Tunable Pseudo-Magnetic Fields for Polaritons in Strained Metasurfaces

Tunable pseudo-magnetic ﬁelds for polaritons in strained metasurfaces

Charlie-Ray Mann,1 Simon A. R. Horsley,1 and Eros Mariani1 1School of Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, United Kingdom. Artificial magnetic fields are revolutionizing our ability to manipulate neutral particles, by enabling the em- ulation of exotic phenomena once thought to be exclusive to charged particles [1–18]. In particular, pseudo- magnetic fields generated by nonuniform strain in artificial lattices have attracted considerable interest because of their simple geometrical origin [11–18]. However, to date, these strain-induced pseudo-magnetic fields have failed to emulate the tunability of real magnetic fields because they are dictated solely by the strain configuration. Here, we overcome this apparent limitation for polaritons supported by strained metasurfaces, which can be realized with classical dipole antennas or quantum dipole emitters. Without altering the strain configuration, we unveil how one can tune the pseudo-magnetic field by modifying the electromagnetic environment via an enclosing photonic cavity which modifies the nature of the interactions between the dipoles. Remarkably, due to the competition between short-range Coulomb interactions and long-range photon-mediated interactions, we find that the pseudo-magnetic field can be entirely switched off at a critical cavity height for any strain configuration. Consequently, by varying only the cavity height, we demonstrate a tunable Lorentz-like force that can be switched on/off and an unprecedented collapse and revival of polariton Landau levels. Unlocking this tunable pseudo-magnetism for the first time poses new intriguing questions beyond the paradigm of conventional tight-binding physics.

Unfortunately, neutral particles do not directly couple to the configuration one can tune the artificial magnetic field by electromagnetic gauge potentials. Therefore, exotic phenom- modifying the real electromagnetic environment. ena exhibited by charged particles in magnetic fields, such as To demonstrate this tunability we embed the metasurface the Lorentz force, Aharonov-Bohm effect, and Landau quan- inside a planar photonic cavity, where one can modify the tization, remain elusive for neutral particles. However, it was nature of the long-range dipole-dipole interactions by sim- recently demonstrated that nonuniform strain in graphene can ply varying the cavity height. As a result, one can tune the generate pseudo-magnetic fields which can mimic some of strength of the pseudo-magnetic field acting on the polari- the properties of real ones [19–22]. This tantalizing prospect tons and even switch it off entirely at a critical cavity height, inspired many attempts to emulate strain-induced pseudo- despite the strain configuration being fixed — this highly magnetic fields for neutral particles in acoustics [15, 16], non-trivial result is impossible to achieve with conventional phononics [13, 14], and photonics [11, 12], by judiciously tight-binding systems. Consequently, we demonstrate a tun- engineering aperiodicity in artificial lattices that mimic the able Lorentz-like force that can be switched on and off, de- tight-binding physics of graphene. However, these strain- flecting polariton wavepackets into effective cyclotron orbits induced pseudo-magnetic fields have failed to emulate one whose radius can be controlled via the cavity height. For large key property of real magnetic fields: tunability. While real strains, we also demonstrate Landau quantization for the po- magnetic fields can be tuned by varying external parameters laritons where progressively decreasing the cavity height can in the lab, these emergent pseudo-magnetic fields are dictated induce an unprecedented collapse and revival of the polariton solely by the strain configuration, rendering them fixed by de- Landau levels. sign. Therefore, a fundamental question arises: can we over- We model the resonant dipole scatterers with a generic bare come this seemingly intrinsic lack of tunability? 2 2 −1 polarizability α0(ω) = 2ω0Ω(ω0 − ω − iωΓ) that is ap- Here we show that tunable pseudo-magnetic fields for plicable to both classical antennas and quantum emitters in neutral particles are indeed possible by going beyond the their linear regime, where Ω characterizes the strength of the paradigm of conventional tight-binding physics. Specifically, polarizability, Γ accounts for non-radiative losses, and ω0 is we consider polaritons supported by a strained honeycomb the free-space resonant frequency. The metasurface is formed metasurface composed of resonant dipole scatterers. This by arranging the scatterers into a honeycomb array consist- ing of two inequivalent hexagonal sublattices. We assume arXiv:2001.11931v1 [physics.optics] 31 Jan 2020 simple model can be realized in a variety of experimental platforms, ranging from microwave [23], plasmonic [23–25], the scatterers to be anisotropic such that the induced dipole or dielectric [26] metamaterials composed of classical dipole moments point normal to the plane of the lattice. Further- antennas, to arrays of excitonic nanostructures [28] or atom- more, we consider subwavelength lattice spacing so that the like quantum emitters [27]. Crucially, the resonant interac- metasurface supports evanescently bound surface polaritons. tion with light results in long-range interactions between the Finally, the metasurface is embedded at the center of a pla- dipoles which qualitatively depend on the surrounding elec- nar photonic cavity of height L, as schematically depicted in tromagnetic environment. Consequently, previous results de- figure 1a, where the cavity walls are assumed to be perfectly rived from conventional tight-binding models do not trivially reflecting mirrors. extend to lattices of interacting dipoles [23]. By exploiting The interactions between the dipoles are mediated by the this key difference, we show that without altering the strain cavity Green’s function which can be decomposed into its lon- 2

