Spontaneous topological transitions in exciton-polariton condensate graphene due to spin bifurcations H. Sigurdsson,1, 2 Y. S. Krivosenko,3 I. V. Iorsh,3 I. A. Shelykh,4, 3 and A. V. Nalitov4, 3 1School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom 2Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow 143026, Russian Federation 3ITMO University, St. Petersburg 197101, Russia 4Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland (Dated: July 26, 2021) We theoretically study the spontaneous formation of the quantum anomalous Hall effect in a graphene system of spin-bifurcated exciton-polariton condensates under nonresonant pumping. We demonstrate that, depending on the parameters of the structure, such as intensity of the pump and coupling strength between condensates, the system shows rich variety of macroscopic magnetic ordering, including analogs of ferromagnetic, antiferromagnetic, and resonant valence bond phases. Transitions between these magnetic polarized phases are associated with dramatic reshaping of the spectrum of the system connected with spontaneous appearance of topological order. Introduction. Recent decades have witnessed a shift are honeycomb [18, 21, 30] and Kagome [19, 24, 25] lat- in attention from both the condensed matter and opti- tices which can be obtained either by controllable etching cal community in investigation on the properties of bulk of a planar microcavity or by using spatial light modu- of materials to instead the properties of their interfaces. lator to control the profile of the external optical pump. It is now well known that there exists a particular class The band inversion and opening of the topological gap is of materials with inverted structure of the bands, which then achieved by cumulative action of the TE-TM split- possess protected states propagating on the system sur- ting of the photonic mode and Zeeman splitting of the face, referred to as topological insulators [1{3]. The en- excitonic mode induced by the application of an external ergy of these edge states lies within the bandgap, and magnetic field [18]. However, in conventional semicon- thus they are protected with respect to scattering into ductor materials excitonic g-factors are extremely small, the bulk. Depending on the dimensionality of a sys- and one needs magnetic fields of tens of Tesla to open the tem, one should distinguish between three-dimensional topological bandgap of at least several meV. The situa- (3D) topological insulators where topological states ap- tion can be potentially improved by using diluted mag- pear on the two-dimensional (2D) surface boundary of netic microcavities [31, 32]. However, the technology of the bulk [4{10], and 2D topological insulators where chi- producing a high quality patterned semimagnetic cavity ral 1D channels form on the system boundary [11, 12]. is still only in its initial stages. It has been recently shown that optical analogs of topo- In the present article we develop an alternative ap- logically nontrivial phases, in 2D systems, may arise in proach for the realization of the quantum anomalous Hall purely photonic structures [13{15] or when driven into effect in a 2D polariton Z-topological insulator without the strong light-matter coupling regime, where hybrid application of any external magnetic fields. Our idea quasiparticles known as cavity exciton-polaritons (from is based on the concept of the spontaneous spin bifur- here on polaritons) are formed [16{21]. Polaritons com- cation in a system of interacting polariton condensates bine the advantages of photons, such as extremely low forming a net magnetic polarization, first proposed in effective mass and long coherence length, with those of Ref. [33] and developed further in the works [34{37]. excitons, namely the possibility of control by external Here we consider a honeycomb polariton condensate lat- electric and magnetic fields together with strong non- tice (polariton graphene) under nonresonant pumping. linear response stemming from interparticle interactions. We demonstrate that, depending on the pump intensity The spin structure of the exciton (or rather, the polari- and coupling strength between the nodes (condensates) ton) is then directly related to the circular polarization of the lattice, the spin bifurcation mechanism can result degree of the cavity photonic mode. Such a union then in spontaneous formation of distinct spin-ordered lattice leads to a rich interplay between nonlinear and topologi- phases analogous to ferromagnetic (FM), antiferromag- cal properties [22{28] with optical lasing in topologically netic (AFM), and resonance valence bond states. Tran- protected edge modes [21, 29]. sition between different phases is associated with cardi- The vast majority of the current proposals on 2D po- nal reshaping