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Constraints on the energy spectrum of non-Hermitian models in open environments

  • Jonatan Melkær MidtgaardEmail author
  • Zhigang Wu
  • Yu Chen
Regular Article
  • 26 Downloads

Abstract

Motivated by recent progress on non-Hermitian topological band theories, we study the energy spectrum of a generic two-band non-Hermitian Hamiltonian. We prove rigorously that the complex energy spectrum of such a non-Hermitian Hamiltonian is restricted to the lower complex plane, provided that the parameters of the Hamiltonian satisfy a certain constraint. Furthermore, we consider one specific scenario where such a non-Hermitian Hamiltonian can arise, namely a two-band model coupled to an environment, and show that this aforementioned constraint originates from very general physical considerations. Using this construction we extract the real-space behaviour of the non-hermitian terms. Our findings are relevant in the definition of the energy gap in non-Hermitian topological band theories and also have implications on simulations of such theories using quantum systems.

Graphical abstract

Keywords

Solid State and Materials 

References

  1. 1.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010) ADSGoogle Scholar
  2. 2.
    X.L. Qi, S.C. Zhang, Rev. Mod. Phys. 83, 1057 (2011) ADSGoogle Scholar
  3. 3.
    A. Bansil, H. Lin, T. Das, Rev. Mod. Phys. 88, 021004 (2016) ADSCrossRefGoogle Scholar
  4. 4.
    C.K. Chiu, J.C.Y. Teo, A.P. Schnyder, S. Ryu, Rev. Mod. Phys. 88, 035005 (2016) ADSCrossRefGoogle Scholar
  5. 5.
    N.P. Armitage, E.J. Mele, A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018) ADSCrossRefGoogle Scholar
  6. 6.
    N. Goldman, J.C. Budich, P. Zoller, Nat. Phys. 12, 639 (2016) CrossRefGoogle Scholar
  7. 7.
    N.R. Cooper, J. Dalibard, I.B. Spielman, Rev. Mod. Phys. 91, 015005 (2019) ADSCrossRefGoogle Scholar
  8. 8.
    T. Ozawa, H.M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto, Rev. Mod. Phys. 91, 015006 (2019) ADSCrossRefGoogle Scholar
  9. 9.
    M. Aidelsburger, S. Nascimbene, N. Goldman, C.R. Physique 19, 394 (2018) ADSCrossRefGoogle Scholar
  10. 10.
    M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 102, 065703 (2009) ADSCrossRefGoogle Scholar
  11. 11.
    M.S. Rudner, L.S. Levitov, Phys. Rev. B 82, 155418 (2010) ADSCrossRefGoogle Scholar
  12. 12.
    Y.C. Hu, T.L. Hughes, Phys. Rev. B 84, 153101 (2011) ADSCrossRefGoogle Scholar
  13. 13.
    K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, Phys. Rev. B 84, 205128 (2011) ADSCrossRefGoogle Scholar
  14. 14.
    M. Sato, K. Hasebe, K. Esaki, M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012) ADSCrossRefGoogle Scholar
  15. 15.
    D.I. Pikulin, Y.V. Nazarov, Phys. Rev. B 87, 235421 (2013) ADSCrossRefGoogle Scholar
  16. 16.
    S.-D. Liang, G.-Y. Huang, Phys. Rev. A 87, 012118 (2013) ADSCrossRefGoogle Scholar
  17. 17.
    H. Schomerus, Opt. Lett. 38, 1912 (2013) ADSCrossRefGoogle Scholar
  18. 18.
    S. Malzard, C. Poli, H. Schomerus, Phys. Rev. Lett. 115, 200402 (2015) ADSCrossRefGoogle Scholar
  19. 19.
    P. San-Jose, J. Cayao, E. Prada, R. Aguado, Sci. Rep. 6, 21427 (2016) ADSCrossRefGoogle Scholar
  20. 20.
    J. González, R.A. Molina, Phys. Rev. Lett. 116, 156803 (2016) ADSCrossRefGoogle Scholar
  21. 21.
