Advertisement

Topological states of non-Hermitian systems

  • V. M. Martinez Alvarez
  • J. E. Barrios Vargas
  • M. Berdakin
  • L. E. F. Foa Torres
Regular Article
  • 75 Downloads
Part of the following topical collections:
  1. Topological States of Matter: Theory and Applications

Abstract

Recently, the search for topological states of matter has turned to non-Hermitian systems, which exhibit a rich variety of unique properties without Hermitian counterparts. Lattices modeled through non-Hermitian Hamiltonians appear in the context of photonic systems, where one needs to account for gain and loss, circuits of resonators, and also when modeling the lifetime due to interactions in condensed matter systems. Here we provide a brief overview of this rapidly growing subject, the search for topological states and a bulk-boundary correspondence in non-Hermitian systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010) ADSCrossRefGoogle Scholar
  2. 2.
    K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980) ADSCrossRefGoogle Scholar
  3. 3.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982) ADSCrossRefGoogle Scholar
  4. 4.
    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988) ADSCrossRefGoogle Scholar
  5. 5.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005). ADSCrossRefGoogle Scholar
  6. 6.
    B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) ADSCrossRefGoogle Scholar
  7. 7.
    M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi, S.C. Zhang, Science 318, 766 (2007) ADSCrossRefGoogle Scholar
  8. 8.
    D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature 452, 970 (2008) ADSCrossRefGoogle Scholar
  9. 9.
    O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014) ADSCrossRefGoogle Scholar
  10. 10.
    T. Oka, H. Aoki, Phys. Rev. B 79, 081406 (2009) ADSCrossRefGoogle Scholar
  11. 11.
    N.H. Lindner, G. Refael, V. Galitski, Nat. Phys. 7, 490 (2011) CrossRefGoogle Scholar
  12. 12.
    L.E.F. Foa Torres, P.M. Perez-Piskunow, C.A. Balseiro, G. Usaj, Phys. Rev. Lett. 113, 266801 (2014) ADSCrossRefGoogle Scholar
  13. 13.
    M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 102, 065703 (2009) ADSCrossRefGoogle Scholar
  14. 14.
    S. Diehl, E. Rico, M.A. Baranov, P. Zoller, Nat. Phys. 7, 971 (2011) CrossRefGoogle Scholar
  15. 15.
    C. Yuce, Phys. Lett. A 379, 1213 (2015) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    P. San-Jose, J. Cayao, E. Prada, R. Aguado, Sci. Rep. 6, 21427 (2016) ADSCrossRefGoogle Scholar
  17. 17.
    T.E. Lee, Phys. Rev. Lett. 116, 133903 (2016) ADSCrossRefGoogle Scholar
  18. 18.
    D. Leykam, K.Y. Bliokh, C. Huang, Y. Chong, F. Nori, Phys. Rev. Lett. 118, 040401 (2017) ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Kawabata, Y. Ashida, H. Katsura, M. Ueda, Phys. Rev. B 98, 085116 (2018) ADSCrossRefGoogle Scholar
  20. 20.
    C. Yuce, Phys. Rev. A 93, 062130 (2016) ADSCrossRefGoogle Scholar
  21. 21.
    I. Rotter, J. Phys. A: Math. Theor. 42, 153001 (2009) zbMATHGoogle Scholar
  22. 22.
    N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, New York, 2011) Google Scholar
  23. 23.
    C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998) ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010) CrossRefGoogle Scholar
  25. 25.
    S. Longhi, Europhys. Lett. 120, 64001 (2017) ADSCrossRefGoogle Scholar
  26. 26.
    T. Ozawa, H.M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto, https://doi.org/arXiv:1802.04173 (2018)
  27. 27.
    V. Kozii, L. Fu, https://doi.org/arXiv:1708.05841 (2017)
  28. 28.
    T. Yoshida, R. Peters, N. Kawakami, Phys. Rev. B 98, 035141 (2018) ADSCrossRefGoogle Scholar
  29. 29.
