Topological phases of a non-Hermitian coupled SSH ladder

doi:10.1088/1674-1056/ab3f94

摘要/Abstract

摘要： We study topological phases of a non-Hermitian coupled Su-Schrieffer-Heeger (SSH) ladder. The model originates from the brick-wall lattices in the two-row limit. The Hamiltonian can be brought into block off-diagonal form and the winding number can be defined with the determine of the block off-diagonal matrix. We find the determine of the off-diagonal matrix has nothing to do with the interleg hopping of the ladder. So the topological phases of the model are the same as those of the chains. Further numerical simulations verify the analysis.

关键词: topological phase, non-Hermitian, Su-Schrieffer-Heeger (SSH) ladder, winding number

Abstract: We study topological phases of a non-Hermitian coupled Su-Schrieffer-Heeger (SSH) ladder. The model originates from the brick-wall lattices in the two-row limit. The Hamiltonian can be brought into block off-diagonal form and the winding number can be defined with the determine of the block off-diagonal matrix. We find the determine of the off-diagonal matrix has nothing to do with the interleg hopping of the ladder. So the topological phases of the model are the same as those of the chains. Further numerical simulations verify the analysis.

Key words: topological phase, non-Hermitian, Su-Schrieffer-Heeger (SSH) ladder, winding number

中图分类号: (Phases: geometric; dynamic or topological)

03.65.Vf

05.70.Fh (Phase transitions: general studies) 64.70.Tg (Quantum phase transitions)

刘建森, 韩炎桢, 刘承师. Topological phases of a non-Hermitian coupled SSH ladder[J]. 中国物理B, 2019, 28(10): 100304-100304.

J S Liu(刘建森), Y Z Han(韩炎桢), C S Liu(刘承师). Topological phases of a non-Hermitian coupled SSH ladder[J]. Chin. Phys. B, 2019, 28(10): 100304-100304.

