Citation: | LIU Hui, WANG Hao-nan, XIE Bo-yang, CHENG Hua, TIAN Jian-guo, CHEN Shu-qi. Progress of two-dimensional photonic topological insulators[J].Chinese Optics, 2021, 14(4): 935-954.doi:10.37188/CO.2021-0076 |
[1] |
CHEN L, RONG Y W. Digital topological method for computing genus and the Betti numbers[J].
Topology and its Applications, 2010, 157(12): 1931-1936.
doi:10.1016/j.topol.2010.04.006
|
[2] |
KLITZING K V, DORDA G, PEPPER M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance[J].
Physical Review Letters, 1980, 45(6): 494-497.
doi:10.1103/PhysRevLett.45.494
|
[3] |
DEN NIJS M. Quantized Hall conductance in a two dimensional periodic potential[J].
Physica A:
Statistical Mechanics and its Applications, 1984, 124(1-3): 199-210.
doi:10.1016/0378-4371(84)90239-5
|
[4] |
HALDANE F D M, RAGHU S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry[J].
Physical Review Letters, 2008, 100(1): 013904.
doi:10.1103/PhysRevLett.100.013904
|
[5] |
RAGHU S, HALDANE F D M. Analogs of quantum-Hall-effect edge states in photonic crystals[J].
Physical Review A, 2008, 78(3): 033834.
doi:10.1103/PhysRevA.78.033834
|
[6] |
WANG ZH, CHONG Y D, JOANNOPOULOS J D,
et al. Reflection-free one-way edge modes in a gyromagnetic photonic crystal[J].
Physical Review Letters, 2008, 100(1): 013905.
doi:10.1103/PhysRevLett.100.013905
|
[7] |
WANG ZH, CHONG Y D, JOANNOPOULOS J D,
et al. Observation of unidirectional backscattering-immune topological electromagnetic states[J].
Nature, 2009, 461(7265): 772-775.
doi:10.1038/nature08293
|
[8] |
LIU SH Y, LU W L, LIN ZH F,
et al. Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials[J].
Applied Physics Letters, 2010, 97(20): 201113.
doi:10.1063/1.3520141
|
[9] |
HE CH, CHEN X L, LU M H,
et al. Left-handed and right-handed one-way edge modes in a gyromagnetic photonic crystal[J].
Journal of Applied Physics, 2010, 107(12): 123117.
doi:10.1063/1.3374470
|
[10] |
QIU W J, WANG ZH, SOLJAČIĆ M. Broadband circulators based on directional coupling of one-way waveguides[J].
Optics Express, 2011, 19(22): 22248-22257.
doi:10.1364/OE.19.022248
|
[11] |
WANG ZH Y, SHEN L F, YU Z H,
et al. Highly efficient photonic-crystal splitters based on one-way waveguiding[J].
Journal of the Optical Society of America B, 2013, 30(1): 173-176.
doi:10.1364/JOSAB.30.000173
|
[12] |
BAHARI B, TELLEZ-LIMON R, KANTÉ B. Topological terahertz circuits using semiconductors[J].
Applied Physics Letters, 2016, 109(14): 143501.
doi:10.1063/1.4963789
|
[13] |
WU Y, LI CH, HU X Y,
et al. Applications of topological photonics in integrated photonic devices[J].
Advanced Optical Materials, 2017, 5(18): 1700357.
doi:10.1002/adom.201700357
|
[14] |
NI X, HE CH, SUN X CH,
et al. Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow[J].
New Journal of Physics, 2015, 17(5): 053016.
doi:10.1088/1367-2630/17/5/053016
|
[15] |
DING Y J, PENG Y G, ZHU Y F,
et al. Experimental demonstration of acoustic chern insulators[J].
Physical Review Letters, 2019, 122(1): 014302.
doi:10.1103/PhysRevLett.122.014302
|
[16] |
JO G B, GUZMAN J, THOMAS C K,
et al. Ultracold atoms in a tunable optical kagome lattice[J].
Physical Review Letters, 2012, 108(4): 045305.
doi:10.1103/PhysRevLett.108.045305
|
[17] |
SOLTAN-PANAHI P, STRUCK J, HAUKE P,
et al. Multi-component quantum gases in spin-dependent hexagonal lattices[J].
Nature Physics, 2011, 7(5): 434-440.
doi:10.1038/nphys1916
|
[18] |
NAKAJIMA S, TOMITA T, TAIE S,
et al. Topological thouless pumping of ultracold fermions[J].
Nature Physics, 2016, 12(4): 296-300.
doi:10.1038/nphys3622
|
[19] |
HUBER S D. Topological mechanics[J].
Nature Physics, 2016, 12(7): 621-623.
doi:10.1038/nphys3801
|
[20] |
WANG P, LU L, BERTOLDI K. Topological phononic crystals with one-way elastic edge waves[J].