    N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996) ADSCrossRefGoogle Scholar
  30. 30.
    N. Hatano, D.R. Nelson, Phys. Rev. B 56, 8651 (1997) ADSCrossRefGoogle Scholar
  31. 31.
    N. Hatano, D.R. Nelson, Phys. Rev. B 58, 8384 (1998) ADSCrossRefGoogle Scholar
  32. 32.
    D.R. Nelson, N.M. Shnerb, Phys. Rev. E 58, 1383 (1998) ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Amir, N. Hatano, D.R. Nelson, Phys. Rev. E 93, 042310 (2016) ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Murugan, S. Vaikuntanathan, Nat. Commun. 8, 13881 (2017) ADSCrossRefGoogle Scholar
  35. 35.
    Y. Xu, S.T. Wang, L.M. Duan, Phys. Rev. Lett. 118, 045701 (2017) ADSCrossRefGoogle Scholar
  36. 36.
    A.A. Zyuzin, A.Y. Zyuzin, Phys. Rev. B 97, 041203 (2018) ADSCrossRefGoogle Scholar
  37. 37.
    A. Cerjan, M. Xiao, L. Yuan, S. Fan, Phys. Rev. B 97, 075128 (2018) ADSCrossRefGoogle Scholar
  38. 38.
    J. González, R.A. Molina, Phys. Rev. B 96, 045437 (2017) ADSCrossRefGoogle Scholar
  39. 39.
    R.A. Molina, J. González, Phys. Rev. Lett. 120, 146601 (2018) ADSCrossRefGoogle Scholar
  40. 40.
    C.E. Bardyn, M.A. Baranov, C.V. Kraus, E. Rico, A. İmamoğlu, P. Zoller, S. Diehl, New J. Phys. 15, 085001 (2013) ADSCrossRefGoogle Scholar
  41. 41.
    J.C. Budich, S. Diehl, Phys. Rev. B 91, 165140 (2015) ADSCrossRefGoogle Scholar
  42. 42.
    J.C. Budich, P. Zoller, S. Diehl, Phys. Rev. A 91, 042117 (2015) ADSCrossRefGoogle Scholar
  43. 43.
    W.T. Silfvast, Laser Fundamentals (Cambridge University Press, New York, 2004) Google Scholar
  44. 44.
    A.V. Smilga, J. Phys. A: Math. Theor. 42, 095301 (2009) ADSCrossRefGoogle Scholar
  45. 45.
    J. Feinberg, A. Zee, Phys. Rev. E 59, 6433 (1999) ADSCrossRefGoogle Scholar
  46. 46.
    T. Kato, Perturbation Theory of Linear Operators (Springer, 1966) Google Scholar
  47. 47.
    W.D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012) ADSCrossRefGoogle Scholar
  48. 48.
    J. von Neuman, E. Wigner, Phys. Zeitschrift 30, 467 (1929) ADSGoogle Scholar
  49. 49.
    M. Berry, Curr. Sci. 67, 220 (1994) Google Scholar
  50. 50.
    M. Berry, Nat. Phys. 6, 148 (2010) CrossRefGoogle Scholar
  51. 51.
    W.D. Heiss, H.L. Harney, Eur. Phys. J. D 17, 149 (2001) ADSCrossRefGoogle Scholar
  52. 52.
    A.U. Hassan, G.L. Galmiche, G. Harari, P. LiKamWa, M. Khajavikhan, M. Segev, D.N. Christodoulides, Phys. Rev. A 96, 052129 (2017) ADSCrossRefGoogle Scholar
  53. 53.
    A.U. Hassan, G.L. Galmiche, G. Harari, P. LiKamWa, M. Khajavikhan, M. Segev, D.N. Christodoulides, Phys. Rev. A 96, 069908 (2017) ADSCrossRefGoogle Scholar
  54. 54.
    Q. Zhong, M. Khajavikhan, D. Christodoulides, R. El-Ganainy, https://doi.org/arXiv:1805.07620 [quant-ph] (2018)
  55. 55.
