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Photonic Versus Electronic Quantum Anomalous Hall Effect O. Bleu, D. D. Solnyshkov, G. Malpuech Institut Pascal, PHOTON-N2, Clermont Auvergne University, CNRS, 4 avenue Blaise Pascal, 63178 Aubière Cedex, France. WederivethediagramofthetopologicalphasesaccessiblewithinagenericHamiltoniandescribing quantum anomalous Hall effect for photons and electrons in honeycomb lattices in presence of a ZeemanfieldandSpin-OrbitCoupling(SOC).Thetwocasesdiffercruciallybythewindingnumber oftheirSOC,whichis1fortheRashbaSOCofelectrons,and2forthephotonSOCinducedbythe energysplittingbetweentheTEandTMmodes. Asaconsequence,thetwomodelsexhibitopposite 7 Chernnumbers±2atlowfield. Moreover,thephotonicsystemshowsatopologicaltransitionabsent 1 intheelectroniccase. Ifthephotonicstatesaremixedwithexcitonicresonancestoforminteracting 0 exciton-polaritons,theeffectiveZeemanfieldcanbeinducedandcontrolledbyacircularlypolarized 2 pump. This new feature allows an all-optical control of the topological phase transitions. n PACSnumbers: a J 3 The discovery of the quantum Hall effect [1] and its the study of interacting particles in artificial topologi- 1 explanationintermsoftopology[2,3]haverefreshedthe cally non-trivial bands could allow direct measurements interest to the band theory in condensed matter physics of Laughlin wavefunctions (WFs) [36] and give access to ] l leadingtothedefinitionofanewclassofinsulators[4,5]. a wide variety of strongly interacting fermionic [37] and l a They include quantum anomalous Hall (QAH) phase [6] bosonic phases [38]. In that framework, the use of inter- h with broken time reversal (TR) symmetry [7–9] (also acting photons, such as cavity polaritons, for which high - s called Chern or Z insulators) and Quantum Spin Hall quality 2D lattices have been realized [39, 40], showing e (QSH or Z ) Topological Insulators with conserved TR collective properties, such as macroscopic quantum co- m 2 symmetry[10–12]. TheQSHeffectwasinitiallypredicted herence and superfluidity [41], could allow to study the . t to occur in honeycomb lattices because of the intrinsic behaviour of bosonic spinor quantum fluids [42, 43] in a m Spin-OrbitCoupling(SOC)oftheatomsformingthelat- topologically non-trivial bands. In photonics, a Rashba- tice, whereas the extrinsic Rashba SOC is detrimental typeSOCcannotbeimplementedforsymmetryreasons, - d for QSH [11]. On the other hand, the classical anoma- but another effective in-plane SOC is induced by the en- n lous Hall effect is now known to arise from a combina- ergysplittingbetweentheTEandTMmodes. Inplanar o tionofextrinsic RashbaSOCandof aneffectiveZeeman cavities,therelatedeffectivemagneticfieldhasawinding c field [13]. In a 2D lattice with Dirac cones it leads to number 2 (instead of 1 for Rashba). It is at the origin [ the formation of a QAH phase, for which the intrinsic of a very large variety of spin-related effects, such as the 1 SOC is detrimental [14–16]. In the large Rashba SOC optical spin Hall effect [44, 45], half-integer topological v limit, this description was found to converge towards an defects [46, 47], Berry phase for photons [48], and the 0 8 extended Haldane model [14]. Another field, which has generationoftopologicallyprotectedspincurrentsinpo- 6 considerably grown these last years, is the emulation of laritonic molecules [49]. The combination of a TE-TM 3 such topological insulators with different types of par- SOC and a Zeeman field in a honeycomb lattice has in- 0 ticles, such as fermions (either charged, as electrons in deed been found to yield a QAH phase [29, 50–55], and . 