ORE Open Research Exeter

TITLE Manipulating Type-I and Type-II Dirac in Cavity-Embedded Honeycomb Metasurfaces

AUTHORS Mann, C-R; Sturges, TJ; Weick, G; et al.

JOURNAL Nature Communications

DEPOSITED IN ORE 21 March 2018

This version available at

http://hdl.handle.net/10871/33238

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The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication ARTICLE DOI: 10.1038/s41467-018-03982-7 OPEN Manipulating type-I and type-II Dirac polaritons 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 Tquasiparticles in a myriad of distinct artificial 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 H = 3,14–16,20–23 pol 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 H and H , respectively. We employ the – mat ph 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 included through the principle of minimal-coupling46. – 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 predicted . However, since their existence is predicated on eigenmode with resonant frequency ω0. 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 the Brillouin zone edge above ω , ensuring the existence of – 0 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- ~ talline metasurfaces consisting of a honeycomb array of reso- Hmat hω~0 aqy aq bqy bq hΩ fqbqy aq H:c: ; 1 nant, dipolar meta-atoms. Despite the elementary nature of ¼ q þ þ q þ ð Þ  these metasurfaces, we unveil the simultaneous existence of X X both type-I and type-II massless Dirac polaritons which have where, for brevity, we have not presented the non-resonant terms distinct physical origins. Crucially, the existence of the latter is (see Methods for derivation). In Eq. (1), ω~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 that by embedding the honeycomb metasurface inside a planar bosonic operators aqy and bqy create quanta of the quasistatic

