Topological states of non-Hermitian systems

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.

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References

  1. 1.

    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

    ADS  Google Scholar 

  2. 2.

    K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)

    ADS  Google Scholar 

  3. 3.

    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)

    ADS  Google Scholar 

  4. 4.

    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)

    ADS  Google Scholar 

  5. 5.

    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005).

    ADS  Google Scholar 

  6. 6.

    B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

    ADS  Google 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)

    ADS  Google Scholar 

  8. 8.

    D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature 452, 970 (2008)

    ADS  Google Scholar 

  9. 9.

    O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014)

    ADS  Google Scholar 

  10. 10.

    T. Oka, H. Aoki, Phys. Rev. B 79, 081406 (2009)

    ADS  Google Scholar 

  11. 11.

    N.H. Lindner, G. Refael, V. Galitski, Nat. Phys. 7, 490 (2011)

    Google Scholar 

  12. 12.

    L.E.F. Foa Torres, P.M. Perez-Piskunow, C.A. Balseiro, G. Usaj, Phys. Rev. Lett. 113, 266801 (2014)

    ADS  Google Scholar 

  13. 13.

    M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 102, 065703 (2009)

    ADS  Google Scholar 

  14. 14.

    S. Diehl, E. Rico, M.A. Baranov, P. Zoller, Nat. Phys. 7, 971 (2011)

    Google Scholar 

  15. 15.

    C. Yuce, Phys. Lett. A 379, 1213 (2015)

    ADS  MathSciNet  Google Scholar 

  16. 16.

    P. San-Jose, J. Cayao, E. Prada, R. Aguado, Sci. Rep. 6, 21427 (2016)

    ADS  Google Scholar 

  17. 17.

    T.E. Lee, Phys. Rev. Lett. 116, 133903 (2016)

    ADS  Google Scholar 

  18. 18.

    D. Leykam, K.Y. Bliokh, C. Huang, Y. Chong, F. Nori, Phys. Rev. Lett. 118, 040401 (2017)

    ADS  MathSciNet  Google Scholar 

  19. 19.

    K. Kawabata, Y. Ashida, H. Katsura, M. Ueda, Phys. Rev. B 98, 085116 (2018)

    ADS  Google Scholar 

  20. 20.

    C. Yuce, Phys. Rev. A 93, 062130 (2016)

    ADS  Google Scholar 

  21. 21.

    I. Rotter, J. Phys. A: Math. Theor. 42, 153001 (2009)

    MATH  Google Scholar 

  22. 22.

    N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, New York, 2011)

  23. 23.

    C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

    ADS  MathSciNet  Google Scholar 

  24. 24.

    C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010)

    Google Scholar 

  25. 25.

    S. Longhi, Europhys. Lett. 120, 64001 (2017)

    ADS  Google 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)

    ADS  Google Scholar 

  29. 29.

    N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996)

    ADS  Google Scholar 

  30. 30.

    N. Hatano, D.R. Nelson, Phys. Rev. B 56, 8651 (1997)

    ADS  Google Scholar 

  31. 31.

    N. Hatano, D.R. Nelson, Phys. Rev. B 58, 8384 (1998)

    ADS  Google Scholar 

  32. 32.

    D.R. Nelson, N.M. Shnerb, Phys. Rev. E 58, 1383 (1998)

    ADS  MathSciNet  Google Scholar 

  33. 33.

    A. Amir, N. Hatano, D.R. Nelson, Phys. Rev. E 93, 042310 (2016)

    ADS  MathSciNet  Google Scholar 

  34. 34.

    A. Murugan, S. Vaikuntanathan, Nat. Commun. 8, 13881 (2017)

    ADS  Google Scholar 

  35. 35.

    Y. Xu, S.T. Wang, L.M. Duan, Phys. Rev. Lett. 118, 045701 (2017)

    ADS  Google Scholar 

  36. 36.

    A.A. Zyuzin, A.Y. Zyuzin, Phys. Rev. B 97, 041203 (2018)

    ADS  Google Scholar 

  37. 37.

    A. Cerjan, M. Xiao, L. Yuan, S. Fan, Phys. Rev. B 97, 075128 (2018)

    ADS  Google Scholar 

  38. 38.

    J. González, R.A. Molina, Phys. Rev. B 96, 045437 (2017)

    ADS  Google Scholar 

  39. 39.

    R.A. Molina, J. González, Phys. Rev. Lett. 120, 146601 (2018)

    ADS  Google 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)

    ADS  Google Scholar 

  41. 41.

    J.C. Budich, S. Diehl, Phys. Rev. B 91, 165140 (2015)

    ADS  Google Scholar 

  42. 42.

    J.C. Budich, P. Zoller, S. Diehl, Phys. Rev. A 91, 042117 (2015)

    ADS  Google Scholar 

  43. 43.

    W.T. Silfvast, Laser Fundamentals (Cambridge University Press, New York, 2004)

  44. 44.

    A.V. Smilga, J. Phys. A: Math. Theor. 42, 095301 (2009)

    ADS  Google Scholar 

  45. 45.