Figure 1. Cavity-tunable pseudo-gauge potentials for polaritons in strained metasurfaces. a Schematic of a trigonally strained honeycomb metasurface composed of two inequivalent hexagonal sublattices of resonant dipole scatterers (e.g., classical antennas or quantum emitters), which is embedded inside a planar photonic cavity with height L. We assume that the scatterers have an anisotropic polarizability such that the induced dipole moments point in the zˆ direction (see inset). The unstrained metasurface has a subwavelength nearest-neighbor separation a λ0 (λ0 = 2πc/ω0) and therefore supports evanescently bound surface polaritons. b Shows how the real part of the cavity Green’s function G (evaluated at the free-space resonant frequency ω0) varies with the distance r from the source (black lines) for cavity heights of a/L = 0.2 (dotted lines), a/L = 0.5 (dashed lines), and a/L = 1 (solid lines). We also show the separate longitudinal Gk (blue lines) and transverse G⊥ (orange lines) components which describe the short-range Coulomb interactions and long-range photon-mediated interactions, respectively. The unstrained metasurface√ supports massless Dirac polaritons characterized by linear Dirac cones near the high symmetry K/K0 points located at ±K = ±(4π/3 3a, 0) in the ﬁrst Brillouin zone as depicted in c. The applied strain leads to a shift of the Dirac cones in frequency and momentum which is described by a pseudo-scalar and pseudo-vector potential, respectively. d-e Show how the strain- independent parameters in the pseudo-scalar potential (Φ0) and pseudo-vector potential (A0) vary as a function of the cavity height (black solid lines), where we also show the separate contributions emerging from the Coulomb interactions (blue dotted lines) and the photon-mediated interactions (orange dashed lines). At critical cavity heights (LΦ and LA) there is perfect cancellation between these contributions resulting in the pseudo-gauge potentials being switched off for any strain conﬁguration. Plots obtained with parameters ω0 = c|K|/2.2 and Ω = 0.01ω0.

gitudinal (k) and transverse (⊥) components G = Gk + G⊥, valley). Therefore, the polaritons behave like massless Dirac where Gk describes short-range Coulomb interactions and G⊥ quasiparticles with a linear Dirac cone dispersion [23], where describes long-range interactions mediated by the cavity pho- the Dirac frequency ωD(L) and the Dirac velocity vD(L) can tons (see Methods for the expressions). In figure 1b, we show be tuned by varying the cavity height (see Methods for the ex- how one can modify the Green’s function by varying the cav- pressions). As schematically shown in figure 1c, the strain ity height. Naively, it is tempting to assume that the near- leads to a spatially-varying shift of the Dirac cone in fre- field Coulomb interactions dominate the physics given the quency and momentum which is effectively described by a subwavelength spacing of the metasurface. However, as the pseudo-scalar potential Φ(r,L) = Φ0(L)[εxx(r) + εyy(r)], cavity height is reduced the Coulomb interactions (blue lines) and a pseudo-vector potential A(r,L) = A0(L)[εxx(r) − are exponentially suppressed and the photon-mediated inter- εyy(r), −2εxy(r)], respectively. In figures 1d-e we show how actions (orange lines) become increasingly dominant, even the strain-independent parameters Φ0(L) and A0(L) can be between nearest-neighbors. tuned by varying only the cavity height (see Methods for To gain analytical insight, we have used a coupled-dipole the expressions). Remarkably, there exists critical cavity model to derive an effective Hamiltonian describing the po- heights (LΦ and LA) where these parameters vanish identi- laritons near the K/K0 points, which is valid to leading order cally, thereby switching off the pseudo-gauge potentials en- in the strain tensor εij(r) = (∂uj/∂ri + ∂ui/∂rj)/2 where tirely for any strain configuration. u(r) is the slowly-varying displacement field. For the K valley, the effective Hamiltonian reads (~ = 1) To elucidate the physical origin of the vanishing pseudo- gauge potentials, one can notice that the pseudo-vector

HK = ωD(L)1+ivD(L)σ ·∇+Φ(r,L)1+σ ·A(r,L) , (1) (scalar) potential is equal to the change in intersublattice (intrasublattice) interaction energy induced by the strain at the where 1 is the identity matrix and σ = (σx, σy) is the vec- high-symmetry points. Interestingly, the change in energy tor of Pauli matrices acting in the sublattice space (see Meth- arising from the Coulomb interactions and photon-mediated ods for the derivation and equivalent Hamiltonian for the K0 interactions have opposite signs and thus tend to compen- 3

