Abstract

Exceptional points in non-Hermitian systems have recently been shown to possess nontrivial topological properties and to give rise to many exotic physical phenomena. However, most studies thus far have focused on isolated exceptional points or one-dimensional lines of exceptional points. Here, we substantially expand the space of exceptional systems by designing two-dimensional surfaces of exceptional points, and find that symmetries are a key element to protect such exceptional surfaces. We construct them using symmetry-preserving non-Hermitian deformations of topological nodal lines, and analyze the associated symmetry, topology, and physical consequences. As a potential realization, we simulate a parity-time-symmetric 3D photonic crystal and indeed find the emergence of exceptional surfaces. Our work paves the way for future explorations of systems of exceptional points in higher dimensions, and applications in emission control and sensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

12 February 2019: Minor corrections were made to the citations.


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References

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2018 (12)

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, Nat. Phys. 14, 11 (2018).
[Crossref]

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

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
[Crossref]

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
[Crossref]

R. A. Molina and J. González, Phys. Rev. Lett. 120, 146601 (2018).
[Crossref]

H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009 (2018).
[Crossref]

S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
[Crossref]

H.-K. Lau and A. A. Clerk, Nat. Commun. 9, 4320 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Science 359, eaar4005 (2018).
[Crossref]

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

Q. Yan, R. Liu, Z. Yan, B. Liu, H. Chen, Z. Wang, and L. Lu, Nat. Phys. 14, 461 (2018).
[Crossref]

B. Qi, L. Zhang, and L. Ge, Phys. Rev. Lett. 120, 093901 (2018).
[Crossref]

2017 (4)

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

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

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. Makris, M. Segev, M. Rechtsman, and A. Szameit, Nat. Mater. 16, 433 (2017).
[Crossref]

2016 (5)

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

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537, 76 (2016).
[Crossref]

A. Cerjan, A. Raman, and S. Fan, Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
[Crossref]

C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
[Crossref]

2015 (1)

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Nature 525, 354 (2015).
[Crossref]

2013 (1)

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, Phys. Rev. Lett. 110, 243902 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[Crossref]

W. D. Heiss, J. Phys. A 45, 444016 (2012).
[Crossref]

2011 (1)

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

2010 (1)

C. Rüter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[Crossref]

2001 (1)

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

1998 (1)

C. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Ashida, Y.

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
[Crossref]

Bandres, M. A.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Science 359, eaar4005 (2018).
[Crossref]

Bender, C.

C. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Bergholtz, E. J.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
[Crossref]

J. C. Budich, J. Carlström, F. K. Kunst, and E. J. Bergholtz, “Symmetry-protected nodal phases in non-Hermitian systems,” arXiv:1810.00914 (2018).

Bernard, D.

D. Bernard and A. LeClair, Statistical Field Theories, A. Cappelli and G. Mussardo, eds. (Springer, 2002).

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[Crossref]

Boettcher, S.

C. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537, 76 (2016).
[Crossref]

Budich, J. C.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
[Crossref]

J. C. Budich, J. Carlström, F. K. Kunst, and E. J. Bergholtz, “Symmetry-protected nodal phases in non-Hermitian systems,” arXiv:1810.00914 (2018).

Cao, H.

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

Carlström, J.

J. C. Budich, J. Carlström, F. K. Kunst, and E. J. Bergholtz, “Symmetry-protected nodal phases in non-Hermitian systems,” arXiv:1810.00914 (2018).

Cerjan, A.

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

A. Cerjan, A. Raman, and S. Fan, Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

A. Cerjan, S. Huang, K. P. Chen, Y. Chong, and M. C. Rechtsman, “Experimental realization of a Weyl exceptional ring,” arXiv:1808.09541 (2018).

Chen, H.

Q. Yan, R. Liu, Z. Yan, B. Liu, H. Chen, Z. Wang, and L. Lu, Nat. Phys. 14, 461 (2018).
[Crossref]

Chen, K. P.

A. Cerjan, S. Huang, K. P. Chen, Y. Chong, and M. C. Rechtsman, “Experimental realization of a Weyl exceptional ring,” arXiv:1808.09541 (2018).

Chen, W.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017).
[Crossref]

Chiu, C.-K.

C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
[Crossref]

Chong, Y.

A. Cerjan, S. Huang, K. P. Chen, Y. Chong, and M. C. Rechtsman, “Experimental realization of a Weyl exceptional ring,” arXiv:1808.09541 (2018).

Christodoulides, D.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, Nat. Phys. 14, 11 (2018).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[Crossref]

C. Rüter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. Christodoulides, S. Ozdemir, and R. El-Ganainy, “Sensing with exceptional surfaces: combining sensitivity with robustness,” arXiv:1810.09417 (2018).