    C. Dembowski, H.D. Gräf, H.L. Harney, A. Heine, W.D. Heiss, H. Rehfeld, A. Richter, Phys. Rev. Lett. 86, 787 (2001) ADSCrossRefGoogle Scholar
  56. 56.
    T. Stehmann, W.D. Heiss, F.G. Scholtz, J. Phys. A: Math. Gen. 37, 7813 (2004) ADSCrossRefGoogle Scholar
  57. 57.
    S.B. Lee, J. Yang, S. Moon, S.Y. Lee, J.B. Shim, S.W. Kim, J.H. Lee, K. An, Phys. Rev. Lett. 103, 134101 (2009) ADSCrossRefGoogle Scholar
  58. 58.
    W. Hu, H. Wang, P.P. Shum, Y.D. Chong, Phys. Rev. B 95, 184306 (2017) ADSCrossRefGoogle Scholar
  59. 59.
    G.A. Álvarez, E.P. Danieli, P.R. Levstein, H.M. Pastawski, J. Chem. Phys. 124, 194507 (2006) ADSCrossRefGoogle Scholar
  60. 60.
    S. Ghosh, Y. Chong, Sci. Rep. 6, 19837 (2016) ADSCrossRefGoogle Scholar
  61. 61.
    J. Doppler, A.A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T.J. Milburn, P. Rabl, N. Moiseyev, S. Rotter, Nature 537, 76 (2016) ADSCrossRefGoogle Scholar
  62. 62.
    H. Xu, D. Mason, L. Jiang, J.G.E. Harris, Nature 537, 80 (2016) ADSCrossRefGoogle Scholar
  63. 63.
    X.L. Zhang, S. Wang, B. Hou, C. Chan, Phys. Rev. X 8, 021066 (2018) Google Scholar
  64. 64.
    J.W. Ryu, S.Y. Lee, S.W. Kim, Phys. Rev. A 85, 042101 (2012) ADSCrossRefGoogle Scholar
  65. 65.
    H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016) ADSCrossRefGoogle Scholar
  66. 66.
    H. Jing, S.K. Özdemir, H. Lü, F. Nori, Sci. Rep. 7, 3386 (2017) ADSCrossRefGoogle Scholar
  67. 67.
    A. Pick, B. Zhen, O.D. Miller, C.W. Hsu, F. Hernandez, A.W. Rodriguez, M. Soljačić, S.G. Johnson, Opt. Express 25, 12325 (2017) ADSCrossRefGoogle Scholar
  68. 68.
    H. Hodaei, A.U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D.N. Christodoulides, M. Khajavikhan, Nature 548, 187 (2017) ADSCrossRefGoogle Scholar
  69. 69.
    E.M. Graefe, U. Günther, H.J. Korsch, A.E. Niederle, J. Phys. A: MathTheor. 41, 255206 (2008) ADSCrossRefGoogle Scholar
  70. 70.
    K. Ding, Z.Q. Zhang, C.T. Chan, Phys. Rev. B 92, 235310 (2015) ADSCrossRefGoogle Scholar
  71. 71.
    V.M. Martinez Alvarez, J.E. Barrios Vargas, L.E.F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018) ADSCrossRefGoogle Scholar
  72. 72.
    Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D.N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011) ADSCrossRefGoogle Scholar
  73. 73.
    L. Feng, Z.J. Wong, R.M. Ma, Y. Wang, X. Zhang, Science 346, 972 (2014) ADSCrossRefGoogle Scholar
  74. 74.
    H. Hodaei, M.A. Miri, M. Heinrich, D.N. Christodoulides, M. Khajavikhan, Science 346, 975 (2014) ADSCrossRefGoogle Scholar
  75. 75.
    W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, L. Yang, Nature 548, 192 (2017) ADSCrossRefGoogle Scholar
  76. 76.
    M.C. Rechtsman, Nature 548, 161 (2017) ADSCrossRefGoogle Scholar
  77. 77.
