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  • Letter
  • Open Access

Braid-protected topological band structures with unpaired exceptional points

J. Lukas K. König1, Kang Yang1,2, Jan Carl Budich3, and Emil J. Bergholtz1,*

  • 1Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden
  • 2Dahlem Center for Complex Quantum Systems and Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
  • 3Institute of Theoretical Physics, Technische Universität Dresden and Würzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany

  • *emil.bergholtz@fysik.su.se
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Phys. Rev. Research 5, L042010 – Published 11 October, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L042010

Abstract

We demonstrate the existence of topologically stable unpaired exceptional points (EPs), and construct simple non-Hermitian (NH) tight-binding models exemplifying such remarkable nodal phases. While fermion doubling, i.e., the necessity of compensating the topological charge of a stable nodal point by an antidote, rules out a direct counterpart of our findings in the realm of Hermitian semimetals, here we derive how nonommuting braids of complex energy levels may stabilize unpaired EPs. Drawing on this insight, we reveal the occurrence of a single, unpaired EP, manifested as a non-Abelian monopole in the Brillouin zone of a minimal three-band model. This third-order degeneracy represents a sweet spot within a larger topological phase that cannot be fully gapped by any local perturbation. Instead, it may only split into simpler (second-order) degeneracies that can only gap out by pairwise annihilation after having moved around inequivalent large circles of the Brillouin zone. Our results imply the incompleteness of a topological classification based on winding numbers, due to non-Abelian representations of the braid group intertwining three or more complex energy levels, and provide insights into the topological robustness of non-Hermitian systems and their non-Abelian phase transitions.

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References (56)