Physical Review Letters, 2015, 115(10): 104302.
doi:10.1103/PhysRevLett.115.104302
|
[21] |
SÜSSTRUNK R, HUBER S D. Observation of phononic helical edge states in a mechanical topological insulator[J].
Science, 2015, 349(6243): 47-50.
doi:10.1126/science.aab0239
|
[22] |
KANE C L, MELE E J.
Z
2topological order and the quantum spin hall effect[J].
Physical Review Letters, 2005, 95(14): 146802.
doi:10.1103/PhysRevLett.95.146802
|
[23] |
KANE C L, MELE E J. Quantum Spin hall effect in graphene[J].
Physical Review Letters, 2005, 95(22): 226801.
doi:10.1103/PhysRevLett.95.226801
|
[24] |
BERNEVIG B A, HUGHES T L, ZHANG SH CH. Quantum spin hall effect and topological phase transition in HgTe quantum wells[J].
Science, 2006, 314(5806): 1757-1761.
doi:10.1126/science.1133734
|
[25] |
KÖNIG M, WIEDMANN S, BRÜNE C,
et al. Quantum Spin Hall Insulator State in HgTe Quantum Wells[J].
Science, 2007, 318(5851): 766-770.
doi:10.1126/science.1148047
|
[26] |
HAFEZI M, DEMLER E A, LUKIN M D,
et al. Robust optical delay lines with topological protection[J].
Nature Physics, 2011, 7(11): 907-912.
doi:10.1038/nphys2063
|
[27] |
HAFEZI M, MITTAL S, FAN J,
et al. Imaging topological edge states in silicon photonics[J].
Nature Photonics, 2013, 7(12): 1001-1005.
doi:10.1038/nphoton.2013.274
|
[28] |
WU L H, HU X. Scheme for achieving a topological photonic crystal by using dielectric material[J].
Physical Review Letters, 2015, 114(22): 223901.
doi:10.1103/PhysRevLett.114.223901
|
[29] |
YANG Y T, XU Y F, XU T,
et al. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials[J].
Physical Review Letters, 2018, 120(21): 217401.
doi:10.1103/PhysRevLett.120.217401
|
[30] |
ZHU X, WANG H X, XU CH Q,
et al. Topological transitions in continuously deformed photonic crystals[J].
Physical Review B, 2018, 97(8): 085148.
doi:10.1103/PhysRevB.97.085148
|
[31] |
JIA N Y, OWENS C, SOMMER A,
et al. Time- and site-resolved dynamics in a topological circuit[J].
Physical Review X, 2015, 5(2): 021031.
doi:10.1103/PhysRevX.5.021031
|
[32] |
KITAGAWA T, BERG E, RUDNER M,
et al. Topological characterization of periodically driven quantum systems[J].
Physical Review B, 2010, 82(23): 235114.
doi:10.1103/PhysRevB.82.235114
|
[33] |
LINDNER N H, REFAEL G, GALITSKI V. Floquet topological insulator in semiconductor quantum wells[J].
Nature Physics, 2011, 7(6): 490-495.
doi:10.1038/nphys1926
|
[34] |
RECHTSMAN M C, ZEUNER J M, PLOTNIK Y,
et al. Photonic Floquet topological insulators[J].
Nature, 2013, 496(7444): 196-200.
doi:10.1038/nature12066
|
[35] |
RUDNER M S, LINDNER N H, BERG E,
et al. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems[J].
Physical Review X, 2013, 3(3): 031005.
doi:10.1103/PhysRevX.3.031005
|
[36] |
NATHAN F, RUDNER M S. Topological singularities and the general classification of Floquet-Bloch systems[J].
New Journal of Physics, 2015, 17(12): 125014.
doi:10.1088/1367-2630/17/12/125014
|
[37] |
MUKHERJEE S, SPRACKLEN A, VALIENTE M,
et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice[J].
Nature Communications, 2017, 8(1): 3918.
|
[38] |
MACZEWSKY L J, ZEUNER J M, NOLTE S,
et al. Observation of photonic anomalous Floquet topological insulators[J].
Nature Communications, 2017, 8(1): 13756.
doi:10.1038/ncomms13756
|
[39] |
LUMER Y, PLOTNIK Y, RECHTSMAN M C,
et al. Self-localized states in photonic topological insulators[J].
Physical Review Letters, 2013, 111(24): 243905.
doi:10.1103/PhysRevLett.111.243905
|
[40] |
LEYKAM D, RECHTSMAN M C, CHONG Y D. Anomalous topological phases and unpaired dirac cones in photonic floquet topological insulators[J].
Physical Review Letters, 2016, 117(1): 013902.
doi:10.1103/PhysRevLett.117.013902
|
[41] |
KRAUS Y E, LAHINI Y, RINGEL Z,
et al. Topological states and adiabatic pumping in quasicrystals[J].
