Abstract
The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk topological invariants defined for infinite periodic systems and the occurrence of protected zero-energy surface states in finite samples, is a ubiquitous and widely observed phenomenon in topological matter. In non-Hermitian generalizations of topological systems, however, this fundamental correspondence has recently been found to be qualitatively altered, largely owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary conditions. In this work, we report on two contributions towards comprehensively explaining this remarkable behavior unique to non-Hermitian systems with theory. First, we analytically solve paradigmatic non-Hermitian topological models for their zero-energy modes in the presence of generalized boundary conditions interpolating between open and periodic boundary conditions, thus explicitly following the breakdown of the conventional BBC. Second, addressing the aforementioned spectral fragility of non-Hermitian matrices, we investigate as to what extent the modified non-Hermitian BBC represents a robust and generically observable phenomenon.
Graphical abstract
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Szameit, M.C. Rechtsman, O. Bahat-Treidel, M. Segev, Phys. Rev. A: At. Mol. Opt. Phys. 84, 021806(R) (2011)
A. Regensburger, C. Bersch, M.A. Miri, G. Onishchukov, D.N. Christodoulides, U. Peschel, Nature 488, 167 (2012)
J. Wiersig, Phys. Rev. Lett. 112, 203901 (2014)
H. Hodaei, A.U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D.N. Christodoulides, M. Khajavikhan, Nature 548, 187 (2017)
L. Feng, R. El-Ganainy, L. Ge, Nat. Photonics 11, 752 (2017)
D.L. Sounas, A. Alù, Nat. Photonics 11, 774 (2017)
G. Harari, M.A. Bandres, Y. Lumer, M.C. Rechtsman, Y.D. Chong, M. Khajavikhan, D.N. Christodoulides, M. Segev, Science 359, eaar4003 (2018)
M.A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D.N. Christodoulides, M. Khajavikhan, Science 359, eaar4005 (2018)
H. Zhao, P. Miao, M.H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, L. Feng, Nat. Commun. 9, 981 (2018)
M. Kremer, T. Biesenthal, L.J. Maczewsky, M. Heinrich, R. Thomale, A. Szameit, Nat. Commun. 10, 435 (2019)
K. Özdemir, S. Rotter, F. Nori, L. Yang, Nat. Mater. 18, 783 (2019)
J. Ningyuan, C. Owens, A. Sommer, D. Schuster, J. Simon, Phys. Rev. X 5, 021031 (2015)
V.V. Albert, L.I. Glazman, L. Jiang, Phys. Rev. Lett. 114, 173902 (2015)
C.H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L.W. Molenkamp, T. Kiessling, R. Thomale, Commun. Phys. 1, 1 (2018)
M. Ezawa, Phys. Rev. B 99, 201411 (2019)
T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L.W. Molenkamp, C.H. Lee, A. Szameit, M. Greiter, R. Thomale, arXiv:1907.11562 (2019)
T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabaši, C.H. Lee, A. Bilušić, R. Thomale, T. Neupert, arXiv:1908.02759 (2019)
L.M. Nash, D. Kleckner, A. Read, V. Vitelli, A.M. Turner, W.T.M. Irvine, Proc. Natl. Acad. Sci. USA 112, 14495 (2015)
M. Brandenbourger, X. Locsin, E. Lerner, C. Coulais, Nat. Commun. 10, 4608 (2019)
A. Ghatak, M. Brandenbourger, J. van Wezel, C. Coulais, arXiv:1907.11619 (2019)
E.J. Bergholtz, J.C. Budich, Phys. Rev. Res. 1, 012003 (2019)
F. Song, S. Yao, Z. Wang, Phys. Rev. Lett. 123, 170401 (2019)
Z. Wang, Y. Chong, J.D. Joannopoulos, M. Soljačić, Nature 461, 772 (2009)
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)
B. Peng, Ş.K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Ylmaz, J. Wiersig, S. Rotter, L. Yang, Proc. Natl. Acad. Sci. USA 113, 6845 (2016)
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)
W. Chen, Ş.K. Özdemir, G. Zhao, J. Wiersig, L. Yang, Nature 548, 192 (2017)
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)
H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, Z. Liu, Nature 560, 61 (2018)
