ARTICLE

DOI: 10.1038/s41467-018-03982-7 OPEN Manipulating type-I and type-II Dirac in cavity-embedded honeycomb metasurfaces

Charlie-Ray Mann 1, Thomas J. Sturges 1, Guillaume Weick2, William L. Barnes 1 & Eros Mariani1

Pseudorelativistic Dirac quasiparticles have emerged in a plethora of artificial systems that mimic the underlying honeycomb symmetry of graphene. However, it is notoriously difficult to manipulate their properties without modifying the lattice structure.

1234567890():,; Here we theoretically investigate polaritons supported by honeycomb metasurfaces and, despite the trivial nature of the resonant elements, we unveil rich Dirac physics stemming from a non-trivial winding in the light–matter interaction. The metasurfaces simultaneously exhibit two distinct species of massless Dirac polaritons, namely type-I and type-II. By modifying only the photonic environment via an enclosing cavity, one can manipulate the location of the type-II Dirac points, leading to qualitatively different phases. This enables one to alter the fundamental properties of the emergent Dirac polaritons while preserving the lattice structure—a unique scenario which has no analog in real or artificial graphene systems. Exploiting the photonic environment will thus give rise to unexplored Dirac physics at the subwavelength scale.

1 EPSRC Centre for Doctoral Training in Metamaterials (XM2), Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, UK. 2 Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France. Correspondence and requests for materials should be addressed to C.-R.M. (email: [email protected]) or to E.M. (email: [email protected])

NATURE COMMUNICATIONS | (2018) 9:2194 | DOI: 10.1038/s41467-018-03982-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03982-7

he groundbreaking discovery of monolayer graphene1 has photonic cavity and simply changing the cavity height, one can inspired an extensive quest to emulate massless Dirac induce multiple phase transitions including the multimerging of T fi quasiparticles in a myriad of distinct arti cial graphene type-I and type-II Dirac points and the annihilation of type-II Dirac – systems2 11, ranging from ultracold atoms in optical lattices3 to points. This striking tunability results in qualitatively different evanescently coupled photonic waveguide arrays4. Owing to their polariton phases, despite the preserved lattice structure. In parti- honeycomb symmetry, linear band-degeneracies manifest in the cular, we unveil a morphing between a linear and a parabolic quasiparticle spectrum which we call conventional Dirac points spectrum accompanied by a change in the topological Berry phase, (CDPs). These belong to the ubiquitous type-I class of two- and an environment-induced inversion of chirality, all of which dimensional (2D) Dirac points that are characterized by Dirac have no analog in graphene or artificial graphene systems studied cones with closed isofrequency contours. As a result, the corre- thus far. Therefore, this unique paradigm will give rise to unex- sponding quasiparticles are described by the rather exotic 2D plored Dirac-related phenomena at the subwavelength scale, such as massless Dirac Hamiltonian12, and thus offer fundamental insight anomalous Klein tunneling, negative refraction, and pseudomag- into pseudorelativistic phenomena such as the iconic Klein netic Landau levels, which can all be tuned via the photonic paradox13. The latter is responsible for the suppression of back- environment alone. scattering and for the antilocalization of massless Dirac quasi- particles, which are highly desirable properties for efficient Results quasiparticle propagation in novel devices. Hamiltonian formulation. While metamaterials have tradition- Since the existence of type-I CDPs is intrinsically linked to the ally been described in terms of macroscopic effective honeycomb structure, the fundamental properties of the massless properties30,33,43, the importance of crystallinity is becoming Dirac quasiparticles are notoriously robust and difficult to increasingly apparent44. Therefore, to capture the essential manipulate. However, by exploiting meticulous control over the physics related to complex non-local effects that arise from lattice structure, artificial graphene systems have enabled the strong multiple-scattering45, here we study the properties of the exploration of Dirac quasiparticles in new regimes that are dif- – cavity-embedded honeycomb metasurface by means of a ficult, if not impossible to achieve in graphene itself14 19. Among microscopic Hamiltonian formalism. This allows us to clearly others, the archetypal example which has attracted considerable identify the distinct physical origins of the type-I and type-II interest is the paradigm of strain-engineering, where it has been Dirac points. = shown that lattice anisotropy can induce the merging and anni- The full polariton Hamiltonian of this system reads Hpol 3,14–16,20–23 + + hilation of type-I CDPs , and that aperiodicity can Hmat Hph Hint, where the interaction Hamiltonian Hint generate large pseudomagnetic fields17,24. couples the matter and photonic subspaces whose free dynamics Moreover, the recent discovery of type-II Dirac/Weyl semi- – are governed by Hmat and Hph, respectively. We employ the metals25 29 sparked a burgeoning exploration into the prospects Coulomb gauge, where the instantaneous Coulomb interaction of a rarer type-II class of three-dimensional Dirac/Weyl points. between the meta-atoms is incorporated within the matter As the latter are characterized by critically tilted Dirac/Weyl Hamiltonian Hmat, and the effects of the dynamic photonic cones with open, hyperbolic isofrequency contours, the corre- environment—described by the transverse vector potential—are sponding Lorentz-violating Dirac/Weyl quasiparticles exhibit 46 – included through the principle of minimal-coupling . markedly different properties from their type-I counterparts25 29. – A schematic of a cavity-embedded honeycomb metasurface is Soon after their realization, electromagnetic analogs emerged30 depicted in Fig. 1. We model each subwavelength meta-atom by a 34, and this exploration has recently been extended to 2D systems single dynamical degree of freedom describing the electric-dipole where a distinct type-II class of 2D Dirac points were theoretically moment associated with its (non-degenerate) fundamental 35,36 ω predicted . However, since their existence is predicated on eigenmode with resonant frequency 0. These meta-atoms are strong anisotropy in judiciously engineered photonic structures, then oriented such that their dipole moments point normal to the one cannot manipulate their properties without modifying the plane of the lattice. Furthermore, we consider subwavelength lattice structure. nearest-neighbor separation a such that the light cone intersects This hunt for exotic quasiparticles has recently entered the ω – the Brillouin zone edge above 0, ensuring the existence of realm of polaritonics37 42. The true potential of polaritons lies in evanescently bound, subwavelength polaritons. The strength of their hybrid nature, where their light and matter constituents can the Coulomb dipole–dipole interaction between neighboring be manipulated independently, thereby providing additional meta-atoms is parametrized by Ω. Finally, the metasurface is tunable degrees of freedom. Among other examples, recent works embedded at the center of a planar photonic cavity of height L, have shown the tantalizing prospect of engineering novel topo- where the cavity walls are assumed to be lossless and perfectly logical polaritons by introducing a winding coupling between conducting metallic plates. Such a structure is imminently ordinary photons and excitons39,41. realizable across the electromagnetic spectrum from arrays of In this work, we exploit the hybrid nature of polaritons in a plasmonic nanoparticles to microwave helical resonators (see different setting, namely metasurfaces, and we unveil unique Fig. 1). Dirac physics by shifting the focus from the lattice structure and its deformations to the effect of manipulating the sur- Emergence of type-I Dirac points. The matter Hamiltonian rounding photonic environment. In particular, we theoretically within the nearest-neighbor approximation reads study the polaritons supported by imminently realizable, crys- XX ¼ ω~ y þ y þ Ω~ y þ : : ; talline metasurfaces consisting of a honeycomb array of reso- Hmat h 0 aqaq bqbq h fqbqaq H c ð1Þ nant, dipolar meta-atoms. Despitetheelementarynatureof q q these metasurfaces, we unveil the simultaneous existence of both type-I and type-II massless Dirac polaritons which have where, for brevity, we have not presented the non-resonant terms ω~ distinct physical origins. Crucially, the existence of the latter is (see Methods for derivation). In Eq. (1), 0 is the renormalized not a result of anisotropy but is intrinsically linked to the resonant frequency and Ω~ is the renormalized Coulombic inter- hybrid nature of the polaritons, emerging from a non-trivial action strength due to the cavity-induced image dipoles (see winding in the light–matter interaction. Furthermore, we show Methods for their dependence on the cavity height). The y y that by embedding the honeycomb metasurface inside a planar bosonic operators aq and bq create quanta of the quasistatic