Physical Review Letters, 2012, 109(10): 106402.
doi:10.1103/PhysRevLett.109.106402
|
[42] |
ZILBERBERG O, HUANG SH, GUGLIELMON J,
et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics[J].
Nature, 2018, 553(7686): 59-62.
doi:10.1038/nature25011
|
[43] |
VERBIN M, ZILBERBERG O, LAHINI Y,
et al. Topological pumping over a photonic Fibonacci quasicrystal[J].
Physical Review B, 2015, 91(6): 064201.
doi:10.1103/PhysRevB.91.064201
|
[44] |
KE Y G, QIN X ZH, MEI F,
et al. Topological phase transitions and thouless pumping of light in photonic waveguide arrays[J].
Laser&
Photonics Reviews, 2016, 10(6): 995-1001.
|
[45] |
BENALCAZAR W A, BERNEVIG B A, HUGHES T L. Quantized electric multipole insulators[J].
Science, 2017, 357(6346): 61-66.
doi:10.1126/science.aah6442
|
[46] |
BENALCAZAR W A, BERNEVIG B A, HUGHES T L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators[J].
Physical Review B, 2017, 96(24): 245115.
doi:10.1103/PhysRevB.96.245115
|
[47] |
PETERSON C W, BENALCAZAR W A, HUGHES T L,
et al. A quantized microwave quadrupole insulator with topologically protected corner states[J].
Nature, 2018, 555(7696): 346-350.
doi:10.1038/nature25777
|
[48] |
LI M Y, ZHIRIHIN D, GORLACH M,
et al. Higher-order topological states in photonic kagome crystals with long-range interactions[J].
Nature Photonics, 2020, 14(2): 89-94.
doi:10.1038/s41566-019-0561-9
|
[49] |
EZAWA M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices[J].
Physical Review Letters, 2018, 120(2): 026801.
doi:10.1103/PhysRevLett.120.026801
|
[50] |
BENALCAZAR W A, LI T H, HUGHES T L. Quantization of fractional corner charge in
Cn-symmetric higher-order topological crystalline insulators[J].
Physical Review B, 2019, 99(24): 245151.
doi:10.1103/PhysRevB.99.245151
|
[51] |
ZHANG X J, XIAO M, CHENG Y,
et al. Topological sound[J].
Communications Physics, 2018, 1(1): 97.
doi:10.1038/s42005-018-0094-4
|
[52] |
YANG ZH J, GAO F, SHI X H,
et al. Topological acoustics[J].
Physical Review Letters, 2015, 114(11): 114301.
doi:10.1103/PhysRevLett.114.114301
|
[53] |
HE CH, NI X, GE H,
et al. Acoustic topological insulator and robust one-way sound transport[J].
Nature Physics, 2016, 12(12): 1124-1129.
doi:10.1038/nphys3867
|
[54] |
XIAO M, CHEN W J, HE W Y,
et al. Synthetic gauge flux and Weyl points in acoustic systems[J].
Nature Physics, 2015, 11(11): 920-924.
doi:10.1038/nphys3458
|
[55] |
XIE B Y, LIU H, CHENG H,
et al. Experimental realization of type-II weyl points and fermi arcs in phononic crystal[J].
Physical Review Letters, 2019, 122(10): 104302.
doi:10.1103/PhysRevLett.122.104302
|
[56] |
XIE B Y, LIU H, CHENG H,
et al. Acoustic topological transport and refraction in a Kekulé Lattice[J].
Physical Review Applied, 2019, 11(4): 044086.
doi:10.1103/PhysRevApplied.11.044086
|
[57] |
ROCKLIN D Z, ZHOU SH N, SUN K,
et al. Transformable topological mechanical metamaterials[J].
Nature Communications, 2017, 8(1): 14201.
doi:10.1038/ncomms14201
|
[58] |
LU L, GAO H ZH, WANG ZH. Topological one-way fiber of second Chern number[J].
Nature Communications, 2018, 9(1): 5384.
doi:10.1038/s41467-018-07817-3
|
[59] |
YANG Y H, GAO ZH, XUE H R,
et al. Realization of a three-dimensional photonic topological insulator[J].
Nature, 2019, 565(7741): 622-626.
doi:10.1038/s41586-018-0829-0
|
[60] |
SLOBOZHANYUK A, MOUSAVI S H, NI X,
et al. Three-dimensional all-dielectric photonic topological insulator[J].
Nature Photonics, 2017, 11(2): 130-136.
doi:10.1038/nphoton.2016.253
|
[61] |
LU L, FANG CH, FU L,
et al. Symmetry-protected topological photonic crystal in three dimensions[J].
Nature Physics, 2016, 12(4): 337-340.
doi:10.1038/nphys3611
|
[62] |
YOUNG S M, ZAHEER S, TEO J C Y,
et al. Dirac Semimetal in Three Dimensions[J].
