Abstract

We classify symmetry-protected and symmetry-breaking dynamical solutions for nonlinear saturable bosonic systems that display a non-hermitian charge-conjugation symmetry, as realized in a series of recent groundbreaking experiments with lasers and exciton polaritons. In particular, we show that these systems support stable symmetry-protected modes that mirror the concept of zero-modes in topological quantum systems, as well as symmetry-protected power-oscillations with no counterpart in the linear case. In analogy to topological phases in linear systems, the number and nature of symmetry-protected solutions can change. The spectral degeneracies signalling phase transitions in linear counterparts extend to bifurcations in the nonlinear context. As bifurcations relate to qualitative changes in the linear stability against changes of the initial conditions, the symmetry-protected solutions and phase transitions can also be characterized by topological excitations, which set them apart from symmetry-breaking solutions. The stipulated symmetry appears naturally when one introduces nonlinear gain or loss into spectrally symmetric bosonic systems, as we illustrate for one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.

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  1. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
    [Crossref]
  2. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
    [Crossref]
  3. C. W. J. Beenakker, “Random-matrix theory of Majorana fermions and topological superconductors,” Rev. Mod. Phys. 87, 1037–1066 (2015).
    [Crossref]
  4. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
    [Crossref]
  5. J. C. Y. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010).
    [Crossref]
  6. P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
    [Crossref]
  7. L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
    [Crossref]
  8. J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
    [Crossref] [PubMed]
  9. M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
    [Crossref]
  10. C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
    [Crossref]
  11. A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44 (suppl.), 131–136 (2001).
    [Crossref]
  12. L. Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-hall-insulator/superconductor junction,” Phys. Rev. B 79, 161408 (2009).
    [Crossref]
  13. C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
    [Crossref] [PubMed]
  14. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
    [Crossref]
  15. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
    [Crossref]
  16. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
    [Crossref] [PubMed]
  17. V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
    [Crossref] [PubMed]
  18. V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).
  19. V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).
  20. R. Süsstrunk and S. D. Huber, “Observation of phononic helical edge states in a mechanical topological insulator,” Science 349, 47–50 (2015).
    [Crossref]
  21. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
    [Crossref] [PubMed]
  22. C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10, 39–45 (2014).
    [Crossref]
  23. N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
    [Crossref]
  24. V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).
  25. T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).
  26. A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
    [Crossref]
  27. L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, Oxford, 2003).
  28. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
    [Crossref]
  29. C. Ciuti and I. Carusotto, “Quantum fluid effects and parametric instabilities in microcavities,” Phys. Status Solidi B 242, 2224–2245 (2005).
    [Crossref]
  30. Y. Kawaguchi and M. Ueda, “Spinor Bose-Einstein condensates,” Phys. Rep. 520, 253–381 (2012).
    [Crossref]
  31. R. Barnett, “Edge-state instabilities of bosons in a topological band,” Phys. Rev. A 88, 063631 (2013).
    [Crossref]
  32. R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
    [Crossref]
  33. G. Engelhardt and T. Brandes, “Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons,” Phys. Rev. A 91, 053621 (2015).
    [Crossref]
  34. C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
    [Crossref]
  35. S. Furukawa and M. Ueda, “Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices,” New J. Phys. 17, 115014 (2015).
    [Crossref]
  36. B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
    [Crossref] [PubMed]
  37. G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
    [Crossref] [PubMed]
  38. M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
    [Crossref] [PubMed]
  39. H. Schomerus and N. Y. Halpern, “Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices,” Phys. Rev. Lett. 110, 013903 (2013).
    [Crossref] [PubMed]
  40. H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38, 1912–1914 (2013).
    [Crossref] [PubMed]
  41. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
    [Crossref] [PubMed]
  42. D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
    [Crossref]
  43. D. I. Pikulin and Yu. V. Nazarov, “Topological properties of superconducting junctions,” JETP Lett. 94, 693–697 (2012).
    [Crossref]
  44. D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B 87, 235421 (2013).
    [Crossref]
  45. S. Malzard, C. Poli, and H. Schomerus, “Topologically protected defect states in open photonic systems with non-hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
    [Crossref] [PubMed]
  46. P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
    [Crossref] [PubMed]
  47. J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).
  48. P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
    [Crossref]
  49. C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
    [Crossref]
  50. H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
    [Crossref] [PubMed]
  51. M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
    [Crossref] [PubMed]
  52. R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).
  53. These considerations can also be extended to include an inhomogeneity g(t) = −Xg*(t).
  54. This exploits the U(1) gauge freedom.
  55. At the juncture of cases (a) and (b), the nonlinearities can also stabilize symmetry-breaking modes with vanishing frequency, which then form a degenerate pair.
  56. This again exploits the U(1) gauge freedom to fix the overall phase factor exp(iα) of the wavefunction, up to an overall sign. If a self-symmetric state is periodic this can therefore be realized in two variants Ψ(T) = Ψ(0) and Ψ(T) = −Ψ(0).
  57. B. Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B 34, 5208–5211 (1986).
    [Crossref]
  58. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
    [Crossref]
  59. S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89, 077002 (2002).
    [Crossref] [PubMed]
  60. E. H. Lieb, “Two theorems on the Hubbard model,” Phys. Rev. Lett. 62, 1201–1204 (1989).
    [Crossref] [PubMed]
  61. H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
    [Crossref]
  62. N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34, 1633–1635 (2009).
    [Crossref] [PubMed]
  63. R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
    [Crossref]
  64. V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
    [Crossref]
  65. N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
    [Crossref]
  66. M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
    [Crossref]
  67. R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
    [Crossref] [PubMed]
  68. S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
    [Crossref] [PubMed]
  69. J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
    [Crossref] [PubMed]
  70. F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
    [Crossref] [PubMed]
  71. H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
    [Crossref] [PubMed]
  72. C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
    [Crossref]
  73. M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
    [Crossref]
  74. G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
    [Crossref]
  75. T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
    [Crossref]
  76. T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
    [Crossref]
  77. J. Keeling and N. G. Berloff, “Spontaneous rotating vortex lattices in a pumped decaying condensate,” Phys. Rev. Lett. 100, 250401 (2008).
    [Crossref] [PubMed]
  78. S. Malzard and H. Schomerus, “Nonlinear mode competition and symmetry-protected power oscillations in topological lasers,” New J. Phys. 20, 063044 (2018).
    [Crossref]
  79. S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
    [Crossref]