2 NATURE COMMUNICATIONS | (2018) 9:2194 | DOI: 10.1038/s41467-018-03982-7 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03982-7 ARTICLE z y To quadratic order in k = q−K (ka  1),P the effective matter eff ¼ ψy Heff ψ Hamiltonian near the K point is HK k k K;k k, with x ψy ¼ y ; y spinor creation operator k ak bk and Bloch Hamiltonian Heff ¼ ω~ 1 À ~σ Á þ ~ðÞσà Á 2 ð Þ K;k h 0 2 hv k ht k 2 1 σ = σ σ σ* = Here, 2 is the 2 × 2 identity matrix, ( x, y) and σ −σ °2 ( x, y) are vectors of Pauli matrices, and represents the Hadamard (element-wise) square. Note that the image dipoles do e3 e2 B not qualitatively affect the physics, but simply lead to a e1 ~ L A renormalization of the group velocity ~v ¼ 3Ωa=2 and trigonal a a ~ 1 2 warping parameter ~t ¼ 3Ωa2=8. Apart from a global energy shift, Meta-atom Eq. (2) is equivalent to a 2D massless Dirac Hamiltonian to leading order in k, with an isotropic spectrum ωmat ¼ ω~ þ τ~jj Possible experimental Microwave Plasmonic kτ 0 v k that is characterized by closed isofrequency realizations helical resonators nanorods contours. Therefore, as expected from the honeycomb symmetry, the CDP belongs to the type-I class of 2D Diracpffiffiffi points, and the θ T jψ i¼ð ; Àτ i k Þ = θ = Fig. 1 Schematic of a cavity-embedded honeycomb metasurface. The corresponding spinors kτ 1 e 2, where k arctan(ky/kx), represent massless Dirac collective-dipoles with a honeycomb array of meta-atoms is composed of two inequivalentp (Affiffi π and B) hexagonal sublattices—defined by lattice vectors a ¼ aðÀ 3 ; 3Þ topological Berry phase of . The effective Hamiltonian near the pffiffi 1 2 2 ′ Heff ¼ðHeff Þà ¼ að 3 ; 3Þ— K point is given by K′;k K;Àk , where the corresponding and a2 2 2 whichpffiffi are connected bypffiffi nearest-neighbor vectors e = a(0, −1), e ¼ að 3 ; 1Þ, and e ¼ aðÀ 3 ; 1Þ, where a is the massless Dirac collective-dipoles have a topological Berry phase 1 2 2 2 3 2 2 −π subwavelength nearest-neighbor separation. Each subwavelength meta- of , as required by time-reversal symmetry. atom is modeled as an electric dipole, oriented normal to the plane of the lattice. The honeycomb metasurface is then embedded inside a photonic Hybridization with the photonic environment. Given the sub- cavity of height L, which is composed of two perfectly conducting metallic wavelength nearest-neighbor separation, it is tempting to assert fi plates, enabling one to modify the photonic environment while preserving that the near- eld Coulomb interactions in Hmat capture the the lattice structure. This general model can be readily realized across the essential physics. In fact, we will show that this quasistatic electromagnetic spectrum, from arrays of plasmonic nanorods to description misses the profound influence of the surrounding microwave helical resonators photonic environment, which has a remarkably non-trivial effect on the Berry curvature and, therefore, on the corresponding nature of the polaritons. collective-dipole modes that extend across the A and B sub- Crucially, the metallic cavity supports a fundamental transverse fi lattices, respectively, with wavevector q inP the rst Brillouin zone electromagnetic (TEM) mode whose polarization (parallel to ¼ 3 ð Á Þ (see Fig. 2a). Finally, the function fq j¼1 exp iq ej encodes the dipole moments) and linear dispersion (see Fig. 2b) are the honeycomb geometry of the lattice with nearest-neighbor independent of the cavity height. For brevity, in what follows we vectors ej (see Fig. 1). do not present the contributions from the other cavity modes WeP diagonalizeP the matter Hamiltonian (Eq. (1)) as since the essential physics emerges from the interaction with the ¼ ωmatβy β Hmat τ¼ ± q h qτ qτ qτ where the bosonic operators fundamental TEM mode (see Methods for the full expressions). βy ¼ ψy jψ i In fact, the higher order cavity modes become increasingly qτ q qτ create quasistatic collective-dipole normal modes negligible for smaller cavities as they are progressively detuned with dispersion ωmat ¼ ω~ þ τΩ~jf j. Here, τ indexes the upper qτ 0 q from the dipole resonances. τ =+ τ = − ψy ¼ð y ; y Þ ( 1) and lower ( 1) bands and q aq bqpffiffiffiis a spinor The effects of the photonic environment are encoded in the φ T jψ i¼ð ; τ i q Þ = T free photonic Hamiltonian creation operator. The spinors qτ 1 e 2, where X denotes the transpose, describe an emergent pseudospin degree ¼  ωph y Hph h qncqncqn ð3Þ of freedom where the two components encode the relative qn amplitude and phase of the dipolar oscillations on the two inequivalent A and B sublattices, respectively, with φq = arg(fq). – and inð Þ theð Þ light matter interaction Hamiltonian These spinors can be represented by a pseudospin vector on the H ¼ H 1 þ H 2 , with = τ φ φ int int int  Bloch sphere which reads Sqτ (cos q, sin q, 0). P ′ ð1Þ ¼  ξ ϕà y þ y y At the high symmetry K and K points (see Fig. 2a), the Hint h i qn n aqcqn aqcÀqn sublattices decouple with no well-defined relative phase (i.e., f = 0), qn q P ð4Þ giving rise to two inequivalent CDPs located at ± K ¼ ± ð p4πffiffi ; 0Þ þ ξ ϕ y þ y y þ : : 3 3a h i qn n bqcqn bqcÀqn H c as observed in Fig. 2b. These CDPs correspond to vortices in the qn pseudospin vector field Sqτ, which give rise to topological singularities in the Berry curvature47. Therefore, the CDPs are and ð Þ P ξ ξ ÀÁ sources of quantized Berry wπ, where w = ±1 is the topological 2 ¼  2 qn qn′ ϕ ϕà Hint h ω Re n n′ charge of the Dirac point corresponding to the winding ′ 0 qnn ð Þ number of the vortex. As expected from the symmetry of the 5 y þ y y þ : : metasurface, the existence of the CDPs is robust against long- cqncqn′ cqncÀqn′ H c range Coulomb interactions as shown in Supplementary Note 1. ξ / À1=2 – In fact, for small cavity heights, the image dipoles quench long- where qn L parametrizes the strength of the light matter range Coulomb interactions and the nearest-neighbor approx- interaction (see Methods for analytical expression). The bosonic y fi imation becomes increasingly accurate as shown in Supplemen- operator cqn creates a TEM photon with wavevector q in the rst ωph ¼ jjÀ tary Figure 1. Brillouin zone and dispersion qn c q Gn ,wheren indexes the

