Topological properties of artificial bandgap materials

Sun Xiao-Chen; He Cheng; Lu Ming-Hui; Chen Yan-Feng

doi:10.7498/aps.66.224203

Abstract

Recently, artificial bandgap materials (such as photonic crystals and phononic crystals) have been becoming the research hotspot of the next generation intelligent materials, because of its extremely designable, tunable and controllable capacity of classical waves. On the other hand, topological material phase, originally proposed and first demonstrated in Fermionic electronic systems, has been proposed in more and more Bosonic systems. In this review paper, we first focus on some of the representative photonic/phononic topological models, and four common types of topological photonic system are discussed:1) photonic/phononic quantum Hall effect with broken time-reversal symmetry; 2) photonic topological insulator and the associated pseudo-time-reversal symmetry protected mechanism; 3) time/space periodically modulated photonic Floquet topological insulator; 4) a summary and outlook including a brief introduction of Zak phase in one-dimensional systems and Weyl point in three-dimensional systems. Finally, the underlying Dirac model is analyzed.

Keywords:

Authors and contacts

1.
National Laboratory of Solid State Microstructures, Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China;
2.
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

Corresponding author: Lu Ming-Hui, luminghui@nju.edu.cn

Funds: Project supported by the National Key RD Program of China (Grant No. 2017YFA0303702), the National Natural Science Foundation of China (Grant Nos. 11134006, 11474158, 11404164, 11625418), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140019), and the support from Academic Program Development of Jiangsu Higher Education (Grant No. PAPD).