1 nanocrystals[17,18],orneutral,suchasfermionicatoms the related model represents a generalization of the sem- 0 in optical lattices [19, 20]) and bosons (atoms, photons, inalHaldane-Raghuproposal[56]ofphotonictopological 7 or mixed light-matter quasiparticles) [21–29]. The main insulator, recovered for large TE-TM SOC. 1 advantage of artificial analogs is the possibility to tune : v the parameters [30], to obtain inaccessible regimes, and In this manuscript, we demonstrate the role played by i thewindingnumberoftheSOContheQAHphases. We X to measure quantities out of reach in the original sys- establish the complete phase diagram for both the pho- tems. These analogs also call for their own applications, r tonic and electronic graphene. In addition to opposite a beyond those of the originals. Photonic systems have in- C in the low-field limit, we find the photonic case to be deed allowed the first demonstration the QAHE [31, 32], n morecomplex,showingatopologicalphasetransitionab- later implemented in electronic [33] and atomic systems sentintheelectronicsystem. Wethenproposearealistic [34]. They have allowed the realization of topological experimental scheme to observe this transition based on bands with high Chern numbers (C ) [35], making pos- n spin-anisotropic interactions in a macro-occupied cavity sible to work with superpositions of chiral edge states. polariton mode. We consider a driven-dissipative model From an applied point of view, they open the way to and demonstrate an all-optical control of these topolog- non-reciprocalphotonictransport,highlydesirabletoim- ical transitions and of the propagation direction of the plement logical photonic circuits. On the other hand, edgemodes. Oneofthestrikingfeaturesisthatthetopo- 2 logicalinversioncanbeachievedatnon-zerovaluesofthe TR-symmetry breaking term, allowing chirality control by weak modulation of the pump intensity. Phase diagram of the photonic and electronic QAH. We recall the linear tight-binding Hamiltonian of a honeycomb lattice in presence of Zeeman splitting and SOC of Rashba [57] and photonic type respectively [58]. It is a4by4matrixwrittenonthebasis(Ψ+,Ψ−,Ψ+,Ψ−)T, A A B B whereAandB standforthelatticeatomtypeand±for the particle spin: (cid:18)∆σ F (cid:19) (cid:18) f J f+λ (cid:19) H = z k,i , F =− k k,i i . (1) ki F† ∆σ k,i f−λ f J k,i z k,i i k J is the tunnelling coefficient between nearest neighbour micropillars (A/B). ∆ is the Zeeman splitting. λ (i = i e,p) are the magnitude for the Rashba (electronic) and TE-TM (photonic) induced SOC respectively [59]. The complex coefficients f and f± are defined by: k k,i Figure 1: (Color online) (a)-(b) Spin polarization textures in presence of TE-TM and Rashba SOC respectively (second (cid:88)3 (cid:88)3 branch). White arrows – the in-plane spin projection. (c)- fk = e(−ikdφj), fk±,e =± e(−i[kdφj∓φj]) (2) (d) Dispersions for TE-TM and Rashba SOC. The trigonal j=1 j=1 warpingappearsindifferentdirections. (∆=0,λe,p =0.2J). 3 (cid:88) f± = e(−i[kdφj∓2φj]) k,p j=1 whered arethelinksbetweennearestneighbourpillars φj (atoms)andφ =2π(j−1)/3theiranglewithrespectto j the horizontal axis. Qualitatively, the crucially different φdependenciesofthetunnelingf± areduetothediffer- k,i entwindingnumbersoftheRashbaandTE-TMeffective fields in the bare 2D systems. Without Zeeman field (∆ = 0), the diagonalization of these two Hamiltonians gives 4 branches of dispersion. NearK andK(cid:48) points, twobranchessplit, andtwo others intersect, giving rise to a so-called trigonal warp- Figure2: (Coloronline)Phasediagrams(a)fortheTE-TM ing effect, namely the appearance of three extra crossing SOC and (b) for the Rashba SOC with an applied field ∆. points (see (Fig. 1(c,d) and Fig. 3(a)). The differences Each phase is marked by the C of the bands. n between the two Hamitonians are clearly visible on the panelsofFig. 