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a M Γ K' K L q b1 y b2 qx Subcritical phaseL > Lc Critical phaseL = Lc Supercritical phase L < Lc 1.05 b c d e Full polariton Quasistatic Hamiltonian dispersion Type-I CDP Two-band 1 2 Quadratic Hamiltonian band-degeneracy 1.00

0 3 4 

/ Type-I CDP q 

0.95 TEM mode dispersion

Type-II SDP Light-cone

0.90 Γ K(K') M Γ Γ K(K') M Γ Γ K(K') M Γ Γ K(K') M Γ fi ¼ Fig.ÀÁ 2pEvolutionffiffiffi of the polaritonÀÁpffiffiffi dispersion as the cavity height is reduced. a First Brillouin zone de ned by primitive reciprocal lattice vectors b1 2π À ; À ¼ 2π ; À 3a 3 1 and b2 3a 3 1 . b Quasistatic dispersion of the collective-dipole normal modes, where the upper band corresponds to a bright, symmetric dipole configuration (↑↑) and the lower band corresponds to a dark, antisymmetric dipole configuration (↑↓). The light-cone (shaded region) is bounded by the linear dispersion of the TEM mode. Due to the non-trivial winding in the light–matter interaction (see Fig. 3), the band crossings are expected to result in large (band crossings ‘1’ and ‘2’) or small (band crossings ‘3’ and ‘4’) direction-dependent anticrossings in the polariton spectrum. c–e H H Polariton dispersion obtained from the polariton Hamiltonian pol (solid black lines) and the two-band Hamiltonian mat (orange dashed lines), for c subcritical (L = 5a), d critical (L = Lc = 1.75a), and e supercritical (L = a) cavity heights, respectively. While type-I CDPs with an isotropic Dirac cone (see inset of c) exist even in the quasistatic dispersion (see b), new type-II SDPs with a critically tilted Dirac cone (see inset in c) emerge due to the vanishing light–matter interaction for the dark quasistatic band along the Γ−K(K′) directions (see Fig. 3). At the critical cavity height Lc, three type-II SDPs merge with the type-I CDP (see Fig. 5) resulting in a quadratic band-degeneracy at K(K′) (see inset in d). After criticality, the type-II SDPs annihilate one another and the massless Dirac cone re-emerges at the type-I CDPs (see inset in e) accompanied by an inversion of chirality (see Fig. 5). Plots obtained with ωph ¼ : ω Ω = ω parameters K0 2 5 0 and 0.01 0 set of reciprocal lattice vectors Gn. The complex phase factors ϕ ¼ mediated by the electromagnetic field. However, by expressing the ðÞÁ ^ n β βy exp iaGn y are associated with Umklapp processes that arise due interaction Hamiltonian (Eq. (4)) in terms of the qτ and qτ to the discrete, in-plane translational symmetry of the metasurface, operators that diagonalize the matter Hamiltonian,  and must be retained as they are critical for maintaining the point- X X ð1Þ ¼  Λ βy þ βy y þ : :; group symmetry of the polariton Hamiltonian. Hint h i qnτ qτcqn qτcÀqn H c ð6Þ fi – τ¼ ± qn We diagonalize Hpol using a generalized Hop eld Bogoliubov 48 – transformation (see Methods for details), and in Fig. 2c e, fi we present the resulting polariton dispersion for different cavity we nd that complex non-local interactions, which arise from strong multiple-scattering in the bipartite structure, result in a heights. Also, in Supplementary Figure 2, we present the full – polariton dispersion which includes long-range Coulomb inter- non-trivial winding of the light matter coupling as a function of wavevector direction actions. For small cavity heights, the full polariton dispersion ÀÁ Ã φ Λ / ξ ϕ i q þ τϕ : ð Þ is almost indistinguishable from that obtained in the nearest- qnτ qn ne n 7 neighbor approximation, and therefore one can conclude that long-range Coulomb interactions do not qualitatively affect Naively, one may expect all of the band crossings in Fig. 2bto the physics presented here. It is important to stress that our be avoided as a result of the hybridization between the collective- general model captures the essential physics that will emerge in a dipole and photonic modes, as it is a characteristic feature of variety of different experimental setups. To show this, in polaritonic systems48,49. Indeed, this is the case forÀÁ the crossings φ Supplementary Figure 3 and Supplementary Figure 4 we present Λ / i q þ fi with the upper quasistatic band where q0þ e 1 (see the polariton dispersions obtained from nite element simula- red line in Fig. 3a) due to the constructive interference between tions of a honeycomb array of plasmonic nanorods and the sublattices of this bright (↑↑) configuration (see Fig. 3b, c). microwave helical resonators, respectively. Indeed, these entirely This results in a large anticrossing for all wavevector directions, different physical realizations show the same evolution of the – as observed in Fig. 2c. In stark contrast, for the lower polariton spectrum as presented in Fig. 2c e and Supplementary quasistaticÀÁ band the coupling constant is significantly reduced φ Figure 2. Λ / i q À q0À e 1 (see blue line in Fig. 3a) due to the destructive interference between the sublattices of this dark (↑↓) configura- Emergence of type-II Dirac points. Given the elementary nature tion (see Fig. 3e). Consequently, this results in a small of the individual resonant elements, one may be tempted to anticrossing for a general wavevector direction. assume that nothing peculiar could emerge from the ordinary Crucially, however, the light–matter interaction for the – Λ = dipole dipole interactions between the meta-atoms which are lower quasistatic band completely vanishes ( q0− 0) along the

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1 a 2.05

2 1 2 2.00 Inversion of chirality

| M  +

q 0.10 q

 Γ i 4 K(K') 4

|e 0 0.05 M 3 0 v/v Γ→M Γ→K(K') Γ→M 1.00 Wavevector direction 2  –1 1–D/ 0 bcde   0/ 0 t/t