    J.P. Bird, Y. Ochiai, Science 303, 1621 (2004) CrossRefGoogle Scholar
  78. 78.
    Y. Yoon, M.G. Kang, T. Morimoto, M. Kida, N. Aoki, J.L. Reno, Y. Ochiai, L. Mourokh, J. Fransson, J.P. Bird, Phys. Rev. X 2, 021003 (2012) Google Scholar
  79. 79.
    I. Rotter, J.P. Bird, Rep. Prog. Phys. 78, 114001 (2015) ADSCrossRefGoogle Scholar
  80. 80.
    N. Auerbach, V. Zelevinsky, Rep. Prog. Phys. 74, 106301 (2011) ADSCrossRefGoogle Scholar
  81. 81.
    H. Eleuch, I. Rotter, Phys. Rev. A 95, 022117 (2017) ADSCrossRefGoogle Scholar
  82. 82.
    M. Hafezi, E. Demler, M. Lukin, J. Taylor, Nat. Phys. 7, 907 (2011) CrossRefGoogle Scholar
  83. 83.
    S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K.G. Makris, M. Segev, M.C. Rechtsman, A. Szameit, Nat. Mater. 16, 433 (2017) ADSCrossRefGoogle Scholar
  84. 84.
    H. Shen, B. Zhen, L. Fu, Phys. Rev. Lett. 120, 146402 (2018) ADSMathSciNetCrossRefGoogle Scholar
  85. 85.
    C. Yin, H. Jiang, L. Li, R. Lü, S. Chen, Phys. Rev. A 97, 052115 (2018) ADSCrossRefGoogle Scholar
  86. 86.
    L. Zhou, Q.h. Wang, H. Wang, J. Gong, Phys. Rev. A 98, 022129 (2018) ADSCrossRefGoogle Scholar
  87. 87.
    J.W. Ryu, N. Myoung, H.C. Park, Phys. Rev. B 96, 125421 (2017) ADSCrossRefGoogle Scholar
  88. 88.
    J.W. Ryu, N. Myoung, H.C. Park, Sci. Rep. 7, 8746 (2017) ADSCrossRefGoogle Scholar
  89. 89.
    C. Yuce, Phys. Rev. A 97, 042118 (2018) ADSCrossRefGoogle Scholar
  90. 90.
    S. Yao, Z. Wang, Phys. Rev. Lett. 121, 086803 (2018) ADSCrossRefGoogle Scholar
  91. 91.
    S. Lieu, Phys. Rev. B 97, 045106 (2018) ADSCrossRefGoogle Scholar
  92. 92.
    B. Zhu, R. Lü, S. Chen, Phys. Rev. A 89, 062102 (2014) ADSCrossRefGoogle Scholar
  93. 93.
    M. Klett, H. Cartarius, D. Dast, J. Main, G. Wunner, Phys. Rev. A 95, 053626 (2017) ADSCrossRefGoogle Scholar
  94. 94.
    F. Dangel, M. Wagner, H. Cartarius, J. Main, G. Wunner, Phys. Rev. A 98, 013628 (2018) ADSCrossRefGoogle Scholar
  95. 95.
    M. Klett, H. Cartarius, D. Dast, J. Main, G. Wunner, https://doi.org/arXiv:1802.06128 (2018)
  96. 96.
    L. Jin, P. Wang, Z. Song, Sci. Rep. 7, 5903 (2017) ADSCrossRefGoogle Scholar
  97. 97.
    C. Yuce, Eur. Phys. J. D 69, 184 (2015) ADSCrossRefGoogle Scholar
  98. 98.
    Z. Turker, S. Tombuloglu, C. Yuce, Phys. Lett. A 382, 2013 (2018) ADSCrossRefGoogle Scholar
  99. 99.
    L. Zhou, J. Gong, https://doi.org/arXiv:1807.00988 [cond-mat. mes-hall] (2018)
  100. 100.
