Abstract

Weyl points, as linearly double degenerated point of band structures, have been extensively researched in electronic and classical wave systems. However, Weyl points’ realization is always accompanied with delicate “lattice structures”. In this work, frequency-tunable terahertz (THz) generalized Weyl points inside the parameter space have been investigated and displayed by a specially designed photonic crystal with polydimethylsiloxane (PDMS) immersed in 4-cyano’-pentylbipenyl (5CB) liquid crystals (LCs). The reflective phase vortices as a signature of the generalized Weyl points are observed through our numerically simulations. Besides, interface states between photonic crystals and any reflective substrates are fulfilled too. Meanwhile, we could also change the orientation of LC molecule by the external magnetic field so as to tune the frequency of the first two bands’ Weyl point from 0.27698THz to 0.30013THz. This band lies in the short-range wireless communication. Thus, our proposal may be beneficial to the investigation and application of Weyl points’ properties and strongly localized states.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (4)

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

H. Ge, X. Ni, Y. Tian, S. K. Gupta, M. H. Lu, X. Lin, W. D. Huang, C. T. Chan, and Y. F. Chen, “Experimental observation of acoustic Weyl points and topological surface states,” Phys. Rev. Appl. 10(1), 014017 (2018).
[Crossref]

H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, “Topological negative refraction of surface acoustic waves in a Weyl phononic crystal,” Nature 560(7716), 61–64 (2018).
[Crossref] [PubMed]

Z. Guo, X. Yang, F. Shen, Q. Zhou, J. Gao, and K. Guo, “Active-tuning and polarization-independent absorber and sensor in the infrared region based on the phase change material of Ge2Sb2Te5 (GST),” Sci. Rep. 8(1), 12433 (2018).
[Crossref] [PubMed]

2017 (8)

Q. Wang, M. Xiao, H. Liu, S. N. Zhu, and C. T. Chan, “Optical interface states protected by synthetic weyl points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

P. Zhu, W. Yang, R. Wang, S. Gao, B. Li, and Q. Li, “Direct writing of flexible barium titanate/polydimethylsiloxane 3D photonic crystals with mechanically tunable terahertz properties,” Adv. Opt. Mater. 5(7), 1600977 (2017).
[Crossref]

Z. Yang, M. Xiao, F. Gao, L. Lu, Y. Chong, and B. Zhang, “Weyl points in a magnetic tetrahedral photonic crystal,” Opt. Express 25(14), 15772–15777 (2017).
[Crossref] [PubMed]

F. Li, X. Q. Huang, J. Y. Lu, J. H. Ma, and Z. Y. Liu, “Weyl points and Fermi arcs in a chiral phononic crystal,” Nat. Phys. 14(1), 30–34 (2017).
[Crossref]

M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017).
[Crossref]

J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
[Crossref]

M. Zhou, L. Ying, L. Lu, L. Shi, J. Zi, and Z. Yu, “Electromagnetic scattering laws in Weyl systems,” Nat. Commun. 8(1), 1388 (2017).
[Crossref] [PubMed]

B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
[Crossref] [PubMed]

2016 (12)

L. Y. Wang, S. K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” Phys. Rev. A (Coll. Park) 93(6), 061801 (2016).
[Crossref] [PubMed]

M. Xiao, Q. Lin, and S. Fan, “Hyperbolic Weyl point in reciprocal chiral metamaterials,” Phys. Rev. Lett. 117(5), 057401 (2016).
[Crossref] [PubMed]

W. J. Chen, M. Xiao, and C. T. Chan, “Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states,” Nat. Commun. 7(1), 13038 (2016).
[Crossref] [PubMed]

Z. Yang and B. Zhang, “Acoustic Type-II Weyl nodes from stacking dimerized chains,” Phys. Rev. Lett. 117(22), 224301 (2016).
[Crossref] [PubMed]

R. P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov, “Multi-terminal Josephson junctions as topological matter,” Nat. Commun. 7(1), 11167 (2016).
[Crossref] [PubMed]

W. Gao, B. Yang, M. Lawrence, F. Fang, B. Béri, and S. Zhang, “Photonic Weyl degeneracies in magnetized plasma,” Nat. Commun. 7(1), 12435 (2016).
[Crossref] [PubMed]

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
[Crossref] [PubMed]

L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41(4), 741–744 (2016).
[Crossref] [PubMed]

Y. Ke, I. Balin, N. Wang, Q. Lu, A. I. Tok, T. J. White, S. Magdassi, I. Abdulhalim, and Y. Long, “Two-dimensional SiO2/VO2 photonic crystals with statically visible and dynamically infrared modulated for smart window deployment,” ACS Appl. Mater. Interfaces 8(48), 33112–33120 (2016).
[Crossref] [PubMed]

