Abstract

In this work, we first discuss systematically three general approaches to construct a non-Hermitian flat band, defined by its dispersionless real part. These approaches resort to, respectively, spontaneous restoration of non-Hermitian particle-hole symmetry, a persisting flat band from the underlying Hermitian system, and a compact Wannier function that is an eigenstate of the entire system. For the last approach in particular, we show the simplest lattice structure where it can be applied, and we further identify a special case of such a flat band where every point in the Brillouin zone is an exceptional point of order 3. A localized excitation in this “EP3 flat band” can display either a conserved power, quadratic power increase, or even quartic power increase, depending on whether the localized eigenstate or one of the two generalized eigenvectors is initially excited. Nevertheless, the asymptotic wave function in the long time limit is always given by the eigenstate, in this case, the compact Wannier function or its superposition in two or more unit cells.

© 2018 Chinese Laser Press

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Corrections

27 March 2018: Typographical corrections were made to the body text.


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References

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  1. V. Apaja, M. Hyrkäs, and M. Manninen, “Flat bands, Dirac cones, and atom dynamics in an optical lattice,” Phys. Rev. A 82, 041402(R) (2010).
    [Crossref]
  2. M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
    [Crossref]
  3. M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
    [Crossref]
  4. 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]
  5. 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]
  6. M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
    [Crossref]
  7. C. L. Kane and E. J. Mele, “Size, shape, and low energy electronic structure of carbon nanotubes,” Phys. Rev. Lett. 78, 1932–1935 (1997).
    [Crossref]
  8. F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
    [Crossref]
  9. A. Simon, “Supraleitung und Chemie,” Angew. Chem. 109, 1873–1891 (1997).
    [Crossref]
  10. S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
    [Crossref]
  11. S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
    [Crossref]
  12. M. Imada and M. Kohno, “Superconductivity from flat dispersion designed in doped Mott insulators,” Phys. Rev. Lett. 84, 143–146 (2000).
    [Crossref]
  13. E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
    [Crossref]
  14. T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
    [Crossref]
  15. S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
    [Crossref]
  16. T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
    [Crossref]
  17. 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]
  18. M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
    [Crossref]
  19. J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
    [Crossref]
  20. J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
    [Crossref]
  21. D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
    [Crossref]
  22. S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
    [Crossref]
  23. L. Ge, “Anomalous minimum and scaling behavior of localization length near an isolated flat band,” Ann. Phys. 527, 201600182 (2017).
  24. S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
    [Crossref]
  25. L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (2015).
    [Crossref]
  26. G.-W. Chern and A. Saxena, “PT-symmetric phase in kagome-based photonic lattices,” Opt. Lett. 40, 5806–5809 (2015).
    [Crossref]
  27. M. I. Molina, “Flatbands and PT-symmetry in quasi-one-dimensional lattices,” Phys. Rev. A 92, 063813 (2015).
    [Crossref]
  28. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).
  29. A. V. Yulin and V. V. Konotop, “Conservative and PT-symmetric compactons in waveguide networks,” Opt. Lett. 38, 4880–4883 (2013).
    [Crossref]
  30. H. Ramezani, “Non-hermiticity-induced flat band,” Phys. Rev. A 96, 011802 (2017).
    [Crossref]
  31. D. Leykam, S. Flach, and Y. D. Chong, “Flat bands in lattices with non-Hermitian coupling,” Phys. Rev. B 96, 064305 (2017).
    [Crossref]
  32. B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
    [Crossref]
  33. G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
    [Crossref]
  34. L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
    [Crossref]
  35. 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]
  36. L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A 95, 023812 (2017).
    [Crossref]
  37. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  38. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
    [Crossref]
  39. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref]
  40. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref]
  41. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
  42. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  43. L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
    [Crossref]
  44. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  45. L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
  46. J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
    [Crossref]
  47. W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A 37, 2455–2464 (2004).
    [Crossref]
  48. M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
    [Crossref]
  49. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge, 2011).
  50. C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
    [Crossref]
  51. J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
    [Crossref]
  52. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
    [Crossref]
  53. M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
    [Crossref]
  54. R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
    [Crossref]
  55. P. D. Lax, Linear Algebra and its Applications (Wiley, 2007).
  56. S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89, 053132 (2014).
    [Crossref]
  57. Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
    [Crossref]
  58. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
    [Crossref]
  59. W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
    [Crossref]

2017 (9)

L. Ge, “Anomalous minimum and scaling behavior of localization length near an isolated flat band,” Ann. Phys. 527, 201600182 (2017).