Physical Review Letters, 2012, 108(14): 140405.
doi:10.1103/PhysRevLett.108.140405
|
[63] |
LU L, FU L, JOANNOPOULOS J D,
et al. Weyl points and line nodes in gyroid photonic crystals[J].
Nature Photonics, 2013, 7(4): 294-299.
doi:10.1038/nphoton.2013.42
|
[64] |
GUO Q H, YOU O B, YANG B,
et al. Observation of three-dimensional photonic dirac points and spin-polarized surface arcs[J].
Physical Review Letters, 2019, 122(20): 203903.
doi:10.1103/PhysRevLett.122.203903
|
[65] |
CHENG H, GAO W L, BI Y G,
et al. Vortical reflection and spiraling fermi arcs with weyl metamaterials[J].
Physical Review Letters, 2020, 125(9): 093904.
doi:10.1103/PhysRevLett.125.093904
|
[66] |
OZAWA T, PRICE H M, GOLDMAN N,
et al. Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics[J].
Physical Review A, 2016, 93(4): 043827.
doi:10.1103/PhysRevA.93.043827
|
[67] |
LIN Q, SUN X Q, XIAO M,
et al. A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension[J].
Science Advances, 2018, 4(10): eaat2774.
doi:10.1126/sciadv.aat2774
|
[68] |
YUAN L Q, LIN Q, XIAO M,
et al. Synthetic dimension in photonics[J].
Optica, 2018, 5(11): 1396-1405.
doi:10.1364/OPTICA.5.001396
|
[69] |
LUSTIG E, WEIMANN S, PLOTNIK Y,
et al. Photonic topological insulator in synthetic dimensions[J].
Nature, 2019, 567(7748): 356-360.
doi:10.1038/s41586-019-0943-7
|
[70] |
CHEN Y, ZHANG Y L, SHEN ZH,
et al. Synthetic gauge fields in a single optomechanical resonator[J].
Physical Review Letters, 2021, 126(12): 123603.
doi:10.1103/PhysRevLett.126.123603
|
[71] |
LI G ZH, ZHENG Y L, DUTT A,
et al. Dynamic band structure measurement in the synthetic space[J].
Science Advances, 2021, 7(2): eabe4335.
doi:10.1126/sciadv.abe4335
|
[72] |
WANG Q, XIAO M, LIU H,
et al. Optical interface states protected by synthetic weyl points[J].
Physical Review X, 2017, 7(3): 031032.
doi:10.1103/PhysRevX.7.031032
|
[73] |
LIN Q, XIAO M, YUAN L Q,
et al. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension[J].
Nature Communications, 2016, 7(1): 13731.
doi:10.1038/ncomms13731
|
[74] |
SUN B Y, LUO X W, GONG M,
et al. Weyl semimetal phases and implementation in degenerate optical cavities[J].
Physical Review A, 2017, 96(1): 013857.
doi:10.1103/PhysRevA.96.013857
|
[75] |
ZHANG Y, ZHU Y Y. Generation of Weyl points in coupled optical microdisk-resonator arrays via external modulation[J].
Physical Review A, 2017, 96(1): 013811.
doi:10.1103/PhysRevA.96.013811
|
[76] |
LIU ZH ZH, ZHANG Q, QIN F F,
et al. Surface states ensured by a synthetic Weyl point in one-dimensional plasmonic dielectric crystals with broken inversion symmetry[J].
Physical Review B, 2019, 99(8): 085441.
doi:10.1103/PhysRevB.99.085441
|
[77] |
LEYKAM D, CHONG Y D. Edge solitons in nonlinear-photonic topological insulators[J].
Physical Review Letters, 2016, 117(14): 143901.
doi:10.1103/PhysRevLett.117.143901
|
[78] |
PODDUBNY A N, SMIRNOVA D A. Ring Dirac solitons in nonlinear topological systems[J].
Physical Review A, 2018, 98(1): 013827.
doi:10.1103/PhysRevA.98.013827
|
[79] |
HADAD Y, KHANIKAEV A B, ALÙ A. Self-induced topological transitions and edge states supported by nonlinear staggered potentials[J].
Physical Review B, 2016, 93(15): 155112.
doi:10.1103/PhysRevB.93.155112
|
[80] |
GREENTREE A D, TAHAN C, COLE J H,
et al. Quantum phase transitions of light[J].
Nature Physics, 2006, 2(12): 856-861.
doi:10.1038/nphys466
|
[81] |
HARTMANN M J, BRANDÃO F G S L, PLENIO M B. Strongly interacting polaritons in coupled arrays of cavities[J].
Nature Physics, 2006, 2(12): 849-855.
doi:10.1038/nphys462
|
[82] |
ANGELAKIS D G, SANTOS M F, BOSE S. Photon-blockade-induced Mott transitions and
XYspin models in coupled cavity arrays[J].