2018 (5)

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

S. Malzard and H. Schomerus, “Nonlinear mode competition and symmetry-protected power oscillations in topological lasers,” New J. Phys. 20, 063044 (2018).
[Crossref]

S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
[Crossref]

2017 (3)

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
[Crossref]

2016 (8)

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

2015 (16)

S. Furukawa and M. Ueda, “Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices,” New J. Phys. 17, 115014 (2015).
[Crossref]

B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
[Crossref] [PubMed]

G. Engelhardt and T. Brandes, “Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons,” Phys. Rev. A 91, 053621 (2015).
[Crossref]

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

R. Süsstrunk and S. D. Huber, “Observation of phononic helical edge states in a mechanical topological insulator,” Science 349, 47–50 (2015).
[Crossref]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

C. W. J. Beenakker, “Random-matrix theory of Majorana fermions and topological superconductors,” Rev. Mod. Phys. 87, 1037–1066 (2015).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref] [PubMed]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

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

2014 (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10, 39–45 (2014).
[Crossref]

2013 (8)

R. Barnett, “Edge-state instabilities of bosons in a topological band,” Phys. Rev. A 88, 063631 (2013).
[Crossref]

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
[Crossref]

D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B 87, 235421 (2013).
[Crossref]

H. Schomerus and N. Y. Halpern, “Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices,” Phys. Rev. Lett. 110, 013903 (2013).
[Crossref] [PubMed]

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38, 1912–1914 (2013).
[Crossref] [PubMed]

2012 (5)

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

D. I. Pikulin and Yu. V. Nazarov, “Topological properties of superconducting junctions,” JETP Lett. 94, 693–697 (2012).
[Crossref]

J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
[Crossref] [PubMed]

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Y. Kawaguchi and M. Ueda, “Spinor Bose-Einstein condensates,” Phys. Rep. 520, 253–381 (2012).
[Crossref]

2011 (3)

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

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
[Crossref]