0.92 0.2 1.0 /L 1 2 3 4 –2 0.2 0.4 0.6 0.8 1.0 Fig. 3 Non-trivial winding in the light–matter interaction. a Dependence of /L φ the magnitude of the light–matter coupling constant jΛ j/j i q þ τj on q0τ e Fig. 4 Tunable parameters in the effective Hamiltonian. Dependence of the the direction of q from Γ−MtoΓ−K(K′) (see inset), for the upper (red parameters in the effective polariton Hamiltonian (Eq. (9)) on the inverse line) and lower (blue line) quasistatic bands. Plots obtained with |q| = |K|/ cavity height. The blue dashed line shows the variation of the group velocity v 2. b–e Schematics of the bright (↑↑) and dark (↑↓) configurations of the two which changes sign at the critical cavity height L , leading to the inversion of sublattices interacting with the photonic mode which is indicated by the c chirality. The orange dot-dashed line shows the variation of the trigonal field profile. Panels b and d represent the crossings labeled ‘1’ and ‘3’ in warping parameter t which becomes dominant close to criticality. These Fig. 2 along the Γ−K(K′) directions, respectively, while c and e represent parameters have been normalized to v = 3Ωa/2 and t = 3Ωa2/8 which are the crossings labeled ‘2’ and ‘4’ along the Γ−M directions, respectively. the group velocity and trigonal warping parameters, respectively, in the Crucially, the light–matter interaction strength for the dark mode vanishes absence of image dipoles and light–matter interactions. The orange dot- (|Λ −| = 0) along the Γ−K(K′) directions due to the complete destructive q0 dashed line in the inset shows the variation of the CDP frequency ω , while interference between the two sublattices (see d), leading to the emergence 0 the blue dashed line in the inset shows the variation of the wavevector- of six inequivalent type-II Dirac points in the polariton spectrum D ωph ¼ : ω Ω = ω dependent diagonal term . Plots obtained with K0 2 5 0 and 0.01 0

high-symmetry Γ−K(K′) directions, where φq = 0, due to the Similarly, the effective Hamiltonian near the K′ point is given by Heff ¼ðHeff Þà ω complete destructive interference between the two sublattices (see K′;k K;Àk . In Eq. (9), the resonant frequency 0, group Fig. 3d). As a result, along these high-symmetry directions the velocity v, and trigonal warping parameter t, now encode non- crossings are protected, leading to six inequivalent Dirac points trivial contributions from the hybridization with the photonic emerging in the polariton spectrum—we call these satellite Dirac environment. There is also an additional wavevector-dependent points (SDPs) to distinguish them from the CDPs. As we will see diagonal term parametrized by D, which breaks the symmetry below, these SDPs belong to the type-II class of 2D Dirac points between the upper and lower polariton bands. The dependence of where the dispersion takes the form of a critically tilted Dirac these parameters on the cavity height is shown in Fig. 4 (see cone (see inset of Fig. 2c), characterized by open, hyperbolic Methods for analytical expressions). To leading order in k, one isofrequency contours. can observe that the effective Hamiltonian (Eq. (9)) near the CDP is equivalent to a 2D massless Dirac Hamiltonian. Therefore, the Effective Hamiltonian in the matter subspace. To explore the polariton CDPs remain in the type-I class and are robust against nature of the polaritons in the vicinity of the different Dirac the coupling with the photonic environment—this is not sur- points, we first neglect non-resonant terms in the matter prising given that their physical origin is intrinsically linked to the Hamiltonian and perform a unitary Schrieffer–Wolff transfor- lattice structure alone, which is preserved here. 50 mation on Hpol to integrate out the photonic degrees of freedom To elucidate the nature of the SDPs, we expand the effective (see Methods for details). Finally, we extract the two-band Hamiltonian (Eq. (9)) near one of the SDPs located at ¼ ðÞ= ; Hamiltonian in the matter sublattice space KS v t 0 and obtain  2 X ξ2 ωph eff Dv 2Dv à qn qn H ; ′ ¼ h ω À À k′ 1 þ hσ Á v Á k′ ; ð10Þ H ¼ H À 2h  KS k 0 2 x 2 mat mat 2 t t ωph Àω~2  qn qn ð8Þ 0 10 where k′ measures the deviation from K and v ¼ v is y þ y þ ϕ2 y þ ϕÃ2 y : S 03 aqaq bqbq nbqaq n aqbq the velocity tensor. Apart from a global energy shift, the effective Hamiltonian (Eq. (10)) near the SDP takes the form Diagonalizing the two-band Hamiltonian (Eq. (8)) leads to an of aP generalizedP 2D massless Dirac Hamiltonian effective dispersion (see Methods) which provides an excellent H ¼ ¼ ; hu k 1 þ ¼ ; hv k σ . If the parameters u and description of the polaritons as indicated by the orange dashed k i x y i i 2 i x y i i i i 2= 2 þ 2= 2 lines in Fig. 2c–e. Finally, we expand the two-band Hamiltonian vi satisfy the condition ux vx uy vy <1, then the Dirac cone becomes tilted and anisotropic51 but still belongs to the type-I (Eq. (8)) up to quadratic order in k Pand obtain the effective eff ¼ ψy Heff ψ class with closed isofrequency contours. However, the condition Hamiltonian near the K point HK k k K;k k (see Sup- 2= 2 þ 2= 2 fi ux vx uy vy >1denes a distinct type-II class of 2D Dirac plementary Note 2 for derivation) with Bloch Hamiltonian points, giving rise to a critically tilted Dirac cone with open,  hyperbolic isofrequency contours. Hence, the type-I and type-II Heff ¼ ω 1 À  σ Á þ  ðÞσà Á 2À jj21 : ð9Þ K;k h 0 2 hv k ht k hD k 2 classes are related via a Lifshitz transition in the topology of the

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a b c Subcritical phase L >Lc Critical phase L =Lc Supercritical phase L < Lc 1 1 1 High –π Γ–K –π π K–M +π – –π +π +π -–ππ 0.5 0.5 0.5 –π –π  k

π π /

y +π y y 0 – 0 –2 0 +π  k k k 0

–0.5 –0.5 –0.5 –π Low

–1 –1 –1 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1

kx kx kx  d k e  f  k k

kx ky

kx ky kx ky

Massless Dirac polaritons Massless Dirac polaritons Massive chiral polaritons with inverted chirality

Fig. 5 Merging of type-I and type-II Dirac points with chirality inversion. a–c Pseudospin vector field and isofrequency contours (see Methods for the specific isofrequency values) for the upper polariton band near the K point for a subcritical (L = 2.3a), b critical (L = Lc = 1.78a), and c supercritical (L = a H 1.4 ) cavity heights, respectively, as obtained from mat. The Dirac points (corresponding to vortices) are depicted by orange circles along with their associated Berry flux. Before criticality, three type-II SDPs (−π Berry flux) are driven towards the type-I CDP (π Berry flux) along the Γ−K directions as the cavity height is reduced (see inset in a). At the critical cavity height Lc, they merge together forming a quadratic band-degeneracy with combined Berry flux of −2π (see b). After criticality, the type-II SDPs re-emerge and are driven past the type-I CDP along the K−M directions (see inset in c). After a small decrease in cavity height, these SDPs annihilate other SDPs that migrate along the opposite direction and have opposite Berry flux, leaving only the type-I CDP remaining in the spectrum with π Berry flux (see c). d-f, Effective polariton spectrum near the K point to leading order in k. The colors of the bands correspond to the chirality of the Dirac polaritons as defined in the main text, where the orange and blue bands indicate a chirality of +1 and −1, respectively. The spinor eigenstates, represented by pseudospin vectors (gold arrows), describe d massless Dirac polaritons with a linear dispersion and Berry phase of π, e massive chiral polaritons with a parabolic dispersion and Berry phase of −2π, and f massless Dirac polaritons with a linear dispersion π ωph ¼ : ω Ω = ω and Berry phase of , but with inverted chirality. All pseudospin and contour plots are obtained with K0 2 5 0 and 0.01 0