    H. Li, T. Kottos, B. Shapiro, Phys. Rev. Appl. 9, 044031 (2018) ADSCrossRefGoogle Scholar
  101. 101.
    T.T. Koutserimpas, R. Fleury, Phys. Rev. Lett. 120, 087401 (2018) ADSCrossRefGoogle Scholar
  102. 102.
    S. Longhi, EPL 117, 10005 (2017) ADSCrossRefGoogle Scholar
  103. 103.
    M. Li, X. Ni, M. Weiner, A. Alù, A.B. Khanikaev, https://doi.org/arXiv:1807.00913 (2018)
  104. 104.
    V. Dal Lago, E. Suárez Morell, L.E.F. Foa Torres, Phys. Rev. B 96, 235409 (2017) ADSCrossRefGoogle Scholar
  105. 105.
    X. Ni, D. Smirnova, A. Poddubny, D. Leykam, Y. Chong, A.B. Khanikaev, https://doi.org/arXiv:1801.04689 (2018)
  106. 106.
    L.E.F. Foa Torres, V. Dal Lago, E. Suárez Morell, Phys. Rev. B 93, 075438 (2016) ADSCrossRefGoogle Scholar
  107. 107.
    B.A. Bernevig, T.L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013) Google Scholar
  108. 108.
    F. Ortmann, S. Roche, S.O. Valenzuela, L.W. Molenkamp (eds.), Topological Insulators: Fundamentals and Perspectives (Wiley, 2015) Google Scholar
  109. 109.
    J.K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators (Springer, Berlin, 2016) Google Scholar
  110. 110.
    J.W. Rhim, J. Behrends, J.H. Bardarson, Phys. Rev. B 95, 035421 (2017) ADSCrossRefGoogle Scholar
  111. 111.
    J.W. Rhim, J.H. Bardarson, R.J. Slager, Phys. Rev. B 97, 115143 (2018) ADSCrossRefGoogle Scholar
  112. 112.
    Y.C. Hu, T.L. Hughes, Phys. Rev. B 84, 153101 (2011) ADSCrossRefGoogle Scholar
  113. 113.
    K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, Phys. Rev. B 84, 205128 (2011) ADSCrossRefGoogle Scholar
  114. 114.
    Y. Xiong, J. Phys. Commun. 2, 035043 (2018) CrossRefGoogle Scholar
  115. 115.
    Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda, https://doi.org/arXiv:1802.07964 (2018)
  116. 116.
    B. Kramer, A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993) ADSCrossRefGoogle Scholar
  117. 117.
    F. Evers, A.D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008) ADSCrossRefGoogle Scholar
  118. 118.
    M.S. Rudner, M. Levin, L.S. Levitov, https://doi.org/arXiv:1605.07652 (2016)
  119. 119.
    S. Yao, F. Song, Z. Wang, https://doi.org/arXiv:1804.04672 (2018)
  120. 120.
    F.K. Kunst, E. Edvardsson, J.C. Budich, E.J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018) ADSCrossRefGoogle Scholar
  121. 121.
    C. Hahn, Y. Choi, J.W. Yoon, S.H. Song, C.H. Oh, P. Berini, Nat. Commun. 7, 12201 (2016) ADSCrossRefGoogle Scholar
  122. 122.
    F. Nazari, N. Bender, H. Ramezani, M. Moravvej-Farshi, D. Christodoulides, T. Kottos, Opt. Express 22, 9574 (2014) ADSCrossRefGoogle Scholar
  123. 123.
    L. Xiao, X. Zhan, Z. Bian, K. Wang, X. Zhang, X. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami et al., Nat. Phys. 13, 1117 (2017) CrossRefGoogle Scholar
  124. 124.
    M. Pan, H. Zhao, P. Miao, S. Longhi, L. Feng, Nat. Commun. 9, 1308 (2018) ADSCrossRefGoogle Scholar
  125. 125.