2010 (5)

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
[Crossref]

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

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

J. C. Y. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010).
[Crossref]

2009 (3)

L. Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-hall-insulator/superconductor junction,” Phys. Rev. B 79, 161408 (2009).
[Crossref]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34, 1633–1635 (2009).
[Crossref] [PubMed]

M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
[Crossref] [PubMed]

2008 (1)

J. Keeling and N. G. Berloff, “Spontaneous rotating vortex lattices in a pumped decaying condensate,” Phys. Rev. Lett. 100, 250401 (2008).
[Crossref] [PubMed]

2006 (1)

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
[Crossref]

2005 (3)

C. Ciuti and I. Carusotto, “Quantum fluid effects and parametric instabilities in microcavities,” Phys. Status Solidi B 242, 2224–2245 (2005).
[Crossref]

P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
[Crossref]

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[Crossref]

2002 (2)

G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
[Crossref]

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

2001 (1)

A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44 (suppl.), 131–136 (2001).
[Crossref]

1997 (1)

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[Crossref]

1996 (1)

H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
[Crossref]

1989 (1)

E. H. Lieb, “Two theorems on the Hubbard model,” Phys. Rev. Lett. 62, 1201–1204 (1989).
[Crossref] [PubMed]

1986 (1)

B. Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B 34, 5208–5211 (1986).
[Crossref]

1979 (1)

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

Abanin, D.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Abbarchi, M.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Aguado, R.

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

Aidelsburger, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Alicea, J.

J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
[Crossref] [PubMed]

Amo, A.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

An, S.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Andersson, E.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

Ando, M.

H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
[Crossref]

Aoki, H.

H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
[Crossref]

Apaja, V.

V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
[Crossref]

Aspuru-Guzik, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Atala, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Avila, J.

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

Aycock, L. M.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Baboux, F.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Bandres, M. A.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Bardyn, C.-E.

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

Barnett, R.

B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
[Crossref] [PubMed]

R. Barnett, “Edge-state instabilities of bosons in a topological band,” Phys. Rev. A 88, 063631 (2013).
[Crossref]

Barreiro, J. T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of Majorana fermions and topological superconductors,” Rev. Mod. Phys. 87, 1037–1066 (2015).
[Crossref]

C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
[Crossref]

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

Bellec, M.

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Benito, M.

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

Bercioux, D.

N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
[Crossref]

Berg, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Berloff, N. G.

J. Keeling and N. G. Berloff, “Spontaneous rotating vortex lattices in a pumped decaying condensate,” Phys. Rev. Lett. 100, 250401 (2008).
[Crossref] [PubMed]

Biondi, M.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Bliokh, K. Y.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref] [PubMed]

Bloch, I.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Bloch, J.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Brandes, T.

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

G. Engelhardt and T. Brandes, “Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons,” Phys. Rev. A 91, 053621 (2015).
[Crossref]

Brendel, C.

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

Broome, M. A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Bryant, G.

Budich, J. C.

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

Cancellieri, E.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Cantillano, C.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Carusotto, I.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

C. Ciuti and I. Carusotto, “Quantum fluid effects and parametric instabilities in microcavities,” Phys. Status Solidi B 242, 2224–2245 (2005).
[Crossref]

Cayao, J.

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

Chen, Z.

Chong, Y.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Chong, Y. D.

D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
[Crossref]

Choudhury, D.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

Christodoulides, D. N.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Ciuti, C.

C. Ciuti and I. Carusotto, “Quantum fluid effects and parametric instabilities in microcavities,” Phys. Status Solidi B 242, 2224–2245 (2005).
[Crossref]

Clarke, E.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Clerk, A. A.

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

Dahlhaus, J. P.

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

Demler, E.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Ding, J.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

El-Ganainy, R.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Engelhardt, G.

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

G. Engelhardt and T. Brandes, “Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons,” Phys. Rev. A 91, 053621 (2015).
[Crossref]

Fedrizzi, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Feng, L.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Fidkowski, L.

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

Flach, S.

D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
[Crossref]

Flensberg, K.

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Fu, L.

L. Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-hall-insulator/superconductor junction,” Phys. Rev. B 79, 161408 (2009).
[Crossref]

Furukawa, S.