References

[1]	Pancharatnam S 1956 Proc. Indian Acad. Sci. Sect. A 44 398
[2]	Berry M V 1984 Proc. Royal Soci. London A:Math. Phys. Engineer. Sci. 392 45
[3]	Tomita A, Chiao R Y 1986 Phys. Rev. Lett. 57 937
[4]	Asorey M 2016 Nat. Phys. 12 616
[5]	Hall E H 1879 Am. J. Math. 2 287
[6]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[7]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[8]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015
[9]	Shen S Q 2012 Topological Insulator. (Springer, Berlin)
[10]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[11]	Knig M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[12]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[13]	Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602
[14]	Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438
[15]	Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178
[16]	Fu L 2011 Phys. Rev. Lett. 106 106802
[17]	Hsieh T H, Lin H, Liu J, Duan W, Bansil A, Fu L 2012 Nat. Commun. 3 982
[18]	Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101
[19]	Cayssol J, Dora B, Simon F, Moessner R 2013 Phys. Status Solidi Rapid Res. Lett. 7 101
[20]	Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167
[21]	Lu L, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 626
[22]	Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
[23]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905
[24]	Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772
[25]	Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
[26]	Yannopapas V 2011 Phys. Rev. B 84 195126
[27]	Liu K, Shen L, He S 2012 Opt. Lett. 37 4110
[28]	Asatryan A A, Botten L C, Fang K, Fan S, McPhedran R C 2013 Phys. Rev. B 88 035127
[29]	Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljačić M 2015 Phys. Rev. Lett. 115 253901
[30]	Fleury R, Sounas D L, Sieck C F, Haberman M R, Al A 2014 Science 343 516
[31]	Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B 2015 Phys. Rev. Lett. 114 114301
[32]	Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016
[33]	Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233
[34]	He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. USA 113 4924
[35]	Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782
[36]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901
[37]	He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124
[38]	Susstrunk R, Huber S D 2015 Science 349 47
[39]	Fang K, Yu Z, Fan S 2012 Phys. Rev. Lett. 108 153901
[40]	Fang K, Yu Z, Fan S 2012 Nat. Photon. 6 782
[41]	Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196
[42]	Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907
[43]	Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photon. 7 1001
[44]	Liang G Q, Chong Y D 2013 Phys. Rev. Lett. 110 203904
[45]	Petrescu A, Houck A A, Le Hur K 2012 Phys. Rev. A 86 053804
[46]	Rechtsman M C, Zeuner J M, Tunnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photon. 7 153
[47]	Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453
[48]	Guzmn-Silva D, Meja-Corts C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061
[49]	Lin Q, Fan S 2014 Phys. Rev. X 4 031031
[50]	Mittal S, Fan J, Faez S, Migdall A, Taylor J M, Hafezi M 2014 Phys. Rev. Lett. 113 087403
[51]	Ozawa T, Carusotto I 2014 Phys. Rev. Lett. 112 133902
[52]	Teras H, Flayac H, Solnyshkov D D, Malpuech G 2014 Phys. Rev. Lett. 112 066402
[53]	Tzuang L D, Fang K, Nussenzveig P, Fan S, Lipson M 2014 Nat. Photon. 8 701
[54]	Hu W, Pillay J C, Wu K, Pasek M, Shum P P, Chong Y D 2015 Phys. Rev. X 5 011012
[55]	Schmidt M, Kessler S, Peano V, Painter O, Marquardt F 2015 Optica 2 635
[56]	Gao F, Gao Z, Shi X, Yang Z, Lin X, Xu H, Joannopoulos J D, Soljacic M, Chen H, Lu L, Chong Y, Zhang B 2016 Nat.Commun. 7 11619
[57]	Fleury R, Khanikaev A B, Al A 2016 Nat.Commun. 7 11744
[58]	He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904
[59]	Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368
[60]	Harper P G 1955 Proc. Phys. Soci. Sec. A 68 874
[61]	Aubry S, Andr G 1980 Ann. Israel Phys. Soc. 3 18
[62]	Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev.Lett. 50 1870
[63]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402
[64]	Verbin M, Zilberberg O, Kraus Y E, Lahini Y, Silberberg Y 2013 Phys. Rev. Lett. 110 076403
[65]	Zak J 1989 Phys. Rev. Lett. 62 2747
[66]	Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698
[67]	Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017
[68]	Xiao M, Ma G, Yang Z, Sheng P, Zhang Z Q, Chan C T 2015 Nat. Phys. 11 240
[69]	Kitaev A Y 2001 Physics-Uspekhi 44 131
[70]	Poddubny A, Miroshnichenko A, Slobozhanyuk A, Kivshar Y 2014 ACS Photon. 1 101
[71]	Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 337
[72]	Wang H, Xu L, Chen H, Jiang J H 2016 Phys. Rev. B 93 235155
[73]	Lu L, Fu L, Joannopoulos J D, Soljacic M 2013 Nat. Photon. 7 294
[74]	Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622
[75]	Xiao M, Chen WJ, He W Y, Chan C T 2015 Nat. Phys. 11 920
[76]	Yang Z, Zhang B 2016 Phys. Rev. Lett. 117 224301
[77]	Noh J, Huang S, Leykam D, Chong Y D, Chen K P, Rechtsman M C 2017 Nat. Phys. 13 611

