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Topological invariants for anomalous Floquet higher-order topological insulators

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Abstract

We review the recent development in constructing higher-order topological band insulators under strong periodic drivings. In particular, we focus on various approaches in formulating the anomalous Floquet topological invariants beyond (quasi-)static band topology, and compare their different physical consequences.

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References

  1. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X 3(3), 031005 (2013)

    Google Scholar 

  2. R. Roy and F. Harper, Periodic table for Floquet topological insulators, Phys. Rev. B 96(15), 155118 (2017)

    Article  ADS  Google Scholar 

  3. S. Yao, Z. Yan, and Z. Wang, Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects, Phys. Rev. B 96(19), 195303 (2017)

    Article  ADS  Google Scholar 

  4. A. C. Potter, T. Morimoto, and A. Vishwanath, Classification of interacting topological Floquet phases in one dimension, Phys. Rev. X 6(4), 041001 (2016)

    Google Scholar 

  5. H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, and A. Vishwanath, Chiral Floquet phases of many-body localized bosons, Phys. Rev. X 6(4), 041070 (2016)

    Google Scholar 

  6. T. Morimoto, H. C. Po, and A. Vishwanath, Floquet topological phases protected by time glide symmetry, Phys. Rev. B 95(19), 195155 (2017)

    Article  ADS  Google Scholar 

  7. I. D. Potirniche, A. C. Potter, M. Schleier-Smith, A. Vishwanath, and N. Y. Yao, Floquet symmetry-protected topological phases in cold-atom systems, Phys. Rev. Lett. 119(12), 123601 (2017)

    Article  ADS  Google Scholar 

  8. A. C. Potter, A. Vishwanath, and L. Fidkowski, Infinite family of three-dimensional Floquet topological paramagnets, Phys. Rev. B 97(24), 245106 (2018)

    Article  ADS  Google Scholar 

  9. H. C. Po, L. Fidkowski, A. Vishwanath, and A. C. Potter, Radical chiral Floquet phases in a periodically driven Kitaev model and beyond, Phys. Rev. B 96(24), 245116 (2017)

    Article  ADS  Google Scholar 

  10. L. Fidkowski, H. C. Po, A. C. Potter, and A. Vishwanath, Interacting invariants for Floquet phases of fermions in two dimensions, Phys. Rev. B 99(8), 085115 (2019)

    Article  ADS  Google Scholar 

  11. K. Sacha and J. Zakrzewski, Time crystals: A review, Rep. Prog. Phys. 81(1), 016401 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  12. V. Khemani, R. Moessner, and S. L. Sondhi, A brief history of time crystals, arXiv: 1910.10745 (2019)

  13. D. V. Else, C. Monroe, C. Nayak, and N. Y. Yao, Discrete time crystals, Annu. Rev. Condens. Matter Phys. 11(1), 467 (2020)

    Article  ADS  Google Scholar 

  14. T. Oka and S. Kitamura, Floquet engineering of quantum materials, Annu. Rev. Condens. Matter Phys. 10(1), 387 (2019)

    Article  ADS  Google Scholar 

  15. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultracold fermions, Nature 515(7526), 237 (2014)

    Article  ADS  Google Scholar 

  16. A. Eckardt, Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89(1), 011004 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Goldman, Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nat. Phys. 11(2), 162 (2015)

    Article  Google Scholar 

  18. D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27(10), 6083 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  19. M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys. 12(4), 350 (2016)

    Article  Google Scholar 

  20. S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological thouless pumping of ultracold fermions, Nat. Phys. 12(4), 296 (2016)

    Article  Google Scholar 

  21. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topological characterization of periodically driven quantum systems, Phys. Rev. B 82(23), 235114 (2010)

    Article  ADS  Google Scholar 

  22. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357(6346), 61 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. 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(24), 245115 (2017)

    Article  ADS  Google Scholar 

  24. 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(7), 408 (2018)

    Article  ADS  Google Scholar 

  25. 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 555(7696), 342 (2018)

    Article  ADS  Google Scholar 

  26. C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, A quantized microwave quadrupole insulator with topologically protected corner states, Nature 555(7696), 346 (2018)

    Article  ADS  Google Scholar 

  27. S. Imhof, C. Berger, F. Bayer, J. Brehm, L. Molenkamp, T. Kiessling, F. Schindler, H. L. Ching, M. Greiter, T. Neupert, and R. Thomale, Topolectrical circuit realization of topological corner modes, Nat. Phys. 14(9), 925 (2018)