S. Furukawa and M. Ueda, “Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices,” New J. Phys. 17, 115014 (2015).
[Crossref]

Furusaki, A.

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

Galilo, B.

B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
[Crossref] [PubMed]

Galopin, E.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Gao, F.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Gao, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Ge, L.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Genkina, D.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Goblot, V.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

Goldman, N.

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
[Crossref]

Gulevich, D. R.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Guo, W.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Haelterman, M.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[Crossref]

Halpern, N. Y.

H. Schomerus and N. Y. Halpern, “Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices,” Phys. Rev. Lett. 110, 013903 (2013).
[Crossref] [PubMed]

Han, X.

G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
[Crossref]

Harari, G.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Harayama, T.

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 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, 077002 (2002).
[Crossref] [PubMed]

Heeger, A. J.

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

Heinzner, P.

P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
[Crossref]

Hodaei, H.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Houde, M.

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

Hromada, I.

Huber, S. D.

R. Süsstrunk and S. D. Huber, “Observation of phononic helical edge states in a mechanical topological insulator,” Science 349, 47–50 (2015).
[Crossref]

Huckleberry, A.

P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
[Crossref]

Hyart, T.

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

Hyrkäs, M.

V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
[Crossref]

Ikeda, K. S.

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

Iorsh, I. V.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Jacqmin, T.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

Kane, C. L.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10, 39–45 (2014).
[Crossref]

J. C. Y. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010).
[Crossref]

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

L. Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-hall-insulator/superconductor junction,” Phys. Rev. B 79, 161408 (2009).
[Crossref]

Karzig, T.

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

Kassal, I.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Kawaguchi, Y.

Y. Kawaguchi and M. Ueda, “Spinor Bose-Einstein condensates,” Phys. Rep. 520, 253–381 (2012).
[Crossref]

Keeling, J.

J. Keeling and N. G. Berloff, “Spontaneous rotating vortex lattices in a pumped decaying condensate,” Phys. Rev. Lett. 100, 250401 (2008).
[Crossref] [PubMed]

Khajavikhan, M.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Kitaev, A.

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

Kitaev, A. Yu.

A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44 (suppl.), 131–136 (2001).
[Crossref]

Kitagawa, T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Kominis, Y.

S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
[Crossref]

Kovanis, V.

S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
[Crossref]

Krizhanovskii, D. N.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Kuhl, U.

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Le Gratiet, L.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Lee, C.-S.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Lee, D. K. K.

B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
[Crossref] [PubMed]

Leijnse, M.

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Lemaître, A.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Levitov, L. S.

M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
[Crossref] [PubMed]

Leykam, D.

D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
[Crossref]

Li, H.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Lieb, E. H.

E. H. Lieb, “Two theorems on the Hubbard model,” Phys. Rev. Lett. 62, 1201–1204 (1989).
[Crossref] [PubMed]

Liew, T. C. H.

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

Lin, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Lindner, N. H.

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

Longhi, S.

S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
[Crossref]

Lu, H.-I.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

Lubensky, T. C.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10, 39–45 (2014).
[Crossref]

Ludwig, A. W. W.

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

Lumer, Y.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Malkova, N.

Malpuech, G.

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Malzard, S.

S. Malzard and H. Schomerus, “Nonlinear mode competition and symmetry-protected power oscillations in topological lasers,” New J. Phys. 20, 063044 (2018).
[Crossref]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

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

Mandel, P.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[Crossref]

Manninen, M.

V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
[Crossref]

Marquardt, F.

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

Matsumoto, R.

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

Matsumura, H.

H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
[Crossref]

Mejía-Cortés, C.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Miao, P.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Molina, M. I.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Morales-Inostroza, L.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Morsch, O.

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
[Crossref]

Mortessagne, F.

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Mukherjee, S.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

Murakami, S.

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

Nalitov, A.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Nalitov, A. V.

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

Nazarov, Y. V.

D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B 87, 235421 (2013).
[Crossref]

Nazarov, Yu. V.

D. I. Pikulin and Yu. V. Nazarov, “Topological properties of superconducting junctions,” JETP Lett. 94, 693–697 (2012).
[Crossref]

Nolte, S.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Nori, F.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref] [PubMed]

Oberthaler, M.