Figure 2. Cavity-tunable Lorentz-like force and cyclotron motion of polariton wavepackets. a Schematic depiction of the cyclotron motion exhibited by polariton wavepackets due to a Lorentz-like force generated by a pseudo-magnetic field which has opposite signs in the K/K0 valleys. b Shows how the cyclotron orbit radius varies with cavity height for a fixed frequency relative to the Dirac point and fixed strain magnitude. The dotted line indicates the region of cavity heights where the linear Dirac cone approximation breaks down (see Supplementary Information for more details). c Centre-of-mass trajectories of Gaussian wavepackets propagating in a trigonally strained metasurface with fixed strain magnitude, but different cavity heights (see Methods for details). Wavepackets in the K (solid lines) and K0 (dotted lines) valleys undergo cyclotron motion in opposite directions (e.g., see snapshots along trajectory 5), and the calculated radii match very well with the analytical predictions (see crosses in b). For subcritical cavity heights L > LA, the cyclotron radius expands as the cavity height is reduced (trajectories 1-3), until a critical height L = LA, where the pseudo-magnetic field is switched off and the wavepackets feel no Lorentz-like force (trajectory 4). For supercritical cavity heights L < LA, the cyclotron orbits reemerge and the orbit radius shrinks as the cavity height is −5 reduced further (trajectories 5-6). Plots obtained with parameters ∆ = 2 × 10 , ω0 = c|K|/2.2, Ω = 0.01ω0, and δω = −0.001ω0. sate each other in the pseudo-gauge potentials (see dotted and rect analogy with charged particles in real magnetic fields. dashed lines, respectively, in figure 1d-e). At the critical cav- Crucially, by modifying the nature of the dipole-dipole in- ity heights, the change in Coulomb interaction energy is per- teractions via the cavity, one can tune the magnitude of the fectly cancelled by the change in photon-mediated interaction Lorentz-like force and also the effective cyclotron mass of 2 energy, resulting in no variation of the total interaction energy, the Dirac polaritons mc(L) = δω/vD, where δω = ω − ωD despite the distances between the dipoles being modified. We is the frequency relative to the Dirac point. As a result, in wish to emphasize the novelty of this phenomenon: within figure 2b we show how the corresponding cyclotron orbit ra- 2 a nearest-neighbor tight-binding model, a vanishing pseudo- dius Rc(L) = |mc|vD/|Fτ | can be tuned by varying only the vector potential would demand a hopping parameter that does cavity height. The dotted line indicates the region of cav- not vary with distance, which is impossible to achieve with ity heights where the linear approximation of the Dirac cone graphene and its artificial analogs. Therefore, this ability to breaks down (see Supplementary Information for more de- switch off and tune the magnitude of the pseudo-gauge poten- tails). tials without altering the lattice structure opens up new per- To verify the tunability of the cyclotron orbits, we simu- spectives beyond conventional tight-binding models. lated the evolution of Gaussian wavepackets using the effec- In what follows, we investigate some of the implications tive Hamiltonian (1), and in figure 2c we plot the trajectories of the tunable pseudo-vector potential. Specifically, we con- of the centre-of-mass for a fixed strain magnitude, but differ- sider a strain configuration described by the displacement field ent cavity heights (see Methods for details). As expected, the 2 2 u(r) = (∆/a)(2xy , x − y ) [19], where ∆ is a measure polariton wavepackets in the K/K0 valleys undergo cyclotron of the strain magnitude. This trigonal strain configuration motion in opposite directions due to time-reversal symmetry gives rise to a vanishing pseudo-scalar potential and a pseudo- (e.g., see snapshots along trajectory 5), and the orbit radii vector potential A = 4(∆/a)A0(y , −x), leading to a uni- agree very well with the analytical predictions (see crosses in form pseudo-magnetic field Bτ = τ∇×A = −τ8(∆/a)A0zˆ figure 2b). Note, the direction of the orbits depend on the signs which, by virtue of time-reversal symmetry, has opposite of the Dirac velocity, cyclotron mass, and pseudo-magnetic 0 signs for the K (τ = +1) and K (τ = −1) valleys. field (see Supplementary Information for more details). For Therefore, in the ‘semiclassical’ limit [29], polariton subcritical cavity heights L > LA where the short-range wavepackets propagating through the strained metasurface be- Coulomb interactions are dominant, the cyclotron radius ex- have as if they were subjected to a Lorentz-like force Fτ (L) = pands as the cavity height is reduced (see trajectories 1-3). At sign(vD)vˆ × Bτ which acts perpendicular to the group ve- the critical cavity height L = LA, the pseudo-magnetic field locity direction vˆ. Consequently, the polaritons exhibit cy- is switched off and, as a result, the polariton wavepackets feel clotron motion, as schematically depicted in figure 2a, in di- no Lorentz-like force despite the metasurface being strained 4

