Abstract

Zak phase and topological protected edge state are usually studied in one-dimensional (1D) photonic systems with spatial inversion symmetry (SIS). In this work, we find that specific classes of 1D structure without SIS can be mapped to a system with SIS and also exhibit novel topology, which manifest as phase cut lines (PCLs) in our specially designed synthetic photonic crystals (SPCs). Zak phase defined in SIS is extended to depict the topology of PCLs after redefinition, and a topological protected edge state is also achieved in our 1D structure without SIS. In our SPCs, the relationship between Chern numbers in two dimensions (2D) and the extended Zak phases of 1D PCLs is given, which are bound by the first type singularities. Higher Chern numbers and multi-chiral edge states are achieved utilizing the concept of synthetic dimensions. The effective Hamiltonian is given, based on which we find that the band edges of each PCL play a role analogous to the valley pseudospin, and our SPC is actually a new type of valley photonic crystal that is usually studied in graphene-like honeycomb lattice. The chiral valley edge transport is also demonstrated. In higher dimensions, the shift of the first type singularity in expanded parameter space will lead to the Weyl point topological transition, which we proposed in our previous work. In this paper, we find a second type of singularity that manifests as a singular surface in our expanded parameter space. The shift of the singular surface will lead to the nodal line topological transition. We find the states on the singular surface possess extremely high robustness against certain randomness, based on which a topological wave filter is constructed.

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

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References

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  1. K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
    [Crossref]
  2. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
    [Crossref]
  3. M. Z. Hasan and C. L. Kane, “Colloquium,” Rev. Mod. Phys. 82, 3045–3067 (2010).
    [Crossref]
  4. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
    [Crossref]
  5. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
    [Crossref] [PubMed]
  6. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
  7. W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
    [Crossref]
  8. P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
    [Crossref]
  9. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [Crossref] [PubMed]
  10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
    [Crossref]
  11. Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
    [Crossref]
  12. L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
    [Crossref] [PubMed]
  13. Y. Hatsugai, “Chern number and edge states in the integer quantum hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
    [Crossref] [PubMed]
  14. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7, 294 (2013).
    [Crossref]
  15. M. Xiao, Q. Lin, and S. Fan, “Hyperbolic weyl point in reciprocal chiral metamaterials,” Phys. Rev. Lett. 117, 057401 (2016).
    [Crossref] [PubMed]
  16. 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, 125136 (2017).
    [Crossref]
  17. Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical interface states protected by synthetic weyl points,” Phys. Rev. X 7, 031032 (2017).
  18. 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, 241 (2014).
    [Crossref] [PubMed]
  19. 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, 050402 (2014).
    [Crossref] [PubMed]
  20. W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
    [Crossref] [PubMed]
  21. L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
    [Crossref] [PubMed]
  22. A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
    [Crossref] [PubMed]
  23. A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
    [Crossref]
  24. L. Pilozzi and C. Conti, “Topological lasing in resonant photonic structures,” Phys. Rev. B 93, 195317 (2016).
    [Crossref]
  25. Q. Li and X. Jiang, “The connection of topology between systems with different dimensions: 1D Zak phases to 2D chern number, weyl point as the jumping channel for one singularity and nodal line to merge all singularities,” ArXiv, 1810.12550 (2018).
  26. M. Ezawa, “Spin valleytronics in silicene: Quantum spin hall–quantum anomalous hall insulators and single-valley semimetals,” Phys. Rev. B 87, 155415 (2013).
    [Crossref]

2018 (3)

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

2017 (3)

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (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, 125136 (2017).
[Crossref]

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

2016 (2)

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

L. Pilozzi and C. Conti, “Topological lasing in resonant photonic structures,” Phys. Rev. B 93, 195317 (2016).
[Crossref]

2015 (2)

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

2014 (5)

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

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, 241 (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, 050402 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

2013 (2)

M. Ezawa, “Spin valleytronics in silicene: Quantum spin hall–quantum anomalous hall insulators and single-valley semimetals,” Phys. Rev. B 87, 155415 (2013).
[Crossref]

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

2012 (1)

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref] [PubMed]

2011 (1)

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

2008 (2)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

1993 (1)

Y. Hatsugai, “Chern number and edge states in the integer quantum hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
[Crossref] [PubMed]

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

1980 (1)

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

Achilleos, V.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

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, 241 (2014).
[Crossref] [PubMed]

Cai, X.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref] [PubMed]

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, 241 (2014).
[Crossref] [PubMed]

Chan, C. T.

