inCoherence and Quantum Optics VII(eds Eberly, J. H. et al.) 313–322 (Springer, Boston, MA, 1996).

Hughes, T. L., Leigh, R. G.&Parrikar, O. Torsional anomalies, Hall viscosity, and bulk-boundary correspondence in topological states.Phys. Rev. D88, 025040 (2013)..

Parameswaran, S. A.&Wan, Y. Topological insulators turn a corner.Physics10, 132 (2017)..

Benalcazar, W. A., Bernevig, B. A.&Hughes, T. L. Quantized electric multipole insulators.Science357, 61–66 (2017)..

Benalcazar, W. A., Bernevig, B. A.&Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators.Phys. Rev. B96, 245115 (2017)..

Langbehn, J. et al. Reflection-symmetric second-order topological insulators and superconductors.Phys. Rev. Lett.119, 246401 (2017)..

Song, Z. D., Fang, Z.&Fang, C. (d-2)-dimensional edge states of rotation symmetry protected topological states.Phys. Rev. Lett.119, 246402 (2017)..

Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices.Phys. Rev. Lett.120, 026801 (2018)..

Lin, M.&Hughes, T. L. Topological quadrupolar semimetals.Phys. Rev. B98, 241103 (2018)..

Noh, J. et al. Topological protection of photonic mid-gap defect modes.Nat. Photonics12, 408–415 (2018)..

Kunst, F. K., Van Miert, G.&Bergholtz, E. J. Lattice models with exactly solvable topological hinge and corner states.Phys. Rev. B97, 241405 (2018)..

El Hassan, A. et al. Corner states of light in photonic waveguides.Nat. Photonics13, 697–700 (2019)..

Li, M. Y. et al. Higher-order topological states in photonic kagome crystals with long-range interactions.Nat. Photonics14, 89–94 (2020)..

Mittal, S. et al. Photonic quadrupole topological phases.Nat. Photonics13, 692–696 (2019)..

Xie, B. Y. et al. Second-order photonic topological insulator with corner states.Phys. Rev. B98, 205147 (2018)..

Su, W. P., Schrieffer, J. R.&Heeger, A. J. Solitons in polyacetylene.Phys. Rev. Lett.42, 1698–1701 (1979)..

Chen, X. D. et al. Direct observation of corner states in second-order topological photonic crystal slabs.Phys. Rev. Lett.122, 233902 (2019)..

Xie, B. Y. et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals.Phys. Rev. Lett.122, 233903 (2019)..

Kim, M.&Rho, J. Topological edge and corner states in a two-dimensional photonic Su-Schrieffer-Heeger lattice.Nanophotonics(2019).https://doi.org/10.1515/nanoph-2019-0451https://doi.org/10.1515/nanoph-2019-0451..

Peterson, C. W. et al. A quantized microwave quadrupole insulator with topologically protected corner states.Nature555, 346–350 (2018)..

He, L. et al. Quadrupole topological photonic crystals.Nat. Commun.11, 3119 (2020)..

Liu, T. et al. Second-order topological phases in non-hermitian systems.Phys. Rev. Lett.122, 076801 (2019)..

Jing, H. et al. High-order exceptional points in optomechanics.Sci. Rep.7, 3386 (2017)..

Ota, Y. et al. Photonic crystal nanocavity based on a topological corner state.Optica6, 786–789 (2019)..

Zhang, L. et al. Higher-order photonic topological states in surface-wave photonic crystals.Adv. Sci.7, 1902724 (2020)..

Yang, S. A., Pan, H.&Zhang, F. Chirality-dependent hall effect in Weyl semimetals.Phys. Rev. Lett.115, 156603 (2015)..

Jiang, Q. D. et al. Topological imbert-fedorov shift in Weyl semimetals.Phys. Rev. Lett.115, 156602 (2015)..

Wang, L. Y.&Jian, S. K. Imbert-Fedorov shift in Weyl semimetals: dependence on monopole charge and intervalley scattering.Phys. Rev. B96, 115448 (2017)..

Jiang, Q. D. et al. Chiral wave-packet scattering in Weyl semimetals.Phys. Rev. B93, 195165 (2016)..

Chattopadhyay, U. et al. Fermi-Arc-induced vortex structure in Weyl beam shifts.Phys. Rev. Lett.122, 066602 (2019)..

Wang, Q. et al. Optical interface states protected by synthetic Weyl points.Phys. Rev. X7, 031032 (2017)..

Slobozhanyuk, A. P. et al. Enhanced photonic spin Hall effect with subwavelength topological edge states.Laser Photonics Rev.10, 656–664 (2016)..

Ye, W. M. et al. Photonic Hall effect and helicalZitterbewegungin a synthetic Weyl system.Light Sci. Appl.8, 49 (2019)..

Zhou, M. et al. Electromagnetic scattering laws in Weyl systems.Nat. Commun.8, 1388 (2017)..

Peano, V. et al. Topological quantum fluctuations and traveling wave amplifiers.Phys. Rev. X6, 041026 (2016)..