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).

H. Ramezani, “Non-hermiticity-induced flat band,” Phys. Rev. A 96, 011802 (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]

B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
[Crossref]

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A 95, 023812 (2017).
[Crossref]

Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

2016 (1)

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]

2015 (8)

L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (2015).
[Crossref]

G.-W. Chern and A. Saxena, “PT-symmetric phase in kagome-based photonic lattices,” Opt. Lett. 40, 5806–5809 (2015).
[Crossref]

M. I. Molina, “Flatbands and PT-symmetry in quasi-one-dimensional lattices,” Phys. Rev. A 92, 063813 (2015).
[Crossref]

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]

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]

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[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]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

2014 (6)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89, 053132 (2014).
[Crossref]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

2013 (5)

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

A. V. Yulin and V. V. Konotop, “Conservative and PT-symmetric compactons in waveguide networks,” Opt. Lett. 38, 4880–4883 (2013).
[Crossref]

M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

2012 (4)

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[Crossref]

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

2011 (2)

E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
[Crossref]

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

2010 (4)

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

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
[Crossref]

J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

2008 (4)

J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

2007 (1)

2006 (1)

M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
[Crossref]

2005 (1)

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

2004 (2)

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A 37, 2455–2464 (2004).
[Crossref]

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

2003 (2)

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
[Crossref]

S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
[Crossref]

2001 (1)

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

2000 (1)

M. Imada and M. Kohno, “Superconductivity from flat dispersion designed in doped Mott insulators,” Phys. Rev. Lett. 84, 143–146 (2000).
[Crossref]

1998 (1)

S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
[Crossref]

1997 (2)

A. Simon, “Supraleitung und Chemie,” Angew. Chem. 109, 1873–1891 (1997).
[Crossref]

C. L. Kane and E. J. Mele, “Size, shape, and low energy electronic structure of carbon nanotubes,” Phys. Rev. Lett. 78, 1932–1935 (1997).
[Crossref]

Abbarchi, M.

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Amo, A.

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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

An, K.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

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]

Apaja, V.

M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
[Crossref]

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

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]

Bahat-Treidel, O.

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

Berry, M. V.

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

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]

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[Crossref]

Blatter, G.

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[Crossref]

Bloch, J.

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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Bodyfelt, J. D.

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[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]

Carusotto, I.

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Cerjan, A.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Chalker, J. T.

J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
[Crossref]

Chamon, C.

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

Chen, W.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Chern, G.-W.

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]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[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]

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Danieli, C.

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

Das Sarma, S.

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[Crossref]

Della Valle, G.

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89, 053132 (2014).
[Crossref]

Demange, G.

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

Dembowski, C.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Deng, S.

S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
[Crossref]

S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
[Crossref]

Desyatnikov, A. S.

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

El-Ganainy, R.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Feng, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).

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]

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

Furukawa, N.

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

Galopin, 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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Garcia-Gracia, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Ge, L.

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A 95, 023812 (2017).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).

B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
[Crossref]

L. Ge, “Anomalous minimum and scaling behavior of localization length near an isolated flat band,” Ann. Phys. 527, 201600182 (2017).

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]

L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (2015).
[Crossref]

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[Crossref]

Geim, A. K.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
[Crossref]

Goda, M.

M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
[Crossref]

Goldman, N.

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]

Gong, Z.

Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
[Crossref]

Graefe, E.-M.

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

Gräf, H.-D.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Gu, Z.-C.

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[Crossref]

Guinea, F.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
[Crossref]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Harney, H. L.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Hassan, A. U.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Heine, A.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Heiss, W. D.

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A 37, 2455–2464 (2004).
[Crossref]

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Hentschel, M.

J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

Higashikawa, S.

Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
[Crossref]

Hodaei, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Hsu, C. W.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Hua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Huber, S. D.

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[Crossref]

Hyrkäs, M.

M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
[Crossref]

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

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Imada, M.

M. Imada and M. Kohno, “Superconductivity from flat dispersion designed in doped Mott insulators,” Phys. Rev. Lett. 84, 143–146 (2000).
[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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Joannopoulos, J. D.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Kane, C. L.

C. L. Kane and E. J. Mele, “Size, shape, and low energy electronic structure of carbon nanotubes,” Phys. Rev. Lett. 78, 1932–1935 (1997).
[Crossref]

Katsnelson, M. I.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
[Crossref]

Khajavikhan, M.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

Kim, S. W.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Kim, S.-W.