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

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, 125136 (2017).
[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, 021017 (2014).

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, 125136 (2017).
[Crossref]

Chen, H.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Chen, S.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref] [PubMed]

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, 125136 (2017).
[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, 241 (2014).
[Crossref] [PubMed]

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, 241 (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, 241 (2014).
[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, 241 (2014).
[Crossref] [PubMed]

Conti, C.

L. Pilozzi and C. Conti, “Topological lasing in resonant photonic structures,” Phys. Rev. B 93, 195317 (2016).
[Crossref]

Ding, Y.-Q.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

Dong, J.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Dorda, G.

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

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, 241 (2014).
[Crossref] [PubMed]

Ezawa, M.

M. Ezawa, “Spin valleytronics in silicene: Quantum spin hall–quantum anomalous hall insulators and single-valley semimetals,” Phys. Rev. B 87, 155415 (2013).
[Crossref]

Fan, S.

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

Fang, M.

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, 241 (2014).
[Crossref] [PubMed]

Fang, X.

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Félix, S.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Fu, L.

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

Gao, F.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Gao, J.

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, 050402 (2014).
[Crossref] [PubMed]

Gao, Z.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Hafezi, M.

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Hatsugai, Y.

Y. Hatsugai, “Chern number and edge states in the integer quantum hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
[Crossref] [PubMed]

Hu, X.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Ivchenko, E. L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Jeffrey, E.

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, 241 (2014).
[Crossref] [PubMed]

Jiang, H.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

Jiang, X.

Q. Li and X. Jiang, “The connection of topology between systems with different dimensions: 1D Zak phases to 2D chern number, weyl point as the jumping channel for one singularity and nodal line to merge all singularities,” ArXiv, 1810.12550 (2018).

Jing, Y.

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
[Crossref]

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

Kalozoumis, P. A.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Kelly, J.

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, 241 (2014).
[Crossref] [PubMed]

Kindel, W. F.

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, 050402 (2014).
[Crossref] [PubMed]

Klitzing, K. V.

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

Kolodrubetz, M.

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, 241 (2014).
[Crossref] [PubMed]

Kolodrubetz, M. H.

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, 050402 (2014).
[Crossref] [PubMed]

Lang, L.-J.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref] [PubMed]

Lehnert, K. W.

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, 050402 (2014).
[Crossref] [PubMed]

Leung, 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, 241 (2014).
[Crossref] [PubMed]

Li, D.

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Li, Q.

Q. Li and X. Jiang, “The connection of topology between systems with different dimensions: 1D Zak phases to 2D chern number, weyl point as the jumping channel for one singularity and nodal line to merge all singularities,” ArXiv, 1810.12550 (2018).

Li, Y.

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

Lin, Q.

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

Liu, H.

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

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
[Crossref]

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

Martinis, J. M.

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, 241 (2014).
[Crossref] [PubMed]

Megrant, 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, 241 (2014).
[Crossref] [PubMed]

Mutus, J.

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, 241 (2014).
[Crossref] [PubMed]

Neill, C.

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, 241 (2014).
[Crossref] [PubMed]

O Malley, P. J. J.

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, 241 (2014).
[Crossref] [PubMed]

Pagneux, V.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Pappas, D. P.

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, 050402 (2014).
[Crossref] [PubMed]

Pepper, M.

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

Pilozzi, L.

L. Pilozzi and C. Conti, “Topological lasing in resonant photonic structures,” Phys. Rev. B 93, 195317 (2016).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Poddubny, A. N.

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Polkovnikov, A.

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, 050402 (2014).
[Crossref] [PubMed]

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, 241 (2014).
[Crossref] [PubMed]

Poshakinskiy, A. V.