Physical Review A, 2007, 76(3): 031805(R).
doi:10.1103/PhysRevA.76.031805
|
[83] |
ZHAO H, QIAO X D, WU T W,
et al. Non-Hermitian topological light steering[J].
Science, 2019, 365(6458): 1163-1166.
doi:10.1126/science.aay1064
|
[84] |
ZHEN B, HSU C W, IGARASHI Y,
et al. Spawning rings of exceptional points out of Dirac cones[J].
Nature, 2015, 525(7569): 354-358.
doi:10.1038/nature14889
|
[85] |
LEE T E. Anomalous edge state in a non-hermitian lattice[J].
Physical Review Letters, 2016, 116(13): 133903.
doi:10.1103/PhysRevLett.116.133903
|
[86] |
SEPKHANOV R A, NILSSON J, BEENAKKER C W J. Proposed method for detection of the pseudospin- Berry phase in a photonic crystal with a Dirac spectrum[J].
Physical Review B, 2008, 78(4): 045122.
doi:10.1103/PhysRevB.78.045122
|
[87] |
XIAO D, CHANG M C, NIU Q. Berry phase effects on electronic properties[J].
Reviews of Modern Physics, 2010, 82(3): 1959-2007.
doi:10.1103/RevModPhys.82.1959
|
[88] |
THOULESS D J, KOHMOTO M, NIGHTINGALE M P,
et al. Quantized hall conductance in a two-dimensional periodic potential[J].
Physical Review Letters, 1982, 49(6): 405-408.
doi:10.1103/PhysRevLett.49.405
|
[89] |
SKIRLO S A, LU L, SOLJAČIĆ M. Multimode one-way waveguides of large chern numbers[J].
Physical Review Letters, 2014, 113(11): 113904.
doi:10.1103/PhysRevLett.113.113904
|
[90] |
SKIRLO S A, LU L, IGARASHI Y,
et al. Experimental observation of large chern numbers in photonic crystals[J].
Physical Review Letters, 2015, 115(25): 253901.
doi:10.1103/PhysRevLett.115.253901
|
[91] |
YANG Y, POO Y, WU R X,
et al. Experimental demonstration of one-way slow wave in waveguide involving gyromagnetic photonic crystals[J].
Applied Physics Letters, 2013, 102(23): 231113.
doi:10.1063/1.4809956
|
[92] |
FU J X, LIU R J, LI Z Y. Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces[J].
Applied Physics Letters, 2010, 97(4): 041112.
doi:10.1063/1.3470873
|
[93] |
WANG D L, QIU CH W, RAKICH P T,
et al.. Guide-wave photonic pulling force using one-way photonic chiral edge states[C].
CLEO: QELS_Fundamental Science 2015, OSA, 2015: FM2D. 7.
|
[94] |
RYCERZ A, TWORZYDŁO J, BEENAKKER C W J. Valley filter and valley valve in graphene[J].
Nature Physics, 2007, 3(3): 172-175.
doi:10.1038/nphys547
|
[95] |
JU L, SHI ZH W, NAIR N,
et al. Topological valley transport at bilayer graphene domain walls[J].
Nature, 2015, 520(7549): 650-655.
doi:10.1038/nature14364
|
[96] |
DONG J W, CHEN X D, ZHU H Y,
et al. Valley photonic crystals for control of spin and topology[J].
Nature Materials, 2017, 16(3): 298-302.
doi:10.1038/nmat4807
|
[97] |
WU X X, MENG Y, TIAN J X,
et al. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals[J].
Nature Communications, 2017, 8(1): 1304.
doi:10.1038/s41467-017-01515-2
|
[98] |
NOH J, HUANG SH, CHEN K P,
et al. Observation of photonic topological valley hall edge states[J].
Physical Review Letters, 2018, 120(6): 063902.
doi:10.1103/PhysRevLett.120.063902
|
[99] |
CHEN Q L, ZHANG L, HE M J,
et al. Valley-hall photonic topological insulators with dual-band kink states[J].
Advanced Optical Materials, 2019, 7(15): 1900036.
doi:10.1002/adom.201900036
|
[100] |
HE X T, LIANG E T, YUAN J J,
et al. A silicon-on-insulator slab for topological valley transport[J].
Nature Communications, 2019, 10(1): 872.
doi:10.1038/s41467-019-08881-z
|
[101] |
LU J Y, QIU CH Y, YE L P,
et al. Observation of topological valley transport of sound in sonic crystals[J].
Nature Physics, 2017, 13(4): 369-374.
doi:10.1038/nphys3999
|
[102] |
LU J Y, QIU CH Y, DENG W Y,
et al. Valley topological phases in bilayer sonic crystals[J].
Physical Review Letters, 2018, 120(11): 116802.
doi:10.1103/PhysRevLett.120.116802
|
[103] |
ZHANG X J, LIU L, LU M H,
et al. Valley-selective topological corner states in sonic crystals[J].