1whichshowa2Dviewofthe2ndbranch spin polarizations (a,b) and energies (c,d). On the pan- els (a,b), we see the difference of the in-plane winding thetopologicalphasediagram. Thetopologicalcharacter number around Γ (w =1 for Rashba and w =2 for of these Hamiltonians with the appearance of the QAH Γ,e Γ,p TE-TM SOC). Around K points, the TE-TM SOC tex- effect has already been discussed by deriving an effective turebecomesDresselhaus-likewithawindingw =−1 Hamiltonian close to the K point in different limits for K,p whereas Rashba remains Rashba with w =1. In each both the electronic [9, 14] and photonic cases [29, 50]. K,e case, the winding numbers around the K and K(cid:48) points However, the presence of other topological phase transi- have the same sign and add to give ±2 C for the elec- tions due to additional degeneracies appearing in other n tronicandphotoniccaserespectivelywhenTRisbroken. points of the first Brillouin zone was not checked. Onthepanels(c,d),onecanclearlyobservetheformation Figure 2 shows the diagram of topological phases of of small triangles near the Dirac points, the vertices of both models versus the SOC and Zeeman field strength. these triangles corresponding to the crossing points with The different phases are characterized by the band C n thethirdenergybands. Wecanobservethatthevertices that we calculate using the standard gauge-independent areorientedalongtheK−K(cid:48) directionforTE-TMSOC and stable technique of [60]. We remind that change of androtatedby60◦ (K−Γdirection)fortheRashbaSOC C isnecessarilyaccompaniedbygapclosing. Obviously, n case, a small detail, which has crucial consequences for thesephasediagramsaresymmetricwithrespectto∆= 3 0(withinvertedsignsofC forthenegativepart). Atlow Γ point for ∆ = 3J and then reopens as a trivial gap, n 3 ∆,bothmodelsarecharacterizedbyC =±2. However, whereas the two other bandgaps are still topological. n their C signs are opposite due to the opposite winding All-optical control of topological phase transitions. In n of their SOC around K. what follows, we propose a practical way to implement Figure 3(b) shows the corresponding band structure thephotonictopologicalphasesanalyzedabove. Wecon- for the photonic case, where the double peak structure centrate on the experimentally realistic configuration of aroundKandK’,arisingfromthetrigonalwarpingeffect aresonantlydrivenphotonic(polaritonic)lattice[39,40], and responsible for the C value, is clearly visible. In- including finite particle lifetime, without any applied n creasing either the SOC or the Zeeman field shifts these magneticfield,anddemonstratetheall-opticalcontrolof band extrema. In the photonic case, the band extrema thebandtopology. Weshowthatthetopologicallytrivial finally meet at the M point, which makes the gap close, bandstructurebecomesnon-trivialunderresonantcircu- as shown on the figure 3(c). The critical Zeeman field larly polarized pumping at the Γ point of the dispersion. value at which this transition takes place can be found A self-induced topological gap opens in the dispersion (cid:113) of the elementary excitations. The tuning of the pump analytically: ∆ = J2−4λ2. Increasing the fields fur- 1 p intensity allows to go through several topological transi- ther leads to an immediate re-opening of the gap with tions demonstrating the chirality inversion. the C passing from +2 to -1 for the valence band. This n A coherent macro-occupied state of exciton-polaritons case is shown on the figure 3(d), where the number of is usually created by resonant optical excitation. This band extrema is twice smaller than on 3(b). This phase regime is well