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Qi, X.-L.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

Quintana, C.

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, 241 (2014).
[Crossref] [PubMed]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

Ren, J.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

Richoux, O.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Roushan, P.

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, 241 (2014).
[Crossref] [PubMed]

Sandberg, M.

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, 050402 (2014).
[Crossref] [PubMed]

Sank, D.

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, 241 (2014).
[Crossref] [PubMed]

Schroer, M. D.

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, 050402 (2014).
[Crossref] [PubMed]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
[Crossref]

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

Sun, Y.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Theocharis, G.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Vainsencher, 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, 241 (2014).
[Crossref] [PubMed]

Vissers, M. R.

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, 050402 (2014).
[Crossref] [PubMed]

Wang, Q.

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

Wenner, J.

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, 241 (2014).
[Crossref] [PubMed]

White, T.

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, 241 (2014).
[Crossref] [PubMed]

Wu, L.-H.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Xiao, M.

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

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, 125136 (2017).
[Crossref]

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

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

Xue, H.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Yang, Y.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Yang, Z.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Zak, J.

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

Zhang, B.

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Zhang, S.-C.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

Zhang, Z. Q.

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

Zhu, S.

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

Zhu, W.

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

Nat. Photonics (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821 (2014).
[Crossref]

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

Nature (1)

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, 241 (2014).
[Crossref] [PubMed]

Phys. Rev. A (3)

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

Phys. Rev. B (5)

L. Pilozzi and C. Conti, “Topological lasing in resonant photonic structures,” Phys. Rev. B 93, 195317 (2016).
[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, 125136 (2017).
[Crossref]

W. Zhu, Y.-Q. Ding, J. Ren, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials,” Phys. Rev. B 97, 195307 (2018).
[Crossref]

M. Ezawa, “Spin valleytronics in silicene: Quantum spin hall–quantum anomalous hall insulators and single-valley semimetals,” Phys. Rev. B 87, 155415 (2013).
[Crossref]

Z. Gao, Z. Yang, F. Gao, H. Xue, Y. Yang, J. Dong, and B. Zhang, “Valley surface-wave photonic crystal and its bulk/edge transport,” Phys. Rev. B 96, 201402 (2017).
[Crossref]

Phys. Rev. Lett. (10)

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Y. Hatsugai, “Chern number and edge states in the integer quantum hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
[Crossref] [PubMed]

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

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

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

M. Xiao, Q. Lin, and S. Fan, “Hyperbolic weyl point in reciprocal chiral metamaterials,” Phys. Rev. Lett. 117, 057401 (2016).
[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, 050402 (2014).
[Crossref] [PubMed]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref] [PubMed]

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref] [PubMed]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Phys. Rev. X (2)

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

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

Rev. Mod. Phys. (2)

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

M. Z. Hasan and C. L. Kane, “Colloquium,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Other (1)

Q. Li and X. Jiang, “The connection of topology between systems with different dimensions: 1D Zak phases to 2D chern number, weyl point as the jumping channel for one singularity and nodal line to merge all singularities,” ArXiv, 1810.12550 (2018).

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Figures (9)