B. Lian and S. C. Zhang, “Five-dimensional generalization of the topological Weyl semimetal,” Phys. Rev. B 94(4), 041105 (2016).
[Crossref]

F. Mei, Z. Y. Xue, D. W. Zhang, L. Tian, C. Lee, and S. L. Zhu, “Witnessing topological Weyl semimetal phase in a minimal circuit-qed lattice,” Quantum Sci. Technol. 1(1), 015006 (2016).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7(1), 13731 (2016).
[Crossref] [PubMed]

2015 (8)

X. C. Huang, L. X. Zhao, Y. J. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
[Crossref]

M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, and T. Miyake, “Weyl node and spin texture in trigonal tellurium and selenium,” Phys. Rev. Lett. 114(20), 206401 (2015).
[Crossref] [PubMed]

B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
[Crossref]

S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
[Crossref]

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527(7579), 495–498 (2015).
[Crossref] [PubMed]

X. G. Wan, “Topological Weyl semimetals,” Phsics 44(7), 427–439 (2015).

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

2014 (6)

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, “Measuring a topological transition in an artificial spin-1/2 system,” Phys. Rev. Lett. 113(5), 050402 (2014).
[Crossref] [PubMed]

J. Liu and D. Vanderbilt, “Weyl semimetals from noncentrosymmetric topological insulators,” Phys. Rev. B Condens. Matter Mater. Phys. 90(15), 155316 (2014).
[Crossref]

D. Bulmash, C. X. Liu, and X. L. Qi, “Prediction of a Weyl semimetal in Hg1−x−yCdxMnyTe,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 081106 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

T. B. Zhou, F. R. Hu, J. Xiao, L. H. Zhang, F. Liu, T. Chen, J. H. Niu, and X. M. Xiong, “Design of a polarization-insensitive and broadband terahertz absorber using metamaterials,” Wuli Xuebao 63(17), 178103 (2014).

2013 (1)

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

2012 (3)

G. B. Halász and L. Balents, “Time-reversal invariant realization of the Weyl semimetal phase,” Phys. Rev. B Condens. Matter Mater. Phys. 85(3), 035103 (2012).
[Crossref]

A. A. Zyuzin, S. Wu, and A. A. Burkov, “Weyl semimetal with broken time reversal and inversion symmetries,” Phys. Rev. B Condens. Matter Mater. Phys. 85(16), 165110 (2012).
[Crossref]

O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, “Quantum simulation of an extra dimension,” Phys. Rev. Lett. 108(13), 133001 (2012).
[Crossref] [PubMed]

2011 (3)

X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, “Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates,” Phys. Rev. B Condens. Matter Mater. Phys. 83(20), 205101 (2011).
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A. A. Burkov and L. Balents, “Weyl semimetal in a topological insulator multilayer,” Phys. Rev. Lett. 107(12), 127205 (2011).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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2003 (1)

1993 (1)

K. C. Lim, J. D. Margerum, and A. M. Lackner, “Liquid crystal millimeter wave electronic phase shifter,” Appl. Phys. Lett. 62(10), 1065–1067 (1993).
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1983 (1)

H. B. Nielsen and M. Ninomiya, “The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal,” Phys. Lett. B 130(6), 389–396 (1983).
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1981 (1)

B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice,” Nucl. Phys. B 193(1), 173–194 (1981).
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1979 (1)

N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51(3), 591–648 (1979).
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1929 (1)

H. Weyl, “Elektron und Gravitation. I,” Z. Phys. 56(5), 330–352 (1929).
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1928 (1)

P. A. M. Dirac, “The quantum theory of the electron,” Proc. R. Soc. Lond. A Math. Phys. Sci. 117(778), 610–624 (1928).
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Abdulhalim, I.

Y. Ke, I. Balin, N. Wang, Q. Lu, A. I. Tok, T. J. White, S. Magdassi, I. Abdulhalim, and Y. Long, “Two-dimensional SiO2/VO2 photonic crystals with statically visible and dynamically infrared modulated for smart window deployment,” ACS Appl. Mater. Interfaces 8(48), 33112–33120 (2016).
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Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Alidoust, N.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
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S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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Balents, L.

G. B. Halász and L. Balents, “Time-reversal invariant realization of the Weyl semimetal phase,” Phys. Rev. B Condens. Matter Mater. Phys. 85(3), 035103 (2012).
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A. A. Burkov and L. Balents, “Weyl semimetal in a topological insulator multilayer,” Phys. Rev. Lett. 107(12), 127205 (2011).
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Balin, I.