参考文献 73

[32]	Choi Y W, Kang S S, Lim S, Kim W, Kim J, Lee J and An K 2010 Phys. Rev. Lett. 104 153601 doi: 10.1103/PhysRevLett.104.153601
[1]	Esaki K, Sato M, Kazuki H and Kohmoto M 2011 Phys. Rev. B 84 205128 doi: 10.1103/PhysRevB.84.205128
[33]	Cao H and Wiersig Jan 2015 Rev. Mod. Phys. 87 61 doi: 10.1103/RevModPhys.87.61
[2]	Lee T E 2016 Phys. Rev. Lett. 116 133903 doi: 10.1103/PhysRevLett.116.133903
[34]	Malzard S, Poli C and Schomerus H 2015 Phys. Rev. Lett. 115 200402 doi: 10.1103/PhysRevLett.115.200402
[3]	Leykam D, Bliokh K Y, Huang C, Chong Y D and Nori F 2017 Phys. Rev. Lett. 118 040401 doi: 10.1103/PhysRevLett.118.040401
[35]	Carmichael H J 1993 Phys. Rev. Lett 70 2273 doi: 10.1103/PhysRevLett.70.2273
[4]	Lieu S 2018 Phys. Rev. B 97 045106 doi: 10.1103/PhysRevB.97.045106
[36]	Lee T E and Chan C K 2014 Phys. Rev. X 23 094206
[5]	Li C, Zhang X Z, Zhang G and Song Z 2018 Phys. Rev. B 97 115436 doi: 10.1103/PhysRevB.97.115436
[37]	Lee T E, Reiter F and Moiseyev N 2014 Phys. Rev. Lett 113 250401 doi: 10.1103/PhysRevLett.113.250401
[6]	Yin C H and Jiang H, Li L H, Lü R and Chen S 2018 Phys. Rev. A 97 052115 doi: 10.1103/PhysRevA.97.052115
[38]	Diehl S, Rico E, Baranov M A and Zoller P 2011 Nat. Phys. 7 971 doi: 10.1038/nphys2106
[7]	Shen H T, Zhen B and Fu L 2018 Phys. Rev. Lett. 120 146402 doi: 10.1103/PhysRevLett.120.146402
[39]	Choi Y, Kang S, Lim S, Kim W, Kim J R, Lee J H and An K 2010 Phys. Rev. Lett. 104 153601 doi: 10.1103/PhysRevLett.104.153601
[8]	Liang S D and Huang G Y 2013 Phys. Rev. A 27 012118
[40]	Han Y Z and Liu C S 2019 Physica E 108 116 doi: 10.1016/j.physe.2018.12.017
[9]	Rudner M S and Levitov L S 2009 Phys. Rev. Lett. 102 065703 doi: 10.1103/PhysRevLett.102.065703
[41]	Deng H, Haug H and Yamamoto Y 2010 Rev. Mod.Phys. 82 1489 doi: 10.1103/RevModPhys.82.1489
[10]	Kawabata K, Ashida Y, Katsura H and Ueda M 2018 Phys. Rev. B 98 085116 doi: 10.1103/PhysRevB.98.085116
[42]	Dattoli G, Hermsen T, Mezi L, Renieri A and Torre A 1988 Phys. Rev. A 37 4334 doi: 10.1103/PhysRevA.37.4334
[11]	Gong J B and Wang Q H 2010 Phys. Rev. A 82 012103 doi: 10.1103/PhysRevA.82.012103
[43]	Makris K G, El-Ganainy R, Christodoulides D N and Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 doi: 10.1103/PhysRevLett.100.103904
[12]	Gong Z P, Higashikawa S and Ueda M 2017 Phys. Rev. Lett. 118 200401 doi: 10.1103/PhysRevLett.118.200401
[44]	Longhi S 2009 Phys. Rev. Lett. 103 123601 doi: 10.1103/PhysRevLett.103.123601
[13]	Hu Y C and Hughes T L 2011 Phys. Rev. B 84 153101 doi: 10.1103/PhysRevB.84.153101
[45]	Klaiman S, Gunther U and Moiseyev N 2008 Phys. Rev. Lett. 101 084002
[14]	Gonzalez J and Molina R A 2017 Phys. Rev. B 96 045437 doi: 10.1103/PhysRevB.96.045437
[46]	Bittner S, Dietz B, Gunther U, Harney H L, Miski-Oglu M, Richter A and Schafer F 2012 Phys. Rev. Lett. 108 024101 doi: 10.1103/PhysRevLett.108.024101
[15]	Cerjan A, Xiao M, Yuan L and Fan S H 2018 Phys. Rev. B 97 075128 doi: 10.1103/PhysRevB.97.075128
[47]	Liertzer M, Ge Li, Cerjan A, Stone A D, Tureci H E and Rotter S 2012 Phys. Rev. Lett 108 173901 doi: 10.1103/PhysRevLett.108.173901
[16]	Zyuzin A A and Zyuzin A Y 2018 Phys. Rev. B 97 041203 doi: 10.1103/PhysRevB.97.041203
[48]	Menke H and Hirschmann M M 2017 Phys. Rev. B 95 174506 doi: 10.1103/PhysRevB.95.174506
[49]	Yuce C 2016 Phys. Rev. A 93 062130 doi: 10.1103/PhysRevA.93.062130
[17]	Rivolta N X A, Benisty H and Maes, B 201 Phys. Rev. A 96 023864
[50]	Klett M, Cartarius H, Dast D, Main J and Wunner G 2018 Eur. Phys. J. D 72 214 doi: 10.1140/epjd/e2018-90502-3
[18]	Wang X H, Liu T T, Xiong Y and Tong, P Q 2015 Phys. Rev. A 92 012116 doi: 10.1103/PhysRevA.92.012116
[51]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167171
[19]	Klett M, Cartarius H, Dast D, Main J and Wunner G 2017 Phys. Rev. A 95 053626 doi: 10.1103/PhysRevA.95.053626