Figure 3. Cavity-induced collapse and revival of polariton Landau levels. a Schematic illustration of a massless Dirac cone splitting into quantized polariton Landau levels due to a large pseudo-magnetic field induced by strain. b Shows the squareroot dependence of the polariton Landau levels on the strain magnitude and Landau index for a fixed cavity height (a/L = 0.2). c Shows the collapse and revival of the polariton Landau levels as the cavity height is reduced for a fixed strain magnitude (∆ = 0.05), as predicted by the effective Hamiltonian (1). The dotted lines indicate the region of cavity heights where the linear Dirac cone approximation breaks down (see Supplementary Information for more details). d-f Local spectral function at the centre of the metasurface (see Methods for details) for a fixed cavity height (a/L = 0.2), but different strain magnitudes of ∆ = 0, ∆ = 0.001, and ∆ = 0.002, respectively. For the strained metasurface (see e), we observe a series of resonant peaks within the Dirac region which correspond to the polariton Landau levels (labeled according to their Landau index) which are not present in the unstrained case (compare with d). As the strain is increased (see f), the Landau level spacing increases in accordance with the analytical prediction (see b). g-i Local spectral function at the centre of the metasurface for a fixed strain magnitude (∆ = 0.002), but different cavity heights of a/L = 0.2, a/L = 0.68, and a/L = 1, respectively. For subcritical cavity heights L > LA (see g), we observe Landau level peaks which are not present in the unstrained case (see inset). At the critical cavity height L = LA (see h), the pseudo-magnetic field is switched off and therefore no Landau level peaks are observed within the Dirac region; in fact, there is very little difference when compared to the corresponding unstrained metasurface (see inset). For supercritical cavity heights L < LA (see i), we observe Landau level peaks within the Dirac region which are not present in the unstrained metasurface (see inset), thus verifying the collapse and revival of the polariton Landau levels. Plots obtained with parameters Ω = 0.01ω0, Γ = 0.025Ω, and ω0 = c|K|/2.2. For b, d-f we use ω0 = c|K|/3.5 to maximize the size of the Dirac region.

