Topological invariants for anomalous Floquet higher-order topological insulators

doi:10.1007/s11467-022-1209-7

Front. Phys.

2023, Vol. 18

Issue (1) : 13601 https://doi.org/10.1007/s11467-022-1209-7

TOPICAL REVIEW

Topological invariants for anomalous Floquet higher-order topological insulators

Biao Huang(

)

Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

Download: PDF(17174 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

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.

Keywords topological Floquet higher-order

Corresponding Author(s): Biao Huang

Issue Date: 27 October 2022

Cite this article:

Biao Huang. Topological invariants for anomalous Floquet higher-order topological insulators[J]. Front. Phys. , 2023, 18(1): 13601.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1209-7
https://academic.hep.com.cn/fop/EN/Y2023/V18/I1/13601

Fig.1 (a) Phase diagram parameterized by dimensionless

(γ, λ) ≡ (γ ′ T / (2 ?), λ ′ T / (2 ?))

. (b) One representative in-gap corner state amplitude in phase ④, with others in different corners. (c, d) Open-boundary spectrum (

L x × L y = 20 × 20

) in different phases. Parameters are

2 (γ, λ) = (π 4, π 2), (π 2, π 4),

(3 π 4, π 2), (π 2, 3 π 4)

for phases ①?④ respectively. Figures taken from Ref. [52].

Fig.2 Schematic illustration of dynamical polarization. Figure taken from Ref. [52].

Fig.3 Toy scenario showing the need for

x^m e a n (t)

Fig.4 Figures taken from Ref. [52] for the model in Eq. (5). Solid/dashed lines denotes +/? sign for the hopping amplitude so each plaquette carries

π

-fluxes.

Fig.5 (a) Exemplary first-order dynamical branches

ν x, μ (k y, t)

in phase ④ at

t = T / 2

. Four branches for each

ε

-gap separate into two doubly degenerate sets

± ν x

. Other phases/instants

t

exhibit

ν x, μ (k y, t)

of similar structures. (b) Quadrupolar motions

? ν y, μ ~ (ν x) ? (t)

in a cycle, with the same parameters as in Fig.1 for each phase respectively. Generically the 1/2 crossing needs not to occur at

t = T / 2

if the chiral symmetry is broken. Figures taken from Ref. [52].

Fig.6 Dynamical singularity when the parameters

γ, λ

in the model of Fig.4 takes various values in the phase diagram. (a) Parametrization for different values of

γ, λ

, where the contour across all 4 phases is parametrized by

g

. (b) The size of

π

-gap for

U (k, t)

, where we see that the phase transition is associated with destroying or creating a dynamical singularity at

t = T

. (c) The size of

π

-gap for the periodized

U ε = π (k, t)

. Due to chiral symmetry for

U ε = π

, the dynamical singularity always occur at

t = T / 2

. In (b) and (c), the red-color (for vanishing

π

-gap) marks the instants where dynamical singularities exist.

Fig.7 Representative instant spectrum for three points in Fig.6(c). Each band is doubly degenerate.

Fig.8 The mirror graded winding numbers within the reduced Brillouin zone defined by

M x y (k) = k

Tab.1 Comparison between different approaches to characterize anomalous Floquet higher-order topology.

Fig.9 Quasienergy spectrum of the Floquet operator under different boundary conditions. Here

(γ y, λ) =

(π / 2) (1, 0.25)

is fixed. Should

γ x = γ y

(green dashed line), it corresponds to phase ③ in Fig.1(a) with

π

corner modes. Decreasing

γ x

below

γ x (c) ≈ 1.2

destroys these corner modes. (a) and (b) clearly show that an edge (rather than bulk) topological phase transition occurs around

γ x (c)

where the edge

π

-gap closes while the bulk gap remains open.

Fig.10 DP for the extended model. We see that the corner states in Fig.9(c) is characterized by DP across the edge topological phase transition.

Fig.11 Dynamical singularity for the extended model is immediately gapped out when the

M x y

is broken by

γ x ≠ γ y

, although corner states still exist for certain ranges as shown by Fig.9(c).