= – isofrequency contours. Indeed, since we have uy 0 and of the two-band Hamiltonian (Eq. (8)). In Fig. 5a cweplotthe 2= 2 ¼ 2= 2 fi ux vx 4D t >1, the SDPs belong to the type-II class of 2D pseudospin vector eld near the K point for different cavity heights Dirac points. Furthermore, since the Hamiltonian (Eq. (10)) is and schematically depict the location of the Dirac points, along with expressed in terms of σ*, the pseudospin winds in the opposite their associated Berry flux. Finally, in Fig. 5d–f, we illustrate the direction around the SDPs as compared to the CDP, and corresponding effective polariton spectrum to leading order in k. therefore the SDPs located along the Γ−K directions are sources NotethatsimilaranalysiscanbeperformedneartheK′ point. −π fl of Berry ux. As required by time-reversal symmetry, the In the subcritical phase (L > Lc), three type-II SDPs are located SDPs located along the Γ−K′ directions are sources of π Berry along the Γ−K directions, each with −π Berry flux surrounding flux (opposite to the CDP located at the K′ point). a type-I CDP with π Berry flux (see Fig. 5a). To leading order in k, the polariton spectrum disperses linearly about the type-I CDPs (see Fig. 2c) forming an isotropic Dirac cone with a group Manipulation of type-I and type-II Dirac points.Wehavethus velocity v that is tunable via the cavity height (see Fig. 4). demonstrated that the honeycomb metasurface simultaneously Here, the effective Hamiltonian (Eq. (9)) is equivalent to a exhibits two distinct species of massless Dirac polaritons, namely 2D massless Diracpffiffiffi Hamiltonian with spinor eigenstates iθ T type-I and type-II. In contrast to the type-I CDPs, the existence of jψ τi¼ð1; Àτe k Þ = 2. These represent massless Dirac polar- k hψ jσ Á ^jψ i¼Àτ the type-II SDPs is intrinsically linked to the hybridization between itons with chirality kτ k kτ , resulting in a pseudos- the light and matter degrees of freedom, and thus one can pin that winds once around the CDP and a topological Berry manipulate their location within the Brillouin zone by simply phase of π (see Fig. 5d). – = modifying the light matter interaction via the cavity height. As a At the critical cavity height (L Lc), the group velocity of the ð Þ¼ result, the polariton spectrum evolves into qualitatively distinct massless Dirac polaritons vanishes v Lc 0 (see Fig. 4) as the phases as highlighted in Fig. 2c–e. To elucidate the differences type-II SDPs merge with the type-I CDP, forming a quadratic between these phases, we study the spinor eigenstates (see Methods) band-degeneracy (see Fig. 2d) with combined −2π Berry flux (see

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Fig. 5b). The leading order term in the effective Hamiltonian As a final remark, we briefly comment on how one might (Eq. (9)) is now quadratic in k pwithffiffiffi corresponding spinor probe the Dirac physics presented in this work. Given that the À θ T jψ i¼ð ; Àτ i2 k Þ = eigenstates kτ 1 e 2. Therefore, during this Dirac points exist in a polaritonic excitation spectrum, one critical merging transition, the massless Dirac polaritons morph must drive the system with photons at the required frequency in into massive chiral polaritons with qualitatively distinct physical order to probe them. In fact, both of the type-I and type-II Dirac properties. These include a parabolic spectrum and chirality points lie outside of the light-line and therefore one must hψ jðσà Á ^Þ2jψ i¼Àτ fi kτ k kτ , resulting in a pseudospin that winds overcome the momentum mismatch with photons. The speci c twice as fast compared to the subcritical phase and a topological experimental technique that one would employ will depend on Berry phase of −2π (see Fig. 5e). the nature of the metasurface and the corresponding frequency Since the point-group symmetry is preserved, the type-II SDPs regime. For example, techniques for plasmonic systems have do not annihilate the type-I CDP, but they re-emerge and traditionally involved coupling via evanescent waves with prisms, continue to migrate along the K−M directions as the cavity gratings, and local scatterers62, or more recent techniques such as 63 height is reduced past criticality (L < Lc) (see inset of Fig. 5c). non-linear wave-mixing . In contrast, realizations in the After a small decrease in cavity height, these SDPs annihilate with microwave regime can be probed using point-like antenna other SDPs that migrate along the opposite direction and have sources and detectors33. In fact, microwave metamaterials are opposite Berry flux. This topological transition leaves only the proving to be a versatile platform for exploring Dirac/Weyl type-I CDP remaining in the polariton spectrum with π Berry flux physics, as one can directly probe the field distributions using (see Figs. 2e and 5c). near-field scanning techniques33, and thus one could directly In this supercritical phase, we recover the linear dispersion probe the environment-induced chirality inversion predicted near the type-I CDP to leading order in k (see Fig. 2e), and here. the effective Hamiltonian (Eq. (9)) is equivalent to a 2D massless Dirac Hamiltonianpffiffiffi with corresponding spinor eigen- θ T Discussion jψ i¼ð ; τ i k Þ = states kτ 1 e 2. Remarkably, massless Dirac polar- To conclude, we have revealed rich and unique Dirac physics that itons thus re-emerge past criticality with an environment-induced emerges even in the most elementary honeycomb metasurfaces. hψ jσ Á ^jψ i¼τ inversion of chirality kτ k kτ (see Fig. 5f). Physically, In particular, we have unveiled the simultaneous existence of both π this corresponds to a -rotation in the relative phase between the type-I and type-II massless Dirac polaritons, where the latter dipole oscillations on the two inequivalent sublattices, which is emerge from a non-trivial winding in the light–matter interac- π also accompanied by a -rotation in the isofrequency domains tion. We would like to emphasize that it is this unique physical (compare Fig. 5a and Fig. 5c). origin of the type-II SDPs, together with the truly 2D nature of We emphasize that it is the chirality of massless Dirac the metasurface, that enables one to qualitatively modify the that is responsible for most of the remarkable properties of 13 fundamental properties of these emergent Dirac polaritons by monolayer graphene, including the Klein tunneling phenomenon . manipulating the surrounding photonic environment alone. This Consequently, this unique environment-induced inversion of stands in stark contrast to conventional artificial graphene sys- chirality could give rise to unconventional phenomena such as tems where the fundamental properties are dictated by the lattice anomalous Klein transport. For example, near the K point, the structure. Therefore, exploiting the rich tunability of the polariton right-propagating polaritons correspondpffiffiffi to an antisymmetric dipole spectrum with the environment offers a new paradigm that opens fi jψ i¼ð ; À ÞT= con guration kτ 1 1 2 in the subcriticalpffiffiffi phase and to a variety of opportunities to explore unique Dirac-related physics fi jψ i¼ð ; ÞT= a symmetric con guration kτ 1 1 2 in the supercritical at the subwavelength scale. phase. Thus, due to the orthogonality between these two For example, one can simultaneously probe the dynamics spinor eigenstates, the inversion of chirality removes the channel of type-I and type-II massless Dirac quasiparticles, where responsible for the perfect transmission in the conventional Klein the latter are predicted to exhibit intriguing anomalous refraction tunneling effect13 (see Fig. 5d, f). Such a scenario could be realized behavior34,35. Furthermore, the environment-induced redshift ω in a simple setup characterized by two neighboring regions with of the CDP frequency 0 (see Fig. 4) will allow the investigation different cavity heights. of polaritonic Klein tunneling through interfaces separating As a side remark, we note that the polariton spectrum near regions with different cavity heights. Consequently, negative criticality bears some resemblance with the low-energy spectrum refraction can be induced by simple variations in the cavity of bilayer graphene with its central Dirac point and three satellite height, which could be exploited in novel schemes for guiding and Dirac points, which all belong to the type-I class52,53. However, manipulating light at the subwavelength scale, including polari- given the type-II nature of the polariton SDPs, the topology of the tonic Veselago lensing64,65. Moreover, the tunable group velocity polariton isofrequency contours are markedly different from that will enable the exploration of velocity barriers for the unprece- of the bilayer spectrum. This is further highlighted at criticality dented guiding and localization of massless Dirac where the polariton bands have the same curvature, which is in quasiparticles66,67, which is extremely difficult to achieve in real stark contrast to the electronic bands in bilayer graphene. graphene. One could also combine the effects of the environment We also note that recent works explored the possibility to with inhomogeneous strain deformations, giving rise to unique manipulate the (3+1) type-I Dirac points in bilayer graphene – pseudomagnetic-related effects, including the intriguing ability to through the application of lattice deformations54 57, leading to induce a pseudo-Landau level spectrum for polaritons that can be the merging and annihilation of pairs of Dirac points. In addition, qualitatively tuned via the cavity height. Finally, the ability to a multimerging transition of all (3+1) type-I Dirac points controllably invert the chirality of the massless Dirac polaritons has been proposed theoretically within tight-binding models opens new perspectives for anomalous pseudorelativistic trans- involving the artificial tuning of third-nearest-neighbor hopping – port through interfaces separating regions in distinct polaritonic amplitudes in a graphene-like honeycomb structure58 61. How- phases. ever, these proposals have no physical realization so far. In stark contrast, the imminently realizable metasurfaces in our work Methods enable the exploration of rich Dirac phases with ease by simply Derivation of the polaritonic Hamiltonian. The cavity-embedded metasurface modifying the photonic environment via an enclosing cavity. is composed of a honeycomb array of identical meta-atoms located at ¼ þ ^ þ L ^ ¼ À ^ þ L ^ RA R ay 2 z and RB R ay 2 z on the A and B sublattices, respectively.