(see trajectory 4). For supercritical cavity heights L < LA frequency. In figure 3b we show the characteristic square- where the long-range photon-mediated interactions become root dependence on the strain magnitude and Landau index dominant, the cyclotron orbits reemerge and the cyclotron ra- [20], which is a manifestation of the pseudo-relativistic na- dius now shrinks as the cavity height is reduced further (see ture of the massless Dirac polaritons. In previous works, if trajectories 5-6). one wanted to modify the Landau level spectrum in artificial lattices one had to fabricate an entirely new structure with a As the pseudo-magnetic field is increased one reaches the different strain configuration [12, 16]. In stark contrast, here ‘quantum’ limit, where the polariton cyclotron orbits undergo one can qualitatively tune the polariton Landau level spectrum Landau quantization in direct analogy with charged particles by varying only the cavity height as shown in figure 3c, where in real magnetic fields [30]. Therefore, as schematically de- again the dotted line indicates the region of cavity heights picted in figure 3a, the massless Dirac cone collapses into where the linear Dirac cone approximation breaks down (see ω (∆,L) = ω (L) ± a quantized√ Landau level spectrum n D Supplementary Information for more details). Remarkably, ωc(∆,L) n, where n = 0, 1, 2 ... is the Landau level in- p despite the strain configuration being fixed, we predict an un- dex, and ωc(∆,L) = 2|vD||Bτ | is the effective cyclotron 5 precedented collapse and revival of the polariton Landau lev- spontaneous emission and strong coupling effects for emitters els by progressively decreasing the cavity height. in subwavelength photonic structures. Furthermore, while we To verify these analytical predictions, we go beyond the ap- have demonstrated a tunable Lorentz-like force and tunable proximations of the effective Hamiltonian and calculate the Landau levels, one should also be able to observe Aharonov- effective polarizability αeff(ω) of a scatterer which fully ac- Bohm-like interference patterns which will depend qualita- counts for the strong multiple scattering between the dipoles tively on the surrounding photonic environment. Finally, here (see Methods for details). From this, we define a local spectral we have focused on some of the implications of a tunable function Im[αeff(ω)] which is related to the local density of pseudo-magnetic field, but what are the implications of a tun- states and characterizes the full spectral response of the meta- able pseudo-electric field arising from the pseudo-scalar po- surface. Figures 3d-f show the local spectral function at the tential? To conclude, while intense efforts are devoted to- centre of the metasurface for a fixed cavity height (L > LA), wards designing systems that emulate tight-binding models, but different strain magnitudes (see Methods for details). For this work hints towards a richer landscape of physics emerg- the strained metasurface (see figure 3e), a series of resonant ing from long-range interactions which are prevalent in elec- peaks emerge within the Dirac region which are not present tromagnetic systems. in the unstrained case (see figure 3d), and these directly correspond to the polariton Landau levels. As the strain is increased further (see figure 3f), the spacing between the Landau level METHODS peaks increases in accordance with the analytical prediction. Interestingly, the Landau level peaks are much sharper than The unstrained metasurface is composed of a honeycomb array of resonant point-dipole scatterers located at periodic positions RA = R − d/2 and RB = R + d/2 the resonance peak for a single scatterer, and their width is which form the A and B hexagonal sublattices, respectively. Here, d = a(0, 1) is the vector connecting the A and B sites in the unit cell, and R = l1√a1 + l2a2 are√ the limited only by non-radiative losses. This is because the local- set of lattice vectors, where l and l are integers, and a = ( 3a/2)(−1, 3) √ √ 1 2 1 ized Landau level states are composed of large Fourier com- and a2 = ( 3a/2)(1, 3) are the primitive lattice vectors. The corresponding ponents that lie outside the light-line, which suppresses any reciprocal lattice vectors are√ G = n1b1 + n2b2, where√ n1 and n2 are integers, and b1 = (2π/3a)(− 3, 1) and b2 = (2π/3a)( 3, 1) are the primitive re- resonant coupling to free-space photons, rendering the polari- ciprocal lattice vectors. The nearest-neighbor vectors which connect dipoles√ located on different sublattices are given by e = a(0, −1), e = (a/2)( 3, 1), and ton Landau level states optically dark (see Supplementary In- √ 1 2 e = (a/2)(− 3, 1) formation for more details). 3 , and the next-nearest-neighbor vectors which connect dipoles located on the same sublattice are given by a1, a2, a3 = a2 − a1, a4 = −a1, a5 = −a2, and a6 = −a3. When strain is applied to the metasurface the dipoles To verify the collapse and revival of the polariton Landau ¯ are displaced to aperiodic positions given by RA/B = RA/B + u(RA/B). Finally, levels, in figures 3g-i we show the local spectral function at the the metasurface is embedded at the centre of a planar photonic cavity formed by two centre of the metasurface for a fixed strain configuration, but perfectly reflecting mirrors located at z = ±L/2. different cavity heights (see Methods for details). For subcritical cavity heights L > LA where the short-range Coulomb interactions are dominant (see figure 3g), we clearly observe the Effective polariton Hamiltonian presence of Landau level peaks within the Dirac region which In this section we use a coupled-dipole model to derive an effective Hamiltonian de- are not present in the unstrained case (see inset). At the criti- scribing the polaritons near the K/K0 points, which is valid to leading order in strain. The scatterers are assumed to be anisotropic such that the induced dipole moments cal cavity height L = LA (see figure 3h), the pseudo-magnetic point normal to the plane of the metasurface, i.e., their bare polarizability tensor reads field is switched off and, as a result, no Landau level peaks can α0(ω) = α0(ω)zˆzˆ, where α0(ω) is defined in the main text. To describe a scat- be observed within the Dirac region despite the applied strain terer’s response to an external field, one has to modify this bare polarizability to take into account the interaction with its own scattered field. The renormalized polarizability (compare with inset showing the strain-free case), thus veri- of a scatterer inside the cavity therefore reads fying the collapse of the polariton Landau levels. For super- α−1(ω) = α−1(ω) − Σ(ω) , (2) critical cavity heights L < LA where the long-range photon- 0 mediated interactions become dominant (see figure 3i), we in- where the polarizability correction is given by deed observe the revival of the polariton Landau level peaks 3 3 3 within the Dirac region which are not present in the unstrained 2a kω 4a h ikω L ikω Li Σ(ω) = i + Li3 e − ikω L Li2 e , (3) case (see inset). 3 L3 This work poses new intriguing questions beyond the realm P∞ l n where kω = ω/c and Lin(z) = l=1 z /l is the polylogarithm of order n (see of conventional tight-binding models, which can be explored Supplementary Information for the derivation). The first term in equation (3) is the usual radiative-reaction correction in free-space [32], whereas the other terms encode the cor- in a variety of experimental platforms across the electromag- rections due to the cavity. In the absence of a driving field, the induced dipole moments p = p zˆ and p = p zˆ on the A and B sublattices are given by the netic spectrum. For example, do quantum-Hall-like edge RA RA RB RB states associated with the polariton Landau levels exist in the self-consistent coupled-dipole equations presence of long-range interactions? Can one controllably 1 X ¯ ¯ 0 X ¯ ¯ pR (ω) = G(RA−RA, ω)pR0 (ω)+ G(RA−RB, ω)pR (ω) α(ω) A A B switch them on and off, or tune their properties, by modify- 0 R RA6=RA B ing the nature of the dipole-dipole interactions? Moreover, (4) do novel edge states emerge at interfaces separating regions and with different photonic environments? The polariton Landau 1 X ¯ ¯ 0 X ¯ ¯ pR (ω) = G(RB−RB, ω)pR0 (ω)+ G(RB−RA, ω)pR (ω) , α(ω) B B A levels also provide a novel way of sculpting the local density 0 R RB6=RB A of states — this could be exploited for enhancing/suppressing (5) 6