García-Elcano, I., González-Tudela, A.&Bravo-Abad, J. Quantum electrodynamics near photonic Weyl points. Preprint at:https://arxiv.org/abs/1903.07513https://arxiv.org/abs/1903.07513(2019).

Zhou, X. et al. Optical isolation with nonlinear topological photonics.New J. Phys.19, 095002 (2017)..

Guo, J. et al. Enhanced nonlinear optical responses of graphene in multi-frequency topological edge modes.Opt. Express27, 32746–32763 (2019)..

Wang, Y. et al. Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial.Nat. Commun.10, 1102 (2019)..

Kruk, S. et al. Third-harmonic generation from photonic topological states in zigzag arrays of silicon nanodisks. InProc. Frontiers in OpticsFTh4B.2 (2017).

Yang, Y. H. et al. Terahertz topological photonics for on-chip communication.Nat. Photonics14, 446–451 (2020)..

Zhang, L. et al. Manipulation of topological valley kink states in an ultrathin substrate-integrated photonic circuitry.Laser Photonics Rev.13, 1900159 (2019)..

Shalaev, M. I. et al. Reconfigurable topological photonic crystal.New J. Phys.20, 023040 (2018)..

Li, C. et al. Thermo‐optical tunable ultracompact chip‐integrated 1D photonic topological insulator.Adv. Opt. Mater.6, 1701071 (2018)..

Cao, T. et al. Dynamically reconfigurable topological edge state in phase change photonic crystals.Sci. Bull.64, 814–822 (2019)..

Kudyshev, Z. A. et al. Tuning topology of photonic systems with transparent conducting oxides.ACS Photonics6, 1922–1930 (2019)..

Shalaev, M. I., Walasik, W.&Litchinitser, N. M. Optically tunable topological photonic crystal.Optica6, 839–844 (2019)..

Hayran, Z., Gangaraj, S. A. H.&Monticone, F. Topologically protected broadband rerouting of propagating waves around complex objects.Nanophotonics8, 1371–1378 (2019)..

Zhang, W. F. et al. Finite-dimensional bistable topological insulators: from small to large.Laser Photonics Rev.13, 1900198 (2019)..

Pilozzi, L. et al. Machine learning inverse problem for topological photonics.Commun. Phys.1, 57 (2018)..

Araki, H., Mizoguchi, T.&Hatsugai, Y. Phase diagram of a disordered higher-order topological insulator: a machine learning study.Phys. Rev. B99, 085406 (2019)..

Zhang, P. F., Shen, H. T.&Zhai, H. Machine learning topological invariants with neural networks.Phys. Rev. Lett.120, 066401 (2018)..

Deng, D. L., Li, X. P.&Das Sarma, S. Machine learning topological states.Phys. Rev. B96, 195145 (2017)..

Sun, N. et al. Deep learning topological invariants of band insulators.Phys. Rev. B98, 085402 (2018)..

Carrasquilla, J. Neural networks identify topological phases.Physics10, 56 (2017)..

Schnyder, A. P. et al. Classification of topological insulators and superconductors in three spatial dimensions.Phys. Rev. B78, 195125 (2008)..

De Nittis, G.&Lein, M. On the role of symmetries in the theory of photonic crystals.Ann. Phys.350, 568–587 (2014)..

Zhao, Y. X.&Wang, Z. D. Topological classification and stability of fermi surfaces.Phys. Rev. Lett.110, 240404 (2013)..

Watanabe, H.&Lu, L. Space group theory of photonic bands.Phys. Rev. Lett.121, 263903 (2018)..

Zhou, H. Y. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points.Science359, 1009–1012 (2018)..

Malzard, S.&Schomerus, H. Bulk and edge-state arcs in non-Hermitian coupled-resonator arrays.Phys. Rev. A98, 033807 (2018)..

Cerjan, A. et al. Experimental realization of a Weyl exceptional ring.Nat. Photonics13, 623–628 (2019)..

Okugawa, R.&Yokoyama, T. Topological exceptional surfaces in non-Hermitian systems with parity-time and parity-particle-hole symmetries.Phys. Rev. B99, 041202 (2019)..

Budich, J. C. et al. Symmetry-protected nodal phases in non-Hermitian systems.Phys. Rev. B99, 041406 (2019)..

Ozawa, T. et al. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics.Phys. Rev. A93, 043827 (2016)..

Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics.Nature553, 59–62 (2018)..

Maczewsky, L. J. et al. Synthesizing multi-dimensional excitation dynamics and localization transition in one-dimensional lattices.Nat. Photonics14, 76–81 (2020)..

Umucalılar, R. O.&Carusotto, I. Fractional quantum hall states of photons in an array of dissipative coupled cavities.Phys. Rev. Lett.108, 206809 (2012)..

Carusotto, I.&Ciuti, C. Quantum fluids of light.Rev. Mod. Phys.85, 299–366 (2013)..

Hafezi, M., Lukin, M. D.&Taylor, J. M. Non-equilibrium fractional quantum Hall state of light.New J. Phys.15, 063001 (2013)..

Hormozi, L., Möller, G.&Simon, S. H. Fractional quantum hall effect of lattice bosons near commensurate flux.Phys. Rev. Lett.108, 256809 (2012)..