described in the mean-field approximation transitionisentirelyabsentintheelectroniccasebecause [41, 61]. We can derive the driven tight-binding Gross- of the different orientations of the trigonal warping. Pitaevskii equation in this honeycomb lattice for a ho- Increasingthefieldevenfurtherleadstoasecondtopo- mogeneous laser pump F ((cid:126)=1). logicaltransitionthistimepresentinbothmodelsandas- sociatedwiththeopeningoftwoadditionalgapsbetween ∂ (cid:88) the two lower and two upper branches (in the middle of i∂tΨi = Hij(k)Ψj +Fiei((kp.r−ωpt) j the”conduction”bandandofthe”valence”band, corre- spondingly),asshownonthefigure3(d). Thistransition +(α1|Ψi|2+α2(cid:12)(cid:12)Ψi+(−1)(i+1) mod2(cid:12)(cid:12)2)Ψi (3) arises, when the minimum energy of the second branch where i,j =1..4 correspond to the four WF components at the Γ point is equal to the maximal energy of the (Ψ+,Ψ−,Ψ+,Ψ−). H are the matrix elements of the lowest band at the K point, and thus the system of 2 A A B B ij tight-binding Hamiltonian defined above (eq. 1) without bands (each containing 2 branches) is split into 4 bands the Zeeman term on the diagonal (∆ = 0). α and α (each containing a single branch). The corresponding 1 2 are the interaction constants between particles with the transition in the photonic case occurs when the Zeeman same and opposite spins, respectively. For polaritons, splitting is: ∆ = 3(J2 −λ2)/2J. The last topological 2 p the latter is suppressed [62] because it involves interme- phasetransitionoccurswhenthemiddlegapclosesatthe diate dark (biexciton) states, which are energetically far from the polariton states. Thus |α | (cid:28) α [63, 64] and 2 1 we neglect it. F is the pump amplitude. In the follow- i ing, we consider a homogeneous pump at k = 0 (pumping beam perpendicular to the cavity plane), which implies that its amplitude on A and B pillars is the same. However, the spin projections Fσ and F−σ, determining s s the spin polarization of the pump, can be different (s - sublattice, σ - spin). The quasi-stationary driven solu- tionhasthesamefrequencyandwavevectorasthepump (Ψσ =ei(kp.r−ωpt)Ψσ ) and satisfies the equations: s p,s (ω +iγ −α |Ψσ |2−α |Ψ−σ|2)Ψσ p p 1 p,s 2 p,s p,s +f JΨσ +fσ λ Ψ−σ =Fσ (4) kp p,−s kp p p,−s s where ω is the frequency of the pump mode. γ is the p p linewidth related to polariton lifetime (τ ), which allows p to take the dissipation into account. The tight-binding Figure3: (Coloronline)Dispersionofphotonicgraphenefor terms (f ,fσ ) of the polariton graphene induce a cou- differentZeemanfield. (a)∆=0,(b)∆=0.5J,(c)∆=∆1, kp kp (d) ∆=1.5J. (λ =0.2J). The different gaps are shown in pling between the sublattices and polarizations. Eq. (4) p grey with the values of the associated Cn. is written for an arbitrary pump wave vector kp. In the following, we consider a pump resonant with the energy 4 a) b) C=+2 C=-1 Figure4: (Coloronline)(a)Topologicalphasediagraminthe Figure5: (Coloronline)Calculatedimagesofemissionfrom resonant pump regime versus the TE-TM SOC and SIZ. (b) the surface states (a) Ω = 0.3 meV, C = 2 (b) Ω = 0.6 SI SI Gap sizes and sign evolution along a path of constant SOC meV, C=-1. Arrows mark the propagation direction. λ = 0.2J (dashed line on (a)). The red and blue curves p correspond to the opening of additional gaps. magnitude of the different gaps multiplied by the sign of the C of the valence band (C = (cid:80)n C ) [66] for of the bare lower polariton dispersion branch in the Γ n i=1 n a given value of the SOC, a quantity highly relevant ex- point (ω = −f J = −3J and k = 0), marked with an p Γ p perimentally. In [39, 40] J is of the order or 0.3 meV, arrow in Fig. 3(a) which implies the stability of the ele- whereas the