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z y To quadratic order in k = q−K (ka 1), the effective matter eff eff Hamiltonian near the K point is HK k ψky K;kψk, with x ¼ H spinor creation operator ψy ay ; by and Bloch Hamiltonian k ¼ k k P eff 2 hω~ 1 h~vσ k h~t σà k  2 HK;k ¼ 0 2 À Á þ ðÞÁ ð Þ * Here, 12 is the 2 × 2 identity matrix, σ = (σx, σy) and σ = °2 (σx, −σy) 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 3Ω~a=2 and trigonal a1 a2 warping parameter ~t 3Ω~a2=8. Apart from¼ a global energy shift, Meta-atom Eq. (2) is equivalent¼ to a 2D massless Dirac Hamiltonian to leading order in k, with an isotropic spectrum ωmat ω~ τ~v k that is characterized by closed isofrequency Possible experimental Microwave Plasmonic kτ ¼ 0 þ realizations helical resonators nanorods contours. Therefore,jj as expected from the honeycomb symmetry, the CDP belongs to the type-I class of 2D Dirac points, and the T corresponding spinors ψ 1; τeiθk =p2, where θ = Fig. 1 Schematic of a cavity-embedded honeycomb metasurface. The j kτi ¼ð À Þ k arctan(ky/kx), represent massless Dirac collective-dipoles with a honeycomb array of meta-atoms is composed of two inequivalent (A ffiffiffi and B) hexagonal sublattices—defined by lattice vectors a a p3 ; 3 topological Berry phase of π. The effective Hamiltonian near the 1 ¼ ðÀ 2 2Þ eff eff and a a p3 ; 3 —which are connected by nearest-neighbor vectors K′ point is given by K′;k K; k Ã, where the corresponding 2 ¼ ð 2 2Þ ffiffi H ¼ðH À Þ e = a(0, −1), e a p3 ; 1 , and e a p3 ; 1 , where a is the massless Dirac collective-dipoles have a topological Berry phase 1 ffiffi 2 ¼ ð 2 2Þ 3 ¼ ðÀ 2 2Þ of −π, as required by time-reversal symmetry. subwavelength nearest-neighborffiffi separation.ffiffi Each subwavelength meta- 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 lattices, respectively, with wavevector q in the first Brillouin zone 3 electromagnetic (TEM) mode whose polarization (parallel to (see Fig. 2a). Finally, the function fq j 1 exp iq ej encodes the dipole moments) and linear dispersion (see Fig. 2b) are ¼ ¼ ð Á Þ the honeycomb geometry of the lattice with nearest-neighbor independent of the cavity height. For brevity, in what follows we P vectors ej (see Fig. 1). do not present the contributions from the other cavity modes We diagonalize the matter Hamiltonian (Eq. (1)) as since the essential physics emerges from the interaction with the mat Hmat τ ± q hωqτ βqyτβqτ where the bosonic operators fundamental TEM mode (see Methods for the full expressions). ¼ ¼ βy ψy ψ create quasistatic collective-dipole normal modes In fact, the higher order cavity modes become increasingly qτ ¼ Pqj qτi P negligible for smaller cavities as they are progressively detuned with dispersion ωmat ω~ τΩ~ f . Here, τ indexes the upper qτ ¼ 0 þ j qj from the dipole resonances. (τ =+1) and lower (τ = −1) bands and ψqy aqy ; bqy is a spinor The effects of the photonic environment are encoded in the ¼ð T Þ creation operator. The spinors ψ 1; τeiφq =p2, where T free photonic Hamiltonian j qτi ¼ð Þ denotes the transpose, describe an emergent pseudospin degree ph ffiffiffi Hph h ωqncqyncqn 3 of freedom where the two components encode the relative ¼ qn ð Þ amplitude and phase of the dipolar oscillations on the two X inequivalent A and B sublattices, respectively, with φq = arg(fq). and in the light–matter interaction Hamiltonian These spinors can be represented by a pseudospin vector on the 1 2 Hint Hintð Þ Hintð Þ, with Bloch sphere which reads Sqτ = τ(cosφq, sinφq, 0). ¼ þ 1 At the high symmetry K and K′ points (see Fig. 2a), the Hintð Þ h iξqnϕnà aqy cqn aqy cy qn ¼ qn þ À sublattices decouple with no well-defined relative phase (i.e., fq = 0), P  4 giving rise to two inequivalent CDPs located at ± K ± 4π ; 0 ð Þ 3p3a h iξqnϕn bqy cqn bqy cy qn H:c: ¼ ð Þ þ qn þ À þ as observed in Fig. 2b. These CDPs correspond to vortices in the  fi P pseudospin vector eld Sqτ, which give rise to topologicalffiffi singularities in the Berry curvature47. Therefore, the CDPs are and sources of quantized Berry wπ, where w = ±1 is the topological 2 2ξqnξqn′ Hintð Þ h Re ϕnϕnà charge of the Dirac point corresponding to the winding ¼ ω0 ′ qnn′ 5 number of the vortex. As expected from the symmetry of the P ÀÁ ð Þ metasurface, the existence of the CDPs is robust against long- cqyncqn′ cqyncy qn′ H:c: þ À þ range Coulomb interactions as shown in Supplementary Note 1.  1=2 In fact, for small cavity heights, the image dipoles quench long- where ξqn LÀ parametrizes the strength of the light–matter range Coulomb interactions and the nearest-neighbor approx- interaction/ (see Methods for analytical expression). The bosonic fi imation becomes increasingly accurate as shown in Supplemen- operator cqyn creates a TEM photon with wavevector q in the rst tary Figure 1. Brillouin zone and dispersion ωph c q G , where n indexes the qn ¼ jjÀ n NATURE COMMUNICATIONS | (2018)9:2194 | DOI: 10.1038/s41467-018-03982-7 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03982-7

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 ! /

q Type-I CDP !

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

Fig. 2 Evolution of the polariton dispersion as the cavity height is reduced. a First Brillouin zone defined by primitive reciprocal lattice vectors b 1 ¼ 2π p3; 1 and b 2π p3; 1 . b Quasistatic dispersion of the collective-dipole normal modes, where the upper band corresponds to a bright, 3a À À 2 ¼ 3a À symmetric dipole configuration (↑↑) and the lower band corresponds to a dark, antisymmetric dipole configuration (↑↓). The light-cone (shaded region) is ÀÁffiffiffi ÀÁffiffiffi 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