1	S. Rudner M. , H. Lindner N. , Berg E. , Levin M. . Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X, 2013, 3(3): 031005 https://doi.org/10.1103/PhysRevX.3.031005
2	Roy R. , Harper F. . Periodic table for Floquet topological insulators. Phys. Rev. B, 2017, 96(15): 155118 https://doi.org/10.1103/PhysRevB.96.155118
3	Yao S. , Yan Z. , Wang Z. . Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects. Phys. Rev. B, 2017, 96(19): 195303 https://doi.org/10.1103/PhysRevB.96.195303
4	C. Potter A. , Morimoto T. , Vishwanath A. . Classification of interacting topological Floquet phases in one dimension. Phys. Rev. X, 2016, 6(4): 041001 https://doi.org/10.1103/PhysRevX.6.041001
5	C. Po H. , Fidkowski L. , Morimoto T. , C. Potter A. , Vishwanath A. . Chiral Floquet phases of many-body localized bosons. Phys. Rev. X, 2016, 6(4): 041070 https://doi.org/10.1103/PhysRevX.6.041070
6	Morimoto T. , C. Po H. , Vishwanath A. . Floquet topological phases protected by time glide symmetry. Phys. Rev. B, 2017, 95(19): 195155 https://doi.org/10.1103/PhysRevB.95.195155
7	D. Potirniche I. , C. Potter A. , Schleier-Smith M. , Vishwanath A. , Y. Yao N. . Floquet symmetry-protected topological phases in cold-atom systems. Phys. Rev. Lett., 2017, 119(12): 123601 https://doi.org/10.1103/PhysRevLett.119.123601
8	C. Potter A. , Vishwanath A. , Fidkowski L. . Infinite family of three-dimensional Floquet topological paramagnets. Phys. Rev. B, 2018, 97(24): 245106 https://doi.org/10.1103/PhysRevB.97.245106
9	C. Po H. , Fidkowski L. , Vishwanath A. , C. Potter A. . Radical chiral Floquet phases in a periodically driven Kitaev model and beyond. Phys. Rev. B, 2017, 96(24): 245116 https://doi.org/10.1103/PhysRevB.96.245116
10	Fidkowski L. , C. Po H. , C. Potter A. , Vishwanath A. . Interacting invariants for Floquet phases of fermions in two dimensions. Phys. Rev. B, 2019, 99(8): 085115 https://doi.org/10.1103/PhysRevB.99.085115
11	Sacha K. , Zakrzewski J. . Time crystals: A review. Rep. Prog. Phys., 2018, 81(1): 016401 https://doi.org/10.1088/1361-6633/aa8b38
12	Khemani V.Moessner R.L. Sondhi S., A brief history of time crystals, arXiv: 1910.10745 (2019)
13	V. Else D. , Monroe C. , Nayak C. , Y. Yao N. . Discrete time crystals. Annu. Rev. Condens. Matter Phys., 2020, 11(1): 467 https://doi.org/10.1146/annurev-conmatphys-031119-050658
14	Oka T. , Kitamura S. . Floquet engineering of quantum materials. Annu. Rev. Condens. Matter Phys., 2019, 10(1): 387 https://doi.org/10.1146/annurev-conmatphys-031218-013423
15	Jotzu G. , Messer M. , Desbuquois R. , Lebrat M. , Uehlinger T. , Greif D. , Esslinger T. . Experimental realization of the topological Haldane model with ultracold fermions. Nature, 2014, 515(7526): 237 https://doi.org/10.1038/nature13915
16	Eckardt A. . Atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys., 2017, 89(1): 011004 https://doi.org/10.1103/RevModPhys.89.011004
17	Aidelsburger M. , Lohse M. , Schweizer C. , Atala M. , T. Barreiro J. , Nascimbène S. , R. Cooper N. , Bloch I. , Goldman N. . Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys., 2015, 11(2): 162 https://doi.org/10.1038/nphys3171
18	J. Thouless D. . Quantization of particle transport. Phys. Rev. B, 1983, 27(10): 6083 https://doi.org/10.1103/PhysRevB.27.6083
19	Lohse M. , Schweizer C. , Zilberberg O. , Aidelsburger M. , Bloch I. . A thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys., 2016, 12(4): 350 https://doi.org/10.1038/nphys3584