    H. Xu, D. Mason, L. Jiang, J.G.E. Harris, https://doi.org/arXiv:1703.07374 (2017)
  126. 126.
    C.A. Downing, G. Weick, https://doi.org/arXiv:1803.08872 (2018)
  127. 127.
    H. Schomerus, Opt. Lett. 38, 1912 (2013) ADSCrossRefGoogle Scholar
  128. 128.
    J.M. Zeuner, M.C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M.S. Rudner, M. Segev, A. Szameit, Phys. Rev. Lett. 115, 040402 (2015) ADSCrossRefGoogle Scholar
  129. 129.
    C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus, Nat. Commun. 6, 6710 (2015) ADSCrossRefGoogle Scholar
  130. 130.
    H. Zhao, P. Miao, M.H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, L. Feng, Nat. Commun. 9, 981 (2018) ADSCrossRefGoogle Scholar
  131. 131.
    M. Parto, S. Wittek, H. Hodaei, G. Harari, M.A. Bandres, J. Ren, M.C. Rechtsman, M. Segev, D.N. Christodoulides, M. Khajavikhan, Phys. Rev. Lett. 120, 113901 (2018) ADSCrossRefGoogle Scholar
  132. 132.
    S. Malzard, C. Poli, H. Schomerus, Phys. Rev. Lett. 115, 200402 (2015) ADSCrossRefGoogle Scholar
  133. 133.
    M.A. Gorlach, X. Ni, D.A. Smirnova, D. Korobkin, D. Zhirihin, A.P. Slobozhanyuk, P.A. Belov, A. Alù, A.B. Khanikaev, Nat. Commun. 9, 909 (2018) ADSCrossRefGoogle Scholar
  134. 134.
    D. Leykam, S. Flach, Y.D. Chong, Phys. Rev. B 96, 064305 (2017) ADSCrossRefGoogle Scholar
  135. 135.
    K. Fang, Z. Yu, S. Fan, Nat. Photon. 6, 782 (2012) ADSCrossRefGoogle Scholar
  136. 136.
    S. Longhi, Opt. Lett. 39, 5892 (2014) ADSCrossRefGoogle Scholar
  137. 137.
    S. Mukherjee, M. Di Liberto, P. Öhberg, R.R. Thomson, N. Goldman, Phys. Rev. Lett. 121, 075502 (2018) ADSCrossRefGoogle Scholar
  138. 138.
    P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L.L. Gratiet, I. Sagnes, J. Bloch, A. Amo, Nat. Photon. 11, 651 (2017) ADSCrossRefGoogle Scholar
  139. 139.
    R. Yao, H. Li, B. Zheng, S. An, J. Ding, C.S. Lee, H. Zhang, W. Guo, https://doi.org/arXiv:1804.01587 (2018)
  140. 140.
    G. Harari, M.A. Bandres, Y. Lumer, M.C. Rechtsman, Y.D. Chong, M. Khajavikhan, D.N. Christodoulides, M. Segev, Science 359, eaar4003 (2018) CrossRefGoogle Scholar
  141. 141.
    A. Szameit, M.C. Rechtsman, O. Bahat-Treidel, M. Segev, Phys. Rev. A 84, 021806 (2011) ADSCrossRefGoogle Scholar
  142. 142.
    B. Zhen, C.W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.L. Chua, J.D. Joannopoulos, M. Soljačić, Nature 525, 354 (2015) ADSCrossRefGoogle Scholar
  143. 143.
    H. Zhou, C. Peng, Y. Yoon, C.W. Hsu, K.A. Nelson, L. Fu, J.D. Joannopoulos, M. Soljačić, B. Zhen, Science 359, 1009 (2018) ADSMathSciNetCrossRefGoogle Scholar
  144. 144.
    J. Carlström, E. Bergholtz, https://doi.org/arXiv:1807.03330 (2018)

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Física, Laboratório de Física Teórica e Computacional, Universidade Federal de PernambucoRecifeBrazil
  2. 2.Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de ChileSantiagoChile

Personalised recommendations