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The well-known Su-Schrieffer-Heeger (SSH) model predicts that a chain of sites with alternating coupling constant exhibits two topological distinct phases, and at the truncated edge of the topological nontrivial phase there exists topologically protected edge modes. Such modes are named zero-energy modes as their eigenvalues are located exactly at the midgaps of the corresponding bandstructures. The previous publications have reported a variety of photonic realizations of the SSH model, however, all of these studies have been restricted in the systems of time-reversal-symmetry (TRS), and thus the important question how the breaking of TRS affects the topological edge modes has not been explored. In this work, to the best of our knowledge, we study for the first time the topological zero-energy modes in the systems where the TRS is broken. The system used here is semiconductor microcavities supporting exciton-polariton quasi-particle, in which the interplay between the spin-orbit coupling stemming from the TE-TM energy splitting and the Zeeman effect causes the TRS to break. We first study the topological edge modes occurring at the edge of one-dimensional microcavity array that has alternative coupling strengths between adjacent microcavity, and, by rigorously solving the Schrdinger-like equations (see Eq.(1) or Eq.(2) in the main text), we find that the eigen-energies of topological zero-energy modes are no longer pinned at the midgap position:rather, with the increasing of the spin-orbit coupling, they gradually shift from the original midgap position, with the spin-down edge modes moving toward the lower band while the spin-up edge modes moving towards the upper band. Interestingly enough, the mode profiles of these edge modes remain almost unchanged even they are approaching the bulk transmission bands, which is in sharp contrast to the conventional defect modes that have an origin of bifurcation from the Bloch mode of the upper or lower bands. We also study the edge modes in the two-dimensional microcavity square array, and find that the topological zero modes acquire mobility along the truncated edge due to the coupling from the adjacent arrays. Importantly, owing to the breaking of the TRS, a pair of counterpropagating edge modes, of which one has a momentum k and the other has -k, is no longer of energy degeneracy; as a result the scattering between the forward-and backward-propagating modes is greatly suppressed. Thus, we propose the concept of the one-dimensional topological zero-energy modes that are propagating along the two-dimensional lattice edge, with extremely weak backscattering even on the collisions of the topological zero-energy modes with structural defects or disorder.
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[26] Kartashov Y V, Skryabin D V 2016 Optica 3 1228
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[28] Flayac H 2012 Ph. D. Dissertation (Clermont-Ferrand: Université Blaise Pascal-Clermont-Ferrand Ⅱ)
[29] Joannopoulos J D, Johnson S G, Winn J N, Meade R D 2008 Photonic Crystals: Molding the Flow of Light (2nd Ed.) (New Jersey: Princeton University Press) p25
[30] Peleg O, Bartal G, Freedman B, Manela O, Segev M, Christodoulides D N 2007 Phys. Rev. Lett. 98 103901
[31] Li Y M, Zhou X, Zhang Y Y, Zhang D, Chang K 2017 Phys. Rev. B 96 035406
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[1] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[2] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[3] Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
[4] Wang Z, Chong Y, Joannopoulos J D, Soljačić M 2009 Nature 461 772
[5] Lu L, Joannopoulos J D, Soljačić M 2014 Nat. Photon. 8 821
[6] Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698
[7] Longhi S 2013 Opt. Lett. 38 3716
[8] Cheng Q, Pan Y, Wang Q, Li T, Zhu S 2015 Laser Photon. Rev. 9 392
[9] Ge L, Wang L, Xiao M, Wen W, Chan C T, Han D 2015 Opt. Express 23 21585
[10] Slobozhanyuk A P, Poddubny A N, Miroshnichenko A E, Belov P A, Kivshar Y S 2015 Phys. Rev. Lett. 114 123901
[11] Sinev I S, Mukhin I S, Slobozhanyuk A P, Poddubny A N, Miroshnichenko A E, Samusev A K, Kivshar Y S 2015 Nanoscale 7 11904
[12] Schomerus H 2013 Opt. Lett. 38 1912
[13] Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Opt. Lett. 34 1633
[14] Xiao M, Zhang Z Q, Chan C T Deng H, Chen X, Panoiu N C, Ye F 2016 Opt. Lett. 41 4281
[15] Deng H, Chen X, Panoiu N C, Ye F 2016 Opt. Lett. 41 4281
[16] Christodoulides D N, Lederer F, Silberberg Y 2003 Nature 424 817
[17] Teo J C Y, Hughes T L 2013 Phys. Rev. Lett. 111 047006
[18] Benalcazar W A, Teo J C Y, Hughes T L 2014 Phys. Rev. B 89 224503
[19] Noh J, Benalcazar W A, Huang S, Collins M J, Chen K, Hughes T L, Rechtsman M C 2016 arXiv: 1611.02373v1
[20] Nalitov A V, Solnyshkov D D, Malpuech G 2015 Phys. Rev. Lett. 114 116401
[21] Bardyn C E, Karzig T, Refael G, Liew T C 2015 Phys. Rev. B 91 161413
[22] Karzig T, Bardyn C E, Lindner N H, Refael G Bleu O, Solnyshkov D D, Malpuech G 2016 Phys. Rev. B 93 085438
[23] Bleu O, Solnyshkov D D, Malpuech G 2016 Phys. Rev. B 93 085438
[24] Milićević M, Ozawa T, Andreakou P, Carusotto I, Jacqmin T, Galopin E, Amo A 2015 2D Mater. 2 034012
[25] Sich M, Krizhanovskii D N, Skolnick M S, Gorbach A V, Hartley R, Skryabin D V, Santos P V 2012 Nat. Photon. 6 50
[26] Kartashov Y V, Skryabin D V 2016 Optica 3 1228
[27] Li Y M, Li J, Shi L K, Zhang D, Yang W, Chang K 2015 Phys. Rev. Lett. 115 166804
[28] Flayac H 2012 Ph. D. Dissertation (Clermont-Ferrand: Université Blaise Pascal-Clermont-Ferrand Ⅱ)
[29] Joannopoulos J D, Johnson S G, Winn J N, Meade R D 2008 Photonic Crystals: Molding the Flow of Light (2nd Ed.) (New Jersey: Princeton University Press) p25
[30] Peleg O, Bartal G, Freedman B, Manela O, Segev M, Christodoulides D N 2007 Phys. Rev. Lett. 98 103901
[31] Li Y M, Zhou X, Zhang Y Y, Zhang D, Chang K 2017 Phys. Rev. B 96 035406
[32] Guzmán-Silva D, Mejía-Cortés C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Vicencio R A 2014 New J. Phys. 16 063061
[33] Schulz S A, Upham J, O’Faolain L, Boyd R W 2017 Opt. Lett. 42 3243
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