Christodoulides, D. N.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Science 359, eaar4005 (2018).
[Crossref]

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

Chua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Nature 525, 354 (2015).
[Crossref]

Clerk, A. A.

H.-K. Lau and A. A. Clerk, Nat. Commun. 9, 4320 (2018).
[Crossref]

Dai, X.

C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
[Crossref]

Dembowski, C.

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537, 76 (2016).
[Crossref]

Duan, L. M.

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

Edvardsson, E.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
[Crossref]

Eichelkraut, T.

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

El-Ganainy, R.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, Nat. Phys. 14, 11 (2018).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

C. Rüter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. Christodoulides, S. Ozdemir, and R. El-Ganainy, “Sensing with exceptional surfaces: combining sensitivity with robustness,” arXiv:1810.09417 (2018).

Fan, S.

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

A. Cerjan, A. Raman, and S. Fan, Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

Fang, C.

C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
[Crossref]

Fang, Z.

C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
[Crossref]

Fu, L.

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

H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009 (2018).
[Crossref]

V. Kozii and L. Fu, “Non-Hermitian topological theory of finite-lifetime quasiparticles: prediction of bulk Fermi arc due to exceptional point,” arXiv:1708.05841 (2017).

Garcia-Gracia, H.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

Ge, L.

B. Qi, L. Zhang, and L. Ge, Phys. Rev. Lett. 120, 093901 (2018).
[Crossref]

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537, 76 (2016).
[Crossref]

Gong, Z.

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
[Crossref]

González, J.

R. A. Molina and J. González, Phys. Rev. Lett. 120, 146601 (2018).
[Crossref]

Gräf, H.-D.

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

Harari, G.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Science 359, eaar4005 (2018).
[Crossref]

Harney, H.

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

Harris, J. G. E.

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

Hassan, A.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

Heine, A.

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

Heiss, W.

C. Dembowski, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
[Crossref]

Heiss, W. D.

W. D. Heiss, J. Phys. A 45, 444016 (2012).
[Crossref]

Higashikawa, S.

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
[Crossref]

Hodaei, H.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
[Crossref]

Hsu, C. W.

H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009 (2018).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Nature 525, 354 (2015).
[Crossref]

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point sensors,” arXiv:1805.12001 (2018).

Huang, J.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, Phys. Rev. Lett. 110, 243902 (2013).
[Crossref]

Huang, S.

A. Cerjan, S. Huang, K. P. Chen, Y. Chong, and M. C. Rechtsman, “Experimental realization of a Weyl exceptional ring,” arXiv:1808.09541 (2018).

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Nature 525, 354 (2015).
[Crossref]

Jiang, L.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Nature 537, 80 (2016).
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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. (a)–(c) Spectral degeneracies in Hermitian and non-Hermitian systems from Eq. (2). The Hermitian nodal line (a) is protected by PT symmetry. A symmetry-preserving non-Hermitian perturbation produces an exceptional torus (b), whereas a symmetry-breaking one produces exceptional rings (c). (d) Spectral DOS for increasing non-Hermitian perturbation γ , with the DOS close to the nodal line frequency (red dashed line) increasing significantly.
Fig. 2.
Fig. 2. (a) Unit cell of a periodic metallic mesh structure with PT -symmetric perturbation to realize a nodal chain in the Hermitian limit and an EP surface under non-Hermitian perturbations. (b) The nodal chain crossing point (red dashed lines) can be gapped out by a mass term (blue solid lines). Depending on the non-Hermitian perturbation strength, the gap between the EP tori can remain (c) or they can become connected (d).
Fig. 3.
Fig. 3. (a) Photonic crystal consisting of a cubic array (lattice constant a = 1 ) of dielectric cubes (side length b = 0.6 a ), with a dielectric constant of ϵ = 12 . Under gain-loss modulation, where the blue (red) cube experiences a loss (gain) γ = 10 , an exceptional surface appears. (b) 2D cross section of the band structure near the band-folding line, at a generic k z = 0.13 π / a , in the Hermitian limit γ = 0 . (c), (e) Similar 2D cross sections of (c) the real and (e) the imaginary part of the band structure in the non-Hermitian modulated case γ = 10 . (d) Extracted EP surfaces for the model in (c) and (e).

Equations (2)

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H = i = 1 d c i ( k ) σ i + c 0 ( k ) ,
H ( k ) = ( m 6 + 2 cos ( k x ) + 2 cos ( k y ) + 2 cos ( k z ) ) σ z + 2 λ sin ( k z ) σ x + i γ σ y + E 0 i γ 0 ,

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