Polaritonic dispersion

In Figure5.3we plot the polariton dispersion ωpolq from equation5.28, which corresponds

to the eigenvalues of the matrix M given in equation 5.29. We truncate the matrix M by limiting the sum over the n Umklapp processes such that the resulting dispersion sufficiently converges. In Figure 5.3(a) we plot the dispersion along the symmetry lines Γ − K − M − Γ in the absence of coupling, that is we plot the TEM mode of the empty cavity and the CP dispersion (equation 4.41). We observe four points where the bands cross, labelled by circled numbers. One may expect that in the presence of coupling between these individual modes there will be anticrossings at all four of these band crossings.

Chapter 5 - Chirality inversion in Dirac polaritons 105

Figure 5.3: (a − b) Plots showing (a) the CP dispersion ω_q± (red line) of equation 4.41and the TEM cavity mode ωqn(black line) in the absence of any coupling; and (b)

the polaritonic dispersion (ωpol_q of equation5.28) that arises from the coupling between these modes. In (a) the circled numbers indicate the locations of the band crossings, and the pairs of blue arrows schematically indicate whether that CP band is ‘bright’ (symmetric) or ‘dark’ (asymmetric). In Figure (b) we observe large anticrossings at

1

and 2 due to the strong light-matter coupling with the optically bright CP mode. There is no anticrossing at 3 , and only a very small anticrossing at 4 , due to the vanishing light-matter interaction with the optically dark CP mode. (c − d) Schematics showing the wavefronts (dashed parallel lines) of a photonic mode with wavevector along the direction (c) Γ − K, and (d) Γ − M , superimposed on a schematic of the honeycomb lattice structure. (e) Schematics of a bright and dark mode where the black arrows indicate the dipole moments on an A site (blue circle) and B site (red circle). In a bright mode the dipole moments are in-phase (parallel), whereas in a dark mode they are out-of-phase (anti-parallel). In the figure c/ω0a = 3/4, D= 1 and a = 3r.

However, in Figure5.3(b) we observe large anticrossings at the points 1 and 2 in the polariton dispersion, but no anticrossing at the point 3 and only a very small anticross- ing at the point 4 . This can be understood as due to the different configurations of the lattice in the upper and lower CP bands. In the upper band the dipole moments in the sublattices A and B are in-phase with one another and so constructively interfere (see schematic in Figure 5.3(e)). This symmetric mode is optically ‘bright’ and couples significantly with photonic modes for any wavevector. In the lower band the dipole mo- ments in each sublattice are out-of-phase with one another and destructively interfere, giving rise to an optically ‘dark’ mode that only weakly interacts with light. The largest (but still small) anticrossing for the dark mode is along the direction Γ − M , whilst the

Chapter 5 - Chirality inversion in Dirac polaritons 106

smallest anticrossing is along the direction Γ − K where the interaction with light com- pletely vanishes. The schematics in Figures5.3(c-d) help to explain this by showing the wavefronts of a plane wave with a wavevector along these two high symmetry directions, superimposed on a schematic of the honeycomb lattice. For example in (c) we observe that the direction Γ − K is orthogonal to one of the separation vectors e1 and so the

total field of the dipoles disappears along this direction. This gives rise to an additional degeneracy along the Γ − K direction at 3 .

Before continuing our discussion of the polaritons we need to address the crucial question of how we should truncate the infinite Hopfield matrix M. The function ξqn is inversely

proportional to |q − Gn| so to capture the qualitative effects one might be tempted to

consider just the G = 0 mode. In fact this is the approximation taken by Hopfield in his seminal paper [35] where he treats exciton-polaritons that arise from the coupling between a single exciton band of a three-dimensional crystal and free photons. In this scenario it is acceptable to neglect Umklapp processes as there are no essential symme- tries or degeneracies to preserve, and so their inclusion only qualitatively modifies the physics. Another key difference to note is that as the lattice and photonic modes are of the same dimension, there is no dependence of the light-matter interaction strength on a geometric length scale.