    Article  Google Scholar 

  28. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4(6), eaat0346 (2018)

    Article  ADS  Google Scholar 

  29. F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S. Jeon, I. Drozdov, H. Bouchiat, S. Guéron, A. Yazdani, B. A. Bernevig, and T. Neupert, Higher-order topology in bismuth, Nat. Phys. 14(9), 918 (2018)

    Article  Google Scholar 

  30. F. K. Kunst, G. van Miert, and E. J. Bergholtz, Lattice models with exactly solvable topological hinge and corner states, Phys. Rev. B 97(24), 241405 (2018)

    Article  ADS  Google Scholar 

  31. Z. Song, Z. Fang, and C. Fang, (d-2)-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119(24), 246402 (2017)

    Article  ADS  Google Scholar 

  32. J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection symmetric second-order topological insulators and superconductors, Phys. Rev. Lett. 119(24), 246401 (2017)

    Article  ADS  Google Scholar 

  33. L. Trifunovic and P. Brouwer, Higher-order bulkboundary correspondence for topological crystalline phases, Phys. Rev. X 9(1), 011012 (2019)

    Google Scholar 

  34. D. Calugaru, V. Juricic, and B. Roy, Higher order topological phases: A general principle of construction, Phys. Rev. B 99, 041301(R) (2019)

    Article  ADS  Google Scholar 

  35. R. Queiroz and A. Stern, Splitting the hinge mode of higher-order topological insulators, Phys. Rev. Lett. 123(3), 036802 (2019)

    Article  ADS  Google Scholar 

  36. M. Ezawa, Topological switch between second-order topological insulators and topological crystalline insulators, Phys. Rev. Lett. 121(11), 116801 (2018)

    Article  ADS  Google Scholar 

  37. F. Zhang, C. L. Kane, and E. J. Mele, Surface state magnetization and chiral edge states on topological insulators, Phys. Rev. Lett. 110(4), 046404 (2013)

    Article  ADS  Google Scholar 

  38. B. Seradjeh, C. Weeks, and M. Franz, Fractionalization in a square-lattice model with time-reversal symmetry, Phys. Rev. B 77(3), 033104 (2008)

    Article  ADS  Google Scholar 

  39. M. Lin and T. L. Hughes, Topological quadrupolar semimetals, Phys. Rev. B 98(24), 241103 (2018)

    Article  ADS  Google Scholar 

  40. M. Ezawa, Higher-order topological insulators and semimetalson the breathing kagome and pyrochlore lattices, Phys. Rev. Lett. 120(2), 026801 (2018)

    Article  ADS  Google Scholar 

  41. M. Ezawa, Magnetic second-order topological insulators and semimetals, Phys. Rev. B 97(15), 155305 (2018)

    Article  ADS  Google Scholar 

  42. H. Shapourian, Y. Wang, and S. Ryu, Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials, Phys. Rev. B 97(9), 094508 (2018)

    Article  ADS  Google Scholar 

  43. Y. Wang, M. Lin, and T. L. Hughes, Weak-pairing higher order topological superconductors, Phys. Rev. B 98(16), 165144 (2018)

    Article  ADS  Google Scholar 

  44. E. Khalaf, Higher-order topological insulators and superconductors protected by inversion symmetry, Phys. Rev. B 97(20), 205136 (2018)

    Article  ADS  Google Scholar 

  45. X. Zhu, Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields, Phys. Rev. B 97(20), 205134 (2018)

    Article  ADS  Google Scholar 

  46. Z. Yan, F. Song, and Z. Wang, Majorana corner modes in a high-temperature platform, Phys. Rev. Lett. 121(9), 096803 (2018)

    Article  ADS  Google Scholar 

  47. Q. Wang, C. C. Liu, Y. M. Lu, and F. Zhang, High-temperature Majorana corner states, Phys. Rev. Lett. 121(18), 186801 (2018)

    Article  ADS  Google Scholar 

  48. T. Liu, J. J. He, and F. Nori, Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor, Phys. Rev. B 98(24), 245413 (2018)

    Article  ADS  Google Scholar 

  49. V. Dwivedi, C. Hickey, T. Eschmann, and S. Trebst, Majorana corner modes in a second-order Kitaev spin liquid, Phys. Rev. B 98(5), 054432 (2018)