[52]	Rüter C E, Makris K G, Elganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192 doi: 10.1038/nphys1515
[20]	Ni X, Smirnova D, Poddubny A, Leykam Daniel, Chong Y and Khanikaev A B 2018 Phys. Rev. B 98 165129 doi: 10.1103/PhysRevB.98.165129
[53]	Feng L, Wong Z J, Ma R M, Wang Y and Zhang X 2014 Science 346 972 doi: 10.1126/science.1258479
[21]	Zhou L W, Wang Q H, Wang H L and Gong J B 2018 Phys. Rev. A 98 022129 doi: 10.1103/PhysRevA.98.022129
[54]	Gao T, Estrecho E, Bliokh K Y, Liew T C H, Fraser M D, Brodbeck S, Kamp M, Schneider C, Hfling S and Yamamoto Y 2015 Nature 526 554 doi: 10.1038/nature15522
[22]	Yuce C 2015 Phys. Lett. A 379 1213 doi: 10.1016/j.physleta.2015.02.011
[55]	Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M, Schuster D, Simon J and Zilberberg O 2018 Rev. Mod. Phys. 91 015006
[23]	Ke S, Wang B, Long H, Wang K and Lu P 2017 Opt. Express 25 011132 doi: 10.1364/OE.25.011132
[56]	Ling L, John D J and Marin S 2014 Nat. Photon. 8 821 doi: 10.1038/nphoton.2014.248
[24]	Gong Z P, Ashida Y, Kawabata K, Takasan K, Higashikawa S and Ueda M 2018 Phys. Rev. X 8 031079
[57]	Elganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S and Christodoulides D N 2018 Nat. Phys. 14 11 doi: 10.1038/nphys4323
[25]	Hu W C, Wang H L, Shum P P and Chong Y D 2017 Phys. Rev. B 23 094206
[58]	Ingrid R 2009 J. Phys. A-Math. Theor. 42 153001 doi: 10.1088/1751-8113/42/15/153001
[26]	Parto M, Wittek S, Hodaei H, Harari G, Bandres M, Ren J, Rechtsman M C, Segev M, Christodoulides D N and Khajavikhan M 2018 Phys. Rev. Lett. 120 113901 doi: 10.1103/PhysRevLett.120.113901
[59]	Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243 doi: 10.1103/PhysRevLett.80.5243
[27]	Zhan X, Xiao L, Bian Z H, Wang K K, Qiu X Z, Sanders B C, Yi W and Xue P 2017 Phys. Rev. Lett 119 130501 doi: 10.1103/PhysRevLett.119.130501
[60]	Mostafazadeh A 2002 J. Math. Phys. 43 205 doi: 10.1063/1.1418246
[28]	Zeuner J M, Rechtsman M C, Plotnik Y, Lumer Y, Nolte S, Rudner M S, Segev M and Szameit A 2015 Phys. Rev. Lett. 115 040402 doi: 10.1103/PhysRevLett.115.040402
[61]	Zhu B G, Lu R and Chen S 2014 Phys. Rev. A 89 062102 doi: 10.1103/PhysRevA.89.062102
[29]	Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D and Kawakami N 2017 Nat. Phys. 13 1117 doi: 10.1038/nphys4204
[62]	Wu H C and Jin L and Song Z 2019 arXiv 1905 11576
[30]	Weimann S, Kremer, M. Plotnik, Y, Lumer, Y, Nolte S, Makris K G, Segev M, Rechtsman M C and Szameit A 2017 Nat. Mater. 16 433 doi: 10.1038/nmat4811
[63]	Jin L and Song Z 2019 Phys. Rev. B 99 081103 doi: 10.1103/PhysRevB.99.081103
[31]	Hengyun Z, Chao P, Yoseob Y, Chia W H, Keith A N, Liang F, John D J, Marin S and Bo Z 2018 Science 359 1009 doi: 10.1126/science.aap9859
[64]	Yao S Y and Wang Z 2018 Phys. Rev. Lett. 121 086803 doi: 10.1103/PhysRevLett.121.086803
[32]	Choi Y W, Kang S S, Lim S, Kim W, Kim J, Lee J and An K 2010 Phys. Rev. Lett. 104 153601 doi: 10.1103/PhysRevLett.104.153601
[65]	Kawabata K, Shiozaki K and Ueda M 2018 Phys. Rev. B 98 165148 doi: 10.1103/PhysRevB.98.165148
[33]	Cao H and Wiersig Jan 2015 Rev. Mod. Phys. 87 61 doi: 10.1103/RevModPhys.87.61
[66]	Yao S Y, Song F and Wang Z 2018 Phys. Rev. Lett. 121 136802 doi: 10.1103/PhysRevLett.121.136802
[34]	Malzard S, Poli C and Schomerus H 2015 Phys. Rev. Lett. 115 200402 doi: 10.1103/PhysRevLett.115.200402
[67]	Deng T S and Yi W 2019 Phys. Rev. B 100 035102 doi: 10.1103/PhysRevB.100.035102
[35]	Carmichael H J 1993 Phys. Rev. Lett 70 2273 doi: 10.1103/PhysRevLett.70.2273
[68]	Clay R T and Mazumdar S 2005 Phys. Rev. Lett. 94 207206 doi: 10.1103/PhysRevLett.94.207206
[36]	Lee T E and Chan C K 2014 Phys. Rev. X 23 094206
[69]	Shimizu Y, Aoyama S, Jinno T, Itoh M and Ueda Y 2015 Phys. Rev. Lett. 114 166403 doi: 10.1103/PhysRevLett.114.166403
[37]	Lee T E, Reiter F and Moiseyev N 2014 Phys. Rev. Lett 113 250401 doi: 10.1103/PhysRevLett.113.250401