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= + the local and block-diagonal form Here, R l1a1 l2a2 is an in-plane lattice translation vector with primitive vectors P  a1 and a2 (see Fig. 1) and integers l1 and l2. Each meta-atom is modeled by a single ¼  ðÞω À ΩS y þ y Hmat h 0 aqaq bqbq dynamical degree of freedom h (with dimensions of length), where the electric- q dipole moment associated with its fundamental eigenmode is p ¼ÀQh^z, with P hi þ Ωð ÀIÞ y þ y þ : : effective charge Q. The Coulomb potential energy between two dipole moments h 1 fqbq aq aÀq H c ð17Þ ′ ′ q p and p located at generic positions r and r , respectively, is given by P  À 1 ΩS y y þ y y þ : : : h 2 aqaÀq bqbÀq H c p Á p′ À 3ðp Á n^Þðp′ Á n^Þ q V ¼ ; ð Þ Coul πε j À ′j3 11 4 0 r r y y In the main text, we do not present the non-resonant terms (e.g., bqaÀq), leading to ω~ ¼ ω À ΩS Ω~ ¼ ΩðÞÀI Eq. (1) where 0 0 and 1 . ^ ¼ ðÞÀ ′ =jjÀ ′ ε – where n r r r r and 0 is the vacuum permittivity. In the Coulomb gauge, the light matter interaction is described by the minimal- The presence of the perfectly conducting metallic plates, placed at z = 0 and coupling Hamiltonian46 which, within the dipole approximation, reads = fi X X X X z L, modi es the boundary conditions on the scalar potential and, therefore, the Q Q2 Coulomb interaction between the meta-atoms. Using the method of images to H ¼ Π A ðR Þ þ A2ðR Þ ; int Rs z s z s 68 M ¼ ; 2M ¼ ; ensure the vanishing of the scalar potential at the cavity walls , we introduce an s A B Rs s A B Rs ð18Þ fi þ ^ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} in nite series of image dipoles located outside the cavity at positions Rs lLz, ð Þ ð Þ H 1 H 2 where s = A, B labels the two sublattices and l is a non-zero integer. Noting that the int int Coulomb potential energy between a real and image dipole is 1/2 of that given by Π ¼ Π ^ Eq. (11)69, the matter Hamiltonian within the nearest-neighbor approximation where we have used R Rz. The vector potential can be decomposed into reads transverse electric (TE) and transverse magnetic (TM) modes of the cavity. However, the photons corresponding to the TE modes have an in-plane  P P Π2 polarization, and therefore only TM modes contribute to the z-component of the ¼ Rs þ M ω2 2 Hmat 2M 2 0hR vector potential ¼ ; s s A B Rs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X  P P3 h jjq À G mπ þ Q2 ð ; Þ¼ n 3 h h þ Az r z π cos z 4πε a RB RB ej ph À þ m ^ 0 ¼ ε N NALωq q Gn z L ð Þ RB j 1 qmnhi0 m mn L 19 P P þ1P ð À ÞÁ y À ð À ÞÁ Q2 3 i q Gn r þ i q Gn r ; À ′ a 2 cqmne cqmne 8πε a3 2 lL hR ð Þ 0 ¼ ; ¼À1 s 12 s A B Rs l pffiffiffi P P3 þ1P lL 2 2 2 2jjÀ1 where A¼3 3a =2 is the area of a unit cell and N = 1 + δ . The bosonic À Q ′ a m m0 πε 3 ÀÁ5 hR hR þe y 8 0 a 2 2 B B j operator c creates a TM photon with wavevector q in the first Brillouin zone R j¼1 l¼À1 1þjjlL qmn B a ωph ¼ jjÀ þ ^ π= = + and dispersion qmn c q Gn zm L . Here, Gn n1b1 n2b2 is a reciprocal P P3 þ1P lL 2 2 2jjÀ1 À Q ′ a lattice vector with primitive vectors b1 and b2, where n indexes the set of ordered πε 3 ÀÁ5 h h À 8 a 2 RA RA ej 0 R j¼1 ¼À1 þjjlL 2 pairs of integers (n1, n2), and m is a non-negative integer indexing the different TM A l 1 a cavity modes. Only TM photons with even m couple to the dipoles due to the parity selection rule at the center of the cavity. where the primed summations exclude the l = 0 term. Here, Π is the conjugate Substituting the vector potential (Eq. (19)) into Eq. (18) we obtain the Rs momentum to the dynamical coordinate h corresponding to the meta-atom located light–matter interaction Hamiltonian given by Rs at R ,andM is an effective mass. Next, we introduce the bosonic operators P  s ð1Þ ¼  ξ ϕà y þ y y Hint h i qmn n aqcqmn aqcÀqmn rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qmn ω P  ð20Þ M 0 1 ð Þ y y y a ¼ h þ i Π 13 þh iξ ϕ b c þ b cÀ þ H:c: RA  RA  ω RA qmn n q qmn q qmn 2h 2hM 0 qmn

and and P ξ ξ ÀÁ ð2Þ 2 qmn qm′n′ Ã H ¼ h ω Re ϕ ϕ ′ rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi int 0 n n qmm′nn′ ð Þ Mω 1  21 b ¼ 0h þ i Π ð14Þ y y y RB  RB  ω RB c c ′ ′ þ c cÀ ′ ′ þ H:c: 2h 2hM 0 qmn qm n qmn qm n