mode linewidth is of the order of 0.05 meV. mentary excitations. We compute the dispersion of the Band gaps of the order of 0.2 J should be observable. elementary excitations using the standard WF of a weak TheSIZmagnitudeshownonthex-axis(below1.5meV) perturbation (|u|,|v|(cid:28)|Φ(cid:126) |): p is compatible with the experimentally accessible values. So in practice the topological transition is observable to- Φ(cid:126) =ei(kp.r−ωpt)(Φ(cid:126) +uei(k.r−ωt)+v∗e−i(k.r−ω∗t)) (5) p gether with the specific dispersion of the edge states in thedifferentphaseswhicharepresentedin[59]. Wenote where Φ(cid:126) =(Ψ+ ,Ψ− ,Ψ+ ,Ψ− )T , u and v are vec- p p,A p,A p,B p,B that the emergence of topological effects driven by in- tors of the form (u+A,u−A,u+B,u−B)T [59]. teractions in bosonic systems has already been reported, A circular pump induces circularly polarized macro- such as Berry curvature in a Lieb lattice for atomic con- occupied state (n− = 0), and n = n+ = n+A + n+B = densates [67] and topological Bogoliubov edge modes in |Ψ+p,A|2+|Ψ+p,B|2. Combined with spin anisotropic inter- two different driven schemes based on Kagome lattices actions,itleadstoaSelf-InducedZeeman(SIZ)splitting [23, 68] with scalar particles. whichbreaksTRsymmetry. Asimpleanalyticalformula To confirm our analytical predictions and support the of the k-dependent SIZ splitting between the two lower observability in a realistic pump-probe experiment (see branches is obtained for λp =0: sketch in [59]), we perform a full numerical simulation beyond the tight-binding or Bogoliubov approximations. (cid:113) Ω =ω +|f |+ (ω +|f |J −2α n+ )2−(α n+ )2 We solve the spinor Gross-Pitaevskii equation for polari- SI p k p k 1 A/B 1 A/B tons with quasi-resonant pumping: One of the key differences with respect to the magnetic field induced Zeeman field is the SIZ dependence on the i(cid:126)∂∂ψt± =−2(cid:126)m2 ∆ψ±+α1|ψ±|2ψ±− 2i(cid:126)τψ±+P0+e−iωt(6) wdeanvceevehcatosraslarenaddeynebregeinessohfotwhnebtaorelemadodteos.thTehiinsvdeerpsieonn- +Uψ±+β(cid:16)∂∂x ∓i∂∂y(cid:17)2ψ∓+(cid:80)jPj−e−(t−τt020)2−(r−σr2j)2−iωt of the effective field sign (and thus the inversion of the topology) when both applied and SIZ fields are present where ψ (r,t),ψ (r,t) are the two circular components + − in a Bose-Einstein condensate [51]. of the WF, m = 5 × 10−5m is the polariton mass, el The figure 4(a) shows the diagram of topological τ = 30 ps the lifetime, U is the lattice potential. The phasesunderresonantpumping(versustheSIZ)whichis main pumping term P is circular polarized (σ+) and 0+ quite similar to the one under magnetic field. A method spatiallyhomogeneous, whilethe3pulsedprobesareσ− tocomputetheC oftheBogoliubovmodeshasbeende- and localized on 3 pillars (circles). The results (filtered n veloped in [65]. The procedure we use is detailed in the byenergyandpolarization)areshowninFig. 5. Ascom- supplementary [59]. The only difference with respect to pared with the previously analyzed [50, 51] C = 2 case thelinearcaseconcernstheopeningofthetwoadditional (a), a larger gap of the C = −1 phase (b) demonstrates gapswhichdoesnottakeplaceatthesamepumpingval- a better edge protection, a longer propagation distance, ues, because of the difference between the SIZ fields in andaninverteddirection,allachievedbymodulatingthe the upper and lower bands. The figure 4(b) shows the pump intensity. 5 Conclusions. 