    J. González, R.A. Molina, Phys. Rev. B 96, 045437 (2017) ADSCrossRefGoogle Scholar
  22. 22.
    R.A. Molina, J. González, Phys. Rev. Lett. 120, 146601 (2018) ADSCrossRefGoogle Scholar
  23. 23.
    T.E. Lee, Phys. Rev. Lett. 116, 133903 (2016) ADSCrossRefGoogle Scholar
  24. 24.
    A.K. Harter, T.E. Lee, Y.N. Joglekar, Phys. Rev. A 93, 062101 (2016) ADSCrossRefGoogle Scholar
  25. 25.
    D. Leykam, K.Y. Bliokh, C. Huang, Y.D. Chong, F. Nori, Phys. Rev. Lett. 118, 040401 (2017) ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Y. Xu, S.-T. Wang, L.-M. Duan, Phys. Rev. Lett. 118, 045701 (2017) ADSCrossRefGoogle Scholar
  27. 27.
    H. Menke, M.M. Hirschmann, Phys. Rev. B 95, 174506 (2017) ADSCrossRefGoogle Scholar
  28. 28.
    L. Campos Venuti, Z. Ma, H. Saleur, S. Haas, Phys. Rev. A 96, 053858 (2017) ADSCrossRefGoogle Scholar
  29. 29.
    S. Lieu, Phys. Rev. B 97, 045106 (2018) ADSCrossRefGoogle Scholar
  30. 30.
    A.A. Zyuzin, A. Yu. Zyuzin, Phys. Rev. B 97, 041203(R) (2018) ADSCrossRefGoogle Scholar
  31. 31.
    A. Cerjan, M. Xiao, L. Yuan, S. Fan, Phys. Rev. B 97, 075128 (2018) ADSCrossRefGoogle Scholar
  32. 32.
    V.M. Martinez Alvarez, J.E. Barrios Vargas, L.E.F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018) ADSCrossRefGoogle Scholar
  33. 33.
    H. Shen, B. Zhen, L. Fu, Phys. Rev. Lett. 120, 146402 (2018) ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    H. Shen, L. Fu, Phys. Rev. Lett. 121, 026403 (2018) ADSCrossRefGoogle Scholar
  35. 35.
    Y. Xiong, J. Phys. Commun. 2, 035043 (2018) CrossRefGoogle Scholar
  36. 36.
    C. Yin, H. Jiang, L. Li, R. Lü, S. Chen, Phys. Rev. A 97, 052115 (2018) ADSCrossRefGoogle Scholar
  37. 37.
    V. Kozii, L. Fu, https://arXiv:1708.05841
  38. 38.
    K. Kawabata, Y. Ashida, H. Katsura, M. Ueda, Phys. Rev. B 98, 085116 (2018) ADSCrossRefGoogle Scholar
  39. 39.
    X. Ni, D. Smirnova, A. Poddubny, D. Leykam, Y. Chong, A.B. Khanikaev, Phys. Rev. B 98, 165129 (2018) ADSCrossRefGoogle Scholar
  40. 40.
    M. Papaj, H. Isobe, L. Fu, Phys. Rev. B 99, 201107 (2019) ADSCrossRefGoogle Scholar
  41. 41.
    S. Yao, Z. Wang, Phys. Rev. Lett. 121, 086803 (2018) ADSCrossRefGoogle Scholar
  42. 42.
    S. Yao, F. Song, Z. Wang, Phys. Rev. Lett. 121, 136802 (2018) ADSCrossRefGoogle Scholar
  43. 43.
    K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, M. Ueda, Phys. Rev. X 8, 031079 (2018) Google Scholar
  44. 44.
    F.K. Kunst, E. Edvardsson, J.C. Budich, E.J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018) ADSCrossRefGoogle Scholar
  45. 45.
    J. Carlström, E.J. Bergholtz, Phys. Rev. A 98, 042114 (2018) ADSCrossRefGoogle Scholar
  46. 46.
    K. Kawabata, K. Shiozaki, M. Ueda, Phys. Rev. B 98, 165148 (2018) ADSCrossRefGoogle Scholar
  47. 47.
    K. Takata, M. Notomi, Phys. Rev. Lett. 121, 213902 (2018) ADSCrossRefGoogle Scholar
  48. 48.
    K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, M. Ueda, Nat. Commun. 10, 297 (2019) ADSCrossRefGoogle Scholar
  49. 49.
    Y. Chen, H. Zhai, Phys. Rev. B 98, 245130 (2018) ADSCrossRefGoogle Scholar
  50. 50.
    T. Yoshida, R. Peters, N. Kawakami, Phys. Rev. B 98, 035141 (2018) ADSCrossRefGoogle Scholar
  51. 51.
    X. Qiu, T.-S. Deng, Y. Hu, P. Xue, W. Yi, iScience 20, 392 (2019) CrossRefGoogle Scholar
  52. 52.
    Z. Yang, J. Hu, Phys. Rev. B 99, 081102 (2019) ADSCrossRefGoogle Scholar