J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Köhler, J.

S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
[Crossref]

S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
[Crossref]

Kohno, M.

M. Imada and M. Kohno, “Superconductivity from flat dispersion designed in doped Mott insulators,” Phys. Rev. Lett. 84, 143–146 (2000).
[Crossref]

Konotop, V. V.

Kubo, K.

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

Lax, P. D.

P. D. Lax, Linear Algebra and its Applications (Wiley, 2007).

Le Gratiet, 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]

Lee, J.-H.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Lee, S.-B.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Lee, S.-Y.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Lemaître, A.

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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

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]

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

Liertzer, M.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Longhi, S.

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89, 053132 (2014).
[Crossref]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Malpuech, G.

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Malzard, S.

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]

Manninen, M.

M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
[Crossref]

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

Matsuda, H.

M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
[Crossref]

Matthies, P.

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

Mei, J.-W.

E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
[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]

Mele, E. J.

C. L. Kane and E. J. Mele, “Size, shape, and low energy electronic structure of carbon nanotubes,” Phys. Rev. Lett. 78, 1932–1935 (1997).
[Crossref]

Miyahara, S.

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

Moiseyev, N.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge, 2011).

Molina, M. I.

M. I. Molina, “Flatbands and PT-symmetry in quasi-one-dimensional lattices,” Phys. Rev. A 92, 063813 (2015).
[Crossref]

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]

Moon, S.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

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]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Mudry, C.

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

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]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Neupert, T.

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

Nishino, S.

M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
[Crossref]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[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]

Okolowicz, J.

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
[Crossref]

Ono, H.

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

Özdemir, S. K.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Pick, A.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Pickles, T. S.

J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
[Crossref]

Ploszajczak, M.

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
[Crossref]

Poli, C.

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]

Qi, B.

B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
[Crossref]

Ramezani, H.

H. Ramezani, “Non-hermiticity-induced flat band,” Phys. Rev. A 96, 011802 (2017).
[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]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

Rehfeld, H.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Richter, A.

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Rotter, I.

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
[Crossref]

Rotter, S.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Sagnes, I.

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]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Santos, L.

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

Saxena, A.

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]

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[Crossref]

Schomerus, H.

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]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Shim, J.-B.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Shimomura, Y.

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

Shukla, P.

J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
[Crossref]

Simon, A.

S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
[Crossref]

S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
[Crossref]

A. Simon, “Supraleitung und Chemie,” Angew. Chem. 109, 1873–1891 (1997).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Soljacic, M.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Solnyshkov, D. D.

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

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]

Stone, A. D.

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Sun, K.

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[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]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

Tang, E.

E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
[Crossref]

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]

Tünnermann, A.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[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]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Ueda, M.

Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
[Crossref]

van Nieuwenburg, E. P. L.

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[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]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

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]

Wen, X.-G.

E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
[Crossref]

Wiersig, J.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Yang, J.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Yang, L.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Yang, S.

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[Crossref]

Yu, X.

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

Yulin, A. V.

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

Zhang, L.

B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
[Crossref]

Zhao, G.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Angew. Chem. (2)

A. Simon, “Supraleitung und Chemie,” Angew. Chem. 109, 1873–1891 (1997).
[Crossref]

S. Deng, A. Simon, and J. Köhler, “Supraleitung und chemische Bindung in Quecksilber,” Angew. Chem. 110, 664–666 (1998).
[Crossref]

Ann. Phys. (1)

L. Ge, “Anomalous minimum and scaling behavior of localization length near an isolated flat band,” Ann. Phys. 527, 201600182 (2017).

Czech. J. Phys. (1)

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

Europhys. Lett. (1)

S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, “Detangling flat bands into Fano lattices,” Europhys. Lett. 105, 30001 (2014).
[Crossref]

J. Phys. A (2)

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A 37, 2455–2464 (2004).
[Crossref]

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

J. Phys. Soc. Jpn. (1)

S. Miyahara, K. Kubo, H. Ono, Y. Shimomura, and N. Furukawa, “Flat-bands on partial line graphs Systematic method for generating flat-band lattice structures,” J. Phys. Soc. Jpn. 74, 1918–1921 (2005).
[Crossref]

J. Solid State Chem. (1)

S. Deng, A. Simon, and J. Köhler, “The origin of a flat band,” J. Solid State Chem. 176, 412–416 (2003).
[Crossref]

Nat. Photonics (2)

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures,” Nat. Photonics 7, 153–158 (2013).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).