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
[Crossref]

Öhberg, P.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

Ohe, J.-I.

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

Ozawa, T.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

Parto, M.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Peano, V.

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

Peñaranda, F.

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

Pikulin, D. I.

D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B 87, 235421 (2013).
[Crossref]

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

D. I. Pikulin and Yu. V. Nazarov, “Topological properties of superconducting junctions,” JETP Lett. 94, 693–697 (2012).
[Crossref]

Pitaevskii, L. P.

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, Oxford, 2003).

Platero, G.

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

Plotnik, Y.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Poli, C.

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

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

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[Crossref]

Prada, E.

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

Qi, X.-L.

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

Real, B.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Rechtsman, M. C.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Refael, G.

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

Ren, J.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Royall, B.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Rudner, M. S.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
[Crossref] [PubMed]

Ryu, S.

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

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

Sagnes, I.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Sala, V. G.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

San-Jose, P.

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

Schemmer, M.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Schmidt, M.

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

Schmidt, S.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Schnyder, A. P.

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

Schomerus, H.

S. Malzard and H. Schomerus, “Nonlinear mode competition and symmetry-protected power oscillations in topological lasers,” New J. Phys. 20, 063044 (2018).
[Crossref]

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

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

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

H. Schomerus and N. Y. Halpern, “Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices,” Phys. Rev. Lett. 110, 013903 (2013).
[Crossref] [PubMed]

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38, 1912–1914 (2013).
[Crossref] [PubMed]

Schrieffer, J. R.

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

Segev, M.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Shao, L. B.

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

Shelykh, I. A.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Shen, R.

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

Shi, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Shindou, R.

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

Skolnick, M. S.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Smirnova, D.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref] [PubMed]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

Solnyshkov, D. D.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

Spielman, I. B.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Spracklen, A.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

St-Jean, P.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

Stringari, S.

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, Oxford, 2003).

Su, W. P.

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

Sugawa, S.

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Sunada, S.

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

Süsstrunk, R.

R. Süsstrunk and S. D. Huber, “Observation of phononic helical edge states in a mechanical topological insulator,” Science 349, 47–50 (2015).
[Crossref]

Sutherland, B.

B. Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B 34, 5208–5211 (1986).
[Crossref]

Szameit, A.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Teimourpour, M. H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Teo, J. C. Y.

J. C. Y. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010).
[Crossref]

Terças, H.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Thomson, R. R.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

Tlidi, M.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[Crossref]

Türeci, H. E.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

Ueda, M.

S. Furukawa and M. Ueda, “Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices,” New J. Phys. 17, 115014 (2015).
[Crossref]

Y. Kawaguchi and M. Ueda, “Spinor Bose-Einstein condensates,” Phys. Rep. 520, 253–381 (2012).
[Crossref]

Urban, D. F.

N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
[Crossref]

Vaitiekus, D.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Vicencio, R. A.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Walker, P. M.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Wang, B.

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

Wang, X.

Weimann, S.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

Wen, G.

G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
[Crossref]

White, A. G.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Whittaker, C. E.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Whittaker, D. M.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

Wittek, S.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

Xing, D. Y.

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

Xu, D.

G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
[Crossref]

Yang, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Yao, R.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Zeuner, J. M.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

Zhang, B.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

Zhang, H.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Zhang, S.-C.

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

Zhao, H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Zheng, B.

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

Zirnbauer, M. R.

P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
[Crossref]

Zoller, P.

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

2D Materials (1)

C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Materials 4, 025008 (2017).
[Crossref]

Annu. Rev. Condens. Matter Phys. (1)

C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
[Crossref]

Chaos (1)

G. Wen, D. Xu, and X. Han, “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos 12, 350–355 (2002).
[Crossref]

Commun. Math. Phys. (1)

P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, “Symmetry classes of disordered fermions,” Commun. Math. Phys. 257, 725–771 (2005).
[Crossref]

Europhys. Lett. (1)

S. Longhi, Y. Kominis, and V. Kovanis, “Presence of temporal dynamical instabilities in topological insulator lasers,” Europhys. Lett. 122, 14004 (2018).
[Crossref]

JETP Lett. (1)

