Spontaneous Topological Transitions in Exciton-Polariton Condensate Graphene Due to Spin Bifurcations

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 = (ψn+, ψn−) , corresponding to spin-up and spin- down polaritons at the n-th site: dΨn i + iγ 1 z i = − g(Sn) − σˆx + (¯αSn + αS σˆz)Ψn dt 2 2 2 n Y-polarized 1 X − [J + δJ (cos(2ϕm)ˆσx + sin(2ϕm)ˆσy)] Ψm, (1) Uncondensed 2 hnmi where the summation is taken over the nearest neighbors, (b) ϕm are the angles of links connecting the neighboring (i) (ii)

sites n and m of the honeycomb lattice. We define the polarization Degree of circular n-th node particle population Sn and z-component of the z condensate pseudospin Sn as: 2 2 2 2 |ψn+| + |ψn−| z |ψn+| − |ψn−| 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 −σˆxΨn+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. Thus, the calculated red Wbif = Γ − γ + η , (4) α( − n↑↓J) curve shown in the Fig. 1a serves as a good indicator for 3

FIG. 2. (i) The band structure of the whole Brillouin zone for the four spin phases in the honeycomb lattice of spin bifurcated polariton condensates. Here, J = 1 is taken as the unit of energy. (A, D): all the four bands are displayed, (B, C): only two bands above and two bands under the midgap are presented. Bottom panels show (ii) the eigenenergies along Γ → M → K → Γ pathway, and (iii) band structure of the ribbon with zigzag edges, the edge states are depicted by red and green dots. Parameters for the AFM: δJ = 0.1J,Δ = 1.3J. Dipole: δJ = 0.1J,Δ = 2.2J. Stripe: δJ = 0.1J,Δ = 3J. FM: δJ = 2J/3, Δ = 5J/3. The band structures of the bulk were calculated on a 200×200 mesh grid in k-space. The size of ribbons was 30 (width) by 100 (length) unit cells. the bifurcation threshold of these nontrivial states in the bispinor |A+, A−, B+, B−i inner-cell states, where A presence of TE-TM splitting. We have performed numer- and B indicate the graphene sublattices, and “+(−)” ical calculations of Eq. (1) that verify that this is indeed specifies right (left) circular polarization (i.e., the spin the case. of the polaritons). The total Hamiltonian in momentum ˆ P Band structure and topological states. In the following space is then written H = k |kihk| ⊗ Hk. The first we discuss the excitation spectra of the stable spin bifur- term in Eq. (5) corresponds to the polaritonic graphene cated condensate configurations. We employ an effective with TE-TM splitting. The 2×2 operator Jˆk, dependent field model, treating the effect of spin-polarized lattice on the quasi-wavevector k = (k1, k1), is written, nodes by introduction of local z-directed (out of the cav- ˆ ˆ ˆ −ik1 ˆ −ik2 ity plane) magnetic fields. The idea is based on expansion Jk = J1 + J2 e + J3 e , (6) of the Bogoliubov dispersion in the two spin components p 2 with E±(E± + U±) ≈ E± + U±/2 with U± ≈ α1|ψn±| being the self interaction energy. It yields the effective −2iϕm z ˆ J δJ e | | Jm = 2iϕ , m = 1, 2, 3. (7) magnetic field magnitude Δ = α1 Sn and the validity δJ e m J region of the approximation as E± U±. The effective field model allows clear analytical investigation of and J and δJ as in Eq. (1). Both k1 and k2 can be chosen the topological properties of the lowest band gap open- to vary from −π to π and to cover the whole Brillouin ing at E ∼ J as long as J Δ. Furthermore, as the zone. In the second term of Eq. (5), the on-diagonal | z | − pseudospin strength is given by Sn = Sn = (W Γ)/η blocks µ1(2)Δ σz serve to account for the excitations in for the fully polarized condensates (see Eq. (S17) in [38]) the magnetic patterns depicted in Fig. 1b. The magni- one can adjust the strength of Δ such that effects due tude of the Zeeman splitting induced by the polarized to birefringence are negligible. The effective field model condensate reads Δ = α1S where we have omitted the then obeys J Δ . A more rigorous Bogoliubov index n in Sn since the condensate density is taken equal treatment is addressed in Ref. [38]. at each lattice site. Here σˆz is the z-Pauli matrix, and