The strength of the light–matter interaction is parametrized by that annihilate quanta of the fundamental eigenmode on the meta-atom located at y !1 ½ ; Š¼δ ωph 2 RA and RB, respectively, and satisfy the commutation relations aR aR′ RR′, 8π a Ω y y ξ ¼ ω Fðωph Þ q0n pffiffiffi ; ð22Þ ½b ; b Š¼δ ′,and½a ; b Š¼0. In terms of these operators, the matter qmn 0 qmn R R′ RR R R′ ωph 3 3N L ωph Hamiltonian (Eq. (12)) reads qmn m qmn

P y P y where, to take into account the finite size of the meta-atoms, we have introduced a H ¼ hω a a þ hω b b ph mat 0 RA RA 0 RB RB phenomenological function Fðω Þ that provides a smooth cut-off for the R R qmn A B interaction with short-wavelength photonic modes where the dipole approximation P P3 h   i þΩð ÀIÞ y þ y þ : : breaks down. We choose the phenomenological cut-off function to be of the Fermi- h 1 bR aR þe aR þe H c ¼ B B j B j RB j 1 Dirac distribution form Phi ð15Þ  y y À h ΩS a a þ a þ H:c: ph 1 2 RA RA RA Fðω Þ¼ ; ð Þ R qmn ðωph À ω Þ=ω 23 A hi 1 þ e2 qmn 3 0 0 P y y À h ΩS b b þ b þ H:c: 2 RB RB RB RB which is smooth enough to avoid spurious artifacts appearing in the polariton spectrum. Finally, the free photonic Hamiltonian of the cavity reads X Ω = 2 πε ω 3 ¼  ωph y : where Q /8 0M 0a parametrizes the strength of the nearest-neighbor Hph h qmncqmncqmn ð24Þ Coulomb interaction, and the parameters qmn ÀÁ X1  X1 2 a 3 2 lL À1 S¼ ; I¼ hia In Eqs. (3), (4), (5) in the main text, we only present the contribution from the 4 2 ÀÁ 5 ð16Þ lL 2 2 = l¼1 l¼1 1 þ lL TEM mode (m 0), dropping the corresponding index. In Supplementary Note 1, a we discuss the effect of the higher order (m ≠ 0) TM cavity modes for larger cavities. encode renormalizations due to the cavity-induced image dipoles. We apply Born- N fi – = + von Kármán boundary conditions over a lattice with 1 unitP cells and introduce Hop eld Bogoliubov diagonalization. The polariton Hamiltonian Hpol Hmat À1=2 Á ¼N iq RA + the Fourier transformP of the bosonic operators aR q aqe and Hph Hint, where Hmat is given by Eq. (17), Hph by Eq.P (24), and Hint by Eqs. (20) À1=2 Á A y pol b ¼N b eiq RB , which transforms the matter Hamiltonian (Eq. (15)) into and (21), can be recast into matrix form as H ¼ 1 Ψ H Ψ where RB q q pol 2 q q q q

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Ψy ¼ðψy ; y ; ψT ; T Þ ψy ¼ð y ; y Þ q q Cq Àq CÀq . Here, q aq bq is the spinor creation operator in Hamiltonian (Eq. (32)) reads y ¼ð y ; y ; ¼ ; y ; ¼ ; y Þ the matter sublattice space and Cq cq1 cq2 cqp cqN is the vector of TM hi 1 ð Þ ð Þ photon creation operators, where p indexes the set of ordered triplets of integers H ’ H þ S; H 1 þ H þ H 2 ; pol mat int |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}ph int ð Þ (n1, n2, m), and N is the total number of photonic operators considered. The |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2 35 pol Hermitian [2(N + 2)] × [2(N + 2)] matrix H can be written in block form as H Hph 0 q 1 mat Hþ HÀ ÀW @ q q q A ξ Hpol ¼ y à ; ð25Þ where the matter and photonic subspaces are decoupled to quadratic order in qmn. q HÀ ÀW Hþ q q Àq Calculating the commutator in Eq. (35) and extracting the Hamiltonian within the matter sublattice space, we obtain the two-band Hamiltonian  where P ξ2 ωph ¼ À  qmn qmn y þ y þ ϕ2 y þ ϕÃ2 y : ð Þ Hmat Hmat 2h ðωph Þ2 Àω~2 aqaq bqbq nbqaq n aqbq 36 W ¼  ω ; ω ; ωph; ωph; ¼ ; ωph; ¼ ; ωph ð Þ qmn qmn 0 q hDiag 0 0 q1 q2 qp qN 26 is the (N + 2) × (N + 2) diagonal matrix of resonant frequencies of the free oscil- In Eq. (8) in the main text, we only present the contribution from the TEM lators. The (N + 2) × (N + 2) block matrices H ± can be sub-divided into block q mode (m = 0), dropping the correspondingP index. We can recast the Hamiltonian matrices ¼ ψy Hmatψ 0 1 (Eq. (36)) into matrix form as Hmat q q q q, with Bloch Hamiltonian Hmat Hint ! q q à H ± ¼ @ y A; ð Þ mat Wq Fq q 27 H ¼  : ð Þ Hint Hph q h 37 ± q q Fq Wq P P where the upper-diagonal block Here W ¼ ω~ À Ω Δ and F ¼ Ω~f À Ω Δ ϕ2 with ! q 0 mn qmn q q mn qmn n ~ à ! ω~ Ωf 2 Hmat ¼  0 q ð Þ π ω2 ωph q h ~ 28 Δ ¼ p16ffiffiffi a 0 q0n F 2ðωph Þ: ð Þ Ωf ω~ qmn q 38 q 0 L ðωph Þ2 À ω~2 ωph mn 3 3Nm qmn 0 qmn is the 2 × 2 matrix in the matter subspace, and the lower-diagonal block Hph is the q ωmat ¼ þ τj j N × N matrix in the photonic subspace with components Diagonalizing Hmat leads to the two-band dispersion qτ Wq Fq , which is indicated by the orange-dashedp linesffiffiffi in Fig. 2c–e. The corresponding spinor  ξ ξ no φ ′ jψ i¼ð ; τ i q ÞT= φ ¼ ð Þ ph ph qp qp à eigenstates τ 1 e 2, where arg Fq , can be represented by the H ¼ hω δ ′ þ 4h Re ϕ ϕ ′ : ð29Þ q q q ′ qp pp ω p p ¼ τð φ ; φ ; Þ pp 0 pseudospin vector Sqτ cos q sin q 0 from which we obtain the pseudospin vector field plots in Fig. 5a–c. Hint Finally, the off-diagonal block q in Eq. (27) is the 2 × N interaction matrix, where the pth column reads ! Expansion of the effective two-band Hamiltonian. Near the K point, the func- Ã Δ  ξ ϕ tion qmn, given by Eq. (38), expands as i qp Hint ¼ h p : ð30Þ hi q ξ ϕ ð Þ ð Þ p i qp p Δ ’ Δ 0 À a2Δ 1 ðK À G Þ k þðK À G Þ k kmn KmnhiKmn n x x n y y þ 1 À 2Δð1Þ þ 4Δð2Þ ðÞÀ 2 2 2 a Kmn a Kmn K Gn x kx hið39Þ The polariton Hamiltonian Hpol is diagonalized by a generalized ð Þ ð Þ fi – 48 Ψ = y ¼ðχy ; χT Þ þ 1 À 2Δ 1 þ 4Δ 2 ðÞK À G 2 2 Hop eld Bogoliubov transformation q TqXq, where Xq q Àq and 2 a Kmn a Kmn n y ky χy ¼ðγy ; γy ; ¼ ; γy ; ¼ ; γy Þ q1 q2 ν þ . To ensure the invariance of the bosonic ð Þ q q qN 2 þa4Δ 2 ðÞK À G ðÞK À G k k commutation relations for the transformed operators, Tq must be a [2(N + 2)] × [2 Kmn n x n y x y + 70 fi η y ¼ yη ¼ η η ¼ (N 2)] paraunitary matrix that satis es Tq zTq Tq zTq z, where z σ 1 and σ is the Pauli matrix. The transformed bosonic operators γy ¼ ΔðυÞ υ = z 2 z qν to quadratic order in k, where the real parameters Kmn ( 0, 1, 2) depend only Ψy η jΨ i γ = 〈Ψ η Ψ ph qν and qν qν| z q that diagonalize the polariton Hamiltonian as on the photon frequencies ω at the K point. Collecting the contributions from q z X Kmn y the degenerate photons (see Supplementary Note 2 for details), we obtain the H ¼ h ωpolγ γ ; pol qν qν qν ð31Þ effective Hamiltonian (Eq. (9)), where parameters are given by qν X ω Ω Ω ð Þ 0 ¼ À SÀ Δ 0 ; ð Þ create and annihilate polaritons in the vth band, respectively. The polariton ω 1 ω ω Kmn 40 ωpol – 0 0 0 mn dispersion qν (black solid lines in Fig. 2c e) and the corresponding linearly independent eigenvectors |Ψqν〉 (first two columns of Tq) are determined from the positive eigenvalue solutions to the non-Hermitian eigenvalue equation X v 4π ð Þ η HpoljΨ i¼ωpoljΨ i ¼ ÀIÀ Δ 1 ; z q qν h qν qν . 1 An Kmn ð41Þ v 27 mn Schrieffer–Wolff transformation. To obtain an effective two-band Hamiltonian in the matter sublattice space, we neglect non-resonant terms in the matter 2 X t 8π ð2Þ Hamiltonian (since Ω=ω  1 for practical realizations of the metasurface), but ¼ 1 ÀIÀ B Δ ; ð42Þ 0 t 81 n Kmn not in the light–matter interaction Hamiltonian since the photons are not resonant mn with the collective-dipoles near the corners of the Brillouin zone (see Fig. 2b). Next, we perform a unitary transformation and  À 1 X 2 ¼ S S ¼ þ½ ; Šþ ½ ½ ; ŠŠ þ ¼ ð Þ D Ω 4π ð2Þ 1 ð1Þ Hpol e Hpole Hpol S Hpol S S Hpol 32 ¼ C Δ À Δ ; ð Þ 2 ω 2 ω n Kmn Kmn 43 0a 0 mn 27 2 and impose the Schrieffer–Wolff condition50 hi with ; þ ¼À ð1Þ ð Þ S Hmat Hph Hint 33 pffiffiffi  3 4π A ¼ ðÞ2 À 3n cos ðÞn þ n – fi ξ n 2 1 3 1 2 which eliminates the light matter interaction to rst order in qmn. From Eq. (33),  ð44Þ π the particular form of the anti-Hermitian operator S reads þ 1 ðÞÀ 4 ðÞþ ; 6n2 3n1 sin n1 n2 P ξ 2 3 ¼À i qmn ðϕà y þ ϕ y Þ S ωph Àω~ naq nbq cqmn qmn qmn 0 ð Þ P ξ 34  þ i qmn ðϕà y þ ϕ y Þ y À : :; ÀÁ4π ωph þω~ naq nbq cÀqmn H c ¼ 2 À 2 À þ þ ðÞþ qmn qmn 0 Bn 3n1 6n2 6n1 6n1n2 2 cos n1 n2 3 ð Þ pffiffiffiÀÁπ 45 jωph ω jΩj j 2 4 where we have used the approximation qmn ± 0 fq that is valid near the K þ 3 2n À 4n þ 6n n À 3n sin ðÞn þ n ; ′ ξ 1 2 1 2 1 3 1 2 and K points. Retaining leading-order terms in qmn, the transformed polariton