Polariton dispersion obtained from the polariton Hamiltonian Hpol (solid black lines) and the two-band Hamiltonian Hmat (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 ph parameters ω 2:5ω and Ω = 0.01ω0 K0 ¼ 0 set of reciprocal lattice vectors G . The complex phase factors ϕ mediated by the electromagnetic field. However, by expressing the n n ¼ exp iaGn ^y are associated with Umklapp processes that arise due interaction Hamiltonian (Eq. (4)) in terms of the βqτ and βqyτ to theðÞ discrete,Á 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- 1 group symmetry of the polariton Hamiltonian. Hintð Þ h iΛqnτ βqyτcqn βqyτcy qn H:c:; 6 ¼ þ À þ fi τ ± qn ð Þ We diagonalize Hpol using a generalized Hop eld–Bogoliubov ¼  48 X X 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 heights. Also, in Supplementary Figure 2, we present the full strong multiple-scattering in the bipartite structure, result in a – polariton dispersion which includes long-range Coulomb inter- non-trivial winding of the light matter coupling as a function of actions. For small cavity heights, the full polariton dispersion wavevector direction iφq is almost indistinguishable from that obtained in the nearest- Λqnτ ξqn ϕnÃe τϕn : 7 neighbor approximation, and therefore one can conclude / þ ð Þ ÀÁ that long-range Coulomb interactions do not qualitatively affect Naively, one may expect all of the band crossings in Fig. 2b to 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 for the crossings Supplementary Figure 3 and Supplementary Figure 4 we present iφq fi with the upper quasistatic band where Λq0 e 1 (see the polariton dispersions obtained from nite element simula- red line in Fig. 3a) due to the constructive interferenceþ / þ between tions of a honeycomb array of plasmonic nanorods and the sublattices of this bright (↑↑) configuration (seeÀÁ 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 quasistatic band the coupling constant is significantly reduced Figure 2. iφq Λq0 e 1 (see blue line in Fig. 3a) due to the destructive interferenceÀ / betweenÀ the sublattices of this dark (↑↓) configura- Emergence of type-II Dirac points. Given the elementary nature tion (seeÀÁ 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 dipole–dipole interactions between the meta-atoms which are lower quasistatic band completely vanishes (Λq0− = 0) along the

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1 a 2.05

2 1 2 2.00 Inversion of chirality

| M # +

q 0.10 q " i 4 Γ K(K') 4

|e 0 0.05 M 3 0 v/v M K(K') M Γ→ Γ→ Γ→ 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 the magnitude of the light–matter coupling constant Λ eiφq τ on j q0τ j/j þ j Fig. 4 Tunable parameters in the effective Hamiltonian. Dependence of the the direction of q from −M to −K(K ) (see inset), for the upper (red Γ Γ ′ 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 2 parameters have been normalized to v = 3Ωa/2 and t = 3Ωa /8 which are the crossings labeled ‘2’ and ‘4’ along the −M directions, respectively. Γ 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- (| | = 0) along the −K(K ) directions due to the complete destructive Λq0− Γ ′ 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 ph dependent diagonal term D. Plots obtained with ω 2:5ω and Ω = 0.01ω0 K0 ¼ 0 high-symmetry Γ−K(K′) directions, where φq = 0, due to the Similarly, the effective Hamiltonian near the K′ point is given by complete destructive interference between the two sublattices (see eff eff K′;k K; k Ã. In Eq. (9), the resonant frequency ω0, group Fig. 3d). As a result, along these high-symmetry directions the velocityH ¼ðvH, andÀ Þ 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 Hamiltonian in the matter sublattice space K v=t; 0 and obtain S ¼ ðÞ 2 2 ph eff Dv 2Dv ξqnωqn K ;k′ h ω0 2 k′x 12 hσà v k′ ; 10 Hmat Hmat 2h 2 S ph H ¼ À t À t þ Á Á ð Þ ¼ À qn 2  ωqn ω~0 8 X À ð Þ 10 where k′ measures the deviation from K and v v is 2 2 S 03 aqy aq bqy bq ϕnbqy aq ϕnà aqy bq : ¼ þ þ þ 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 a generalized 2D massless Dirac Hamiltonian effective dispersion (see Methods) which provides an excellent k i x;y huiki12 i x;y hvikiσi. If the parameters ui and description of the polaritons as indicated by the orange dashed H ¼ ¼ þ 2¼ 2 2 2 vi satisfy the condition ux=vx uy=vy <1, then the Dirac cone lines in Fig. 2c–e. Finally, we expand the two-band Hamiltonian P P þ51 (Eq. (8)) up to quadratic order in k and obtain the effective becomes tilted and anisotropic but still belongs to the type-I eff eff class with closed isofrequency contours. However, the condition Hamiltonian near the K point H ψy ψ (see Sup- K ¼ k kHK;k k u2=v2 u2=v2>1 defines a distinct type-II class of 2D Dirac plementary Note 2 for derivation) with Bloch Hamiltonian x x þ y y P points, giving rise to a critically tilted Dirac cone with open, eff 2 2 hyperbolic isofrequency contours. Hence, the type-I and type-II hω 1 hvσ k ht σà k  hD k 1 : 9 HK;k ¼ 0 2 À Á þ ðÞÁ À jj 2 ð Þ 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 K–M –π –π +π +π -–ππ 0.5 0.5 0.5 –π –π ! k