Fig. 1
Fig. 1 The structure of our synthetic PC with χ within (a) [0, db) (b) [db, db + dc). The plane wave normally incidents on it, whose forward and backward coefficients are denoted by E+ and E. The unit cell is marked by black dashed box.
Fig. 2
Fig. 2 (a) The reflection phase ϕ and (b) intensity |r| as a function of frequency ω and synthetic dimension χ. The parameters are given by na = 3.2nb, nc = 2nb, da = db = dc = 0.5dbc and ω0 = c/(nbdb). The PCLs are marked by white dashed line, the order of which are labeled by red and pink numbers for χ within [0, db) and [db, db + dc) respectively. Nonzero extended Zak phase of PCLs are labeled in (a) and Chern number of upper and lower bands are indicated in (b). The symmetric and antisymmetric band edge states are marked by red and black dots respectively, which are plotted in Fig. 3 correspondingly. (c) Band gap structure along 0, −2 and +2 order PCLs with χ within [db, db +dc). (d) Band gap structure along −1 and +1 order PCLs with χ within [db, db +dc). The gap edges are marked by black lines in (a).
Fig. 3
Fig. 3 Absolute value of E field of eight band edge states marked in Fig. 2. The red and yellow strips represent the layer A and B respectively.
Fig. 4
Fig. 4 The parameters are the same as Fig. 2. Reflection phase of left and right side PC along +1(−1) order PCL with χ within [0, db) are plotted in blue and red line respectively in (a). (b) Transmission intensity of attached PC with 10 cells on each side. Edge state is marked by green dashed line.
Fig. 5
Fig. 5 Reflection phase spectra around the fourth gap is calculated by transfer matrix with a = 4b, c = 1.01b and (a) da = 0.4dbc, (b) 0.36dbc, (c) 0.32dbc. Gap edges calculated by our effective Hamiltonian are plotted by black solid lines.
Fig. 6
Fig. 6 Reflection phase spectra around valley P0 with (a) na = 2nb, nc = 1.01nb, da = 0.5dbc and dc = 0.32dbc db = 0.68dbc for PC1, (b) dc = 0.4dbc db = 0.6dbc for PC2. Grey area in (c) represents the band of attached PC with 10 cells on each side. For certain δχ, χ of the PCs on the left and right side are δχ + 0.84dbc and δχ + 0.8dbc respectively. The edge state is marked by red circle, which is confirmed by the transfer matrix.
Fig. 7
Fig. 7 Reflection phase spectra around ω = 2πω0 with da = db = dc = 0.5dbc and (a) na = 3.9nb, (b) na = 4nb, (c) na = 4.1nb. The gap edges and singular lines are plotted by black solid and white dashed lines respectively. In (b), the gap is closed and turns into a degenerated line at ω = 2πω0.
Fig. 8
Fig. 8 The parameters are given by na = 3.7nb, nc = 2nb, db = dc = 0.5dbc and N=80. (c) Reflection intensity of finite PC with unit cell ABCB and da = 0.5dbc. We first apply randomness to the width of layer A with da = 0.5(1 + 0.4w)dbc and w is a random number between [0, 1), whose distribution with respect to position is plotted in (a). (d) The reflection intensity of the da disordered PC. Next, we apply randomness to the χ of each cell, which is randomly distributed over [0, 1) with respect to position, whose distribution is shown in (b). (e) The reflection intensity of the χ disordered PC.
Fig. 9
Fig. 9 The parameters are given by na = 3.7nb, nc = 2nb and db = dc = 0.5dbc. Reflection intensity of semi-infinite PC composed of with unit cell ABCB with (a) da = 0.42dbc, (b) da = 0.54dbc and (c) da = 0.66dbc. (d) Reflection intensity of topological wave filter composed of 80 cells with da gradually increasing from 0.42dbc to 0.66dbc.

Equations (48)