  53. 53.
    H. Wang, J. Ruan, H. Zhang, Phys. Rev. B 99, 075130 (2019) ADSCrossRefGoogle Scholar
  54. 54.
    C.H. Lee, R. Thomale, Phys. Rev. B 99, 201103 (2019) ADSCrossRefGoogle Scholar
  55. 55.
    L. Jin, Z. Song, Phys. Rev. B 99, 081103 (2019) ADSCrossRefGoogle Scholar
  56. 56.
    J.C. Budich, J. Carlström, F.K. Kunst, E.J. Bergholtz, Phys. Rev. B 99, 041406 (2019) ADSCrossRefGoogle Scholar
  57. 57.
    R. Okugawa, T. Yokoyama, Phys. Rev. B 99, 041202 (2019) ADSCrossRefGoogle Scholar
  58. 58.
    T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, F. Nori, Phys. Rev. Lett. 122, 076801 (2019) ADSCrossRefGoogle Scholar
  59. 59.
    H. Zhou, J.Y. Lee, S. Liu, B. Zhen, Optica 6, 190 (2019) ADSCrossRefGoogle Scholar
  60. 60.
    J. Carlström, M. Stálhammar, J.C. Budich, E.J. Bergholtz, Phys. Rev. B 99, 161115 (2019) ADSCrossRefGoogle Scholar
  61. 61.
    E. Edvardsson, F.K. Kunst, E.J. Bergholtz, Phys. Rev. B 99, 081302 (2019) ADSCrossRefGoogle Scholar
  62. 62.
    K. Kawabata, K. Shiozaki, M. Ueda, M. Sato, Phys. Rev. X 9, 041015 (2019) Google Scholar
  63. 63.
    H. Zhou, J.Y. Lee, Phys. Rev. B 99, 235112 (2019) ADSCrossRefGoogle Scholar
  64. 64.
    L. Herviou, J.H. Bardarson, N. Regnault, Phys. Rev. A 99, 052118 (2019) ADSCrossRefGoogle Scholar
  65. 65.
    K.Y. Bliokh, D. Leykam, M. Lein, F. Nori, Nat. Commun. 10, 580 (2019) ADSCrossRefGoogle Scholar
  66. 66.
    M.G. Silveirinha, Phys. Rev. B 99, 125155 (2019) ADSCrossRefGoogle Scholar
  67. 67.
    C. Wang, X.R. Wang, https://arXiv:1901.06982
  68. 68.
    Q. Zeng, Y. Yang, Y. Xu,https://arXiv:1901.08060
  69. 69.
    H. Jiang, L. Lang, C. Yang, S. Zhu, S. Chen, Phys. Rev. B 100, 054301 (2019) ADSCrossRefGoogle Scholar
  70. 70.
    M.R. Hirsbrunner, T.M. Philip, M.J. Gilbert, Phys. Rev. B 100, 081104 (2019) ADSCrossRefGoogle Scholar
  71. 71.
    H.-G. Zirnstein, G. Refael, B. Rosenow, https://arXiv:1901.11241
  72. 72.
    W.B. Rui, Y.X. Zhao, A.P. Schnyder, Phys. Rev. B 99, 241110 (2019) ADSCrossRefGoogle Scholar
  73. 73.
    D.S. Borgnia, A.J. Kruchkov, R.-J. Slager, https://arXiv:1902.07217
  74. 74.
    A. Ghatak, T. Das, J. Phys.: Condens. Matter 31, 263001 (2019) ADSGoogle Scholar
  75. 75.
    K. Kawabata, T. Bessho, M. Sato, Phys. Rev. Lett. 123, 066405 (2019) ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    K. Yokomizo, S. Murakami, Phys. Rev. Lett. 123, 066404 (2019) ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    J.D.H. Rivero, L. Ge, https://arXiv:1903.02231
  78. 78.
    X. Luo, C. Zhang, Phys. Rev. Lett. 123, 073601 (2019) ADSMathSciNetCrossRefGoogle Scholar
  79. 79.
    T. Deng, W. Yi, iScience 20, 392 (2019) CrossRefGoogle Scholar
  80. 80.
    Z. Ge, Y. Zhang, T. Liu, S. Li, H. Fan, F. Nori, Phys. Rev. B 100, 054105 (2019) ADSCrossRefGoogle Scholar
  81. 81.
    E.J. Bergholtz, J.C. Budich, Phys. Rev. Research 1, 012003 (2019) Google Scholar
  82. 82.
    P.A. McClarty, J.G. Rau, Phys. Rev. B 100, 100405 (2019) ADSCrossRefGoogle Scholar
  83. 83.
    X. Yang, Y. Cao, Y. Zhai, https://arXiv:1904.02492
  84. 84.
    G. Dubach, Y. Peled, https://arXiv:1904.04312
  85. 85.
    J. Hou, Z. Li, Q. Gu, C. Zhang, https://arXiv:1904.05260
  86. 86.
    N. Okuma, M. Sato, Phys. Rev. Lett. 123, 097701 (2019) ADSCrossRefGoogle Scholar
  87. 87.