/ y y y 0 –π +π 0 –2π 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 =

1.4a) cavity heights, respectively, as obtained from Hmat. 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 ph and Berry phase of π, but with inverted chirality. All pseudospin and contour plots are obtained with ω 2:5ω and Ω = 0.01ω0 K0 ¼ 0 isofrequency contours. Indeed, since we have uy = 0 and of the two-band Hamiltonian (Eq. (8)). In Fig. 5a–c we plot the 2 2 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 Dirac¼ 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 Note that similar analysis can be performed near the K′ point. of −π Berry flux. 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. We have thus 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 Dirac Hamiltonian with spinor eigenstates T type-I and type-II. In contrast to the type-I CDPs, the existence of ψ 1; τeiθk =p2. These represent massless Dirac polar- j kτi ¼ð À Þ ^ the type-II SDPs is intrinsically linked to the hybridization between itons with chirality ψkτ σ k ψkτ τ, resulting in a pseudos- the light and matter degrees of freedom, and thus one can pin that winds onceh aroundffiffiffij Á j thei CDP¼À 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 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-Ið CDP,Þ¼ forming a quadratic between these phases, we study the spinor eigenstates (see Methods) band-degeneracy (see Fig. 2d) with combined −2π Berry flux (see

6 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

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 with corresponding spinor probe the Dirac physics presented in this work. Given that the i2θk T eigenstates ψkτ 1; τeÀ =p2. Therefore, during this Dirac points exist in a polaritonic excitation spectrum, one critical mergingj i transition,¼ð À the masslessÞ Dirac polaritons morph must drive the system with photons at the required frequency in into massive chiral polaritons with qualitativelyffiffiffi 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 ^ 2 fi ψkτ σà k  ψkτ τ, resulting in a pseudospin that winds overcome the momentum mismatch with photons. The speci c htwicejð as fastÁ Þ comparedj i ¼À 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 Hamiltonian with corresponding spinor eigen- T Discussion states ψ 1; τeiθk =p2. Remarkably, massless Dirac polar- j kτi ¼ð Þ 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. inversion of chirality ψ σffiffiffi k^ ψ τ (see Fig. 5f). Physically, h kτj Á j kτi ¼ In particular, we have unveiled the simultaneous existence of both this corresponds to a π-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- also accompanied by a π-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 correspond to an antisymmetric dipole spectrum with the environment offers a new paradigm that opens fi T con guration ψk 1; 1 =p2 in the subcritical phase and to a variety of opportunities to explore unique Dirac-related physics j fiτi ¼ð À Þ T a symmetric con guration ψkτ 1; 1 =p2 in the supercritical at the subwavelength scale. phase. Thus, due to thej orthogonalityi ¼ðffiffiffi Þ between these two For example, one can simultaneously probe the dynamics spinor eigenstates, the inversion of chirality removesffiffiffi 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 in a simple setup characterized by two neighboring regions with of the CDP frequency ω0 (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 R R a^y L ^z and R R ay^ L ^z on the A and B sublattices, respectively. A ¼ þ þ 2 B ¼ À þ 2 NATURE COMMUNICATIONS | (2018)9:2194 | DOI: 10.1038/s41467-018-03982-7 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03982-7