where the first and second sums in each equation correspond to the field (z-component) Since the Coulomb interactions are short-range, the lattice sums converge rapidly and generated by all the other dipoles belonging to the same and opposite sublattice, re- therefore we retain only the dominant nearest and next-nearest-neighbor Coulomb inter- spectively (see Supplementary Information for the derivation). The interactions between actions. Then, after expanding equation (14) to leading order in the displacement field the dipoles are mediated by the cavity Green’s function which can be expressed in the we obtain spectral domain as 2 3 Z 0 0 AB 0 d r X −i(K+k )·em i(k −k)·r ∞ ! D (k, k ) = G (em)e e πa3k2 k2 q k (2π)2 k ω X m (1) 2 2 m=1 G(r, ω) = i 1 − H0 r kω − km , (6) (15) L k2 " # m=−∞ ω β ei Z × 1 + nn m d2q u (q)1 − e−iq·em eiq·r , 2 ei 2π |em| (1) where km = 2mπ/L and H0 is the Hankel function of zeroth order and first kind (see Supplementary Information for the derivation). Note, equation (6) is the zz- where βnn = |∂ log[Gk(em)]/∂ log(|em|)| encodes how the Coulomb interaction component of the dyadic Green’s function where both the source and observation points strength changes with respect to changes in the nearest-neighbor separation distance, and are located at the centre of the cavity and connected by the in-plane vector r. We decompose the cavity Green’s function (6) into its longitudinal and transverse Z 2 components G = G + G , where the longitudinal component can be expressed in the d r −iq·r k ⊥ ue(q) = u(r)e (16) spectral domain as 2π

3 ∞ is the Fourier transform of the displacement field. Next, performing the spatial integral 4a X 2 in equation (15) we obtain G (r) = − k K0(kmr) , (7) k L m m=1 3 0 AB 0 X −i(K+k )·em Dk (k, k ) = Gk(em)e where K0 is the modified Bessel function of zeroth order and second kind (see Sup- m=1 plementary Information for the derivation). The longitudinal component of the Green’s (17) ( i ) function (7) mediates Coulomb interactions between the dipoles inside the cavity. In 0 0 βnn em 0 −i(k−k )·em contrast, the transverse component G = G − G describes interactions that are me- × δ(k − k) + ui(k − k ) 1 − e . ⊥ k 2π |e |2 e diated by the transverse photonic modes of the cavity. For simplicity, we retain only the m dominant contribution from the fundamental transverse electromagnetic (TEM) mode of 0 the cavity which has a linear dispersion ωk = c|k| and polarization (along zˆ) that are Finally, expanding equation (17) to leading order in k and k we obtain independent of the cavity height. This is a good approximation since we are interested in the regime of small cavity heights where all the other cavity modes are detuned from 3 AB 0 X −iK·em the dipole resonances. The corresponding Green’s function for the TEM mode reads Dk (k, k ) = Gk(em)e m=1 (18) 2a3k2 Z d2k eik·r iπa3k2 " i j # TEM ω ω (1) 0 i 0 βnn emem 0 G⊥ (r, ω) = = H0 (kω r) , (8) × δ(k − k)(1 − ie k ) + ∇u (k − k ) , 2 2 + m i 2 f ij L 2π k − kω − i0 L 2π |em| where the infinitesimal imaginary shift i0+ ensures that the dipoles emit outgoing waves where we have identified ikiuj (k) = ∇fuij (k), which is the Fourier transform of (see Supplementary Information for the derivation). e the displacement gradient tensor defined as ∇uij = ∂uj /∂ri. Performing similar In what follows, we derive the effective Hamiltonian for the K valley since the effec- analysis for the intrasublattice (diagonal) matrix elements in equation (12) we obtain tive Hamiltonian for the K0 valley is related via time-reversal symmetry. To analyze the polaritons near the K point, we write the dipole moments as 6 AA/BB 0 X −iK·am Dk (k, k ) = Gk(am)e iK·RA K iK·RB K p (ω) = e ψ (RA, ω) , p (ω) = e ψ (RB, ω) , (9) m=1 RA A RB B (19) " # β ai aj × δ(k0 − k)(1 − iai k0 ) + nnn m m ∇u (k − k0) , K K m i 2 f ij where ψA and ψB are slowly-varying envelope fields for the A and B sublattices, 2π |am| respectively. Here we have used the crystal reference frame [33] where the dipoles are located at their original unstrained positions. To not overburden notation, we suppress where β = |∂ log[G (a )]/∂ log(|a |)| encodes how the Coulomb interaction the valley index until the end. By introducing the Fourier transform of the envelope fields nnn k m m strength changes with respect to changes in the next-nearest-neighbor separation distance. Z 2 Z 2 d r −ik·r d r −ik·r We will now analyze the transverse dynamical matrix elements in equation (13), ψe (k, ω) = ψ (r, ω)e , ψe (k, ω) = ψ (r, ω)e A 2π A B 2π B where the intersublattice (off-diagonal) matrix elements read (10) we can write the coupled-dipole equations (4) and (5) in matrix form as Z 2 0 d r X DAB(k, k , ω) = GTEM(R − d + u(r) − u(r − R + d), ω) ⊥ (2π)2 ⊥ Z R 1 2 0 0 0 0 ψe(k, ω) = d k Dk(k, k ) + D⊥(k, k , ω) ψe(k , ω) , (11) 0 0 αˇ(ω) × e−i(K+k )·(R−d)ei(k −k)·r . (20) −1 −1 where αˇ (ω) = α0 (ω) − Re[Σ(ω)]. In equation (11), the Fourier transform of T Since the photon-mediated interactions are long-range, we seek to perform the lattice the spinor envelope field reads ψe(k, ω) = [ψeA(k, ω) , ψeB(k, ω)] , the longitudinal dynamical matrix encoding the Coulomb interactions is sums in reciprocal space where they converge rapidly. To do this, we first insert the integral representation of the TEM Green’s function (8) into equation (20) which gives

