Abstract
We consider a breathing Kagome lattice with complex hop**s by imposing \( \mathbb {Z}_{3}\) clock symmetry in the complex-energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We construct topological corner states with the \(\mathbb {Z}_{3}\) clock-symmetric model. It is also shown that the model is realized in an electric circuit properly designed, where corner states are observed by impedance resonance. We also construct the \(\mathbb {Z}_{4}\) and \(\mathbb {Z}_{6}\) symmetric models on breathing square and honeycomb lattices, respectively.
Graphic abstract
Similar content being viewed by others
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All numerical results have been presented as graphical results in the text.]
References
F. Zhang, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 110, 046404 (2013)
W.A. Benalcazar, B.A. Bernevig, T.L. Hughes, Science 357, 61 (2017)
F. Schindler, A. Cook, M. G. Vergniory, T. Neupert, in APS March Meeting (2017)
Y. Peng, Y. Bao, F. von Oppen, Phys. Rev. B 95, 235143 (2017)
J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, P.W. Brouwer, Phys. Rev. Lett. 119, 246401 (2017)
Z. Song, Z. Fang, C. Fang, Phys. Rev. Lett. 119, 246402 (2017)
W.A. Benalcazar, B.A. Bernevig, T.L. Hughes, Phys. Rev. B 96, 245115 (2017)
F. Schindler, A.M. Cook, M.G. Vergniory, Z. Wang, S.S.P. Parkin, B.A. Bernevig, T. Neupert, Science Advances 1, eaat0346 (2018)
C. Fang, L. Fu, Sci. Adv. 5, eaat2374 (2019)
M. Ezawa, Phys. Rev. Lett. 120, 026801 (2018)
M. Ezawa, Phys. Rev. B 98, 045125 (2018)
M. Geier, L. Trifunovic, M. Hoskam, P.W. Brouwer, Phys. Rev. B 97, 205135 (2018)
E. Khalaf, Phys. Rev. B 97, 205136 (2018)
M. Ezawa, Phys. Rev. B 100, 045407 (2019)
H. Xue, Y. Yang, F. Gao, Y. Chong, B. Zhang, Nat. Mater. 18, 108 (2019)
X. Ni, M. Weiner, A. Alu, A.B. Khanikaev, Nat. Mater. 18, 113 (2019)
A.E. Hassan, F.K. Kunst, A. Moritz, G. Andler, E.J. Bergholtz, M. Bourennane, Nat. Photonics 13, 697 (2019)
M. Li, D. Zhirihin, D. Filonov, X. Ni, A. Slobozhanyuk, A. Alu, A.B. Khanikaev, Nat. Photonics 14, 89 (2020)
S.N. Kempkes, M.R. Slot, J.J. van den Broeke, P. Capiod, W.A. Benalcazar, D. Vanmaekelbergh, D. Bercioux, I. Swart, C. Morais Smith, Nat. Mater. 18, 1292–1297 (2019)
M. Ezawa, Nat. Mater. 18, 1266 (2019)
J. Wu, X. Huang, J. Lu, Y. Wu, W. Deng, F. Li, Z. Liu, Phys. Rev. B 102, 104109 (2020)
C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
C.M. Bender, D.C. Brody, H.F. Jones, Phys. Rev. Lett. 89, 270401 (2002)
S. Malzard, C. Poli, H. Schomerus, Phys. Rev. Lett. 115, 200402 (2015)
V.V. Konotop, J. Yang, D.A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016)
T. Rakovszky, J.K. Asboth, A. Alberti, Phys. Rev. B 95, 201407(R) (2017)
R. El-Ganainy, K.G. Makris, M. Khajavikhan, Z.H. Musslimani, S. Rotter, D.N. Christodoulides, Nat. Phys. 14, 11 (2018)
B. Zhu, R. Lu, S. Chen, Phys. Rev. A 89, 062102 (2014)
S. Yao, Z. Wang, Phys. Rev. Lett. 121, 086803 (2018)
L. **, Z. Song, Phys. Rev. B 99, 081103 (2019)
S.-D. Liang, G.-Y. Huang, Phys. Rev. A 87, 012118 (2013)
D. Leykam, K.Y. Bliokh, C. Huang, Y.D. Chong, F. Nori, Phys. Rev. Lett. 118, 040401 (2017)
S. Lieu, Phys. Rev. B 97, 045106 (2018)
Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda, Phys. Rev. X 8, 031079 (2018)
E. Cobanera, A. Alase, G. Ortiz, L. Viola, Phys. Rev. B 98, 245423 (2018)
H. Jiang, C. Yang, S. Chen, Phys. Rev. A 98, 052116 (2018)