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[55] D.R.Gulevich,D.Yudin,I.V.Iorsh,andI.A.Shelykh, Phys.Rev.B94,115437(2016),URLhttp://link.aps. org/doi/10.1103/PhysRevB.94.115437. Optical spin-orbit coupling [56] F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008), URL http://link.aps.org/doi/ 10.1103/PhysRevLett.100.013904. In the main text, we introduce two kind of SOC λe [57] M. Zarea and N. Sandler, Phys. Rev. B 79, and λp for electrons and polaritons respectively. We 165442 (2009), URL http://link.aps.org/doi/10. choose this notation to make clearer the comparison be- 1103/PhysRevB.79.165442. tween the two cases. Indeed, in our precedent works [58] A. V. Nalitov, G. Malpuech, H. Terças, and D. D. on polariton honeycomb lattices, we used the notation Solnyshkov, Phys. Rev. Lett. 114, 026803 (2015), URL λ = δJ [50, 51, 58]. Taking into account the TE- http://link.aps.org/doi/10.1103/PhysRevLett.114. p TM splitting the tunneling coefficients are defined in the 026803. [59] See Supplemental Material at [URL will be inserted by circular-polarization basis as: publisher] for more details on the calculations. (cid:104)A,±|H|B,±(cid:105) = −J (7) [60] T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Physical Society of Japan 74, 1674 (2005). (cid:104)A,±|H|B,∓(cid:105) = −λ e−2iφj p 7 J is the tunneling coefficient without spin inversion, like macro-occupied state reads: in conventional graphene. The SOC coefficient λ is p defined by: λ = δJ = (J − J )/2, where J and p L T L J are the tunneling coefficients for the longitudinally T and transversally-polarized polaritons respectively. The Φ(cid:126) =ei(kp.r−ωpt)(Φ(cid:126) +uei(k.r−ωt)+v∗e−i(k.r−ω∗t)) (8) p difference of phase between Rashba (e±iφ) and TE-TM (e±i2φ) terms comes from the different winding number where Φ(cid:126) =(Ψ+ ,Ψ− ,Ψ+ ,Ψ− )T , u and v are vec- between the two in plane SOC. p p,A p,A p,B p,B torsoftheform(u+,u−,u+,u−)T too. Indeedbecauseof A A B B the non-linear term of the GP equation, the Bloch state Weak excitations dispersions in the resonant pump characterizedbyawave-vectork andfrequencyω iscou- regime pled to its complex conjugated, namely the wave with a wave-vector −k and frequency −ω. Then, inserting this In this section we present the derivation of the Bo- wave function in the driven dissipative Gross-Pitaevskii goliubov excitation of a resonantly pumped interacting equation (main text) and linearizing for u and v, we ob- photon system. The weak perturbation of the pumped tain the following matrix: (d+A−ωp−iγp) α2Ψ−p,∗AΨ+p,A −fkp+kJ −fk+p+kλp α1Ψ+p,2A α2Ψ−p,AΨ+p,A 0 0   α2Ψ+p,∗AΨ−p,A (d−A−ωp−iγp) −fk−p+kλp −fkp+kJ α2Ψ−p,AΨ+p,A α1Ψ−p,2A 0 0   −fkp+kJ −fk−p∗+kλp (d+B−ωp−iγp) α2Ψ−p,∗BΨ+p,B 0 0 α1Ψ+p,2B α2Ψ−p,BΨ+p,B  M = −−αfk1+pΨ∗++pk,2AJ∗ −α−2fΨkp−p,+∗AkΨλ+pp,∗A α2Ψ−p,0BΨ+p,∗B (d−B−ω0p−iγp)(ωp−iγ0p−d+A) −α2Ψ−p0,AΨ+p,∗A α2Ψfk∗−pp,−BkΨJ+p,B fαk+1p∗Ψ−−pk,λ2Bp  (9) −α2Ψ−p,∗AΨp+,∗A −α1Ψp−,2A∗ 0 0 −α2Ψ−p,∗AΨ+p,A (ωp−iγp−d−A) fk−p∗−kλp fk∗p−kJ   0 0 −α1Ψ+p,2B∗ −α2Ψ−p,∗BΨ+p,∗B fk∗p−kJ fk−p∗−kλp (ωp−iγp−d+B) −α2Ψ−p,AΨ+p,∗A  0 0 −α2Ψ−p,∗BΨ+p,∗B −α1Ψ−p,2B∗ fk+p∗−kλp fk∗p−kJ −α2Ψ−p,∗BΨ+p,B (ωp−iγp−d−B) The diagonal elements are defined by: where i index labels the different u (v ) components of i i an eigenstate. dσ =2α |Ψσ |2+α |Ψ−σ|2 (10) s 1 p,s 2 p,s The Bogoliubov eigenenergies and eigenvectors (u+,u−,u+,u−,v+,v−,v+,v−)T are finally obtained by This condition physically signifies that the creation of A A B B A A B B diagonalizing this 8 by 8 matrix. one bogolon corresponds to the creation of a quanta of √Intheexpressionfortheself-inducedfieldΩSI(Γ)/2= energy ω. 3α1n/2 the factor√1/2 comes from the presence of two Chern numbers of Bogoliubov excitations sublattices and the 3 appears from resonant pumping, as compared with a blue shift of an equilibrium condensate µ=αn. The normalisation condition, requires for the Bogoli- ubovtransformationtobecanonical,namelytokeepbo- The standard formula for the computation of the golons as bosons reads [69, 70]: Chern number can be applied, but