the local and block-diagonal form Here, R = l1a1 + l2a2 is an in-plane lattice translation vector with primitive vectors a1 and a2 (see Fig. 1) and integers l1 and l2. Each meta-atom is modeled by a single Hmat h ω0 Ω aqy aq bqy bq dynamical degree of freedom h (with dimensions of length), where the electric- ¼ q ðÞÀ S þ dipole moment associated with its fundamental eigenmode is p Qh^z, with  ¼À P effective charge Q. The Coulomb potential energy between two dipole moments h Ω 1 fqbqy aq ay q H:c: 17 þ q ð À IÞ þ À þ ð Þ p and p′ located at generic positions r and r′, respectively, is given by hi P 1 h 2 Ω aqy ay q bqy by q H:c: : p p′ 3 p n^ p′ n^ À q S À þ À þ VCoul Á À ð Á Þð3 Á Þ ; 11 P  ¼ 4πε0 r r′ ð Þ j À j In the main text, we do not present the non-resonant terms (e.g., bqy ay q), leading to Eq. (1) where ω~ ω Ω and Ω~ Ω 1 . À 0 ¼ 0 À S ¼ ðÞÀ I where n^ r r′ = r r′ and ε0 is the vacuum permittivity. In the Coulomb gauge, the light–matter interaction is described by the minimal- The presence¼ ðÞÀ of thejjÀ perfectly conducting metallic plates, placed at z = 0 and coupling Hamiltonian46 which, within the dipole approximation, reads z = L, modifies the boundary conditions on the scalar potential and, therefore, the 2 Q Q 2 Coulomb interaction between the meta-atoms. Using the method of images to Hint ΠR Az Rs Az Rs ; 68 ¼ M s ð Þ þ 2M ð Þ ensure the vanishing of the scalar potential at the cavity walls , we introduce an s A;B Rs s A;B Rs 18 fi X¼ X X¼ X ð Þ in nite series of image dipoles located outside the cavity at positions Rs lL^z, 1 2 þ Hð Þ Hð Þ 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 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 69 where we have used Π Π ^z. The vector potential can be decomposed into Eq. (11) , the matter Hamiltonian within the nearest-neighbor approximation R ¼ R reads transverse electric (TE) and transverse magnetic (TM) modes of the cavity. However, the photons corresponding to the TE modes have an in-plane Π2 polarization, and therefore only TM modes contribute to the z-component of the H Rs M ω2h2 mat 2M 2 0 Rs vector potential ¼ s A;B R þ ¼ s  P P 3 h q G mπ Q2 A r; z jjÀ n cos z 4πε a3 hR hR e z ph mπ 0 B B þ j ð Þ¼ q G ^z L þ RB j 1 qmnsεffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0Nm Lωqmn n L 19 ¼ X NA À þ  ð Þ 2 P P i q G r i q G r Q þ1 a 3 2 c e ð À nÞÁ cy eÀ ð À nÞÁ ; 3 ′2 h qmn q mn 8πε a lL Rs 12 þ À 0 s A;B R l ¼ s ¼À1 ð Þ hi P P P 2 2 2 3 lL Q þ1 2 a 1 where 3p3a =2 is the area of a unit cell and Nm = 1 + δm0. The bosonic 3 À 5 h h 8πε a ′ jj RB RB ej A ¼ fi 0 lL 2 2 operator cy creates a TM photon with wavevector q in the rst Brillouin zone À RB j 1 l 1 þ qmn ¼ ¼À1 þ a ffiffiffi ph jj and dispersion ω c q Gn ^zmπ=L . Here, Gn = n1b1 + n2b2 is a reciprocal P P3 P 2 qmn 2 2 lL 1 ¼ jjÀ þ Q þ1 ÀÁa À lattice vector with primitive vectors b1 and b2, where n indexes the set of ordered 3 ′ jj 5 hR hR e 8πε0 a 2 2 A A j À R j 1 l 1 lL À pairs of integers (n1, n2), and m is a non-negative integer indexing the different TM A ¼ a P P ¼ÀP1 þjj cavity modes. Only TM photons with even m couple to the dipoles due to the parity ÀÁ 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 Rs, and M is an effective mass. Next, we introduce the bosonic operators 1 Hintð Þ h iξqmnϕnà aqy cqmn aqy cy qmn ¼ qmn þ À Mω 1 P  20 a 0h i Π 13 h i b c b c H:c: ð Þ RA RA RA ξqmnϕn qy qmn qy y qmn ¼ rffiffiffiffiffiffiffiffiffiffi2h þ s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihMω0 ð Þ þ qmn þ À þ P  and and 2 2ξqmn ξqm′n′ Hintð Þ h Re ϕnϕnà ¼ ω0 ′ qmm′nn′ 21 Mω0 1 P ÀÁ ð Þ bR hR i ΠR 14 B ¼ 2h B þ 2hMω B ð Þ cqymncqm′n′ cqymncy qm′n′ H:c: rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 þ À þ  The strength of the light–matter interaction is parametrized by that annihilate quanta of the fundamental eigenmode on the meta-atom located at 1 ph 2 RA and RB, respectively, and satisfy the commutation relations aR ; aRy δRR′, ω 8π a Ω ½ ′Š¼ ph q0n 22 b ; by δ , and a ; by 0. In terms of these operators, the matter ξqmn ω0 ωqmn ; R R′ RR′ R R′ ¼ Fð Þ ph 3p3N L ph ð Þ Hamiltonian½ Š¼ (Eq. (12))½ readsŠ¼ ωqmn m ωqmn! ffiffiffi where, to take into account the finite size of the meta-atoms, we have introduced a H hω ay a hω by b ph mat 0 RA RA 0 RB RB phenomenological function ω that provides a smooth cut-off for the ¼ R þ R qmn A B interaction with short-wavelengthFð photonicÞ modes where the dipole approximation P 3 P hΩ 1 by a ay H:c: breaks down. We choose the phenomenological cut-off function to be of the Fermi- RB RB ej RB ej þ ð À IÞ R j 1 þ þ þ þ Dirac distribution form B ¼ h   i 15 h P P ð Þ Ω ay a ay H:c: ph 1 2 RA RA RA ω ; À S R þ þ qmn ph 23 A Fð Þ¼ 2 ωqmn 3ω0 =ω0 hi 1 e ð À Þ ð Þ h P þ Ω by b by H:c: 2 RB RB RB À S RB þ þ which is smooth enough to avoid spurious artifacts appearing in the polariton Phi spectrum. Finally, the free photonic Hamiltonian of the cavity reads 2 3 ph : where Ω = Q /8πε0Mω0a parametrizes the strength of the nearest-neighbor Hph h ωqmncqymncqmn 24 ¼ qmn ð Þ Coulomb interaction, and the parameters X