    Article  ADS  Google Scholar 

  50. Y. You, T. Devakul, F. J. Burnell, and T. Neupert, Higher order symmetry-protected topological states for interacting bosons and fermions, Phys. Rev. B 98, 235102 (2018)

    Article  ADS  Google Scholar 

  51. A. Rasmussen and Y. M. Lu, Classification and construction of higher-order symmetry protected topological phases of interacting bosons, Phys. Rev. B 101, 085137 (2019)

    Article  ADS  Google Scholar 

  52. B. Huang and W. V. Liu, Floquet higher-order topological insulators with anomalous dynamical polarization, Phys. Rev. Lett. 124(21), 216601 (2020)

    Article  ADS  Google Scholar 

  53. R. W. Bomantara, L. Zhou, J. Pan, and J. Gong, Coupled-wire construction of static and Floquet second-order topological insulators, Phys. Rev. B 99, 045441 (2019)

    Article  ADS  Google Scholar 

  54. M. Rodriguez-Vega, A. Kumar, and B. Seradjeh, Higher-order Floquet topological phases with corner and bulk bound states, Phys. Rev. B 100, 085138 (2019)

    Article  ADS  Google Scholar 

  55. Y. Peng and G. Refael, Floquet second-order topological insulators from nonsymmorphic space—time symmetries, Phys. Rev. Lett. 123(1), 016806 (2019)

    Article  ADS  Google Scholar 

  56. A. K. Ghosh, T. Nag, and A. Saha, Dynamical construction of quadrupolar and octupolar topological superconductors, Phys. Rev. B 105(15), 155406 (2022)

    Article  ADS  Google Scholar 

  57. H. Hu, B. Huang, E. Zhao, and W. V. Liu, Dynamical singularities of Floquet higher-order topological insulators, Phys. Rev. Lett. 124(5), 057001 (2020)

    Article  ADS  Google Scholar 

  58. R. X. Zhang and Z. C. Yang, Theory of anomalous Floquet higher-order topology: Classification, characterization, and bulk-boundary correspondence, arXiv: 2010.07945 (2020)

  59. D. D. Vu, R. X. Zhang, Z. C. Yang, and S. Das Sarma, Superconductors with anomalous Floquet higher-order topology, Phys. Rev. B 104(14), L140502 (2021)

    Article  ADS  Google Scholar 

  60. Y. Peng, Floquet higher-order topological insulators and superconductors with space—time symmetries, Phys. Rev. Res. 2(1), 013124 (2020)

    Article  MathSciNet  Google Scholar 

  61. C. K. Chiu, H. Yao, and S. Ryu, Classification of topological insulators and superconductors in the presence of reflection symmetry, Phys. Rev. B 88(7), 075142 (2013)

    Article  ADS  Google Scholar 

  62. E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Boundary-obstructed topological phases, Phys. Rev. Res. 3(1), 013239 (2021)

    Article  Google Scholar 

  63. R. Resta, Quantum-mechanical position operator in extended systems, Phys. Rev. Lett. 80(9), 1800 (1998)

    Article  ADS  Google Scholar 

  64. D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82(3), 1959 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. K. Wintersperger, C. Braun, F. N. Ünal, A. Eckardt, M. D. Liberto, N. Goldman, I. Bloch, and M. Aidelsburger, Realization of an anomalous Floquet topological system with ultracold atoms, Nat. Phys. 16(10), 1058 (2020)

    Article  Google Scholar 

  66. F. N. Ünal, B. Seradjeh, and A. Eckardt, How to directly measure Floquet topological invariants in optical lattices, Phys. Rev. Lett. 122(25), 253601 (2019)

    Article  ADS  Google Scholar 

  67. M. Rodriguez-Vega, A. Kumar, and B. Seradjeh, Higher-order Floquet topological phases with corner and bulk bound states, Phys. Rev. B 100(8), 085138 (2019)

    Article  ADS  Google Scholar 

  68. R. W. Bomantara, L. Zhou, J. Pan, and J. Gong, Coupled-wire construction of static and Floquet second-order topological insulators, Phys. Rev. B 99(4), 045441 (2019)

    Article  ADS  Google Scholar 

  69. L. Fu and C. L. Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100(9), 096407 (2008)

    Article  ADS  Google Scholar 

  70. M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

    Article  ADS  Google Scholar 

  71. M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Second-order topological insulators and superconductors with an order-two crystalline symmetry, Phys. Rev. B 97(20), 205135 (2018)