  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D.-e. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306, 666 (2004).
  2. S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, Experimental realization of a three-dimensional Dirac semimetal, Phys. Rev. Lett. 113, 027603 (2014).
  3. S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee et al., Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science 349, 613 (2015).
  4. S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez, G. Chang et al., Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide, Nat. Phys. 11, 748 (2015).
  5. H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides, Phys. Rev. X 5, 011029 (2015).
  6. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  7. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, Weyl points and line nodes in gyroid photonic crystals, Nat. Photonics 7, 294 (2013).
  8. L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics, Nat. Photonics 8, 821 (2014).
  9. L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, Experimental observation of Weyl points, Science 349, 622 (2015).
  10. B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5, 031013 (2015).
  11. L. Yang, Z. Liu, Y. Sun, H. Peng, H. Yang, T. Zhang, B. Zhou, Y. Zhang, Y. Guo, M. Rahn et al., Weyl semimetal phase in the non-centrosymmetric compound TaAs, Nat. Phys. 11, 728 (2015).
  12. L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M.-T. Suzuki, R. Arita, Y. Wu, D. Mou, H. Cao, J. Yan et al., Spectroscopic evidence for a type II Weyl semimetallic state in MoTe2, Nat. Mater. 15, 1155 (2016).
  13. G. Volovik, The Universe in a Helium Droplet, International Series of Monographs on Physics (Oxford University Press, Oxford, U.K., 2003).
  14. B. Yan and C. Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
  15. H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory, Nucl. Phys. B 185, 20 (1981).
  16. H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (II). Intuitive topological proof, Nucl. Phys. B 193, 173 (1981).
  17. M. Berry, Physics of nonhermitian degeneracies, Czech. J. Phys. 54, 1039 (2004).
  18. W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012).
  19. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization, Phys. Rev. X 6, 021007 (2016).
  20. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  21. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  22. K. Ding, C. Fang, and G. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022).
  23. Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Fermion doubling theorems in two-dimensional non-Hermitian systems for Fermi points and exceptional points, Phys. Rev. Lett. 126, 086401 (2021).
  24. E. J. Pap, D. Boer, and H. Waalkens, Non-Abelian nature of systems with multiple exceptional points, Phys. Rev. A 98, 023818 (2018).
  25. Z. Li and R. S. K. Mong, Homotopical characterization of non-Hermitian band structures, Phys. Rev. B 103, 155129 (2021).
  26. C. C. Wojcik, X.-Q. Sun, T. Bzdušek, and S. Fan, Homotopy characterization of non-Hermitian Hamiltonians, Phys. Rev. B 101, 205417 (2020).
  27. H. Hu and E. Zhao, Knots and non-Hermitian Bloch bands, Phys. Rev. Lett. 126, 010401 (2021).
  28. K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological complex-energy braiding of non-Hermitian bands, Nature (London) 598, 59 (2021).
  29. Y. S. Patil, J. Höller, P. A. Henry, C. Guria, Y. Zhang, L. Jiang, N. Kralj, N. Read, and J. G. Harris, Measuring the knot of non-Hermitian degeneracies and non-commuting braids, Nature (London) 607, 271 (2022).
  30. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018).
  31. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  32. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  33. P. Delplace, T. Yoshida, and Y. Hatsugai, Symmetry-protected multifold exceptional points and their topological characterization, Phys. Rev. Lett. 127, 186602 (2021).
  34. K. Kawabata, T. Bessho, and M. Sato, Classification of exceptional points and non-Hermitian topological semimetals, Phys. Rev. Lett. 123, 066405 (2019).
  35. N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979).
  36. C. Kassel, O. Dodane, and V. Turaev, Braid Groups, Graduate Texts in Mathematics (Springer, New York, 2008).
  37. X.-Q. Sun, S.-C. Zhang, and T. Bzdušek, Conversion rules for Weyl points and nodal lines in topological media, Phys. Rev. Lett. 121, 106402 (2018).
  38. G. P. Alexander, B. G.-g. Chen, E. A. Matsumoto, and R. D. Kamien, Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals, Rev. Mod. Phys. 84, 497 (2012).
  39. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X 9, 021013 (2019).
  40. Q. Wu, A. A. Soluyanov, and T. Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365, 1273 (2019).
  41. A. Bouhon, Q. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137 (2020).
  42. B. Jiang, A. Bouhon, Z.-K. Lin, X. Zhou, B. Hou, F. Li, R.-J. Slager, and J.-H. Jiang, Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nat. Phys. 17, 1239 (2021).
  43. R.-J. Slager, A. Bouhon, and F. N. Ünal, Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase, arXiv:2208.12824.
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L042010 for technical details.
  45. C. C. Wojcik, K. Wang, A. Dutt, J. Zhong, and S. Fan, Eigenvalue topology of non-Hermitian band structures in two and three dimensions, Phys. Rev. B 106, L161401 (2022).
  46. H. Hu, S. Sun, and S. Chen, Knot topology of exceptional point and non-Hermitian no-go theorem, Phys. Rev. Res. 4, L022064 (2022).
  47. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics Vol. 82 (Springer New York, 1982).
  48. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  49. I. Mandal and E. J. Bergholtz, Symmetry and higher-order exceptional points, Phys. Rev. Lett. 127, 186601 (2021).
  50. S. Sayyad and F. K. Kunst, Realizing exceptional points of any order in the presence of symmetry, Phys. Rev. Res. 4, 023130 (2022).
  51. B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, J. Knot Theory Ramif. 28, 1850082 (2019).
  52. J. Munkres, Topology (Prentice-Hall, Upper Saddle River, NJ, 2000).
  53. J. R. Munkres, Elements of Algebraic Topology (CRC Press, Boca Raton, FL, 2018).
  54. C.-X. Guo, S. Chen, K. Ding, and H. Hu, Exceptional non-Abelian topology in multiband non-Hermitian systems, Phys. Rev. Lett. 130, 157201 (2023).
  55. Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non-Hermitian Bloch braids and associated topological phase transitions, Phys. Rev. Lett. 130, 017201 (2023).
  56. Q. Zhang, L. Zhao, X. Liu, X. Feng, L. Xiong, W. Wu, and C. Qiu, Experimental characterization of three-band braid relations in non-Hermitian acoustic lattices, Phys. Rev. Res. 5, L022050 (2023).

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Phys. Rev. Research 5, L042010– Published 11 October, 2023
  • Received 16 November 2022
  • Revised 7 June 2023
  • Accepted 14 September 2023
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