Cited By

[1]	Pancharatnam S 1956 Proc. Indian Acad. Sci. Sect. A 44 398
[2]	Berry M V 1984 Proc. Royal Soci. London A:Math. Phys. Engineer. Sci. 392 45
[3]	Tomita A, Chiao R Y 1986 Phys. Rev. Lett. 57 937
[4]	Asorey M 2016 Nat. Phys. 12 616
[5]	Hall E H 1879 Am. J. Math. 2 287
[6]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[7]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[8]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015
[9]	Shen S Q 2012 Topological Insulator. (Springer, Berlin)
[10]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[11]	Knig M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[12]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[13]	Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602
[14]	Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438
[15]	Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178
[16]	Fu L 2011 Phys. Rev. Lett. 106 106802
[17]	Hsieh T H, Lin H, Liu J, Duan W, Bansil A, Fu L 2012 Nat. Commun. 3 982
[18]	Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101
[19]	Cayssol J, Dora B, Simon F, Moessner R 2013 Phys. Status Solidi Rapid Res. Lett. 7 101
[20]	Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167
[21]	Lu L, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 626
[22]	Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
[23]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905
[24]	Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772
[25]	Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
[26]	Yannopapas V 2011 Phys. Rev. B 84 195126
[27]	Liu K, Shen L, He S 2012 Opt. Lett. 37 4110
[28]	Asatryan A A, Botten L C, Fang K, Fan S, McPhedran R C 2013 Phys. Rev. B 88 035127
[29]	Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljačić M 2015 Phys. Rev. Lett. 115 253901
[30]	Fleury R, Sounas D L, Sieck C F, Haberman M R, Al A 2014 Science 343 516
[31]	Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B 2015 Phys. Rev. Lett. 114 114301
[32]	Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016
[33]	Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233
[34]	He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. USA 113 4924
[35]	Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782
[36]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901
[37]	He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124
[38]	Susstrunk R, Huber S D 2015 Science 349 47
[39]	Fang K, Yu Z, Fan S 2012 Phys. Rev. Lett. 108 153901
[40]	Fang K, Yu Z, Fan S 2012 Nat. Photon. 6 782
[41]	Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196
[42]	Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907
[43]	Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photon. 7 1001
[44]	Liang G Q, Chong Y D 2013 Phys. Rev. Lett. 110 203904
[45]	Petrescu A, Houck A A, Le Hur K 2012 Phys. Rev. A 86 053804
[46]	Rechtsman M C, Zeuner J M, Tunnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photon. 7 153
[47]	Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453
[48]	Guzmn-Silva D, Meja-Corts C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061
[49]	Lin Q, Fan S 2014 Phys. Rev. X 4 031031
[50]	Mittal S, Fan J, Faez S, Migdall A, Taylor J M, Hafezi M 2014 Phys. Rev. Lett. 113 087403
[51]	Ozawa T, Carusotto I 2014 Phys. Rev. Lett. 112 133902
[52]	Teras H, Flayac H, Solnyshkov D D, Malpuech G 2014 Phys. Rev. Lett. 112 066402
[53]	Tzuang L D, Fang K, Nussenzveig P, Fan S, Lipson M 2014 Nat. Photon. 8 701
[54]	Hu W, Pillay J C, Wu K, Pasek M, Shum P P, Chong Y D 2015 Phys. Rev. X 5 011012
[55]	Schmidt M, Kessler S, Peano V, Painter O, Marquardt F 2015 Optica 2 635
[56]	Gao F, Gao Z, Shi X, Yang Z, Lin X, Xu H, Joannopoulos J D, Soljacic M, Chen H, Lu L, Chong Y, Zhang B 2016 Nat.Commun. 7 11619
[57]	Fleury R, Khanikaev A B, Al A 2016 Nat.Commun. 7 11744
[58]	He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904
[59]	Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368
[60]	Harper P G 1955 Proc. Phys. Soci. Sec. A 68 874
[61]	Aubry S, Andr G 1980 Ann. Israel Phys. Soc. 3 18
[62]	Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev.Lett. 50 1870
[63]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402
[64]	Verbin M, Zilberberg O, Kraus Y E, Lahini Y, Silberberg Y 2013 Phys. Rev. Lett. 110 076403
[65]	Zak J 1989 Phys. Rev. Lett. 62 2747
[66]	Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698
[67]	Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017
[68]	Xiao M, Ma G, Yang Z, Sheng P, Zhang Z Q, Chan C T 2015 Nat. Phys. 11 240
[69]	Kitaev A Y 2001 Physics-Uspekhi 44 131
[70]	Poddubny A, Miroshnichenko A, Slobozhanyuk A, Kivshar Y 2014 ACS Photon. 1 101
[71]	Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 337
[72]	Wang H, Xu L, Chen H, Jiang J H 2016 Phys. Rev. B 93 235155
[73]	Lu L, Fu L, Joannopoulos J D, Soljacic M 2013 Nat. Photon. 7 294
[74]	Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622
[75]	Xiao M, Chen WJ, He W Y, Chan C T 2015 Nat. Phys. 11 920
[76]	Yang Z, Zhang B 2016 Phys. Rev. Lett. 117 224301
[77]	Noh J, Huang S, Leykam D, Chong Y D, Chen K P, Rechtsman M C 2017 Nat. Phys. 13 611

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