G. Monambaux Laboraoire de Phyique de Solide Univerié Pari-Sud, Oray, France G. Monambaux Laboraoire de Phyique de Solide Univerié Pari-Sud, Oray, France M. Milicevic, I. Sagne, E. Galoin, A. Lemaîre, J. Bloch, A. Amo M. Milicevic, I. Sagne, E. Galoin, A. Lemaîre, J. Bloch, A. Amo T. Ozawa, I. Caruoo Treno T. Ozawa, I. Caruoo Treno Auoi, November 7 h 018 C aom 4 emiconducing m caviie elecron olarion and orbial orbial orbial orbial orbial cf z Bragg mirror AlGaA D (MBE grown) eching ingle illar 4 m x + + y Quanum well Bragg mirror I. Caruoo & C. Ciui. Quanum fluid of ligh. Rev. Mod. Phy. 85, 99 (013) 3 4
( z ) orbial armchair armchair orbial + + + + 5 new edge ae + 1 6 Moion and merging of Dirac oin qy E qx Moion and merging of Dirac oin qy E qx Semi-Dirac : 1 : 1 Univeral cenario for merging of Dirac oin Y. Haegawa, R. Konno, H. Nakano, and M. Kohmoo, Phy. Rev. B 74, 033413 006. G. M., F. Piéchon, J.N. Fuch, M.O. Goerbig, Phy. Rev. B 80, 15341 (009) h://uer.l.u-ud.fr/monambaux/ubli-dirac.hm 7 Y. Haegawa, R. Konno, H. Nakano, and M. Kohmoo, Phy. Rev. B 74, 033413 006. G. M., F. Piéchon, J.N. Fuch, M.O. Goerbig, Phy. Rev. B 80, 15341 (009) h://uer.l.u-ud.fr/monambaux/ubli-dirac.hm Phoonic cryal Microwave Ulracold aom in oical laice -(BEDT-TTF) I 3 Phohorene 8
Moion and merging of Dirac oin qy E qx Reminder: -edge ae, oological argumen Wrie he Hamilonian comaible wih he boundary condiion Semi-Dirac M. Milicevic, e al., CN : 1 Y. Haegawa, R. Konno, H. Nakano, and M. Kohmoo, Phy. Rev. B 74, 033413 006. G. M., F. Piéchon, J.N. Fuch, M.O. Goerbig, Phy. Rev. B 80, 15341 (009) h://uer.l.u-ud.fr/monambaux/ubli-dirac.hm Univeral cenario for merging of Dirac oin q x Phoonic cryal Microwave Ulracold aom in oical laice -(BEDT-TTF) I 3 q y Phohorene 9 The number of edge ae i relaed o he winding number W(k k )= 1 Z φ( ~ k) π k W(k k )= 1 iπ Z ln f (k k,k ) k Ryu, Haugai (00), Dellace, Ullmo, G.M (011) Reminder: -edge ae, oological argumen (SSH chain) m =0 m = M +1 A B A B A B Reminder: -edge ae, oological argumen Wrie he Hamilonian comaible wih he boundary condiion Bulk oluion, infinie yem ψ k i = 1 X µ e iφ k ( m, Ai, m, Bi) e ikma0 N 1 m 0 = co K k Bulk oluion, wih aroriae boundary condiion : ψ k i = 1 X µ in(kma0 φ ( m, Ai, m, Bi) k ) N in(kma 0 ) m hm +1,A ψ k i =0 h0,b ψ k i =0 k(m +1)a 0 φ k = κπ, κ =1,,M ha bulk oluion The number of edge ae i relaed o he winding number edge ae W(k k )= 1 Z φ( ~ k) π k W(k k )= 1 iπ Z ln f (k k,k ) k Ryu, Haugai (00), Dellace, Ullmo, G.M (011) Ryu, Haugai (00), Dellace, Ullmo, G.M (011)
Reminder: -edge ae, oological argumen and ae are comlemenary armchair β =0.4 β =1 β =1.5 W(k k )= 1 iπ Z ln f (k k,k ) k β < 1 β > 1 β 6= 1 edge ae a he exene of ae edge ae a he exene of ae aearance of edge ae a he armchair edge W(k k )= 1 Z φ( ~ k) π k f () = e i~ k ~a 1 f () W () =1 W () Nice exerimen : diored microwave laice Polarion honeycomb laice : edge β =0.4 β =1 β =1.5 β =.5 armchair Armchair β < 1 β > 1 edge ae a he exene of ae edge ae a he exene of ae β 1 ξ Maniulaion of edge ae in microwave arificial grahene 0 m M. Bellec, U. Kuhl, G.M., F. Moreagne New J. Phy. 16, 113 03 (014) k k Milicevic e al, D Maer., 03401 (015) 16
/ igh-binding Hamilonian ecrum u u 1 Energy + + + + + + + + C. Wu & S. Da Sarma, x,y-orbial counerar of grahene: cold aom in he honeycomb oical laice PRB 77, 35107 (008) -5-4 -3 - -1 0 1 3 4 5 k y /(π/3 3a) M. Milicevic, e al., CN 18 -band edge ae: igh-binding Toological roerie of he and Hamilonian armchair -fold 4-fold -fold 4-fold Zero energy ae : -ae and -ae are comlemenary Toological decriion One addiional zero energy ae New dierive edge ae 19 C. Kane, T. Lubenky, Na. Phy. (014)
Toological roerie of he and Hamilonian W ( ) =1 W ( ) = W ( ) W ( ) = W ( ) = 1 + W ( ) 1 W () = W () W () = 1 + W () Deformaion of he laice β 6= 1 Correondance -band / -band -band -band edge ae edge ae increae decreae -band edge ae edge ae decreae increae -band W (, β) =1 W (, 1 β ) ε fla (β) ε di. (β) = 3 4 β ε ( 1 β ) 3 W (, β) = W (, 1 β ) 4
Correondance -band / -band -band edge ae edge ae increae decreae -band edge ae edge ae decreae increae W (, β) =1 W (, 1 β ) = W (, 1 β ) W (, β) = W (, 1 β ) = 1 + W (, 1 β ) 5 W (, β) = W (, 1 β ) 6 Bulk edge edge Exerimen E/ l 1.5 0 1.5 1.5 1 E/ l 0 0 1.5 W (, β) = 1 + W (, 1 β ) 7 k y (b) M. Milićević, T. Ozawa, G. Monambaux, I. Caruoo, E. Galoin, A. Lemaîre, L. Le Graie, I. Sagne, J. Bloch, A. Amo Phy. Rev. Le. 118, 107403 (017) k y 7 G.M., Arificial grahene: Dirac maer beyond condened maer, C. R. Phyique 19, 85 (018) arxiv:1810.07505 uer.l.u ud.fr/monambaux/ubli dirac.hm 8