2 3 2 lL 1 1 a 1 a In Eqs. (3), (4), (5) in the main text, we only present the contribution from the 4 ; 2 À 5 16 S ¼ lL I ¼ 2 2 l 1 l 1 1 ÀÁlL ð Þ TEM mode (m = 0), dropping the corresponding index. In Supplementary Note 1, X¼  X¼ þ a we discuss the effect of the higher order (m ≠ 0) TM cavity modes for larger hiÀÁ cavities. encode renormalizations due to the cavity-induced image dipoles. We apply Born- von Kármán boundary conditions over a lattice with 1 unit cells and introduce Hopfield–Bogoliubov diagonalization. The polariton Hamiltonian Hpol = Hmat + 1=2 N  iq RA the Fourier transform of the bosonic operators aR À aqe Á and Hph + Hint, where Hmat is given by Eq. (17), Hph by Eq. (24), and Hint by Eqs. (20) 1=2 A q iq RB ¼ N 1 pol bR À bqe Á , which transforms the matter Hamiltonian (Eq. (15)) into and (21), can be recast into matrix form as Hpol Ψy H Ψq where B ¼ N q P ¼ 2 q q q P P 8 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

T T Ψqy ψqy ; Cqy ; ψ q; C q . Here, ψqy aqy ; bqy is the spinor creation operator in Hamiltonian (Eq. (32)) reads ¼ð À À Þ ¼ð Þ the matter sublattice space and Cqy cqy1; cqy2; ¼ ; cqyp; ¼ ; cqyN is the vector of TM ¼ð Þ 1 1 2 photon creation operators, where p indexes the set of ordered triplets of integers H H S; Hð Þ H Hð Þ ; pol mat 2 int ph int (n1, n2, m), and N is the total number of photonic operators considered. The ’ þ þ þ 35 pol hi ð Þ Hph Hermitian [2(N + 2)] × [2(N + 2)] matrix Hq can be written in block form as Hmat

