Abstract

We study mode coupling of an edge state with bulk states in a coupled waveguide chain in the presence of gain and losses. In a low non-Hermiticity regime, it is found that the edge state associated with the interface between two topologically different waveguide chains is well isolated from the bulk states, sharing the exact same features as the well-known Su-Schrieffer-Heeger Model. As the non-Hermiticity increases, the two bands of the overall waveguide chain merge together, which leads to the overlap in the band diagram between the edge state and states of lower bands. We find that before the PT-symmetry breaking, the edge state is strongly coupled to bulk states of lower bands, evident by the anti-crossing feature in the band diagram. The strong mode hybridization is verified by a non-Hermitian coupled mode theory developed from reaction conservation, and is further examined using phenomenological models. We envisage that the strong coupling between the edge state and the bulk state may be useful in expanding our understanding in topological photonics in non-Hermitian condition, as well as in applications such as mode conversion between edge and bulk states.

© 2017 Optical Society of America

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References

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2016 (3)

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

2015 (6)

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[Crossref] [PubMed]

J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
[Crossref] [PubMed]

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

2014 (5)

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[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).

2013 (1)

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

2012 (3)

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

K. X. Liu, L. F. Shen, and S. He, “One-way edge mode in a gyromagnetic photonic crystal slab,” Opt. Lett. 37(19), 4110–4112 (2012).
[Crossref] [PubMed]

2011 (3)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

2008 (1)

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

2005 (1)

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

2002 (1)

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

1993 (1)

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

1982 (1)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

1980 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Al, A.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Ando, Y.

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Bandres, M. A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Bao, M.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Bertoldi, K.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Chan, C. T.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[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).

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Chen, W. J.

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

Chen, Y.

Chen, Y. B.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Chen, Y. F.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Chong, Y.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Das, P.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Delplace, P.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Den Nijs, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Feng, L.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Fleury, R.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Fung, K. H.

Ge, H.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Guzman-Silva, D.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Hatsugai, Y.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

He, C.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

He, S.

Heeger, A. J.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Ho, S.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Huang, X. Q.

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Jia, H.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Joannopoulos, J.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Khanikaev, A. B.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Kohmoto, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Lee, T. E.

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

Lin, F.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Ling, C. W.

Liu, K. X.

Liu, X. L.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Liu, X. P.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Liu, Z. Z.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Longhi, S.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Lu, L.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Lu, M. H.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Macdonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Maizard, S.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Mejia-Cortes, C.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Mele, E. J.

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

Mong, R.

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

Montambaux, G.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Ni, X.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Nightingale, M. P.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Noh, C.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Nolte, S.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Peng, Y. G.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Poli, C.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Qi, X. L.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Rechtsman, M. c.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Ryu, S.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Schomerus, H.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Schrieffer, J. R.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Segev, M.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Shen, L. F.

Shen, S. Q.

S. Q. Shen, “Topological insulators: Dirac equation in condensed matters,” Springer Series in Solid-State Sciences174 (2012).
[Crossref]

Shen, Y. X.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Shivamoggi, V.

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Soljai, M.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Su, W. P.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Sun, X. C.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Szameit, A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Thouless, D. J.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Tse, W. K.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Ullmo, D.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Vicencio, R. A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Wang, P.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Weimann, S.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Wen, X.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Xiao, J. J.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Xiao, M.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[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).

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Xu, J.

Yao, Y.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Yu, S. F.

Yu, X. Y.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Zhang, Q.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Zhang, Z. Q.

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (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, 021017 (2014).

Zhao, D. G.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Zhao, H.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Zhu, X. F.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

J. High Energy Phys. (1)

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

J. Phys. Soc. Jpn. (1)

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

Nat. Commun. (2)

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Nat. Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Nat. Phys. (1)

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

New J. Phys. (1)

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (4)

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

Phys. Rev. Lett. (9)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Phys. Rev. X (1)

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).

Rev. Mod. Phys (1)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Rev. Mod. Phys. (1)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Sci. Rep. (3)

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Other (3)

COMSOL MULTIPHYSICS 5.2, (2015). COMSOL Multiphysics: a finite element analysis, solver and simulation software for various physics and engineering application, especially coupled phenomena, or multiphysics. URL http://www.comsol.com/ .

S. Q. Shen, “Topological insulators: Dirac equation in condensed matters,” Springer Series in Solid-State Sciences174 (2012).
[Crossref]

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

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Figures (4)

Fig. 1
Fig. 1 Bulk properties of the dimerized PT-symmetric waveguide lattices A and B. Each unit cell contains two identical single-mode waveguides with width and height denoted by w and h. The working wavelength is λ0, and the relative permittivity of single-mode waveguide core is r 0 = 10, embedded in air cladding. (a)–(b)/(d)–(e) show the real part and imaginary part of eigenvalues of band structure with four values of Δ for waveguide lattice A/B. (c)/(f) shows the width of the projected band Δneff and the band-center as function of non-Hermiticity Δ.
Fig. 2
Fig. 2 Edge state in coupled waveguide chains with global PT symmetry and local PT symmetry which consist of seventeen waveguides. (a) is the structure of PT symmetric waveguide chain with the condition n(x) = n*(−x), and x = 0 is the center of the structure. The larger space between two waveguides dL = 0.8λ0, the smaller one dS = 0.4λ0 and the periodicity D = 1.2λ0. (b) is the structure of waveguide chain with local PT symmetry and mirror symmetry. Local PT symmetry is in the unit cell with two waveguides and mirror symmetry means n(x) = n*(−x) with the center of the structure. (c) and (d) show the band dispersion of the structure in (a) and (b), respectively.
Fig. 3
Fig. 3 Band structure and mode evolution sketch of waveguide chain with five waveguides. (a) and (b) show the real and imaginary part of band dispersion in coupled waveguide chains with global PT symmetry which consist of five waveguides, respectively. Blue circles represent the eigenvalues from COMSOL, and red stars represent the eigenvalues from GCMT. (c) The mode evolution in the mode crossing regime, i.e., the black rectangle in (a). (d) The values of k31, k32, k34, k35 and k45 with varying Δ.
Fig. 4
Fig. 4 Band structure and the sketch of mode evolution of waveguide chain with three waveguides. (a/b) shows the real (imaginary) part of band dispersion in coupled waveguide chains. Blue circles represent the eigenvalues from full-wave simulation, and Red dots represent results from phenomenological models. (c) The mode evolution of two mode intersection.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

j a j ( β β 0 , j ) p i j = j a j k i j ,
[ h 11 h 12 h 13 h 14 h 15 h 21 h 22 h 23 h 24 h 25 h 31 h 32 h 33 h 34 h 35 h 41 h 42 h 43 h 44 h 45 h 51 h 52 h 53 h 54 h 55 ] [ a 1 a 2 a 3 a 4 a 5 ] = n eff k 0 [ p 11 p 12 p 13 p 14 p 15 p 21 p 22 p 23 p 24 p 25 p 31 p 32 p 33 p 34 p 35 p 41 p 42 p 43 p 44 p 45 p 51 p 52 p 53 p 54 p 55 ] [ a 1 a 2 a 3 a 5 a 5 ] ,
H = [ β odd i V 1 0 i V 1 β even i V 2 0 i V 2 β single ] ,

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