Physical Review Letters, 2021, 126(15): 156401.
doi:10.1103/PhysRevLett.126.156401
|
[104] |
CHEN W J, JIANG SH J, CHEN X D,
et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide[J].
Nature Communications, 2014, 5(1): 5782.
doi:10.1038/ncomms6782
|
[105] |
LAI K, MA T, BO X,
et al. Experimental realization of a reflections-free compact delay line based on a photonic topological insulator[J].
Scientific Reports, 2016, 6(1): 28453.
doi:10.1038/srep28453
|
[106] |
XIAO B, LAI K, YU Y,
et al. Exciting reflectionless unidirectional edge modes in a reciprocal photonic topological insulator medium[J].
Physical Review B, 2016, 94(19): 195427.
doi:10.1103/PhysRevB.94.195427
|
[107] |
CHENG X J, JOUVAUD C, NI X,
et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator[J].
Nature Materials, 2016, 15(5): 542-548.
doi:10.1038/nmat4573
|
[108] |
MA T, KHANIKAEV A B, MOUSAVI S H,
et al. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides[J].
Physical Review Letters, 2015, 114(12): 127401.
doi:10.1103/PhysRevLett.114.127401
|
[109] |
YVES S, FLEURY R, LEMOULT F,
et al. Topological acoustic polaritons: Robust sound manipulation at the subwavelength scale[J].
New Journal of Physics, 2017, 19(7): 075003.
doi:10.1088/1367-2630/aa66f8
|
[110] |
GORLACH M A, NI X, SMIRNOVA D A,
et al. Far-field probing of leaky topological states in all-dielectric metasurfaces[J].
Nature Communications, 2018, 9(1): 909.
doi:10.1038/s41467-018-03330-9
|
[111] |
BARIK S, KARASAHIN A, FLOWER C,
et al. A topological quantum optics interface[J].
Science, 2018, 359(6376): 666-668.
doi:10.1126/science.aaq0327
|
[112] |
IMHOF S, BERGER C, BAYER F,
et al. Topolectrical-circuit realization of topological corner modes[J].
Nature Physics, 2018, 14(9): 925-929.
doi:10.1038/s41567-018-0246-1
|
[113] |
LEE C H, IMHOF S, BERGER C,
et al. Topolectrical circuits[J].
Communications Physics, 2018, 1(1): 39.
doi:10.1038/s42005-018-0035-2
|
[114] |
LU Y H, JIA N Y, SU L,
et al. Probing the Berry curvature and Fermi arcs of a Weyl circuit[J].
Physical Review B, 2019, 99(2): 020302(R).
doi:10.1103/PhysRevB.99.020302
|
[115] |
WALLRAFF A, SCHUSTER D I, BLAIS A,
et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[J].
Nature, 2004, 431(7005): 162-167.
doi:10.1038/nature02851
|
[116] |
SALA V G, SOLNYSHKOV D D, CARUSOTTO I,
et al. Spin-orbit coupling for photons and polaritons in microstructures[J].
Physical Review X, 2015, 5(1): 011034.
doi:10.1103/PhysRevX.5.011034
|
[117] |
CAYSSOL J, DÓRA B, SIMON F,
et al. Floquet topological insulators[J].
Physica Status Solidi, 2013, 7(1-2): 101-108.
doi:10.1002/pssr.201206451
|
[118] |
FANG K J, YU Z F, FAN SH H. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation[J].
Nature Photonics, 2012, 6(11): 782-787.
doi:10.1038/nphoton.2012.236
|
[119] |
OZAWA T, PRICE H M, AMO A,
et al. Topological photonics[J].
Reviews of Modern Physics, 2019, 91(1): 015006.
doi:10.1103/RevModPhys.91.015006
|
[120] |
PASEK M, CHONG Y D. Network models of photonic Floquet topological insulators[J].
Physical Review B, 2014, 89(7): 075113.
doi:10.1103/PhysRevB.89.075113
|
[121] |
GAO F, GAO ZH, SHI X H,
et al. Probing topological protection using a designer surface plasmon structure[J].
Nature Communications, 2016, 7(48): 11619.
|
[122] |
YANG ZH J, LUSTIG E, LUMER Y,
et al. Photonic Floquet topological insulators in a fractal lattice[J].
Light:
Science&
Applications, 2020, 9(1): 128.
|
[123] |
LI J, CHU R L, JAIN J K,
et al. Topological anderson insulator[J].
Physical Review Letters, 2009, 102(13): 136806.
doi:10.1103/PhysRevLett.102.136806
|
[124] |
GROTH C W, WIMMER M, AKHMEROV A R,
et al. Theory of the topological anderson insulator[J].
