Topological invariants for anomalous Floquet higher-order topological insulators

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.

Graphical abstract

Keywords

topological / Floquet / higher-order

References

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

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

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

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

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

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

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

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

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

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

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

[12] V.KhemaniR.MoessnerS.L. Sondhi, A brief history of time crystals, arXiv: 1910.10745 (2019)

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

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

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

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

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

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

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

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

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

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

[23] W. A. Benalcazar , B. A. Bernevig , T. L. Hughes . Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B, 2017, 96(24): 245115

[24] J. Noh , W. A. Benalcazar , S. Huang , M. J. Collins , K. P. Chen , T. L. Hughes , M. C. Rechtsman . Topological protection of photonic mid-gap defect modes. Nat. Photonics, 2018, 12(7): 408

[25] M. Serra-Garcia , V. Peri , R. Süsstrunk , O. R. Bilal , T. Larsen , L. G. Villanueva , S. D. Huber . Observation of a phononic quadrupole topological insulator. Nature, 2018, 555(7696): 342

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

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

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

[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 , T. Neupert . Higher-order topology in bismuth. Nat. Phys., 2018, 14(9): 918

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

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

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

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

[34] D.CalugaruV.JuricicB.Roy, Higher order topological phases: A general principle of construction, Phys. Rev. B 99, 041301(R) (2019)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[58] R.X. ZhangZ.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 , S. Das Sarma . Superconductors with anomalous Floquet higher-order topology. Phys. Rev. B, 2021, 104(14): L140502

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

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

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

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

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

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

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

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

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

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

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

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

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

[73] Y. Peng . Topological space−time crystal. Phys. Rev. Lett., 2022, 128(18): 186802

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

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

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

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

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

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

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

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

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

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

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

[85] R.ZhangK.HinoN.Maeshima, Floquet-weyl semimetals generated by an optically resonant interband-transition, arXiv: 2201.01578 (2022)

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

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

[88] K.KoorR.W. BomantaraL.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 , B. Zhang . Time-periodic corner states from Floquet higher order topology. Nat. Commun., 2022, 13(1): 11

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

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

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

Acknowledgements

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

RIGHTS & PERMISSIONS

2023 Higher Education Press