[70]	Cheon S, Kim T, Lee S and Yeom H W 2015 Science 350 182 doi: 10.1126/science.aaa7055
[38]	Diehl S, Rico E, Baranov M A and Zoller P 2011 Nat. Phys. 7 971 doi: 10.1038/nphys2106
[71]	Song F, Yao S Y and Wang Z 2019 arXiv 1905 02211
[39]	Choi Y, Kang S, Lim S, Kim W, Kim J R, Lee J H and An K 2010 Phys. Rev. Lett. 104 153601 doi: 10.1103/PhysRevLett.104.153601
[72]	Kazuki Y and Shuichi M 2019 Phys. Rev. Lett. 123 066404 doi: 10.1103/PhysRevLett.123.066404
[73]	Li C, Lin S, Zhang G and Song Z 2019 Phys. Rev. B 96 125418
[40]	Han Y Z and Liu C S 2019 Physica E 108 116 doi: 10.1016/j.physe.2018.12.017
[41]	Deng H, Haug H and Yamamoto Y 2010 Rev. Mod.Phys. 82 1489 doi: 10.1103/RevModPhys.82.1489
[42]	Dattoli G, Hermsen T, Mezi L, Renieri A and Torre A 1988 Phys. Rev. A 37 4334 doi: 10.1103/PhysRevA.37.4334
[43]	Makris K G, El-Ganainy R, Christodoulides D N and Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 doi: 10.1103/PhysRevLett.100.103904
[44]	Longhi S 2009 Phys. Rev. Lett. 103 123601 doi: 10.1103/PhysRevLett.103.123601
[45]	Klaiman S, Gunther U and Moiseyev N 2008 Phys. Rev. Lett. 101 084002
[46]	Bittner S, Dietz B, Gunther U, Harney H L, Miski-Oglu M, Richter A and Schafer F 2012 Phys. Rev. Lett. 108 024101 doi: 10.1103/PhysRevLett.108.024101
[47]	Liertzer M, Ge Li, Cerjan A, Stone A D, Tureci H E and Rotter S 2012 Phys. Rev. Lett 108 173901 doi: 10.1103/PhysRevLett.108.173901
[48]	Menke H and Hirschmann M M 2017 Phys. Rev. B 95 174506 doi: 10.1103/PhysRevB.95.174506
[49]	Yuce C 2016 Phys. Rev. A 93 062130 doi: 10.1103/PhysRevA.93.062130
[50]	Klett M, Cartarius H, Dast D, Main J and Wunner G 2018 Eur. Phys. J. D 72 214 doi: 10.1140/epjd/e2018-90502-3
[51]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167171
[52]	Rüter C E, Makris K G, Elganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192 doi: 10.1038/nphys1515
[53]	Feng L, Wong Z J, Ma R M, Wang Y and Zhang X 2014 Science 346 972 doi: 10.1126/science.1258479
[54]	Gao T, Estrecho E, Bliokh K Y, Liew T C H, Fraser M D, Brodbeck S, Kamp M, Schneider C, Hfling S and Yamamoto Y 2015 Nature 526 554 doi: 10.1038/nature15522
[55]	Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M, Schuster D, Simon J and Zilberberg O 2018 Rev. Mod. Phys. 91 015006
[56]	Ling L, John D J and Marin S 2014 Nat. Photon. 8 821 doi: 10.1038/nphoton.2014.248
[57]	Elganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S and Christodoulides D N 2018 Nat. Phys. 14 11 doi: 10.1038/nphys4323
[58]	Ingrid R 2009 J. Phys. A-Math. Theor. 42 153001 doi: 10.1088/1751-8113/42/15/153001
[59]	Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243 doi: 10.1103/PhysRevLett.80.5243
[60]	Mostafazadeh A 2002 J. Math. Phys. 43 205 doi: 10.1063/1.1418246
[61]	Zhu B G, Lu R and Chen S 2014 Phys. Rev. A 89 062102 doi: 10.1103/PhysRevA.89.062102
[62]	Wu H C and Jin L and Song Z 2019 arXiv 1905 11576
[63]	Jin L and Song Z 2019 Phys. Rev. B 99 081103 doi: 10.1103/PhysRevB.99.081103
[64]	Yao S Y and Wang Z 2018 Phys. Rev. Lett. 121 086803 doi: 10.1103/PhysRevLett.121.086803
[65]	Kawabata K, Shiozaki K and Ueda M 2018 Phys. Rev. B 98 165148 doi: 10.1103/PhysRevB.98.165148
[66]	Yao S Y, Song F and Wang Z 2018 Phys. Rev. Lett. 121 136802 doi: 10.1103/PhysRevLett.121.136802
[67]	Deng T S and Yi W 2019 Phys. Rev. B 100 035102 doi: 10.1103/PhysRevB.100.035102
[68]	Clay R T and Mazumdar S 2005 Phys. Rev. Lett. 94 207206 doi: 10.1103/PhysRevLett.94.207206
[69]	Shimizu Y, Aoyama S, Jinno T, Itoh M and Ueda Y 2015 Phys. Rev. Lett. 114 166403 doi: 10.1103/PhysRevLett.114.166403
[70]	Cheon S, Kim T, Lee S and Yeom H W 2015 Science 350 182 doi: 10.1126/science.aaa7055
[71]	Song F, Yao S Y and Wang Z 2019 arXiv 1905 02211
[72]	Kazuki Y and Shuichi M 2019 Phys. Rev. Lett. 123 066404 doi: 10.1103/PhysRevLett.123.066404
[73]	Li C, Lin S, Zhang G and Song Z 2019 Phys. Rev. B 96 125418