Y. Ke, I. Balin, N. Wang, Q. Lu, A. I. Tok, T. J. White, S. Magdassi, I. Abdulhalim, and Y. Long, “Two-dimensional SiO2/VO2 photonic crystals with statically visible and dynamically infrared modulated for smart window deployment,” ACS Appl. Mater. Interfaces 8(48), 33112–33120 (2016).
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Bansil, A.

S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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Barends, R.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
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Barr, L. E.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
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B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
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Belopolski, I.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
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S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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Béri, B.

B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
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W. Gao, B. Yang, M. Lawrence, F. Fang, B. Béri, and S. Zhang, “Photonic Weyl degeneracies in magnetized plasma,” Nat. Commun. 7(1), 12435 (2016).
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Bernevig, B. A.

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527(7579), 495–498 (2015).
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Bian, G.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
[Crossref] [PubMed]

S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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Bisti, F.

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
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Boada, O.

O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, “Quantum simulation of an extra dimension,” Phys. Rev. Lett. 108(13), 133001 (2012).
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Bulmash, D.

D. Bulmash, C. X. Liu, and X. L. Qi, “Prediction of a Weyl semimetal in Hg1−x−yCdxMnyTe,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 081106 (2014).
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Burkov, A. A.

A. A. Zyuzin, S. Wu, and A. A. Burkov, “Weyl semimetal with broken time reversal and inversion symmetries,” Phys. Rev. B Condens. Matter Mater. Phys. 85(16), 165110 (2012).
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A. A. Burkov and L. Balents, “Weyl semimetal in a topological insulator multilayer,” Phys. Rev. Lett. 107(12), 127205 (2011).
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Cai, X.

H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, “Topological negative refraction of surface acoustic waves in a Weyl phononic crystal,” Nature 560(7716), 61–64 (2018).
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Campbell, B.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Celi, A.

O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, “Quantum simulation of an extra dimension,” Phys. Rev. Lett. 108(13), 133001 (2012).
[Crossref] [PubMed]

Chan, C. T.

H. Ge, X. Ni, Y. Tian, S. K. Gupta, M. H. Lu, X. Lin, W. D. Huang, C. T. Chan, and Y. F. Chen, “Experimental observation of acoustic Weyl points and topological surface states,” Phys. Rev. Appl. 10(1), 014017 (2018).
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M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017).
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Q. Wang, M. Xiao, H. Liu, S. N. Zhu, and C. T. Chan, “Optical interface states protected by synthetic weyl points,” Phys. Rev. X 7(3), 031032 (2017).
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W. J. Chen, M. Xiao, and C. T. Chan, “Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states,” Nat. Commun. 7(1), 13038 (2016).
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M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
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M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
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Chang, G.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
[Crossref] [PubMed]

S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

Chang, M. L.

M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017).
[Crossref]

Chang, T. R.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
[Crossref] [PubMed]

Chen, C. Y.

Chen, D.

X. C. Huang, L. X. Zhao, Y. J. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
[Crossref]

Chen, G.

X. C. Huang, L. X. Zhao, Y. J. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
[Crossref]

Chen, G. F.

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
[Crossref]

B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
[Crossref]

Chen, J.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

Chen, K. P.

J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
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Chen, T.

T. B. Zhou, F. R. Hu, J. Xiao, L. H. Zhang, F. Liu, T. Chen, J. H. Niu, and X. M. Xiong, “Design of a polarization-insensitive and broadband terahertz absorber using metamaterials,” Wuli Xuebao 63(17), 178103 (2014).

Chen, W. J.

M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017).
[Crossref]

W. J. Chen, M. Xiao, and C. T. Chan, “Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states,” Nat. Commun. 7(1), 13038 (2016).
[Crossref] [PubMed]

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

Chen, Y.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

Chen, Y. F.

H. Ge, X. Ni, Y. Tian, S. K. Gupta, M. H. Lu, X. Lin, W. D. Huang, C. T. Chan, and Y. F. Chen, “Experimental observation of acoustic Weyl points and topological surface states,” Phys. Rev. Appl. 10(1), 014017 (2018).
[Crossref]

Chen, Z.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

Chiaro, B.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

Chong, Y.

Chong, Y. D.

J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
[Crossref]

Chou, F.

S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

Cleland, A. N.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
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Dai, X.

B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
[Crossref]

X. C. Huang, L. X. Zhao, Y. J. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
[Crossref]

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527(7579), 495–498 (2015).
[Crossref] [PubMed]

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
[Crossref]

Ding, H.

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
[Crossref]

B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
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Dirac, P. A. M.

P. A. M. Dirac, “The quantum theory of the electron,” Proc. R. Soc. Lond. A Math. Phys. Sci. 117(778), 610–624 (1928).
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Dunsworth, A.

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014).
[Crossref] [PubMed]

Fan, D.