    Article  ADS  Google Scholar 

  72. S. Xu and C. Wu, Space—time crystal and space—time group, Phys. Rev. Lett. 120(9), 096401 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  73. Y. Peng, Topological space—time crystal, Phys. Rev. Lett. 128(18), 186802 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  74. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8(1), 50 (2017)

    Article  ADS  Google Scholar 

  75. E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry indicators and anomalous surface states of topological crystalline insulators, Phys. Rev. X 8(3), 031070 (2018)

    Google Scholar 

  76. J. Yu, R. X. Zhang, and Z. D. Song, Dynamical symmetry indicators for Floquet crystals, Nat. Commun. 12(1), 5985 (2021)

    Article  ADS  Google Scholar 

  77. H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and Wannier obstructions, Phys. Rev. Lett. 121(12), 126402 (2018)

    Article  ADS  Google Scholar 

  78. D. V. Else, H. C. Po, and H. Watanabe, Fragile topological phases in interacting systems, Phys. Rev. B 99(12), 125122 (2019)

    Article  ADS  Google Scholar 

  79. S. Ono, Y. Yanase, and H. Watanabe, Symmetry indicators for topological superconductors, Phys. Rev. Res. 1(1), 013012 (2019)

    Article  Google Scholar 

  80. M. Nakagawa, R. J. Slager, S. Higashikawa, and T. Oka, Wannier representation of Floquet topological states, Phys. Rev. B 101(7), 075108 (2020)

    Article  ADS  Google Scholar 

  81. J. Yu, Y. Ge, and S. Das Sarma, Dynamical fragile topology in Floquet crystals, Phys. Rev. B 104(18), L180303 (2021)

    Article  ADS  Google Scholar 

  82. R. X. Zhang and Z. C. Yang, Tunable fragile topology in Floquet systems, Phys. Rev. B 103(12), L121115 (2021)

    Article  ADS  Google Scholar 

  83. Y. Zhu, T. Qin, X. Yang, G. Xianlong, and Z. Liang, Floquet and anomalous Floquet Weyl semimetals, Phys. Rev. Res. 2(3), 033045 (2020)

    Article  Google Scholar 

  84. W. Zhu, M. Umer, and J. Gong, Floquet higher-order Weyl and nexus semimetals, Phys. Rev. Res. 3(3), L032026 (2021)

    Article  Google Scholar 

  85. R. Zhang, K. Hino, and N. Maeshima, Floquet-weyl semimetals generated by an optically resonant inter-band-transition, arXiv: 2201.01578 (2022)

  86. H. Wu, B. Q. Wang, and J. H. An, Floquet second-order topological insulators in non-Hermitian systems, Phys. Rev. B 103(4), L041115 (2021)

    Article  ADS  Google Scholar 

  87. A. K. Ghosh and T. Nag, Non-hermitian higherorder topological superconductors in two-dimension: statics and dynamics, arXiv: 2205.09915 (2022)

  88. K. Koor, R. W. Bomantara, and L. C. Kwek, Symmetry protected topological corner modes in a periodically driven interacting spin lattice, arXiv: 2206.06660 (2022)

  89. W. Zhu, H. Xue, J. Gong, Y. Chong, and B. Zhang, Time-periodic corner states from Floquet higher order topology, Nat. Commun. 13(1), 11 (2022)

    Article  ADS  Google Scholar 

  90. W. Zhu, Y. D. Chong, and J. Gong, Floquet higher order topological insulator in a periodically driven bipartite lattice, Phys. Rev. B 103(4), L041402 (2021)

    Article  ADS  Google Scholar 

  91. S. Chaudhary, A. Haim, Y. Peng, and G. Refael, Phonon-induced Floquet topological phases protected by space—time symmetries, Phys. Rev. Res. 2(4), 043431 (2020)

    Article  Google Scholar 

  92. B. Bauer, T. Pereg-Barnea, T. Karzig, M. T. Rieder, G. Refael, E. Berg, and Y. Oreg, Topologically protected braiding in a single wire using Floquet Majorana modes, Phys. Rev. B 100(4), 041102 (2019)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12174389).

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Huang, B. Topological invariants for anomalous Floquet higher-order topological insulators. Front. Phys. 18, 13601 (2023). https://doi.org/10.1007/s11467-022-1209-7

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