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  • Letter
  • Open Access

Second-order topological modes in two-dimensional continuous media

Jan Košata and Oded Zilberberg

  • Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

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Phys. Rev. Research 3, L032029 – Published 30 July, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L032029

Abstract

We present a symmetry-based scheme to create zero-dimensional (0D) second-order topological modes in continuous two-dimensional (2D) systems. We show that a metamaterial with a p6m-symmetric pattern exhibits two Dirac cones, which can be gapped in two distinct ways by deforming the pattern. Combining the deformations in a single system then emulates the 2D Jackiw-Rossi model of a topological vortex, where 0D in-gap bound modes are guaranteed to exist. We exemplify our approach with the simple hexagonal, kagome, and honeycomb lattices. We furthermore formulate a quantitative method to extract the topological properties from finite-element simulations, which facilitates further optimization of the bound mode characteristics. Our scheme enables the realization of second-order topology in a wide range of experimental systems.

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References (56)

  1. F.-F. Li, H.-X. Wang, Z. Xiong, Q. Lou, P. Chen, R.-X. Wu, Y. Poo, J.-H. Jiang, and S. John, Topological light-trapping on a dislocation, Nat. Commun. 9, 2462 (2018).
  2. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  3. J. Ningyuan, C. Owens, A. Sommer, D. Schuster, and J. Simon, Time- and Site-Resolved Dynamics in a Topological Circuit, Phys. Rev. X 5, 021031 (2015).
  4. T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, and R. Thomale, Chiral Voltage Propagation and Calibration in a Topolectrical Chern Circuit, Phys. Rev. Lett. 122, 247702 (2019).
  5. L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Topological mechanics of gyroscopic metamaterials, Proc. Natl. Acad. Sci. USA 112, 14495 (2015).
  6. S. H. Mousavi, A. B. Khanikaev, and Z. Wang, Topologically protected elastic waves in phononic metamaterials, Nat. Commun. 6, 8682 (2015).
  7. S. D. Huber, Topological mechanics, Nat. Phys. 12, 621 (2016).
  8. C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, Acoustic topological insulator and robust one-way sound transport, Nat. Phys. 12, 1124 (2016).
  9. G. Ma, M. Xiao, and C. T. Chan, Topological phases in acoustic and mechanical systems, Nat. Rev. Phys. 1, 281 (2019).
  10. N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  11. M. Milićević, T. Ozawa, P. Andreakou, I. Carusotto, T. Jacqmin, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, and J. Bloch, Edge states in polariton honeycomb lattices, 2D Mater. 2, 034012 (2015).
  12. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological States and Adiabatic Pumping in Quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
  13. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, Imaging topological edge states in silicon photonics, Nat. Photonics 7, 1001 (2013).
  14. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature (London) 496, 196 (2013).
  15. L.-H. Wu and X. Hu, Scheme for Achieving a Topological Photonic Crystal by using Dielectric Material, Phys. Rev. Lett. 114, 223901 (2015).
  16. Y. E. Kraus, Z. Ringel, and O. Zilberberg, Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal, Phys. Rev. Lett. 111, 226401 (2013).
  17. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017).
  18. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017).
  19. I. Petrides, H. M. Price, and O. Zilberberg, Six-dimensional quantum Hall effect and three-dimensional topological pumps, Phys. Rev. B 98, 125431 (2018).
  20. O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, Photonic topological boundary pumping as a probe of 4D quantum Hall physics, Nature (London) 553, 59 (2018).
  21. W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators, Phys. Rev. B 99, 245151 (2019).
  22. D. Călugăru, V. Juričić, and B. Roy, Higher-order topological phases: A general principle of construction, Phys. Rev. B 99, 041301(R) (2019).
  23. T. Fukui, Dirac fermion model associated with a second-order topological insulator, Phys. Rev. B 99, 165129 (2019).
  24. I. Petrides and O. Zilberberg, Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions, Phys. Rev. Research 2, 022049(R) (2020).
  25. S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert et al., Topolectrical-circuit realization of topological corner modes, Nat. Phys. 14, 925 (2018).