Nat. Phys. (2)

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30–33 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (3)

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Hua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Opt. Lett. (3)

Phys. Rep. (1)

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374, 271–383 (2003).
[Crossref]

Phys. Rev. A (11)

J. Wiersig, S.-W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[Crossref]

L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (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(R) (2010).
[Crossref]

M. Hyrkäs, V. Apaja, and M. Manninen, “Many-particle dynamics of bosons and fermions in quasi-one-dimensional flat-band lattices,” Phys. Rev. A 87, 023614 (2013).
[Crossref]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A 95, 023812 (2017).
[Crossref]

H. Ramezani, “Non-hermiticity-induced flat band,” Phys. Rev. A 96, 011802 (2017).
[Crossref]

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89, 053132 (2014).
[Crossref]

M. I. Molina, “Flatbands and PT-symmetry in quasi-one-dimensional lattices,” Phys. Rev. A 92, 063813 (2015).
[Crossref]

Phys. Rev. B (4)

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

J. T. Chalker, T. S. Pickles, and P. Shukla, “Anderson localization in tight-binding models with flat bands,” Phys. Rev. B 82, 104209 (2010).
[Crossref]

S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, “Topological flat band models with arbitrary Chern numbers,” Phys. Rev. B 86, 241112(R) (2012).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, “Flat band states: disorder and nonlinearity,” Phys. Rev. B 88, 224203 (2013).
[Crossref]

Phys. Rev. Lett. (21)

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an xceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[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]

M. Goda, S. Nishino, and H. Matsuda, “Inverse Anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
[Crossref]

J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, “Flatbands under correlated perturbations,” Phys. Rev. Lett. 113, 236403 (2014).
[Crossref]

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]

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]

M. Biondi, E. P. L. van Nieuwenburg, G. Blatter, S. D. Huber, and S. Schmidt, “Incompressible polaritons in a flat band,” Phys. Rev. Lett. 115, 143601 (2015).
[Crossref]

C. L. Kane and E. J. Mele, “Size, shape, and low energy electronic structure of carbon nanotubes,” Phys. Rev. Lett. 78, 1932–1935 (1997).
[Crossref]

M. Imada and M. Kohno, “Superconductivity from flat dispersion designed in doped Mott insulators,” Phys. Rev. Lett. 84, 143–146 (2000).
[Crossref]

E. Tang, J.-W. Mei, and X.-G. Wen, “High-temperature fractional quantum Hall states,” Phys. Rev. Lett. 106, 236802 (2011).
[Crossref]

T. Neupert, L. Santos, C. Chamon, and C. Mudry, “Fractional quantum Hall states at zero Magnetic field,” Phys. Rev. Lett. 106, 236804 (2011).
[Crossref]

B. Qi, L. Zhang, and L. Ge, “Defect states emerging from a non-Hermitian flatband of photonic zero modes,” Phys. Rev. Lett. 120, 093901 (2017).
[Crossref]

Z. Gong, S. Higashikawa, and M. Ueda, “Zeno Hall effect,” Phys. Rev. Lett. 118, 200401 (2017).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[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]

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Phys. Rev. X (1)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

Other (2)

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge, 2011).

P. D. Lax, Linear Algebra and its Applications (Wiley, 2007).

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Figures (9)