þ À |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} Hq Hq À Wq Hpol ; 25 where the matter and photonic subspaces are decoupled to quadratic order in ξqmn. q ¼ 0 y à 1 ð Þ qÀ q þq Calculating the commutator in Eq. (35) and extracting the Hamiltonian within the H À W HÀ matter sublattice space, we obtain the two-band Hamiltonian @ A where 2 ph ξqmn ωqmn 2 2 Hmat Hmat 2h ph 2 2 aqy aq bqy bq ϕnbqy aq ϕnà aqy bq : 36 ¼ À ωqmn ω~0 þ þ þ hDiag ω ; ω ; ωph; ωph; ¼ ; ωph; ¼ ; ωph 26 qmn ð Þ À ð Þ Wq ¼ 0 0 q1 q2 qp qN ð Þ P   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 ± can be sub-divided into block q mode (m = 0), dropping the corresponding index. We can recast the Hamiltonian matrices H mat (Eq. (36)) into matrix form as Hmat q ψqy q ψq, with Bloch Hamiltonian mat int ¼ H q q ± H H mat PWq Fqà q ; 27 h : 37 H ¼ 0 ± int y ph 1 ð Þ Hq ¼ F W ð Þ Hq Hq q q ! @  A where the upper-diagonal block Here W ω~ Ω Δ and F Ω~f Ω Δ ϕ2 with q ¼ 0 À mn qmn q ¼ q À mn qmn n ω~ Ω~f à 2 mat 0 q P 2 Pph h 28 16π a ω0 ωq0n 2 ph Hq ¼ ~ ~ ð Þ Δ ω : 38 Ωfq ω0 ! qmn ph 2 2 ph qmn ð Þ ¼ 3p3Nm L ωqmn ω~ ωqmn!F ð Þ ð Þ À 0 is the 2 × 2 matrix in the matter subspace, and the lower-diagonal block ph is the ffiffiffi Hq mat N × N matrix in the photonic subspace with components Diagonalizing Hmat leads to the two-band dispersion ωqτ Wq τ Fq , which is indicated by the orange-dashed lines in Fig. 2c–e. The corresponding¼ þ j spinorj iφ T ξqpξqp q ph ph ′ eigenstates ψqτ 1; τe =p2, where φq arg Fq , can be represented by the q hωqpδpp′ 4h Re ϕpϕpÃ′ : 29 j i ¼ð Þ ¼ ð Þ H pp′¼ þ ω0 ð Þ pseudospin vector Sqτ τ cos φq; sin φq; 0 from which we obtain the pseudospin  no vector field plots in Fig.¼ 5að–c. ffiffiffi Þ Finally, the off-diagonal block int in Eq. (27) is the 2 × N interaction matrix, Hq where the pth column reads Expansion of the effective two-band Hamiltonian. Near the K point, the func- iξ ϕ tion Δqmn, given by Eq. (38), expands as int qp pà q h : 30 H p¼ iξ ϕ ð Þ 0 2 1 qp p ! Δ Δð Þ a Δð Þ K G k K G k  kmn ’ Kmn À Kmn ð À nÞx x þð À nÞy y 1 hi2 2 1 a2Δð Þ a4Δð Þ K G k2 þ 2 À Kmn þ KmnðÞÀ n x x The polariton Hamiltonian Hpol is diagonalized by a generalized 39 hi1 2 2 ð Þ fi 48 T 1 a2 ð Þ a4 ð Þ K G k2 Hop eld–Bogoliubov transformation Ψq = TqXq, where Xqy χqy ; χ q and 2 ΔKmn ΔKmn n y y ¼ð À Þ þ À þ ðÞÀ χqy γqy1; γqy2; ¼ ; γqyν ; ¼ ; γqyN 2 . To ensure the invariance of the bosonic hi2 ¼ð þ Þ a4Δð Þ K G K G k k commutation relations for the transformed operators, Tq must be a [2(N + 2)] × [2 þ KmnðÞÀ n xðÞÀ n y x y 70 fi (N + 2)] paraunitary matrix that satis es TqηzTqy TqyηzTq ηz, where ηz σ 1 and σ is the Pauli matrix. The transformed¼ bosonic operators¼ γ ¼ υ z 2 z qyν to quadratic order in k, where the real parameters ΔKð mnÞ (υ = 0, 1, 2) depend only ¼ ph Ψy η Ψqν and γqν = 〈Ψqν|ηzΨq that diagonalize the polariton Hamiltonian as on the photon frequencies at the K point. Collecting the contributions from q zj i ωKmn pol the degenerate photons (see Supplementary Note 2 for details), we obtain the H h ω γy γ ; pol qν qν qν 31 effective Hamiltonian (Eq. (9)), where parameters are given by ¼ qν ð Þ X ω0 Ω Ω 0 create and annihilate polaritons in the vth band, respectively. The polariton 1 ΔKð mnÞ ; 40 pol ω0 ¼ À ω0 S À ω0 mn ð Þ dispersion ωqν (black solid lines in Fig. 2c–e) and the corresponding linearly X fi independent eigenvectors |Ψqν〉 ( rst two columns of Tq) are determined from the positive eigenvalue solutions to the non-Hermitian eigenvalue equation v 4π 1 pol pol ð Þ ηzHq Ψqν hωqν Ψqν . 