D. I. Pikulin and Yu. V. Nazarov, “Topological properties of superconducting junctions,” JETP Lett. 94, 693–697 (2012).
[Crossref]

Nat. Comm. (1)

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Comm. 3, 882 (2012).
[Crossref]

Nat. Commun. (3)

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2016).
[Crossref] [PubMed]

Nat. Photon. (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a 1D lattice,” Nat. Photon. 11, 651–656 (2017).
[Crossref]

Nat. Phys. (3)

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10, 39–45 (2014).
[Crossref]

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

New J. Phys. (3)

S. Furukawa and M. Ueda, “Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices,” New J. Phys. 17, 115014 (2015).
[Crossref]

S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New J. Phys. 12, 065010 (2010).
[Crossref]

S. Malzard and H. Schomerus, “Nonlinear mode competition and symmetry-protected power oscillations in topological lasers,” New J. Phys. 20, 063044 (2018).
[Crossref]

Opt. Lett. (2)

Phys. Rep. (1)

Y. Kawaguchi and M. Ueda, “Spinor Bose-Einstein condensates,” Phys. Rep. 520, 253–381 (2012).
[Crossref]

Phys. Rev. A (5)

R. Barnett, “Edge-state instabilities of bosons in a topological band,” Phys. Rev. A 88, 063631 (2013).
[Crossref]

G. Engelhardt and T. Brandes, “Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons,” Phys. Rev. A 91, 053621 (2015).
[Crossref]

V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402 (2010).
[Crossref]

N. Goldman, D. F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
[Crossref]

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

Phys. Rev. B (10)

R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, “Single Dirac cone with a flat band touching on line-centered-square optical lattices,” Phys. Rev. B 81, 041410 (2010).
[Crossref]

B. Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B 34, 5208–5211 (1986).
[Crossref]

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral Bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
[Crossref]

D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B 87, 235421 (2013).
[Crossref]

D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
[Crossref]

R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, “Topological chiral magnonic edge mode in a magnonic crystal,” Phys. Rev. B 87, 174427 (2013).
[Crossref]

J. C. Y. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010).
[Crossref]

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

L. Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-hall-insulator/superconductor junction,” Phys. Rev. B 79, 161408 (2009).
[Crossref]

H. Aoki, M. Ando, and H. Matsumura, “Hofstadter butterflies for flat bands,” Phys. Rev. B 54, R17296 (1996).
[Crossref]

Phys. Rev. E (1)

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[Crossref]

Phys. Rev. Lett. (19)

C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, “Fermion-parity anomaly of the critical supercurrent in the quantum spin-hall effect,” Phys. Rev. Lett. 110, 017003 (2013).
[Crossref] [PubMed]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref] [PubMed]

B. Galilo, D. K. K. Lee, and R. Barnett, “Selective population of edge states in a 2D topological band system,” Phys. Rev. Lett. 115, 245302 (2015).
[Crossref] [PubMed]

G. Engelhardt, M. Benito, G. Platero, and T. Brandes, “Topological instabilities in ac-driven bosonic systems,” Phys. Rev. Lett. 117, 045302 (2016).
[Crossref] [PubMed]

M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
[Crossref] [PubMed]

H. Schomerus and N. Y. Halpern, “Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices,” Phys. Rev. Lett. 110, 013903 (2013).
[Crossref] [PubMed]

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton 𝕑 topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polaritons in a two-dimensional Lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120, 113901 (2018).
[Crossref] [PubMed]

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

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

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

E. H. Lieb, “Two theorems on the Hubbard model,” Phys. Rev. Lett. 62, 1201–1204 (1989).
[Crossref] [PubMed]

J. Keeling and N. G. Berloff, “Spontaneous rotating vortex lattices in a pumped decaying condensate,” Phys. Rev. Lett. 100, 250401 (2008).
[Crossref] [PubMed]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized states in Lieb photonic lattices,” Phys. Rev. Lett. 114, 245503 (2015).
[Crossref] [PubMed]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114, 245504 (2015).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref] [PubMed]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref] [PubMed]

H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, “Geometrical pumping with a Bose-Einstein condensate,” Phys. Rev. Lett. 116, 200402 (2016).
[Crossref] [PubMed]

Phys. Rev. X (4)