Firstly, to examine the band structures of the AFM (i) the coefficients µ1 and µ2 define FM (µ1 = µ2 = 1) and and FM (iv) configurations shown in Fig. 1b, we scruti- AFM (µ1 = −µ2 = 1) phases. The translation basis vec- × nize the following 4 4 tight-binding Hamiltonian: tors (a1,2) in real space are chosen to be conventional for the graphene lattice. 1 0 Jˆk 1 µ1Δσ ˆz 0 Hk = − † + . (5) Since FM phase corresponds to the case of a uniform, 2 Jˆ 0 2 0 µ2Δσ ˆz k out-of-plane, external magnetic field, a gap opens (see The above Hamiltonian is written in the basis of the Fig. 2d) between the bands, characterized by different 4

marks the boundary between gapless and gapped phases. The points A and D correspond to Fig. 2a and d, respectively. Figure 3e shows the band structures of FM-phase ribbons and serves to illustrate the effect of Chern numbers of the bands (indicated in Fig. 3d) on the dispersion of the chiral edge states. To examine the band structure of the stripe and dipole phases, the following 8×8 Hamiltonian should be constructed in reciprocal (k1, k2) space: 1 Δ Hk = bJ(k) + · diag (µ1, µ2, µ3, µ4) ⊗ σˆz . (8) 2 2 Note, that a unit cell in these two cases differs from that corresponding to AFM and FM phases and should be constructed as a pair of graphene unit cells taken suc- FIG. 3. (A-D) Color maps of the energy gap Eg , for four spin phases, as a function of δJ and Δ. The solid orange line cessively, with the translational vectors being a1 and separates the gapless (solid dark blue) and gapped (linearly 2a2. The first term in Eq. (8), bJ(k), represents po- fading blue) domains. The dashed orange lines are positive- laritonic graphene with TE-TM splitting and is speci- integer-Eg contours. (E) The band structures of FM-type fied in the Supplementary material [38], the second term ribbons for (α): δJ=0.2, Δ=0.4, (β): δJ=2/3, Δ=5/3, and is responsible for the magnetic patterns: for the dipole (γ): ΔJ=0.5, Δ=3.5. The bands Chern numbers are dis- phase one sets µ = µ = 1, µ = µ = −1, for the played in light yellow boxes. Here J=1 is taken as a unit of 1 4 2 3 − measurement. stripe phase µ1 = µ2 = 1, µ3 = µ4 = 1. Both phases are characterized by topologically trivial band structure and edge states localized within the bulk (see Fig. 2b,c). Figure 3(b,c) shows phase diagram for dipole Chern numbers, and bridged by chiral edge states [18]. and stripe phases with points B and C corresponding to Fig. 2d(i) displays the band structure of this phase, which the Fig. 2b,c. In the trivial case δJ = 0 one√ arrives to the is numerically obtained for δJ = 2/3, Δ = 5/3, with gap opening condition Δ > J and Δ > 3J for dipole energy counted in units of J. We used the convention and stripe spin phases respectively which is plotted in 0 for which K and K points in the first Brillouin zone Fig. 1a indicating that gap opening only takes place at are positioned at (k1, k2) equal to (2π/3, −2π/3), and higher condensate densities (i.e., higher excitation pow- vice versa, Γ is placed at the origin, and M at (π, π). ers). Fig. 2d(ii) shows a slice of the band structure along Γ → To investigate the FM phase in more detail, we per- M → K → Γ pathway shown by the green solid line formed Bogoliubov linearization of Eq. (1), generalized at the bottom of Fig. 2d(i). Figure 2d(iii) illustrates the to all four spin patterns, and applied it to the FM band structure of the graphene stripe (not to be confused one (see [38] for more details). Fig. 4 demonstrates with the ”stripe” phase in Fig. 1b) with zigzag edges (A) the topological phase diagram for the effective spin- revealing the topologically protected edge states, marked orbit interaction strength δJ/J and the effective inter- by red and blue colored lines for each edge respectively. action energy Δ, and (B, C) eigenenergy curves along For AFM phase, it can be shown that a gap opens when Γ → M → K → Γ pathway. In addition, green dotted Δ > 3δJ, the gap value being Eg = 2(Δ− 3δJ) [33]. The and red dashed curves show the dispersions of Eq. (5) gapped spectrum in the AFM lattice is characterized by without and with effective magnetic field respectively, trivial topology (see Fig. 2a) and no edge states connect- with the corresponding value Δ. Note that the bogolon ing the bulk bands. In Fig. 1a the dotted white area in dispersion overlaps with the one obtained in the effective AFM phase corresponds to the bands touching, whereas magnetic field approximation in the region of high en- the yellow area to the bands being gapped. Here, the ergies E Δ and expectedly deviates for low energies boundary separating the two regimes in Fig. 11a is calcu- (in the vicinity of Γ point). This shows that the effec- lated for the case of strong TE-TM splitting δJ/J = 0.5. tive magnetic field model accurately describes the linear We point out that that gapped-ungapped AFM boundary excitations of the condensate in the vicinity of K-point. coincides with the Wbif boundary (red curve in Fig. 1a) Moreover, Fig. 4(A) clearly resembles Fig. 3(D). when δJ → 0. The AFM edge states shown in Fig. 2a(iii), Conclusions. We have proposed an experimen- separated from the bands and marked by red and blue tally friendly geometry for realization of an optical Z- colors, are doubly degenerate and are not topologically topological insulator based on polaritonic graphene in protected. Figures 3(a,d) show phase diagrams for the the spin bifurcation regime. Differently from previous AFM and FM spin phases as a function of effective Zee- works, our proposal does not require application of an man splitting and TE-TM splitting. The dark orange line external magnetic field and the topological order ap- 5

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