Fig. 1.
Fig. 1. (a) Band structure of a Hermitian Lieb lattice. The flat band is shown by the thick line. Inset: schematic of the Lieb lattice, where G (solid lines) is 3/4 of J (dashed lines). Partially transparent dots show the spatial profile of the compact Wannier function. (b) Same as (a) but with gain and loss modulation γ A = 1 , γ B = 0.5 , γ C = 0.1 . Dashed lines show the imaginary parts of the band structure, and those of the two dispersive bands are the same (given by the thin dashed line) due to the relation ω l ( k ) = ω m * ( k ) ( l m ) imposed by NHPH symmetry.
Fig. 2.
Fig. 2. Band structure of a 2D rectangular lattice with loss introduced to the A sites. (a) Schematics of the rectangular lattice and its reciprocal lattice. The unit cell is highlighted by the rounded box. G = 1.2 J > 0 . (b)–(d) Real part of the band structure when γ A / J = 0 , 2 , 4.8 .
Fig. 3.
Fig. 3. Same as Fig. 2 but plotted in three dimensions with G = 0.8 J > 0 and γ A / J = 3.2 in (a) and 4 in (b).
Fig. 4.
Fig. 4. Band structure of the 2D square lattice shown in Fig. 2(a) but with a detuning Δ A = 2 J and γ A / J = 4.8 in (a) and 20 in (b). The effectively Hermitian Lieb lattice in (b) and its band structure are shown in (c) and (d).
Fig. 5.
Fig. 5. Persisting and nonpersisting Hermitian flat bands. (a) Band structure of a Hermitian quasi-1D edge-centered square lattice. Inset: G (solid lines) = J (dashed lines). Partially transparent dots show the spatial profile of the compact Wannier function. (b) Same as (a) but with γ A , B , D , E = 0.5 , γ C = 1 . Again the imaginary parts (dashed lines) of the dispersive bands are identical pairwise due to NHPH symmetry. (c), (d) Same as (a), (b) but for a saw lattice with γ A = 0 , γ B = 2 . Inset in (c):  G (solid lines) = 2 J (dashed lines).
Fig. 6.
Fig. 6. Persisting flat band in a 2D non-Hermitian Lieb lattice with G = 1.2 J > 0 . γ B , C , D = 0.78 , 0.71 , 0.32 in (a) and 2.1 , 0.3 , 3.3 in (b).
Fig. 7.
Fig. 7. (a, b) Two compact Wannier functions (partially transparent dots) for the cross-stitched lattice shown. Couplings are represented by dashed lines ( J ), dash–dotted lines ( J * ), and solid lines ( G ). Gain ( i γ ) and loss ( i γ ) are introduced to B and A sites, respectively. Here, G = | J | , Arg [ J ] = 0.3 θ / 2 , and γ = G / sin θ in (a) and G sin θ in (b). (c,d) Band structure of this cross-stitched lattice for the values of γ in (a) and (b). The flat bands result from the compact Wannier functions in (a) and (b).
Fig. 8.
Fig. 8. Polynomial power dependence for a localized initial excitation in an EP3 flat band. The excitation of (a) a Wannier function in a single unit cell, (b) a C site, and (c) an A site are shown by the red arrows. Note the different scales of the vertical axis in these three panels, even though the initial amplitudes of the excitation all equal 1. (d) Fixed amplitude | Ψ ( C ) | = 1 at the excitation site in (b) (solid line) and linearly increasing amplitude | Ψ ( A ) | next to it (dashed line). (e) Quadratically increasing amplitude | Ψ ( A ) | at the excitation site in (c) (dotted line) and linearly increasing amplitude | Ψ ( C ) | next to it (dashed line).
Fig. 9.
Fig. 9. Relation between the three general approaches to construct a non-Hermitian flat band.

Equations (11)

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Ψ n ( x ; k ) = j e i k a j W n ( x j a ) ,
H ( k ) = [ ω A G 0 G ω B J ( 1 + e i k a ) 0 J ( 1 + e i k a ) ω C ] ,
H ( k ) = [ i γ a J ˜ ( k x ) G ˜ ( k y ) 0 J ˜ * ( k x ) 0 0 G ˜ ( k y ) G ˜ * ( k y ) 0 0 J ˜ ( k x ) 0 G ˜ * ( k y ) J ˜ * ( k x ) 0 ] ,
H ( k ) = [ i γ G J ( 1 + e i k a ) G i γ J * ( 1 + e i k a ) J ( 1 + e i k a ) J * ( 1 + e i k a ) 0 ] ,
H ( k ) = [ i γ A G ( 1 + e i k a ) G ( 1 + e i k a ) i γ B + 2 J cos k a ] .
ω FB = 2 J + i γ B , ω D = 2 J ( 1 + cos k a ) + i γ A .
H ( k ) = [ i G G J ( 1 + e i k a ) G i G i J ( 1 + e i k a ) J ( 1 + e i k a ) i J ( 1 + e i k a ) 0 ] .
[ H ω 0 1 ] Ψ 1 = Ψ 0 ,
e i H t Ψ 1 = e i ω 0 t ( Ψ 1 i t Ψ 0 ) ,
[ H ω 0 1 ] Ψ 2 = Ψ 1 ,
e i H t Ψ 2 = e i ω 0 t ( Ψ 2 i t Ψ 1 t 2 2 Ψ 0 ) .

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