    J. Feinberg, A. Zee, Phys. Rev. E 59, 6433 (1999)

    ADS  Google Scholar 

  46. 46.

    T. Kato, Perturbation Theory of Linear Operators (Springer, 1966)

  47. 47.

    W.D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012)

    ADS  Google Scholar 

  48. 48.

    J. von Neuman, E. Wigner, Phys. Zeitschrift 30, 467 (1929)

    ADS  Google Scholar 

  49. 49.

    M. Berry, Curr. Sci. 67, 220 (1994)

    Google Scholar 

  50. 50.

    M. Berry, Nat. Phys. 6, 148 (2010)

    Google Scholar 

  51. 51.

    W.D. Heiss, H.L. Harney, Eur. Phys. J. D 17, 149 (2001)

    ADS  Google 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)

    ADS  Google 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)

    ADS  Google 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)

    ADS  Google Scholar 

  56. 56.

    T. Stehmann, W.D. Heiss, F.G. Scholtz, J. Phys. A: Math. Gen. 37, 7813 (2004)

    ADS  Google 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)

    ADS  Google Scholar 

  58. 58.

    W. Hu, H. Wang, P.P. Shum, Y.D. Chong, Phys. Rev. B 95, 184306 (2017)

    ADS  Google Scholar 

  59. 59.

    G.A. Álvarez, E.P. Danieli, P.R. Levstein, H.M. Pastawski, J. Chem. Phys. 124, 194507 (2006)

    ADS  Google Scholar 

  60. 60.

    S. Ghosh, Y. Chong, Sci. Rep. 6, 19837 (2016)

    ADS  Google 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)

    ADS  Google Scholar 

  62. 62.

    H. Xu, D. Mason, L. Jiang, J.G.E. Harris, Nature 537, 80 (2016)

    ADS  Google 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)

    ADS  Google Scholar 

  65. 65.

    H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

    ADS  Google Scholar 

  66. 66.

    H. Jing, S.K. Özdemir, H. Lü, F. Nori, Sci. Rep. 7, 3386 (2017)

    ADS  Google 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)

    ADS  Google 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)

    ADS  Google Scholar 

  69. 69.

    E.M. Graefe, U. Günther, H.J. Korsch, A.E. Niederle, J. Phys. A: MathTheor. 41, 255206 (2008)

    ADS  Google Scholar 

  70. 70.

    K. Ding, Z.Q. Zhang, C.T. Chan, Phys. Rev. B 92, 235310 (2015)

    ADS  Google Scholar 

  71. 71.

    V.M. Martinez Alvarez, J.E. Barrios Vargas, L.E.F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018)

    ADS  Google Scholar 

  72. 72.

    Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D.N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011)

    ADS  Google Scholar 

  73. 73.

    L. Feng, Z.J. Wong, R.M. Ma, Y. Wang, X. Zhang, Science 346, 972 (2014)

    ADS  Google Scholar 

  74. 74.

    H. Hodaei, M.A. Miri, M. Heinrich, D.N. Christodoulides, M. Khajavikhan, Science 346, 975 (2014)

    ADS  Google Scholar 

  75. 75.

    W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, L. Yang, Nature 548, 192 (2017)

    ADS  Google Scholar 

  76. 76.

    M.C. Rechtsman, Nature 548, 161 (2017)

    ADS  Google Scholar 

  77. 77.

    J.P. Bird, Y. Ochiai, Science 303, 1621 (2004)

    Google 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)

    ADS  Google Scholar 

  80. 80.

    N. Auerbach, V. Zelevinsky, Rep. Prog. Phys. 74, 106301 (2011)

    ADS  Google Scholar 

  81. 81.

    H. Eleuch, I. Rotter, Phys. Rev. A 95, 022117 (2017)

    ADS  Google Scholar 

  82. 82.

    M. Hafezi, E. Demler, M. Lukin, J. Taylor, Nat. Phys. 7, 907 (2011)

    Google 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)

    ADS  Google Scholar 

  84. 84.

    H. Shen, B. Zhen, L. Fu, Phys. Rev. Lett. 120, 146402 (2018)

    ADS  MathSciNet  Google Scholar 

  85. 85.

    C. Yin, H. Jiang, L. Li, R. Lü, S. Chen, Phys. Rev. A 97, 052115 (2018)

    ADS  Google Scholar 

  86. 86.

    L. Zhou, Q.h. Wang, H. Wang, J. Gong, Phys. Rev. A 98, 022129 (2018)

    ADS  Google Scholar 

  87. 87.

    J.W. Ryu, N. Myoung, H.C. Park, Phys. Rev. B 96, 125421 (2017)

    ADS  Google Scholar 

  88. 88.

    J.W. Ryu, N. Myoung, H.C. Park, Sci. Rep. 7, 8746 (2017)

    ADS  Google Scholar 

  89. 89.

    C. Yuce, Phys. Rev. A 97, 042118 (2018)

    ADS  Google Scholar 

  90. 90.

    S. Yao, Z. Wang, Phys. Rev. Lett. 121, 086803 (2018)

    ADS  Google Scholar 

  91. 91.