20	Nakajima S. , Tomita T. , Taie S. , Ichinose T. , Ozawa H. , Wang L. , Troyer M. , Takahashi Y. . Topological thouless pumping of ultracold fermions. Nat. Phys., 2016, 12(4): 296 https://doi.org/10.1038/nphys3622
21	Kitagawa T. , Berg E. , Rudner M. , Demler E. . Topological characterization of periodically driven quantum systems. Phys. Rev. B, 2010, 82(23): 235114 https://doi.org/10.1103/PhysRevB.82.235114
22	A. Benalcazar W. , A. Bernevig B. , L. Hughes T. . Quantized electric multipole insulators. Science, 2017, 357(6346): 61 https://doi.org/10.1126/science.aah6442
23	A. Benalcazar W. , A. Bernevig B. , L. Hughes T. . Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B, 2017, 96(24): 245115 https://doi.org/10.1103/PhysRevB.96.245115
24	Noh J. , A. Benalcazar W. , Huang S. , J. Collins M. , P. Chen K. , L. Hughes T. , C. Rechtsman M. . Topological protection of photonic mid-gap defect modes. Nat. Photonics, 2018, 12(7): 408 https://doi.org/10.1038/s41566-018-0179-3
25	Serra-Garcia M. , Peri V. , Süsstrunk R. , R. Bilal O. , Larsen T. , G. Villanueva L. , D. Huber S. . Observation of a phononic quadrupole topological insulator. Nature, 2018, 555(7696): 342 https://doi.org/10.1038/nature25156
26	W. Peterson C. , A. Benalcazar W. , L. Hughes T. , Bahl G. . A quantized microwave quadrupole insulator with topologically protected corner states. Nature, 2018, 555(7696): 346 https://doi.org/10.1038/nature25777
27	Imhof S. , Berger C. , Bayer F. , Brehm J. , Molenkamp L. , Kiessling T. , Schindler F. , L. Ching H. , Greiter M. , Neupert T. , Thomale R. . Topolectrical circuit realization of topological corner modes. Nat. Phys., 2018, 14(9): 925 https://doi.org/10.1038/s41567-018-0246-1
28	Schindler F. , M. Cook A. , G. Vergniory M. , Wang Z. , S. P. Parkin S. , A. Bernevig B. , Neupert T. . Higher-order topological insulators. Sci. Adv., 2018, 4(6): eaat0346 https://doi.org/10.1126/sciadv.aat0346
29	Schindler F. , Wang Z. , G. Vergniory M. , M. Cook A. , Murani A. , Sengupta S. , Y. Kasumov A. , Deblock R. , Jeon S. , Drozdov I. , Bouchiat H. , Guéron S. , Yazdani A. , A. Bernevig B. , Neupert T. . Higher-order topology in bismuth. Nat. Phys., 2018, 14(9): 918 https://doi.org/10.1038/s41567-018-0224-7
30	K. Kunst F. , van Miert G. , J. Bergholtz E. . Lattice models with exactly solvable topological hinge and corner states. Phys. Rev. B, 2018, 97(24): 241405 https://doi.org/10.1103/PhysRevB.97.241405
31	Song Z. , Fang Z. , Fang C. . (d-2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett., 2017, 119(24): 246402 https://doi.org/10.1103/PhysRevLett.119.246402
32	Langbehn J. , Peng Y. , Trifunovic L. , von Oppen F. , W. Brouwer P. . Reflection symmetric second-order topological insulators and superconductors. Phys. Rev. Lett., 2017, 119(24): 246401 https://doi.org/10.1103/PhysRevLett.119.246401
33	Trifunovic L. , Brouwer P. . Higher-order bulkboundary correspondence for topological crystalline phases. Phys. Rev. X, 2019, 9(1): 011012 https://doi.org/10.1103/PhysRevX.9.011012
34	Calugaru D.Juricic V.Roy B., Higher order topological phases: A general principle of construction, Phys. Rev. B 99, 041301(R) (2019)
35	Queiroz R. , Stern A. . Splitting the hinge mode of higher-order topological insulators. Phys. Rev. Lett., 2019, 123(3): 036802 https://doi.org/10.1103/PhysRevLett.123.036802