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E ( x ) = E + exp [ i k a ( x + d a / 2 ) ] + E exp [ i k a ( x + d a / 2 ) ]
k b ( d b 1 d b 2 ) = s π , ( 0 χ d b ) d b 1 d b 2 = d b 2 χ
k c ( d c 1 d c 2 ) = p π , ( d b χ d b + d c ) d c 1 d c 2 = d c + 2 d b 2 χ
tan k c d c = F 3 sin k b d b F 2 sin 2 k b d b / 2 F 4 cos 2 k b d b / 2 , sin k c d c 0
F 2 = n b 2 n c n a n c n a n b 2 , F 3 = n b n a n a n b , F 4 = n c n b n b n c
tan k c d c = F 3 sin k b d b F 4 sin 2 k b d b / 2 F 2 cos 2 k b d b / 2 , sin k c d c 0
ϕ L + ϕ R = 2 j π , j
C n = l s n l θ Z a k l / π
N 1 k a cos k a ( x d a 2 ) , ( 0 x d a )
N 1 k b cos k a ( x d a d b c 2 ) , ( d a x d b c + d a )
N 2 k b sin k a ( x d a 2 ) , ( 0 x d a )
N 2 k b sin k b ( x d a d b c 2 ) , ( d a x d b c + d a )
H eff = α σ 0 + β σ
α = k b 2 + t ( p ( χ ) 2 k b d c ) N 1 2 2 + t ( p ( χ ) 2 k b d c ) N 2 2 2
β K , χ = [ t q ( χ ) N 1 N 2 , 2 k b K c 1 , t ( p ( χ ) 2 k b d c ) N 1 2 2 t ( p ( χ ) 2 k b d c ) N 2 2 2 ]
c 1 = [ ( n a d a ) 2 + ( n b d b c ) 2 + ( a b + b a ) n a d a n b d b c ] / D 2
p ( χ ) = sin 2 k b χ sin 2 k b ( χ d b c + d c )
q ( χ ) = cos 2 k b ( χ d b c + d c ) cos 2 k b χ
H w = C σ 0 + δ χ v χ σ x + K v K σ y + M σ z
H w = C σ 0 + δ χ v χ σ x + K v K σ y + M σ z
H w = ( C + D δ d ) σ 0 + δ χ v χ σ x + K v K σ y + δ d v d σ z
n a d a : n b d b : n c d c = l : m : n
[ T exp ( i KD ) ] ( E + E ) = 0
E + = T 12 E = exp ( i K D ) T 11
E ( x ) = E + exp [ i k a ( x + d a / 2 ) ] + E exp [ i k a ( x + d a / 2 )
r = exp ( i K D ) T 11 T 12 exp ( i k a d a )
Re [ T 11 ] = cos k a d a cos k c d c cos k b d b 1 2 M 1 cos k b d b 1 cos k b d b 2 sin k a d a sin k c d c + 1 2 M 2 sin k b d b 1 sin k b d b 2 sin k a d a sin k c d c 1 2 M 3 sin k b d b sin k a d a cos k c d c + 1 2 M 4 sin k b d b sin k a d a sin k c d c
Im [ T 11 ] = sin k a d a cos k c d c cos k b d b + 1 2 M 1 cos k b d b 1 cos k b d b 2 sin k a d a sin k c d c 1 2 M 2 sin k b d b 1 sin k b d b 2 cos k a d a sin k c d c 1 2 M 3 sin k b d b cos k a d a cos k c d c 1 2 M 4 sin k b d b sin k a d a sin k c d c
T 12 = e i k a d a 2 [ i F 1 sin k c d c cos k b d b 1 cos k b d b 2 i F 2 sin k c d c sin k b d b 1 sin k b d b 2 + i F 3 cos k c d c sin k b d b F 4 sin k c d c sin k b ( d b 2 d b 1 ) ]
M 1 = n a n c + n c n a , M 2 = n b 2 n c n a + n c n a n b 2 , M 3 = n a n b + n b n a , M 4 = n b n c + n c n b
F 1 = n c n a n a n c , F 2 = n b 2 n c n a n c n a n b 2 , F 3 = n b n a n a n b , F 4 = n c n b n b n c
Re [ T 11 ] = cos K D
k b ( d b 1 d b 2 ) = 2 l π , l
tan k c d c = F 3 sin k b d b F 2 sin 2 k b d b / 2 F 4 cos 2 k b d b / 2 , sin k c d c 0
tan k c d c = F 3 sin k b d b F 4 sin 2 k b d b / 2 F 2 cos 2 k b d b / 2 , sin k c d c 0
E n ( K , x ) = ± E n ( K , x )
u n ( K , x ) = ± u n ( K , x )
A K n = A K n
u K = f ω ω K + f K
u χ = f ω ω χ + f χ
ω K = F K F ω
ω K = F χ F ω
exp ( i K D ) T 11 𝒪 ( δ K 2 + δ χ 2 )
T 12 exp ( i k a d a ) exp ( i k a d a ) ( A δ χ + B i δ ω )
sin n b d b ω / c = 0 , sin n c d c ω / c = 0
n a d a : n b d b : n c d c = l : m : n
| Re [ T 11 ] | = 1
( Re [ T 11 ] ) / K = 0

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