L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, P. Xue, arXiv:1907.12566 (2019)
A. Cerjan, S. Huang, M. Wang, K.P. Chen, Y. Chong, M.C. Rechtsman, Nat. Photonics 13, 623 (2019)
H. Zhou, J.Y. Lee, S. Liu, B. Zhen, Optica 6, 190 (2019)
K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, Phys. Rev. B 84, 205128 (2011)
Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda, Phys. Rev. X 8, 031079 (2018)
J. Carlström, E.J. Bergholtz, Phys. Rev. A 98, 042114 (2018)
K. Kawabata, K. Shiozaki, M. Ueda, M. Sato, Phys. Rev. X 9, 041015 (2019)
X.W. Luo, C. Zhang, Phys. Rev. Lett. 123, 073601 (2019)
J.C. Budich, J. Carlström, F.K. Kunst, E.J. Bergholtz, Phys. Rev. B 99, 041406 (2019)
T. Yoshida, R. Peters, N. Kawakami, Y. Hatsugai, Phys. Rev. B 99, 121101 (2019)
H. Shen, B. Zhen, L. Fu, Phys. Rev. Lett. 120, 146402 (2018)
D.J. Luitz, F. Piazza, Phys. Rev. Res. 1, 033051 (2019)
J. Carlström, M. Stålhammar, J.C. Budich, E.J. Bergholtz, Phys. Rev. B 99, 161115 (2019)
M. Stålhammar, L. Rødland, G. Arone, J.C. Budich, E.J. Bergholtz, SciPost Phys. 7, 019 (2019)
D. Leykam, K.Y. Bliokh, C. Huang, Y.D. Chong, F. Nori, Phys. Rev. Lett. 118, 40401 (2017)
E. Edvardsson, F.K. Kunst, E.J. Bergholtz, Phys. Rev. B 99, 081302 (2019)
H. Zhou, J.Y. Lee, Phys. Rev. B 99, 235112 (2019)
L.E.F. Fao Torres, J. Phys. Mater. 3, 014002 (2019)
C.H. Lee, L. Li, J. Gong. Phys. Rev. Lett. 123, 016805 (2019)
K. Kawabata, T. Bessho, M. Sato, Phys. Rev. Lett. 123, 066405 (2019)
K. Zhang, Z. Yang, C. Fang, arXiv:1910.01131 (2019)
T.E. Lee, Phys. Rev. Lett. 116, 133903 (2016)
Y. Xiong, T. Wang, X. Wang, P. Tong, arXiv:1610.06275 (2016)
V.M. Martinez Alvarez, J.E. Barrios Vargas, M. Berdakin, L.E.F. Foa Torres, Eur. Phys. J. Special Topics 227, 1295 (2018)
Y. Xiong, J. Phys. Commun. 2, 035043 (2018)
M. Zahid Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)
F.K. Kunst, E. Edvardsson, J.C. Budich, E.J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018)
S. Yao, Z. Wang, Phys. Rev. Lett. 121, 086803 (2018)
H.G. Zirnstein, G. Refael, B. Rosenow, https://arxiv.org/abs/1901.11241 (2019)
L. Herviou, J.H. Bardarson, N. Regnault, Phys. Rev. A 99, 52118 (2019)
L. Jin, Z. Song, Phys. Rev. B 99, 081103 (2019)
F. Song, S. Yao, Z. Wang, Phys. Rev. Lett. 123, 246801 (2019)
C.H. Lee, R. Thomale, Phys. Rev. B 99, 201103 (2019)
K. Yokomizo, S. Murakami, Phys. Rev. Lett. 123, 66404 (2019)
K.I. Imura, Y. Takane, Phys. Rev. B 100, 165430 (2019)
L. Reichel, L.N. Trefethen, Linear Algebra Appl. 162, 153 (1992)
S. Yao, F. Song, Z. Wang, Phys. Rev. Lett. 121, 136802 (2018)
F.K. Kunst, V. Dwivedi, Phys. Rev. B 99, 245116 (2019)
N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996)
W.D. Heiss, Czech. J. Phys. 54, 1091 (2004)
W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22, 2099 (1980)
D.C. Brody, J. Phys. A Math. Theor. 47, 35305 (2013)
F.K. Kunst, G. van Miert, E.J. Bergholtz, Phys. Rev. B 97, 241405 (2018)
V.M. Martinez Alvarez, J.E. Barrios Vargas, L.E.F. Foa Torres, Phys. Rev. B 97, 121401 (2018)
S. Lieu, Phys. Rev. B 97, 045106 (2018)
W. Nolting, Grundkurs Theoretische Physik 7: Viel-Teilchen-Theorie (Springer-Verlag, 2006)
W. Nolting, Grundkurs Theoretische Physik 5/1 (Springer-Verlag, Berlin, 2006)
Acknowledgments
Open access funding provided by Project DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Topological Ultracold Atoms and Photonic Systems”, edited by G. Juzeliunas, R. Ma, Y.-J. Lin and T. Calarco.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Koch, R., Budich, J.C. Bulk-boundary correspondence in non-Hermitian systems: stability analysis for generalized boundary conditions. Eur. Phys. J. D 74, 70 (2020). https://doi.org/10.1140/epjd/e2020-100641-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjd/e2020-100641-y