36	Ezawa M. . Topological switch between second-order topological insulators and topological crystalline insulators. Phys. Rev. Lett., 2018, 121(11): 116801 https://doi.org/10.1103/PhysRevLett.121.116801
37	Zhang F. , L. Kane C. , J. Mele E. . Surface state magnetization and chiral edge states on topological insulators. Phys. Rev. Lett., 2013, 110(4): 046404 https://doi.org/10.1103/PhysRevLett.110.046404
38	Seradjeh B. , Weeks C. , Franz M. . Fractionalization in a square-lattice model with time-reversal symmetry. Phys. Rev. B, 2008, 77(3): 033104 https://doi.org/10.1103/PhysRevB.77.033104
39	Lin M. , L. Hughes T. . Topological quadrupolar semimetals. Phys. Rev. B, 2018, 98(24): 241103 https://doi.org/10.1103/PhysRevB.98.241103
40	Ezawa M. . Higher-order topological insulators and semimetalson the breathing kagome and pyrochlore lattices. Phys. Rev. Lett., 2018, 120(2): 026801 https://doi.org/10.1103/PhysRevLett.120.026801
41	Ezawa M. . Magnetic second-order topological insulators and semimetals. Phys. Rev. B, 2018, 97(15): 155305 https://doi.org/10.1103/PhysRevB.97.155305
42	Shapourian H. , Wang Y. , Ryu S. . Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials. Phys. Rev. B, 2018, 97(9): 094508 https://doi.org/10.1103/PhysRevB.97.094508
43	Wang Y. , Lin M. , L. Hughes T. . Weak-pairing higher order topological superconductors. Phys. Rev. B, 2018, 98(16): 165144 https://doi.org/10.1103/PhysRevB.98.165144
44	Khalaf E. . Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B, 2018, 97(20): 205136 https://doi.org/10.1103/PhysRevB.97.205136
45	Zhu X. . Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields. Phys. Rev. B, 2018, 97(20): 205134 https://doi.org/10.1103/PhysRevB.97.205134
46	Yan Z. , Song F. , Wang Z. . Majorana corner modes in a high-temperature platform. Phys. Rev. Lett., 2018, 121(9): 096803 https://doi.org/10.1103/PhysRevLett.121.096803
47	Wang Q. , C. Liu C. , M. Lu Y. , Zhang F. . High-temperature Majorana corner states. Phys. Rev. Lett., 2018, 121(18): 186801 https://doi.org/10.1103/PhysRevLett.121.186801
48	Liu T. , J. He J. , Nori F. . Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor. Phys. Rev. B, 2018, 98(24): 245413 https://doi.org/10.1103/PhysRevB.98.245413
49	Dwivedi V. , Hickey C. , Eschmann T. , Trebst S. . Majorana corner modes in a second-order Kitaev spin liquid. Phys. Rev. B, 2018, 98(5): 054432 https://doi.org/10.1103/PhysRevB.98.054432
50	You Y. , Devakul T. , J. Burnell F. , Neupert T. . Higher order symmetry-protected topological states for interacting bosons and fermions. Phys. Rev. B, 2018, 98: 235102 https://doi.org/10.1103/PhysRevB.98.235102
51	Rasmussen A. , M. Lu Y. . Classification and construction of higher-order symmetry protected topological phases of interacting bosons. Phys. Rev. B, 2019, 101: 085137 https://doi.org/10.1103/PhysRevB.101.085137
52	Huang B. , V. Liu W. . Floquet higher-order topological insulators with anomalous dynamical polarization. Phys. Rev. Lett., 2020, 124(21): 216601 https://doi.org/10.1103/PhysRevLett.124.216601
53	W. Bomantara R. , Zhou L. , Pan J. , Gong J. . Coupled-wire construction of static and Floquet second-order topological insulators. Phys. Rev. B, 2019, 99: 045441 https://doi.org/10.1103/PhysRevB.99.045441
54	Rodriguez-Vega M. , Kumar A. , Seradjeh B. . Higher-order Floquet topological phases with corner and bulk bound states. Phys. Rev. B, 2019, 100: 085138 https://doi.org/10.1103/PhysRevB.100.085138