    S. Lieu, Phys. Rev. B 100, 085110 (2019) ADSCrossRefGoogle Scholar
  88. 88.
    T. Ohashi, S. Kobayashi, Y. Kawaguchi, https://arXiv:1904.08724
  89. 89.
    K. Snizhko, R. Egger, Y. Gefen, Phys. Rev. B 100, 085303 (2019) ADSCrossRefGoogle Scholar
  90. 90.
    F. Song, S. Yao, Z. Wang, Phys. Rev. Lett. 123, 170401 (2019) ADSCrossRefGoogle Scholar
  91. 91.
    F. Song, S. Yao, Z. Wang, https://arXiv:1905.02211
  92. 92.
    C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus, Nat. Commun. 6, 6710 (2015) ADSCrossRefGoogle Scholar
  93. 93.
    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
  94. 94.
    B. Zhen, C.W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.L. Chua, J.D. Joannopoulos, M. Soljacić, Nature 525, 354 (2015) ADSCrossRefGoogle Scholar
  95. 95.
    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
  96. 96.
    D. Kim, K. Mochizuki, N. Kawakami, H. Obuse, https://arXiv:1609.09650
  97. 97.
    L. Xiao, X. Zhan, Z.H. Bian, K.K. Wang, X. Zhang, X.P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B.C. Sanders, P. Xue, Nat. Phys. 13, 1117 (2017) CrossRefGoogle Scholar
  98. 98.
    X. Zhan, L. Xiao, Z. Bian, K. Wang, X. Qiu, B.C. Sanders, W. Yi, P. Xue, Phys. Rev. Lett. 119, 130501 (2017) ADSMathSciNetCrossRefGoogle Scholar
  99. 99.
    P. St-Jean, V. Goblot, E. Galopin, A. Lemaître , T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, A. Amo, Nat. Photon. 11, 651 (2017) ADSCrossRefGoogle Scholar
  100. 100.
    H. Zhou, C. Peng, Y. Yoon, C.W. Hsu, K.A. Nelson, L. Fu, J.D. Joannopoulos, M. Soljacić, B. Zhen, Science 359, 1009 (2018) ADSMathSciNetCrossRefGoogle Scholar
  101. 101.
    H. Zhao, P. Miao, M.H. Teimourpour, S. Malzard, R. El-Ganainy, Schomerus, L. Feng, Nat. Commun. 9, 981 (2018) ADSCrossRefGoogle Scholar
  102. 102.
    M. Pan, H. Zhao, P. Miao, S. Longhi, L. Feng, Nat. Commun. 9, 1308 (2018) ADSCrossRefGoogle Scholar
  103. 103.
    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
  104. 104.
    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
  105. 105.
    M.A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. Christodoulides, M. Khajavikhan, Science 359, eaar4005 (2018) CrossRefGoogle Scholar
  106. 106.
    K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, B.C. Sanders, W. Yi, P. Xue, Nat. Commun. 10, 2293 (2019) ADSCrossRefGoogle Scholar
  107. 107.
    A. Cerjan, S. Huang, K.P. Chen, Y. Chong, M.C. Rechtsman, https://arXiv:1808.09541
  108. 108.
    L. He, Z. Addison, J. Jin, E.J. Mele, S.G. Johnson, B. Zhen, https://arXiv:1902.08560
  109. 109.
    L. Feng, R. El-Ganainy, L. Ge, Nat. Photon. 11, 752 (2017) ADSCrossRefGoogle Scholar
  110. 110.
    A. Mostafazadeh, J. Math. Phys. 43, 205 (2002) ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jonatan Melkær Midtgaard
    • 1
    Email author
  • Zhigang Wu
    • 2
    • 3
  • Yu Chen
    • 4
  1. 1.Department of Physics and AstronomyAarhus UniversityAarhus CDenmark
  2. 2.Shenzhen Institute for Quantum Science and Engineering (SIQSE) and Department of Physics, Southern University of Science and Technology (SUSTech)ShenzhenP.R. China
  3. 3.Center for Quantum Computing, Peng Cheng LaboratoryShenzhenP.R. China
  4. 4.Center for Theoretical Physics and Department of Physics, Capital Normal UniversityBeijingP.R. China

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