taking into account that bogolons are constituted by two Bloch waves of op- (cid:88) |u |2−|v |2 =1 (11) posite wave vectors: i i 1≤i≤4 8 1 (cid:90)(cid:90) C = ∇ ×(cid:104)Φ(k)|∇ |Φ(k)(cid:105)dk (12) 2iπ k k BZ 1 (cid:90)(cid:90) 1 (cid:90)(cid:90) = ∇ ×(cid:104)u(k)|∇ |u(k)(cid:105)dk+ ∇ ×(cid:104)v(−k)|∇ |v(−k)(cid:105)dk 2iπ k k 2iπ k k BZ BZ 1 (cid:90)(cid:90) 1 (cid:90)(cid:90) = ∇ ×(cid:104)u(k)|∇ |u(k)(cid:105)dk+ ∇ ×(cid:104)v(k)|∇ |v(k)(cid:105)dk 2iπ k k 2iπ −k −k BZ −BZ 1 (cid:90)(cid:90) 1 (cid:90)(cid:90) = ∇ ×(cid:104)u(k)|∇ |u(k)(cid:105)dk− ∇ ×(cid:104)v(k)|∇ |v(k)(cid:105)dk (13) 2iπ k k 2iπ k k BZ BZ where dk = dk dk and we drop the band index n for x y simplicity. We can see that the integration of the v part makes appear a minus sign because the integration takes place over an inverted Brillouin zone (BZ). This fact has been noticed in ref [65], and is commonly used [23, 67, 71, 72]. It is typically formulated by introducing a matrix τ = σ ⊗114 directly in the definition of the z z Berry connexion A=(cid:104)Φ(k)|∇ τ |Φ(k)(cid:105). k z Bogoliubov edge states To demonstrate one-way edge states in tight-binding approach, we derive a 8Nx8N Bogoliubov matrix for a polariton graphene stripe, consisting of N coupled infi- nite zig-zag chains following the procedure of Ref.[51]. For this, we set a basis of Bogoliubov Bloch waves Figure6: (Coloronline)(a,b)Bandstructuresofagraphene (u± ,v± )wherenindexnumeratesstripes,andk ribbon in two different phases. Blue and red colors refer to A/B,n A/B,n y is the quasi-wavevector in the zigzag direction. The di- the states localized on the right and left edges. Parameters: λ = 0.2J and (a) α n = 1J, (b) α n = 4J. (c) Real space agonalblocksdescribecouplingwithinonechainandare p 1 1 sketchoftheexperimentalsetup. Theyellowarrowsrepresent derived in the same fashion as the M matrix in the pre- the edge states when C=+2 (dashed ones when C=-1). vious section (2), coupling between stripes is accounted for in subdiagonal blocks. In Fig. 1(a), there is only one topological gap charac- Figures 1(a,b) show the results of the band structure terized by a Chern number +2 and hence there are two calculationfortwodifferentvaluesofα n. Thedegreeof edge modes on each side of the ribbon. In Fig. 1(b), we 1 localizationonedgesiscalculatedfromthewavefunction can observe three topological gaps with the Chern num- densities on the edge chains |Ψ |2 and |Ψ |2 (left/right, ber of the top and bottom bands being ±1 respectively. R L see inset), and is shown with colour, so that the edge Each of them is characterized by the presence of only states are blue and red. one edge mode on a given edge of the ribbon, and the 9 group velocities of the modes are opposite to the previ- tion. The inversion of chirality of center gap edge states ous phase: the chirality is controlled by the intensity of (Fig. 1(a,b)) should be observable in a pump-probe ex- the pump. This inversion, associated with the change of periment as shown by the numerical simulation in the the topological phase (|C| = 2 → 1), is fundamentally main text. A sketch of the experiment using a σ+ and a differentfromtheoneofRef. [51], observedforthesame σ− polarized lasers (the homogeneous pump and the lo- phase (|C|=2). calizedprobe)ispresentedonFig.1(c). Oneshouldnote This optically-controlled transition allows to observe that we can also obtain the inverted phases more con- the inversion of chirality for weak modulations of a TR- ventionally by inverting the direction of the self-induced symmetry breaking pump around a non-zero constant Zeeman field which is controlled by the circularity of the value, which can also possibly be used for amplifica- homogeneous pump.