1 AnΔKmn; 41 j i ¼ j i v ¼ À I À 27 mn ð Þ X Schrieffer–Wolff transformation. To obtain an effective two-band Hamiltonian in the matter sublattice space, we neglect non-resonant terms in the matter 2 t 8π 2 Hamiltonian (since Ω=ω 1 for practical realizations of the metasurface), but 1 B Δð Þ ; 42 0 t ¼ À I À 81 n Kmn not in the light–matter interaction Hamiltonian since the photons are not resonant mn ð Þ X with the collective-dipoles near the corners of the Brillouin zone (see Fig. 2b). Next, we perform a unitary transformation and 1 2 S S D Ω 4π 2 1 1 Hpol e HpoleÀ Hpol S; Hpol S S; Hpol ¼ 32 C ð Þ ð Þ ; ¼ ¼ þ½ Šþ2 ½ ½ ŠŠ þ ð Þ 2 nΔKmn ΔKmn 43 ω0a ¼ ω0 mn 27 À 2 ð Þ X and impose the Schrieffer–Wolff condition50 with 1 S; Hmat Hph Hintð Þ 33 þ ¼À ð Þ p3 4π hi A 2 3n cos n n fi n ¼ 2 ðÞÀ 1 3 ðÞ1 þ 2 which eliminates the light–matter interaction to rst order in ξqmn. From Eq. (33), ffiffiffi  44 the particular form of the anti-Hermitian operator S reads 1 4π ð Þ 6n 3n sin n n ; þ 2 ðÞ2 À 1 3 ðÞ1 þ 2 iξqmn  S ph ϕnÃaqy ϕnbqy cqmn ¼Àqmn ωqmn ω~0 ð þ Þ À 34 P iξqmn ð Þ ph ϕnÃaqy ϕnbqy cy qmn H:c:; 2 2 4π þ ωqmn ω~0 ð þ Þ À À B 3n 6n 6n 6n n 2 cos n n qmn þ n ¼ 1 À 2 À 1 þ 1 2 þ 3 ðÞ1 þ 2 P 45 ÀÁ ph 2 4π ð Þ where we have used the approximation ωqmn ± ω0 Ω fq that is valid near the K p3 2n 4n 6n n 3n sin n n ; j j  j j þ 1 À 2 þ 1 2 À 1 3 ðÞ1 þ 2 and K′ points. Retaining leading-order terms in ξqmn, the transformed polariton  ffiffiffiÀÁ 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 1 3n n 1 3n n n : 46 n ¼ þ 1ðÞþ1 À 2ðÞ2 À 1 ð Þ 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 υ 196–200 (2013). plots in Fig. 4, where the coefficients Δð Þ in Eq. (39) are given by K0n 19. Ni, X., Purtseladze, D., Smirnova, D. A., Slobozhanyuk, A. & Alù, A. 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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. J. & Von Oppen, F. Fictitious gauge fields in bilayer from the Leverhulme Trust (Research Project Grant RPG-2015-101), and the Royal graphene. Phys. Rev. B 86, 165448 (2012). Society (International Exchange Grant No. IE140367, Newton Mobility Grants 56. Verberck, B., Partoens, B., Peeters, F. M. & Trauzettel, B. Strain-induced band NI160073, and Theo Murphy Award TM160190). gaps in bilayer graphene. Phys. Rev. B 85, 125403 (2012). 57. Pearce, A. J., Cavaliere, F. & Mariani, E. Conductance and shot noise in Author contributions strained bilayer graphene. J. Phys. Condens. Matter 25, 375301 (2013). C.-R.M. performed the theoretical calculations and wrote the manuscript with E.M.; T.J. 58. Bena, C. & Simon, L. Dirac point metamorphosis from third-neighbor S. contributed to the theoretical calculations; E.M. and G.W. conceived the project; and E. couplings in graphene and related materials. Phys. Rev. B 83, 115404 (2011). M. and W.L.B. supervised the project. 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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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