NATURE COMMUNICATIONS | (2018) 9:2194 | DOI: 10.1038/s41467-018-03982-7 | www.nature.com/naturecommunications 9 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03982-7 and 16. Rechtsman, M. C. et al. Topological creation and destruction of edge states in photonic graphene. Phys. Rev. Lett. 111, 103901 (2013). C ¼ 1 þ 3n ðÞþn À 1 3n ðÞn À n : ð46Þ n 1 1 2 2 1 17. Rechtsman, M. C. et al. Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures. Nat. Photonics 7, 153–158 (2013). = 18. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, For brevity, we retain only the dominant (m 0) TEM contribution for the – fi ΔðυÞ 196 200 (2013). plots in Fig. 4, where the coef cients K0n in Eq. (39) are given by "# 19. Ni, X., Purtseladze, D., Smirnova, D. A., Slobozhanyuk, A. & Alù, A. Spin and π  ω2 valley polarized one-way Klein tunneling in photonic topological insulators. 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52. McCann, E. & Fal’ko, V. I. Landau-level degeneracy and C.-R.M acknowledges the EPSRC Centre for Doctoral Training in Metamaterials (Grant in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006). No. EP/L015331/1) and QinetiQ for additional funding. G.W. acknowledges financial 53. McCann, E. & Koshino, M. The electronic properties of bilayer graphene. Rep. support from Agence Nationale de la Recherche (Project ANR-14-CE26-0005 Q-Meta- Prog. Phys. 76, 056503 (2013). Mat) and the Centre National de la Recherche Scientifique through the Projet Interna- 54. Mucha-Kruczyński, M., Aleiner, I. L. & Fal’ko, V. I. Strained bilayer graphene: tional de Coopération Scientifique program (Contract No. 6384 APAG). W.L.B. Band structure topology and Landau level spectrum. Phys. Rev. B 84, acknowledges the financial support from EPSRC (Grant No. EP/K041150/1) and EU 041404(R) (2011). ERC project Photmat (ERC-2016-ADG-742222). E.M. acknowledges financial support 55. Mariani, E., Pearce, A. 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Matter 29, 315301 appropriate credit to the original author(s) and the source, provide a link to the Creative (2017). Commons license, and indicate if changes were made. The images or other third party 68. Jackson, J. D. Classical Electrodynamics, 3rd edn (Wiley: New York, 1999). material in this article are included in the article’s Creative Commons license, unless 69. Power, E. A. & Thirunamachandran, T. Quantum electrodynamics in a cavity. indicated otherwise in a credit line to the material. If material is not included in the Phys. Rev. A 25, 2473–2484 (1982). article’s Creative Commons license and your intended use is not permitted by statutory 70. Colpa, J. H. P. Diagonalization of the quadratic boson Hamiltonian. Physica A regulation or exceeds the permitted use, you will need to obtain permission directly from – 93, 327 353 (1978). the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. Acknowledgements We would like to thank Ian Hooper for useful discussions and help with the numerical © The Author(s) 2018 simulations. C.-R.M. and T.J.S. acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. In addition,

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