AA 0 AB 0 ! 0 Dk (k, k ) Dk (k, k ) Z 2 Z 2 00 D (k, k ) = , (12) AB 0 d r d k X TEM 00 i(k0−k)·r k DAB∗(k0, k) DBB(k, k0) D (k, k , ω) = Ge (k , ω)e k k ⊥ (2π)2 2π ⊥ R (21) 00 0 00 and the transverse dynamical matrix encoding the photon-mediated interactions reads × ei(k −k −K)·(R−d)eik ·[u(r)−u(r−R+d)] ,

DAA(k, k0, ω) DAB(k, k0, ω) TEM 3 2 2 2 −1 0 ⊥ ⊥ where Ge⊥ (k, ω) = (2a kω /L)(k − kω ) is the Fourier transform of the TEM D⊥(k, k , ω) = AB∗ 0 BB 0 . (13) D⊥ (k , k, ω) D⊥ (k, k , ω) Green’s function. Next, the expansion of equation (21) to leading order in the displacement ﬁeld then yields We will ﬁrst analyze the longitudinal dynamical matrix elements in equation (12), Z 2 Z 2 00 where the intersublattice (off-diagonal) matrix elements read AB 0 d r d k X TEM 00 i(k0−k)·r D (k, k , ω) = Ge (k , ω)e ⊥ (2π)2 2π ⊥ R Z 2 0 d r X DAB(k, k ) = G (R − d + u(r) − u(r − R + d)) ( Z ) k 2 k i(k00−k0−K)·(R−d) i 00 2 −iq·(R−d) iq·r (2π) × e 1 + k d q ui(q) 1 − e e , R (14) 2π i e 0 0 × e−i(K+k )·(R−d)ei(k −k)·r . (22) 7

and after performing the spatial integral in equation (22) we obtain and, ﬁnally, the strain-independent parameter in the pseudo-vector potential is

Z 2 00 2 2 2 2 AB 0 d k X TEM 00 i(k00−k0−K)·(R−d) 3Ωβnn|Gk(em)| X 2Ωω ξ c (K − G) φ D (k, k , ω) = Ge (k , ω)e A (L) = − 0 x G . (31) ⊥ 2π ⊥ 0 2 2 2 R 4 (ωK−G − ω0 ) (23) G ( ) 0 i 00 0 −i(k−k0)·(R−d) × δ(k − k) + ki uei(k − k ) 1 − e . Note, to evaluate the Dirac frequency (28) one requires a suitable regularization pro- 2π cedure since the last two terms separately diverge (see Supplementary Information for details). P 0 We now use Poisson’s summation identity R exp i(k − k) · R = 2 P 0 [(2π) /A] G δ(k − k + G) to convert the sum over lattice vectors to a sum over reciprocal lattice vectors which gives Numerical simulation of polariton wavepackets ( AB 0 2π X TEM 0 0 D (k, k , ω) = φ Ge (k + K − G, ω)δ(k − k) ⊥ A G ⊥ G In this section we use the second-order split-operator method [34] to approximate the time evolution of the polariton envelope ﬁelds. After a small time δt has elapsed, the ) i TEM 0 0 0 0 polariton envelope ﬁeld in the K valley is given by + Ge (k + K − G, ω)(k + K − G)i − (k ↔ k) ui(k − k ) , 2π ⊥ e −iHKδt (24) ψK(r, t + δt) = e ψK(r, t) . (32)