Physical Review Letters, 2009, 103(19): 196805.
doi:10.1103/PhysRevLett.103.196805
|
[125] |
TITUM P, LINDNER N H, RECHTSMAN M C,
et al. Disorder-induced Floquet topological insulators[J].
Physical Review Letters, 2015, 114(5): 056801.
doi:10.1103/PhysRevLett.114.056801
|
[126] |
TITUM P, LINDNER N H, REFAEL G. Disorder-induced transitions in resonantly driven Floquet topological insulators[J].
Physical Review B, 2017, 96(5): 054207.
doi:10.1103/PhysRevB.96.054207
|
[127] |
STÜTZER S, PLOTNIK Y, LUMER Y,
et al. Photonic topological Anderson insulators[J].
Nature, 2018, 560(7719): 461-465.
doi:10.1038/s41586-018-0418-2
|
[128] |
LIU G G, YANG Y H, REN X,
et al. Topological anderson insulator in disordered photonic crystals[J].
Physical Review Letters, 2020, 125(13): 133603.
doi:10.1103/PhysRevLett.125.133603
|
[129] |
HUANG H Q, LIU F. Theory of spin Bott index for quantum spin Hall states in nonperiodic systems[J].
Physical Review B, 2018, 98(12): 125130.
doi:10.1103/PhysRevB.98.125130
|
[130] |
HUANG H Q, LIU F. Quantum Spin hall effect and spin bott index in a quasicrystal lattice[J].
Physical Review Letters, 2018, 121(12): 126401.
doi:10.1103/PhysRevLett.121.126401
|
[131] |
SHINDOU R, MURAKAMI S. Effects of disorder in three-dimensional
Z
2quantum spin Hall systems[J].
Physical Review B, 2009, 79(4): 045321.
doi:10.1103/PhysRevB.79.045321
|
[132] |
HUANG X Q, LU J Y, YAN Z B,
et al.. Acoustic corner states in topological insulators with built-in Zeeman-like fields[J]. arXiv: 2008.06272, 2020.
|
[133] |
LORING T A, HASTINGS M B. Disordered topological insulators via C*-algebras[J].
EPL(
Europhysics Letters)
|
[134] |
HASTINGS M B, LORING T A. Topological insulators and C*-algebras: theory and numerical practice[J].
Annals of Physics, 2011, 326(7): 1699-1759.
doi:10.1016/j.aop.2010.12.013
|
[135] |
LORING T A. A guide to the bott index and localizer index[J]. arXiv: 1907.11791, 2019.
|
[136] |
TONIOLO D. On the equivalence of the Bott index and the Chern number on a torus, and the quantization of the Hall conductivity with a real space Kubo formula[J]. arXiv: 1708.05912, 2017.
|
[137] |
MEIER E J, AN F A, DAUPHIN A,
et al. Observation of the topological Anderson insulator in disordered atomic wires[J].
Science, 2018, 362(6417): 929-933.
doi:10.1126/science.aat3406
|
[138] |
GUO H M, ROSENBERG G, REFAEL G,
et al. Topological Anderson insulator in three dimensions[J].
Physical Review Letters, 2010, 105(21): 216601.
doi:10.1103/PhysRevLett.105.216601
|
[139] |
SMIRNOVA D, LEYKAM D, CHONG Y D,
et al. Nonlinear topological photonics[J].
Applied Physics Reviews, 2020, 7(2): 021306.
doi:10.1063/1.5142397
|
[140] |
DU Z Z, WANG C M, LI SH,
et al. Disorder-induced nonlinear Hall effect with time-reversal symmetry[J].
Nature Communications, 2019, 10(1): 3047.
doi:10.1038/s41467-019-10941-3
|
[141] |
ZENG Y Q, CHATTOPADHYAY U, ZHU B F,
et al. Electrically pumped topological laser with valley edge modes[J].
Nature, 2020, 578(7794): 246-250.
doi:10.1038/s41586-020-1981-x
|
[142] |
BANDRES M A, WITTEK S, HARARI G,
et al. Topological insulator laser: Experiments[J].
Science, 2018, 359(6381): eaar4005.
doi:10.1126/science.aar4005
|
[143] |
TANG L ZH, ZHANG L F, ZHANG G Q,
et al. Topological Anderson insulators in two-dimensional non-Hermitian disordered systems[J].
Physical Review A, 2020, 101(6): 063612.
doi:10.1103/PhysRevA.101.063612
|
[144] |
LUO X W, ZHANG CH W. Non-hermitian disorder-induced topological insulators[J]. arXiv: 1912.10652, 2019.
|
[145] |
SILVEIRINHA M G. Proof of the bulk-edge correspondence through a link between topological photonics and fluctuation-electrodynamics[J].
Physical Review X, 2019, 9(1): 011037.
doi:10.1103/PhysRevX.9.011037
|
[146] |
LU L, JOANNOPOULOS J D, SOLJAČIĆ M. Topological photonics[J].