  26. J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, Topological protection of photonic mid-gap defect modes, Nat. Photonics 12, 408 (2018).
  27. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, eaat0346 (2018).
  28. M. Serra-Garcia, V. Peri, R. Süsstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, Observation of a phononic quadrupole topological insulator, Nature (London) 555, 342 (2018).
  29. H. Fan, B. Xia, L. Tong, S. Zheng, and D. Yu, Elastic Higher-Order Topological Insulator with Topologically Protected Corner States, Phys. Rev. Lett. 122, 204301 (2019).
  30. H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, Acoustic higher-order topological insulator on a kagome lattice, Nat. Mater. 18, 108 (2019).
  31. B.-Y. Xie, H.-F. Wang, H.-X. Wang, X.-Y. Zhu, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Second-order photonic topological insulator with corner states, Phys. Rev. B 98, 205147 (2018).
  32. X. Zhang, H.-X. Wang, Z.-K. Lin, Y. Tian, B. Xie, M.-H. Lu, Y.-F. Chen, and J.-H. Jiang, Second-order topology and multidimensional topological transitions in sonic crystals, Nat. Phys. 15, 582 (2019).
  33. X. Zhang, Z.-K. Lin, H.-X. Wang, Z. Xiong, Y. Tian, M.-H. Lu, Y.-F. Chen, and J.-H. Jiang, Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals, Nat. Commun. 11, 65 (2020).
  34. X. Zhou, Z.-K. Lin, W. Lu, Y. Lai, B. Hou, and J.-H. Jiang, Twisted quadrupole topological photonic crystals, Laser Photonics Rev. 14, 2070046 (2020).
  35. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  36. J. C. Y. Teo and C. L. Kane, Topological defects and gapless modes in insulators and superconductors, Phys. Rev. B 82, 115120 (2010).
  37. C.-Y. Hou, C. Chamon, and C. Mudry, Electron Fractionalization in Two-Dimensional Graphenelike Structures, Phys. Rev. Lett. 98, 186809 (2007).
  38. P. Gao, D. Torrent, F. Cervera, P. San-Jose, J. Sánchez-Dehesa, and J. Christensen, Majorana-Like Zero Modes in Kekulé Distorted Sonic Lattices, Phys. Rev. Lett. 123, 196601 (2019).
  39. X. Gao, L. Yang, H. Lin, L. Zhang, J. Li, F. Bo, Z. Wang, and L. Lu, Dirac-vortex topological cavities, Nat. Nanotechnol. 15, 1012 (2020).
  40. Y. Yang, Z. Jia, Y. Wu, R.-C. Xiao, Z. H. Hang, H. Jiang, and X. Xie, Gapped topological kink states and topological corner states in honeycomb lattice, Sci. Bull. 65, 531 (2020).
  41. X. Wu, Y. Meng, Y. Hao, R.-Y. Zhang, J. Li, and X. Zhang, Topological Corner Modes Induced by Dirac Vortices in Arbitrary Geometry, Phys. Rev. Lett. 126, 226802 (2021).
  42. M. Saba, S. Wong, M. Elman, S. S. Oh, and O. Hess, Nature of topological protection in photonic spin and valley Hall insulators, Phys. Rev. B 101, 054307 (2020).
  43. J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Building blocks of topological quantum chemistry: Elementary band representations, Phys. Rev. B 97, 035139 (2018).
  44. R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13, 3398 (1976).
  45. C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005).
  46. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  47. C. Bradley and A. Cracknell, The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups (Oxford University Press, Oxford, U.K., 2009).
  48. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L032029 for the derivations of the representation matrices and symmetry-allowed perturbative terms.
  49. M. Saba, J. M. Hamm, J. J. Baumberg, and O. Hess, Group Theoretical Route to Deterministic Weyl Points in Chiral Photonic Lattices, Phys. Rev. Lett. 119, 227401 (2017).
  50. T. Ma and G. Shvets, All-Si valley-Hall photonic topological insulator, New J. Phys. 18, 025012 (2016).
  51. J. Lu, C. Qiu, L. Ye, X. Fan, M. Ke, F. Zhang, and Z. Liu, Observation of topological valley transport of sound in sonic crystals, Nat. Phys. 13, 369 (2017).
  52. comsol multiphysics v. 5.5, https://www.comsol.com/ (2019).
  53. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7, 021306 (2020).
  54. Y. E. Kraus, O. Zilberberg, and R. Berkovits, Enhanced compressibility due to repulsive interaction in the Harper model, Phys. Rev. B 89, 161106(R) (2014).
  55. H. Lin and L. Lu, Dirac-vortex topological photonic crystal fibre, Light Sci. Appl. 9, 202 (2020).
  56. L. Lu, H. Gao, and Z. Wang, Topological one-way fiber of second Chern number, Nat. Commun. 9, 5384 (2018).

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Phys. Rev. Research 3, L032029– Published 30 July, 2021
  • Received 9 March 2021
  • Accepted 28 June 2021
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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