where φG = exp(iG · d) are non-trivial phase factors that are crucial for maintaining In the second-order split operator method, the evolution operator in equation (32) is ap- 0 the correct symmetry. Finally, expanding equation (24) to leading order in k and k we proximated as obtain −iH δt − i Hε δt −iH0 δt − i Hε δt 3 2 2 ( e K = e 2 K e K e 2 K + O(δt ) , (33) AB 0 X ω ξ φG 0 1 0 D (k, k , ω) = δ(k − k) − ∇fuii(k − k) ⊥ ω2 − ω2 2π G K−G where H0 = ω 1 + iv σ · ∇ and Hε = Φ(r)1 + σ · A(r). Note, the cubic error (25) K D D K 2 ) in δt is due to the noncommutativity of the position and gradient operators. To calculate 2c (K − G)i 0 0 1 0 − k δ(k − k) − (K − G) ∇u (k − k ) , the field after time Ntδt has elapsed, we have to apply the operation (33) iteratively 2 2 i j f ij ωK−G − ω 2π N t ξ2 = 4πa3/AL Y K −1 K K where parameterizes the strength of the light-matter interaction. ψK(r, t + Ntδt) ≈ Mr F Mk FMr ψK(r, t) , (34) Performing similar analysis for the intrasublattice (diagonal) matrix elements in equa- i=1 tion (13) we obtain where F and F −1 represent the direct and inverse Fourier transform operations, respec- 2 2 ( AA/BB 0 X ω ξ 0 1 0 tively. Using the standard identity for the exponential of Pauli matrices, we can write the D (k, k , ω) = δ(k − k) − ∇fuii(k − k) ⊥ ω2 − ω2 2π position-dependent operator in equation (34) as G K−G 2 ) " # 2c (K − G)i 0 0 1 0 K −iδtΦ/2 sin δtA/2 − kiδ(k − k) − (K − G)j ∇fuij (k − k ) 2 2 Mr = e cos δtA/2 1 − i σ · A , (35) ωK−G − ω 2π A TEM 0 − Re[G⊥ (0, ω)]δ(k − k) . (26) and the momentum-dependent operator as " # To obtain an effective Hamiltonian for the polariton envelope fields, we first neglect sin vDkδt K −iωDδt 2 2 Mk = e cos vDkδt 1 + i σ · k . (36) non-radiative losses for simplicity and assume small detuning such that ω0 − ω ' k 2ω0(ω0 − ω). Next, we neglect the frequency dependence of the polarizability correction (3) and the transverse dynamical matrix (13) by evaluating them at the free-space Similarly, the evolution of the polariton envelope field in the K0 valley can be approxi- resonant frequency ω0. This approximation is equivalent to treating the light-matter interactions to second order in perturbation theory [35], and is valid for polaritons near mated as 2 2 2 the corners of the Brillouin zone where ω0Ωξ ωK−G − ω0 . Therefore, by Nt Fourier transforming equation (11) to the real-space and time domains, we obtain the Y K0 −1 K0 K0 ψ (r, t + N δt) ≈ M F M FM ψ (r, t) , equation of motion i∂tψK(r, t) = HKψK(r, t) for the spinor envelope field in K0 t r k r K0 (37) K K T i=1 the K valley ψK(r, t) = [ψA (r, t) , ψB (r, t)] , where the effective Hamiltonian is given by equation (1) in the main text. Similarly, we obtain the equation of mo- 0 K0 K0 tion i∂tψK0 (r, t) = HK0 ψK0 (r, t) for the spinor envelope field in the K valley where the operators Mr and Mk are related to equations (35) and (36) by the re- ∗ K0 K0 T placement σ ↔ σ and k ↔ −k. ψK0 (r, t) = [ψA (r, t) , ψB (r, t)] , where the effective Hamiltonian is related to ∗ For the simulations in figure 2c, we initialize the following Gaussian wavepackets equation (1) by time-reversal symmetry HK0 = HK, and reads

∗ ∗ |r|2 HK0 = ωD(L)1 − ivD(L)σ · ∇ + Φ(r,L)1 + σ · A(r,L) , (27) 1 − 1 ik ·r ψ (r, t = 0) = √ e 2w2 e 0 , (38) K − sign(v ) ∗ 2w 2π D where σ = (σx, −σy ). Note, since time-reversal symmetry is preserved, the pseudo- vector potential, and the corresponding pseudo-magnetic field, couple with opposite and signs in the two valleys. In equations (1) and (27) the Dirac frequency reads |r|2 1 − 1 ik ·r ωD(L) = ω0 − Ω Re[Σ(ω0)] − 3Ω|Gk(am)| 2w2 0 ψK0 (r, t = 0) = √ e e , (39) sign(vD) 2 2 2w 2π TEM X Ωω0 ξ (28) + Ω Re[G (0, ω0)] − , ⊥ ω2 − ω2 0 G K−G 0 in the K and K valleys, respectively. We consider wavepackets that are located in the lower polariton band with a fixed central frequency δω = −0.001ω0 relative to the the Dirac velocity is Dirac point, and an initial central wavevector k0 = −|δω/vD|xˆ (i.e., initial group velocity in the xˆ direction). Furthermore, the wavepackets are initially centred at the 2 2 2 origin with a width of w = 100a. We then track the centre-of-mass trajectory of the 3Ωa|Gk(em)| X 2Ωω0 ξ c (K − G)xφG vD(L) = − , (29) 2 (ω2 − ω2)2 wavepackets for the two valleys which is given by G K−G 0 R d2r |ψ |2r the strain-independent parameter in the pseudo-scalar potential reads K/K0 hri 0 = . (40) K/K R d2r |ψ |2 " 2 2 2 2 2 2 # K/K0 3Ωβnnn|Gk(am)| X Ωω0 ξ 2Ωω0 ξ c (K − G)x Φ0(L) = + − , 2 ω2 − ω2 (ω2 − ω2)2 G K−G 0 K−G 0 In the Supplementary information we go beyond the linear Dirac cone approximation by (30) including second order field gradients in the effective Hamiltonian. 8

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Additional information

Supplementary information accompanies this work. Correspondence and requests for materials should be addressed to C.-R.M (email: [email protected]).

Data availability

All relevant data are available from the corresponding authors upon reason- able request.

Competing interests

The authors declare no competing interests.