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

Phys. Status Solidi B (1)

C. Ciuti and I. Carusotto, “Quantum fluid effects and parametric instabilities in microcavities,” Phys. Status Solidi B 242, 2224–2245 (2005).
[Crossref]

Phys. Usp. (1)

A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44 (suppl.), 131–136 (2001).
[Crossref]

Rep. Prog. Phys. (2)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[Crossref]

J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
[Crossref] [PubMed]

Rev. Mod. Phys. (4)

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

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

C. W. J. Beenakker, “Random-matrix theory of Majorana fermions and topological superconductors,” Rev. Mod. Phys. 87, 1037–1066 (2015).
[Crossref]

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
[Crossref]

Sci. Rep. (1)

P. San-Jose, J. Cayao, E. Prada, and R. Aguado, “Majorana bound states from exceptional points in non-topological superconductors,” Sci. Rep. 6, 21427 (2016).
[Crossref] [PubMed]

Science (2)

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref] [PubMed]

R. Süsstrunk and S. D. Huber, “Observation of phononic helical edge states in a mechanical topological insulator,” Science 349, 47–50 (2015).
[Crossref]

Semicond. Sci. Technol. (1)

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Other (7)

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, Oxford, 2003).

J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, “Non-hermitian topology: a unifying framework for the Andreev versus Majorana states controversy,” arXiv:1807.04677 (2018).

R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.-S. Lee, H. Zhang, and W. Guo, “Electrically tunable and reconfigurable topological edge state lasers,” arXiv:1804.01587 (2018).

These considerations can also be extended to include an inhomogeneity g(t) = −Xg*(t).

This exploits the U(1) gauge freedom.

At the juncture of cases (a) and (b), the nonlinearities can also stabilize symmetry-breaking modes with vanishing frequency, which then form a degenerate pair.

This again exploits the U(1) gauge freedom to fix the overall phase factor exp(iα) of the wavefunction, up to an overall sign. If a self-symmetric state is periodic this can therefore be realized in two variants Ψ(T) = Ψ(0) and Ψ(T) = −Ψ(0).

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Figures (4)