B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
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Fan, S.

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7(1), 13731 (2016).
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M. Xiao, Q. Lin, and S. Fan, “Hyperbolic Weyl point in reciprocal chiral metamaterials,” Phys. Rev. Lett. 117(5), 057401 (2016).
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L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41(4), 741–744 (2016).
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Fan, X.

H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, “Topological negative refraction of surface acoustic waves in a Weyl phononic crystal,” Nature 560(7716), 61–64 (2018).
[Crossref] [PubMed]

Fang, C.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
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[Crossref] [PubMed]

Zahid Hasan, M.

C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016).
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Zhang, C. L.

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Zhang, D. W.

F. Mei, Z. Y. Xue, D. W. Zhang, L. Tian, C. Lee, and S. L. Zhu, “Witnessing topological Weyl semimetal phase in a minimal circuit-qed lattice,” Quantum Sci. Technol. 1(1), 015006 (2016).
[Crossref]

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Zhang, S.

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B. Lian and S. C. Zhang, “Five-dimensional generalization of the topological Weyl semimetal,” Phys. Rev. B 94(4), 041105 (2016).
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Figures (6)

Fig. 1
Fig. 1 Realization of Weyl points in a synthetic space. (a) Photonic crystals (PCs) with different p, q values. p and q form a parameter space, which determines the geometric structure of PCs. The inset shows one unit cell of the PC, where the first and the third layers are made of 5CB LCs (cyan), and the second and the forth layers are made of PDMS (gray). The thickness of each layer is related to its position in the p-q parameter space. (b) The band dispersion of PCs with two layers in one unit cell (red dash line) and four layers in one unit cell (blue solid line). Crossing points appear inside four layers’ dispersion. Here, d a =125um, d b =35um, and p=q=0. (c) The dispersion of PCs in the p-q space with k=0.5 k 0 , and k 0 =π/( d a + d b ). Here, two bands form a conical intersection. Panels (b) and (c) together show that the band dispersions are linear in all directions around the degenerated point in synthetic space, so we call it generalized Weyl point. (d) The equal frequency contours around generalized Weyl point in p-q space and its charge “-1”.
Fig. 2
Fig. 2 (a) The schematic illustration of the terahertz wave’s propagation and polarization, where the electric field is polarized along the z-direction(blue) and the magnetic field is along the y-direction(green). The plane wave propagates along x-direction (red). The LCs are randomly distributed inside the box. (b) The schematic illustration of measuring the reflective phase around the generalized Weyl points under the external magnetic field.
Fig. 3
Fig. 3 The position of generalized Weyl points of band-1 and band-2 when change the direction of external magnetic field. Panel (a) and (b) show three kinds of band dispersion. The blue, red and green lines represent external magnetic along long axis, short axis and no external field respectively. (c), (d) and (e) The dispersion of PCs in p-q parameter space with k=0.5 k 0 . In three cases, two bands all form conical intersection. (c) External field along y- axis. (d) No external field. (e) External field along z-axis.
Fig. 4
Fig. 4 The reflection coefficient and phase around generalized Weyl points of first two bands in p-q parameter space. (a), (d) The external magnetic field along y-axis. (b), (e) No external magnetic field. (c), (f) The external magnetic field along z-axis. Here, the reflection coefficient at p=q=0 is minimal because of the intersection of two bands at the generalized Weyl point. Reflective phase exhibits vortex structure, and continuously change from π to with the increase ofφ.
Fig. 5
Fig. 5 The reflective phases of different position centered on generalized Weyl point in parameter space. Here, (p, q) = (0.75, 0), (0, 0.75), (−0.75, 0) and (0, −0.75) are signed with pentagram, triangle, diamond and square in Fig. 4(e). The reflective phase of these positions covers 2π inside band gap. Gray strips on both sides denote bulk bands.
Fig. 6
Fig. 6 The interface state between the designed PC with generalized Weyl points and polyimide substrate. (a) The reflective phase of PC (blue line) and polyimide substrate (red line). (b) The reflection efficiency with (red line) and without (blue line) polyimide layer on the surface of PC. (c) The distribution of Electric field along the PC. In the interface (indicated by red dashed line), Electric field is the strongest. (d) The structure to measure the interface states. Red dashed line is interface between PC (right side) and substrate (left side, blue).

Equations (4)

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H( k )= i=x,y,z j=0,x,y,z k i A ij σ j ,
H=( P w Q w K w )( 0 0 0.03921 0 0 0.02997 0 0 1.995 0 0 0 )( σ x σ y σ z σ 0 ),
ε ave = n ave 2 = 2 n o 2 + n e 2 3 .
ϕ PC + ϕ S =2mπ( mZ ),

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