55	Peng Y. , Refael G. . Floquet second-order topological insulators from nonsymmorphic space−time symmetries. Phys. Rev. Lett., 2019, 123(1): 016806 https://doi.org/10.1103/PhysRevLett.123.016806
56	K. Ghosh A. , Nag T. , Saha A. . Dynamical construction of quadrupolar and octupolar topological superconductors. Phys. Rev. B, 2022, 105(15): 155406 https://doi.org/10.1103/PhysRevB.105.155406
57	Hu H. , Huang B. , Zhao E. , V. Liu W. . Dynamical singularities of Floquet higher-order topological insulators. Phys. Rev. Lett., 2020, 124(5): 057001 https://doi.org/10.1103/PhysRevLett.124.057001
58	X. Zhang R.C. Yang Z., Theory of anomalous Floquet higher-order topology: Classification, characterization, and bulk-boundary correspondence, arXiv: 2010.07945 (2020)
59	D. Vu D. , X. Zhang R. , C. Yang Z. , Das Sarma S. . Superconductors with anomalous Floquet higher-order topology. Phys. Rev. B, 2021, 104(14): L140502 https://doi.org/10.1103/PhysRevB.104.L140502
60	Peng Y. . Floquet higher-order topological insulators and superconductors with space−time symmetries. Phys. Rev. Res., 2020, 2(1): 013124 https://doi.org/10.1103/PhysRevResearch.2.013124
61	K. Chiu C. , Yao H. , Ryu S. . Classification of topological insulators and superconductors in the presence of reflection symmetry. Phys. Rev. B, 2013, 88(7): 075142 https://doi.org/10.1103/PhysRevB.88.075142
62	Khalaf E. , A. Benalcazar W. , L. Hughes T. , Queiroz R. . Boundary-obstructed topological phases. Phys. Rev. Res., 2021, 3(1): 013239 https://doi.org/10.1103/PhysRevResearch.3.013239
63	Resta R. . Quantum-mechanical position operator in extended systems. Phys. Rev. Lett., 1998, 80(9): 1800 https://doi.org/10.1103/PhysRevLett.80.1800
64	Xiao D. , C. Chang M. , Niu Q. . Berry phase effects on electronic properties. Rev. Mod. Phys., 2010, 82(3): 1959 https://doi.org/10.1103/RevModPhys.82.1959
65	Wintersperger K. , Braun C. , N. Ünal F. , Eckardt A. , D. Liberto M. , Goldman N. , Bloch I. , Aidelsburger M. . Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys., 2020, 16(10): 1058 https://doi.org/10.1038/s41567-020-0949-y
66	N. Ünal F. , Seradjeh B. , Eckardt A. . How to directly measure Floquet topological invariants in optical lattices. Phys. Rev. Lett., 2019, 122(25): 253601 https://doi.org/10.1103/PhysRevLett.122.253601
67	Rodriguez-Vega M. , Kumar A. , Seradjeh B. . Higher-order Floquet topological phases with corner and bulk bound states. Phys. Rev. B, 2019, 100(8): 085138 https://doi.org/10.1103/PhysRevB.100.085138
68	W. Bomantara R. , Zhou L. , Pan J. , Gong J. . Coupled-wire construction of static and Floquet second-order topological insulators. Phys. Rev. B, 2019, 99(4): 045441 https://doi.org/10.1103/PhysRevB.99.045441
69	Fu L. , L. Kane C. . Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett., 2008, 100(9): 096407 https://doi.org/10.1103/PhysRevLett.100.096407
70	Z. Hasan M. , L. Kane C. . Topological insulators. Rev. Mod. Phys., 2010, 82(4): 3045 https://doi.org/10.1103/RevModPhys.82.3045
71	Geier M. , Trifunovic L. , Hoskam M. , W. Brouwer P. . Second-order topological insulators and superconductors with an order-two crystalline symmetry. Phys. Rev. B, 2018, 97(20): 205135 https://doi.org/10.1103/PhysRevB.97.205135
72	Xu S. , Wu C. . Space−time crystal and space−time group. Phys. Rev. Lett., 2018, 120(9): 096401 https://doi.org/10.1103/PhysRevLett.120.096401
73	Peng Y. . Topological space−time crystal. Phys. Rev. Lett., 2022, 128(18): 186802 https://doi.org/10.1103/PhysRevLett.128.186802