    S. Lieu, Phys. Rev. B 97, 045106 (2018)

    ADS  Google Scholar 

  92. 92.

    B. Zhu, R. Lü, S. Chen, Phys. Rev. A 89, 062102 (2014)

    ADS  Google Scholar 

  93. 93.

    M. Klett, H. Cartarius, D. Dast, J. Main, G. Wunner, Phys. Rev. A 95, 053626 (2017)

    ADS  Google Scholar 

  94. 94.

    F. Dangel, M. Wagner, H. Cartarius, J. Main, G. Wunner, Phys. Rev. A 98, 013628 (2018)

    ADS  Google 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)

    ADS  Google Scholar 

  97. 97.

    C. Yuce, Eur. Phys. J. D 69, 184 (2015)

    ADS  Google Scholar 

  98. 98.

    Z. Turker, S. Tombuloglu, C. Yuce, Phys. Lett. A 382, 2013 (2018)

    ADS  Google 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)

    ADS  Google Scholar 

  101. 101.

    T.T. Koutserimpas, R. Fleury, Phys. Rev. Lett. 120, 087401 (2018)

    ADS  Google Scholar 

  102. 102.

    S. Longhi, EPL 117, 10005 (2017)

    ADS  Google 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)

    ADS  Google 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)

    ADS  Google Scholar 

  107. 107.

    B.A. Bernevig, T.L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013)

  108. 108.

    F. Ortmann, S. Roche, S.O. Valenzuela, L.W. Molenkamp (eds.), Topological Insulators: Fundamentals and Perspectives (Wiley, 2015)

  109. 109.

    J.K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators (Springer, Berlin, 2016)

  110. 110.

    J.W. Rhim, J. Behrends, J.H. Bardarson, Phys. Rev. B 95, 035421 (2017)

    ADS  Google Scholar 

  111. 111.

    J.W. Rhim, J.H. Bardarson, R.J. Slager, Phys. Rev. B 97, 115143 (2018)

    ADS  Google Scholar 

  112. 112.

    Y.C. Hu, T.L. Hughes, Phys. Rev. B 84, 153101 (2011)

    ADS  Google Scholar 

  113. 113.

    K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, Phys. Rev. B 84, 205128 (2011)

    ADS  Google Scholar 

  114. 114.

    Y. Xiong, J. Phys. Commun. 2, 035043 (2018)

    Google 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)

    ADS  Google Scholar 

  117. 117.

    F. Evers, A.D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008)

    ADS  Google 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)

    ADS  Google Scholar 

  121. 121.

    C. Hahn, Y. Choi, J.W. Yoon, S.H. Song, C.H. Oh, P. Berini, Nat. Commun. 7, 12201 (2016)

    ADS  Google Scholar 

  122. 122.

    F. Nazari, N. Bender, H. Ramezani, M. Moravvej-Farshi, D. Christodoulides, T. Kottos, Opt. Express 22, 9574 (2014)

    ADS  Google 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)

    Google Scholar 

  124. 124.

    M. Pan, H. Zhao, P. Miao, S. Longhi, L. Feng, Nat. Commun. 9, 1308 (2018)

    ADS  Google 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)

    ADS  Google 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)

    ADS  Google Scholar 

  129. 129.

    C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus, Nat. Commun. 6, 6710 (2015)

    ADS  Google Scholar 

  130. 130.

    H. Zhao, P. Miao, M.H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, L. Feng, Nat. Commun. 9, 981 (2018)

    ADS  Google 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)

    ADS  Google Scholar 

  132. 132.

    S. Malzard, C. Poli, H. Schomerus, Phys. Rev. Lett. 115, 200402 (2015)

    ADS  Google 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)

    ADS  Google Scholar 

  134. 134.

    D. Leykam, S. Flach, Y.D. Chong, Phys. Rev. B 96, 064305 (2017)

    ADS  Google Scholar 

  135. 135.

    K. Fang, Z. Yu, S. Fan, Nat. Photon. 6, 782 (2012)

    ADS  Google Scholar 

  136. 136.

    S. Longhi, Opt. Lett. 39, 5892 (2014)

    ADS  Google Scholar 

  137. 137.

    S. Mukherjee, M. Di Liberto, P. Öhberg, R.R. Thomson, N. Goldman, Phys. Rev. Lett. 121, 075502 (2018)

    ADS  Google 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)

    ADS  Google 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)

    Google Scholar 

  141. 141.

    A. Szameit, M.C. Rechtsman, O. Bahat-Treidel, M. Segev, Phys. Rev. A 84, 021806 (2011)

    ADS  Google 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)

    ADS  Google 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)

    ADS  MathSciNet  Google Scholar 

  144. 144.

    J. Carlström, E. Bergholtz, https://doi.org/arXiv:1807.03330 (2018)

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Martinez Alvarez, V.M., Barrios Vargas, J.E., Berdakin, M. et al. Topological states of non-Hermitian systems. Eur. Phys. J. Spec. Top. 227, 1295–1308 (2018). https://doi.org/10.1140/epjst/e2018-800091-5

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