of the spectrum of the system excitations lariton topological insulators are based on Z (or Chern) and spontaneous appearance/disappearance of topologi- insulators with the following requirements: Polaritons cal order. should be placed into a 2D lattice of a particular symme- Spin bifurcations and phase transitions in polariton try allowing the appearance of Dirac points in the Bril- graphene. A lattice of driven-dissipative connected po- louin zone where the bands touch each other. Examples lariton condensates is conventionally modeled, in the 2 AFM Dipole Stripe FM tight-binding picture, with a set of generalized Gross- (a) Nontrivial Pitaevskii equations for the spinor order parameters T Ψn = ( n+; n−) , corresponding to spin-up and spin- down polaritons at the n-th site: dΨn i + iγ 1 z i = − g(Sn) − σ^x + (¯αSn + αS σ^z)Ψn dt 2 2 2 n Y-polarized 1 X − [J + δJ (cos(2'm)^σx + sin(2'm)^σy)] Ψm; (1) Uncondensed 2 hnmi where the summation is taken over the nearest neighbors, (b) 'm are the angles of links connecting the neighboring (i) (ii) sites n and m of the honeycomb lattice. We define the Degree of circular polarization n-th node particle population Sn and z-component of the z condensate pseudospin Sn as: 2 2 2 2 j n+j + j n−j z j n+j − j n−j Sn ≡ ;S ≡ : (2) 2 n 2 (iv) We also define an effective decay rate g(Sn) = ηSn + (iii) Γ − W with Γ being the polariton decay rate, W the replenishment rate of the condensate non-polarized inco- herent pump, and η is the gain-saturation nonlinearity. The constants and γ define the splitting of the XY - polarized states in both energy and decay respectively FIG. 1. (a) Phase map of the honeycomb polariton sys- due to the inherent cavity birefringence, and α¯ = α +α 1 2 tem. At low pump powers W no condensation occurs (red and α = α1 − α2 are spin-anisotropic interaction pa- area). When crossing the condensation threshold (dashed- rameters. Finally, J > δJ are spin conserving and non- dotted line) the system condenses and settles into Y -polarized conserving (TE-TM splitting) tunneling rates of polari- state (white area). At higher powers the system bifurcates tons between nodes respectively. into an organized spin pattern (above red curve). Bandgap The condensation threshold of the system is defined as opening in each phase takes place in the yellow area whereas the point where an eigenvalue of the linearized Eq. (1) white patterned areas the bandgap is closed. Only in the FM phase the gap is topologically nontrivial. (b) Cutout pieces obtains a positive imaginary component due to increase of the four spin graphene patterns labeled as (i) AFM, (ii) of the laser power W , leading to the triggering of the Dipole, (iii) Stripe, and (iv) FM. Yellow arrows denote the stimulated bosonic scattering into the condensed state at unit cell vectors of each pattern. Parameters are chosen based −1 −1 Wcond = Γ − γ. Due to the splitting γ in the lifetimes of on experiment [33]: η = 0:005 ps ;Γ = 0:1 ps ; = 0:06 −1 −1 the linear polarized states the condensate first forms an, ps ; γ = 0:2; α1 = 0:005 ps ; α2 = −0:1α1. T in-phase, Y -polarized state, i.e., Ψn = Ψn+1 / (1; −1) (white area in Fig. 1a). This Y -polarized state however becomes unstable at higher pumping powers and under- where n"# denotes the number of AFM neighbors. goes a bifurcation into one of two states with high degree In Fig. 1a we plot the minimum of Eq. (4) as a func- of the circular polarization at individual nodes [33, 36]. tion of coupling strength J (red curve) neglecting TE-TM We begin our consideration by presenting a class of splitting. The cusps in the red curve indicate that the stationary solutions which minimize the spin bifurcation lowest bifurcation point is shared between two distinct threshold [37], spin phases. The four spin phases of interest, verified by z z numerical integration of Eq. (1), are shown in Fig. 1b(i- Ψn+1; if Sn = Sn+1; Ψn = z z (3) iv) and are labeled AFM, Dipole, Stripe, and FM phases −σ^xΨn+1; if Sn = −Sn+1: respectively. The ansatz above describes in-phase FM bonds and anti- We point out that the bifurcation threshold for AFM phase AFM bonds between nearest neighbors respec- and FM phases is invariant of δJ whereas for dipole and tively. Plugging Eq. (3) into Eq. (1), and setting the stripe phases, strictly speaking, this is not the case as condition that all nodes have the same number of co- and for them the ansatz given by the Eq. (3) should be mod- counter-polarized nearest neighbours (equivalence crite- ified. However, given that δJ=J 1 (which is usually ria), the coupled set of the equations of motion reduce to the case in micropillar structures in standard semicon- a single equation with a bifurcation threshold, ductor microcavities), it is reasonable to infer that Wbif 2 2 ( − n"#J) + γ is only weakly affected by δJ.