A. Ghatak, T. Das, J. Phys.: Condens. Matter 31, 263001 (2019)
T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, F. Nori, Phys. Rev. Lett. 122, 076801 (2019)
M. Ezawa, Phys. Rev. B 99, 201411(R) (2019)
M. Ezawa, Phys. Rev. B 99, 121411(R) (2019)
E. Edvardsson, F.K. Kunst, E.J. Bergholtz, Phys. Rev. B 99, 081302 (2019)
K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008)
H. Schomerus, Opt. Lett. 38, 1912 (2013)
M. Pan, H. Zhao, P. Miao, S. Longhi, L. Feng, Nat. Commun. 9, 1308 (2018)
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)
C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus, Nat. Commun. 6, 6710 (2015)
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)
M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 102, 065703 (2009)
L. **ao, 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)
H. Hodaei, A.U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D.N. Christodoulides, M. Khajavikhan, Nature 548, 187 (2017)
T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L.W. Molenkamp, C.H. Lee, A. Szameit, M. Greiter, R. Thomale, Nat. Phys. 16, 747 (2020)
A. Stegmaier, S. Imhof, T. Helbig, T. Hofmann, C.H. Lee, M. Kremer, A. Fritzsche, T. Feichtner, S. Klembt, S. Hofling, I. Boettcher, I.C. Fulga, L. Ma, O.G. Schmidt, M. Greiter, T. Kiessling, A. Szameit, R. Thomale, Phys. Rev. Lett. 126, 215302 (2021)
S. Liu, R. Shao, S. Ma, L. Zhang, O. You, H. Wu, Y. J. **ang, T. J. Cui, S. Zhang, Research Article ID 5608038 (2021)
R.J. Baxter, Phys. Lett. A 140, 155 (1989)
R.J. Baxter, J. Stat. Phys. 57, 1 (1989)
P. Fendley, J. Phys. A 47, 075001 (2014)
G. Ortiz, E. Cobanera, Z. Nussinov, Nuc. Phys. B 854, 780 (2012)
J. Alicea, P. Fendley, Annu. Rev. Condens. Matter Phys. 7, 119 (2016)
K. Kawabata, K. Shiozaki, M. Ueda, M. Sato, Phys. Rev. X 9, 041015 (2019)
S. Imhof, C. Berger, F. Bayer, J. Brehm, L. Molenkamp, T. Kiessling, F. Schindler, C.H. Lee, M. Greiter, T. Neupert, R. Thomale, Nat. Phys. 14, 925 (2018)
C.H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L.W. Molenkamp, T. Kiessling, R. Thomale, Commun. Phys. 1, 39 (2018)
T. Helbig, T. Hofmann, C.H. Lee, R. Thomale, S. Imhof, L.W. Molenkamp, T. Kiessling, Phys. Rev. B 99, 161114 (2019)
Y. Lu, N. Jia, L. Su, C. Owens, G. Juzeliunas, D.I. Schuster, J. Simon, Phys. Rev. B 99, 020302 (2019)
Y. Li, Y. Sun, W. Zhu, Z. Guo, J. Jiang, T. Kariyado, H. Chen, X. Hu, Nat. Commun. 9, 4598 (2018)
M. Ezawa, Phys. Rev. B 98, 201402(R) (2018)
M. Serra-Garcia, R. Susstrunk, S.D. Huber, Phys. Rev. B 99, 020304 (2019)
T. Hofmann, T. Helbig, C.H. Lee, M. Greiter, R. Thomale, Phys. Rev. Lett. 122, 247702 (2019)
M. Ezawa, Phys. Rev. B 102, 075424 (2020)
T. Mizoguchi, H. Araki, Y. Hatsugai, J. Phys. Soc. Jpn. 88, 104703 (2019)
G. van Miert, C. Ortix, npj Quantum Mater. 5, 63 (2020)
C.-H. Liu, H. Jiang, S. Chen, Phys. Rev. B 99, 125103 (2019)
Acknowledgements
The author is very much grateful to Y. Tanaka and N. Nagaosa for helpful discussions on the subject. This work is supported by the Grants-in-Aid for Scientific Research from MEXT KAKENHI (Grants No. JP17K05490 and No. JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1 and JPMJCR20T2).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ezawa, M. Clock-symmetric non-Hermitian second-order topological insulator. Eur. Phys. J. B 95, 97 (2022). https://doi.org/10.1140/epjb/s10051-022-00364-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjb/s10051-022-00364-3