74	C. Po H. , Vishwanath A. , Watanabe H. . Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun., 2017, 8(1): 50 https://doi.org/10.1038/s41467-017-00133-2
75	Khalaf E. , C. Po H. , Vishwanath A. , Watanabe H. . Symmetry indicators and anomalous surface states of topological crystalline insulators. Phys. Rev. X, 2018, 8(3): 031070 https://doi.org/10.1103/PhysRevX.8.031070
76	Yu J. , X. Zhang R. , D. Song Z. . Dynamical symmetry indicators for Floquet crystals. Nat. Commun., 2021, 12(1): 5985 https://doi.org/10.1038/s41467-021-26092-3
77	C. Po H. , Watanabe H. , Vishwanath A. . Fragile topology and Wannier obstructions. Phys. Rev. Lett., 2018, 121(12): 126402 https://doi.org/10.1103/PhysRevLett.121.126402
78	V. Else D. , C. Po H. , Watanabe H. . Fragile topological phases in interacting systems. Phys. Rev. B, 2019, 99(12): 125122 https://doi.org/10.1103/PhysRevB.99.125122
79	Ono S. , Yanase Y. , Watanabe H. . Symmetry indicators for topological superconductors. Phys. Rev. Res., 2019, 1(1): 013012 https://doi.org/10.1103/PhysRevResearch.1.013012
80	Nakagawa M. , J. Slager R. , Higashikawa S. , Oka T. . Wannier representation of Floquet topological states. Phys. Rev. B, 2020, 101(7): 075108 https://doi.org/10.1103/PhysRevB.101.075108
81	Yu J. , Ge Y. , Das Sarma S. . Dynamical fragile topology in Floquet crystals. Phys. Rev. B, 2021, 104(18): L180303 https://doi.org/10.1103/PhysRevB.104.L180303
82	X. Zhang R. , C. Yang Z. . Tunable fragile topology in Floquet systems. Phys. Rev. B, 2021, 103(12): L121115 https://doi.org/10.1103/PhysRevB.103.L121115
83	Zhu Y. , Qin T. , Yang X. , Xianlong G. , Liang Z. . Floquet and anomalous Floquet Weyl semimetals. Phys. Rev. Res., 2020, 2(3): 033045 https://doi.org/10.1103/PhysRevResearch.2.033045
84	Zhu W. , Umer M. , Gong J. . Floquet higher-order Weyl and nexus semimetals. Phys. Rev. Res., 2021, 3(3): L032026 https://doi.org/10.1103/PhysRevResearch.3.L032026
85	Zhang R.Hino K.Maeshima N., Floquet-weyl semimetals generated by an optically resonant interband-transition, arXiv: 2201.01578 (2022)
86	Wu H. , Q. Wang B. , H. An J. . Floquet second-order topological insulators in non-Hermitian systems. Phys. Rev. B, 2021, 103(4): L041115 https://doi.org/10.1103/PhysRevB.103.L041115
87	K. Ghosh A.Nag T., Non-hermitian higherorder topological superconductors in two-dimension: statics and dynamics, arXiv: 2205.09915 (2022)
88	Koor K.W. Bomantara R.C. Kwek L., Symmetry protected topological corner modes in a periodically driven interacting spin lattice, arXiv: 2206.06660 (2022)
89	Zhu W. , Xue H. , Gong J. , Chong Y. , Zhang B. . Time-periodic corner states from Floquet higher order topology. Nat. Commun., 2022, 13(1): 11 https://doi.org/10.1038/s41467-021-27552-6
90	Zhu W. , D. Chong Y. , Gong J. . Floquet higher order topological insulator in a periodically driven bipartite lattice. Phys. Rev. B, 2021, 103(4): L041402 https://doi.org/10.1103/PhysRevB.103.L041402
91	Chaudhary S. , Haim A. , Peng Y. , Refael G. . Phonon-induced Floquet topological phases protected by space−time symmetries. Phys. Rev. Res., 2020, 2(4): 043431 https://doi.org/10.1103/PhysRevResearch.2.043431
92	Bauer B. , Pereg-Barnea T. , Karzig T. , T. Rieder M. , Refael G. , Berg E. , Oreg Y. . Topologically protected braiding in a single wire using Floquet Majorana modes. Phys. Rev. B, 2019, 100(4): 041102 https://doi.org/10.1103/PhysRevB.100.041102