Fig. 1
Fig. 1 Illustration of the five types of modes for a topological laser array based on a Su-Schrieffer-Heeger chain with background loss γA = γB = 0.3 and various amounts of saturable gain [see Eqs. (4), (5) and (6)]. (a) Stationary symmetry-breaking mode (gA = 0.4, gB = 0.7). (b) Stationary self-symmetric zero mode (gA = 0.8, gB = 0.0). (c) Oscillating symmetry-breaking mode (gA = 0.7, gB = 0.4). (d) Oscillating self-symmetric mode (gA = 0.7, gB = 0.4 with symmetry-preserving initial conditions). (e) Twisted oscillating mode (gA = 0.8, gB = 0.2). The sketches at the very top symbolize the traces of the solutions in low-dimensional cross-sections of the dynamical phase space, where in (a–d) the symmetry is represented as a reflection and in (e) as a rotation. In the second row, the circles represent the resonators, where the area denotes the intensity (A and B sublattice in red and blue; in the time-dependent case, we show two circles corresponding to the largest and smallest intensity over a cycle.) The third row shows the time traces of the intensities IA = |A|2 (red), IB = |B|2 (blue), and Itot = IA + IB (black). The bottom panels in (a,b) show the stability spectra of the stationary states, while in (c–e) they show the correlation functions C = |〈Ψ(0)|Ψ(t)〉| (orange), = |〈Ψ(0)|Ψ̃(t)〉| (brown) of the oscillating states.
Fig. 2
Fig. 2 Same as Fig. 1 but for a polaritonic flat-band condensate based on a Lieb lattice with linear gain and density-dependent loss [see Eq. (7)]. To demonstrate the generality of our findings we include 50% relative disorder in all parameters, including for the couplings around their mean value tkl = 1, and the losses with average strength γA = γB = 0.3 (see Fig. 3 for details of the configuration). (a) Stationary symmetry-breaking mode (average gain gA = 0.15, gB = 0.3). (b) Stationary self-symmetric zero mode (average background loss gA = −0.2 and gain gB = 0.5). (c) Slowly oscillating symmetry-breaking mode (gA = 0.35, gB = 0.2). (d) Oscillating self-symmetric mode (gA = 0.5, gB = 0.3 with symmetry-preserving initial conditions). (e) Twisted oscillating mode (gA = 0.1, gB = 0.4).
Fig. 3
Fig. 3 Detailed geometry of the two illustrative models investigated in this work. (a) The model based on the SSH chain consists of a linear arrangement of 21 sites (11 A sites and 10 B sites) with alternating couplings 1, 0.7. The centre contains a defect with two consecutive couplings 0.7. This separates two configurations, denoted α and β, which can be characterized by topological features of their band structure. In the hermitian model the coupling defect induces one zero mode, which is spatially localized and exhibits an anomalous response to loss and gain of different strength on the two sublattices. For this model, we introduce nonlinearities in the form of saturable gain. (b) The model based on the Lieb lattice also consists of 21 sites, but these are arranged in two dimensions so that 12 are A sites and 9 are B sites. In the hermitian limit, there are now at least three zero modes, even in the case of disorder in the couplings. In this model we study density-dependent losses, and include disorder in the couplings tkl as well as in the gain and loss parameters gA,B and γA,B. This disorder is generated from independent random numbers rn with a box distribution over [0.75, 1.25], so that pn = prn for any model parameter pn with average p.
Fig. 4
Fig. 4 (a) Phase transition from a stationary zero mode to a twisted oscillating state of period T as signalled by the linear stability excitation spectrum, here shown for the SSH laser array with gA = 0.693, gB = 0.1. At the transition the excitations match up via the relation λ = exp(−iωT). This leads to a three-fold degeneracy of marginally stable excitations with λ = 1. (b) Away from the transition (gA = 0.8, gB = 0.1), the excitations rearrange to describe the stabilization of the oscillation amplitude of the emerging twisted mode (λf), leaving a two-fold degeneracy of marginal excitations λ0 = λt = 1 corresponding to U(1) and time-translation invariance. In the half-step operators, these two excitations are separated at λ′0 = −1, λ′t = 1 and hence structurally stabilized, which provides a signature of twisted states in terms of topological excitations.

Equations (18)

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i d d t Ψ ( t ) = H Ψ ( t ) + F [ Ψ ( t ) , Ψ * ( t ) ] Ψ ( t ) .
Ψ ˜ ( t ) = X Ψ * ( t ) .
Ω n Ψ n = H Ψ n + F [ Ψ n , Ψ n * ] Ψ n .
i d d t A k = < l > t k l B l + f A ( | A k | 2 ) A k ,
i d d t B k = < l > t l k A l + f B ( | B k | 2 ) B k ,
f s ( | Ψ k | 2 ) = i g s 1 + | Ψ k | 2 i γ s ( saturable gain ) ,
f s ( | Ψ k | 2 ) = i ( g s γ s | Ψ k | 2 ) ( density dependent loss )
ω + u + = ( H + f + 2 f | Ψ 0 | 2 ) u + ,
ω u = ( H + f ) u ,
i d d t u + ( t ) = [ H + f ( t ) + 2 f ( t ) | Ψ ( t ) | 2 ] u + ( t ) ,
i d d t u ( t ) = [ H + f ( t ) ] u ( t ) ,
Ψ ( t ) = [ Ψ n + u exp ( i ω t ) + v * exp ( i ω t ) ] exp ( i Ω n t )
ω ψ = ( [ Ψ n ] Σ z Ω n ) ψ
[ Ψ ] = ( H + f + f | Ψ | 2 f Ψ 2 [ f Ψ 2 ] * [ H + f + f | Ψ | 2 ] * ) ,
𝒳 ( [ Ψ n ] Σ z Ω n ) * 𝒳 = ( [ Ψ ˜ n ] + Σ z Ω n )
ω ± u ± = [ H + F + F Ψ Ψ 0 ± F Ψ * Ψ 0 X ] u ± .
i d d t ψ ( t ) = [ Ψ ( t ) ] ψ ( t ) ,
U ( T ) = 𝒳 U * ( T / 2 ) 𝒳 U ( T / 2 ) = 𝒳 Σ x U ( T / 2 ) Σ x 𝒳 U ( T / 2 ) .

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