Nature Photonics, 2014, 8(11): 821-829.
doi:10.1038/nphoton.2014.248
|
[147] |
PARAMESWARAN S A, WAN Y. Topological insulators turn a corner[J].
Physics, 2017, 10: 132.
doi:10.1103/Physics.10.132
|
[148] |
SCHINDLER F, COOK A M, VERGNIORY M G,
et al. Higher-order topological insulators[J].
Science Advances, 2018, 4(6): eaat0346.
doi:10.1126/sciadv.aat0346
|
[149] |
SERRA-GARCIA M, PERI V, SÜSSTRUNK R,
et al. Observation of a phononic quadrupole topological insulator[J].
Nature, 2018, 555(7696): 342-345.
doi:10.1038/nature25156
|
[150] |
MITTAL S, ORRE V V, ZHU G Y,
et al. Photonic quadrupole topological phases[J].
Nature Photonics, 2019, 13(10): 692-696.
doi:10.1038/s41566-019-0452-0
|
[151] |
HE L, ADDISON Z, MELE E J,
et al. Quadrupole topological photonic crystals[J].
Nature Communications, 2020, 11(1): 3119.
doi:10.1038/s41467-020-16916-z
|
[152] |
ZHOU X X, LIN Z K, LU W X,
et al. Twisted quadrupole topological photonic crystals[J].
Laser&
Photonics Reviews, 2020, 14(8): 2000010.
|
[153] |
SU W P, SCHRIEFFER J R, HEEGER A J. Solitons in polyacetylene[J].
Physical Review Letters, 1979, 42(25): 1698-1701.
doi:10.1103/PhysRevLett.42.1698
|
[154] |
XUE H R, YANG Y H, GAO F,
et al. Acoustic higher-order topological insulator on a kagome lattice[J].
Nature Materials, 2019, 18(2): 108-112.
doi:10.1038/s41563-018-0251-x
|
[155] |
LIU F, WAKABAYASHI K. Novel topological phase with a zero berry curvature[J].
Physical Review Letters, 2017, 118(7): 076803.
doi:10.1103/PhysRevLett.118.076803
|
[156] |
XIE B Y, WANG H F, WANG H X,
et al. Second-order photonic topological insulator with corner states[J].
Physical Review B, 2018, 98(20): 205147.
doi:10.1103/PhysRevB.98.205147
|
[157] |
CHEN X D, DENG W M, SHI F L,
et al. Direct observation of corner states in second-order topological photonic crystal slabs[J].
Physical Review Letters, 2019, 122(23): 233902.
doi:10.1103/PhysRevLett.122.233902
|
[158] |
XIE B Y, SU G X, WANG H F,
et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals[J].
Physical Review Letters, 2019, 122(23): 233903.
doi:10.1103/PhysRevLett.122.233903
|
[159] |
KIM M, RHO J. Topological edge and corner states in a two-dimensional photonic Su-Schrieffer-Heeger lattice[J].
Nanophotonics, 2020, 9(10): 3227-3234.
doi:10.1515/nanoph-2019-0451
|
[160] |
OTA Y, LIU F, KATSUMI R,
et al. Photonic crystal nanocavity based on a topological corner state[J].
Optica, 2019, 6(6): 786-789.
doi:10.1364/OPTICA.6.000786
|
[161] |
NOH J, BENALCAZAR W A, HUANG SH,
et al. Topological protection of photonic mid-gap defect modes[J].
Nature Photonics, 2018, 12(7): 408-415.
doi:10.1038/s41566-018-0179-3
|
[162] |
EL HASSAN A, KUNST F K, MORITZ A,
et al. Corner states of light in photonic waveguides[J].
Nature Photonics, 2019, 13(10): 697-700.
doi:10.1038/s41566-019-0519-y
|
[163] |
XIE X, ZHANG W X, HE X W,
et al. Cavity quantum electrodynamics with second-order topological corner state[J].
Laser&
Photonics Reviews, 2020, 14(8): 1900425.
|
[164] |
ZHANG W X, XIE X, HAO H M,
et al. Low-threshold topological nanolasers based on the second-order corner state[J].
Light:
Science&
Applications, 2020, 9(1): 109.
|
[165] |
ZHANG L, YANG Y H, LIN Z K,
et al. Higher-order topological states in surface-wave photonic crystals[J].
Advsnced Science, 2020, 7(6): 1902724.
|
[166] |
LUO X W, ZHANG C W. Higher-order topological corner states induced by gain and loss[J].
Physical Review Letters, 2019, 123(7): 73601.
doi:10.1103/PhysRevLett.123.073601
|
[167] |
LIU T, ZHANG Y R, AI Q,
et al. Second-order topological phases in non-hermitian systems[J].
Physical Review Letters, 2019, 122(7): 76801.
doi:10.1103/PhysRevLett.122.076801
|