[1]	Shengshi Li, Weixiao Ji, Jianping Zhang, Yaping Wang, Changwen Zhang, Shishen Yan. Two-dimensional rectangular bismuth bilayer: A novel dual topological insulator[J]. Front. Phys. , 2023, 18(4): 43301-.
[2]	Xudong Zhu, Yuqian Chen, Zheng Liu, Yulei Han, Zhenhua Qiao. Valley-polarized quantum anomalous Hall effect in van der Waals heterostructures based on monolayer jacutingaite family materials[J]. Front. Phys. , 2023, 18(2): 23302-.
[3]	Lei Shi, Hai-Zhou Lu. Quantum transport in topological semimetals under magnetic fields (III)[J]. Front. Phys. , 2023, 18(2): 21307-.
[4]	Yuan Wang, Fayuan Zhang, Meng Zeng, Hongyi Sun, Zhanyang Hao, Yongqing Cai, Hongtao Rong, Chengcheng Zhang, Cai Liu, Xiaoming Ma, Le Wang, Shu Guo, Junhao Lin, Qihang Liu, Chang Liu, Chaoyu Chen. Intrinsic magnetic topological materials[J]. Front. Phys. , 2023, 18(2): 21304-.
[5]	Liang Kong, Hao Zheng. Categorical computation[J]. Front. Phys. , 2023, 18(2): 21302-.
[6]	Xu-Tao Zeng, Ziyu Chen, Cong Chen, Bin-Bin Liu, Xian-Lei Sheng, Shengyuan A. Yang. Topological hinge modes in Dirac semimetals[J]. Front. Phys. , 2023, 18(1): 13308-.
[7]	Meng Xu, Lei Guo, Lei Chen, Ying Zhang, Shuang-Shuang Li, Weiyao Zhao, Xiaolin Wang, Shuai Dong, Ren-Kui Zheng. Emerging weak antilocalization effect in Ta_0.7Nb_0.3Sb₂semimetal single crystals[J]. Front. Phys. , 2023, 18(1): 13304-.
[8]	Junjie Zeng, Rui Xue, Tao Hou, Yulei Han, Zhenhua Qiao. Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems[J]. Front. Phys. , 2022, 17(6): 63503-.
[9]	Jia-En Yang, Hang Xie. Energy-resolved spin filtering effect and thermoelectric effect in topological-insulator junctions with anisotropic chiral edge states[J]. Front. Phys. , 2022, 17(6): 63504-.
[10]	Jin-Xuan Han, Jin-Lei Wu, Zhong-Hui Yuan, Yan Xia, Yong-Yuan Jiang, Jie Song. Fast topological pumping for the generation of large-scale Greenberger−Horne−Zeilinger states in a superconducting circuit[J]. Front. Phys. , 2022, 17(6): 62504-.
[11]	Yiqing Tian, Yiqi Zhang, Yongdong Li, R. Belić Milivoj. Vector valley Hall edge solitons in the photonic lattice with type-II Dirac cones[J]. Front. Phys. , 2022, 17(5): 53503-.
[12]	Kai-Tong Wang, Fuming Xu, Bin Wang, Yunjin Yu, Yadong Wei. Transport features of topological corner states in honeycomb lattice with multihollow structure[J]. Front. Phys. , 2022, 17(4): 43501-.
[13]	Annan Fan, Shi-Dong Liang. Complex energy plane and topological invariant in non-Hermitian systems[J]. Front. Phys. , 2022, 17(3): 33501-.
[14]	Jiahao Fan, Huaqing Huang. Topological states in quasicrystals[J]. Front. Phys. , 2022, 17(1): 13203-.
[15]	Chengyong Zhong. Predication of topological states in the allotropes of group-IV elements[J]. Front. Phys. , 2021, 16(6): 63503-.

Viewed

Full text

Abstract

Cited

Shared

Discussed