However, for our system of interacting CPs and cavity photons, neglecting Umklapp processes leads to a fundamentally incorrect result as shown in Figure 5.4(a), where for small enough values of the cavity height L the Dirac point is destroyed. This is because at q = K there are three degenerate photonic modes (G1 = 0, G2 = b1 and

G3 = b2− b1) with the same interaction strength ξK1= ξK2= ξK3, so in selecting G1

but neglecting the other two we artificially break the degeneracy of the K point. In Figure 5.4(b) we show that selecting sets of Gn which are degenerate at the K point

keeps the associated Dirac point intact. However, we point out that in this scenario the Dirac point at the K0 point artificially shifts or ceases to exist. Thus, in principle there is no methodical way to maintain all essential symmetries in the dispersion when truncating the Hopfield matrix. In practice, by choosing a large enough number of photonic modes the symmetries are approximately maintained at both K and K0, as shown in Figure5.4(c).

Chapter 5 - Chirality inversion in Dirac polaritons 107

Figure 5.4: Comparison of the polariton dispersions ωpolq obtained from including

different photonic modes in the Hopfield matrix M of equation5.29. We consider (a) only the photonic mode Gn = 0, (b) the three modes which are degenerate with Gn = 0

at the Dirac point, and (c) 48 photonic modes centred around q = 0. The green ticks and red crosses next to the black circles highlight whether the symmetries around the K or K0points have been artificially broken. In the figure a = 3r, D= 1, c/ω0a = 3/5,

and L = 12a.

With these considerations in mind let us examine the actual features of the dispersion ωqpol a little further. In Figure 5.5 we present the evolution of the dispersion with

decreasing cavity height. Note that there is a global negative energy shift of the bands, caused by the larger anticrossing between the photonic mode and the bright CP modes as the light-matter coupling strength increases. There is a large renormalisation of the group velocity around the K point, in fact in panel (d) the bands are practically flat around this point. We also observe that the additional degeneracy along the Γ − K direction approaches and then merges with the degeneracy at K at a critical cavity height Lc (in the figure Lc ≈ 5a). Subsequently, as the cavity height is decreased

further, it appears that the bands have inverted on themselves.

The transition where the degeneracies merge at the K(K0) point must occur for the critical cavity height Lc when the group velocity along the Γ − K(K0) direction vanishes

∂ωpol,±q (Lc) ∂qx    q=K = 0, (5.35)

Chapter 5 - Chirality inversion in Dirac polaritons 108

Figure 5.5: Evolution of the polaritonic dispersion with decreasing cavity height (indicated above each panel). With decreasing cavity height the light-matter interac- tion strengthens which increases the magnitude of the anticrossing with the bright CP modes. This causes a global negative energy shift of the bands degenerate at the K point. Also we notice the additional degeneracy along the Γ − K direction approach the K point and merge with the degeneracy at about L/a = 5 as shown in panel (d). Subsequently, in panels (e-f) it appears that the bands have inverted on themselves. In the figure a = 3r, c/ω0a = 3/5, D= 1, and we take 12 photonic modes centred about

K.

where ωpol,±q indicates the upper (+) and lower (−) polariton bands that are degenerate

at the K(K0) point. Equation 5.35 can be solved numerically, but later we will obtain an analytical expression for Lc accurate to second order in ξqn.

Though we have not proved it yet, we imagine most readers are confident in the existence of a Dirac point at K and K0 in the limit of weak light-matter coupling, which in the following we term a Conventional Dirac Point (CDP). It is perhaps less evident that the CDP should continue to exist after merging with the additional degeneracy along Γ − K, which we term a Satellite Dirac Point (SDP). To better understand this merging and the properties of the eigenstates we now perform a unitary Schrieffer-Wolff transformation [134] to obtain an effective Hamiltonian in the CP subspace, which provides an accurate description of the two bands degenerate at the K and K0 points to second order in the light-matter coupling parameter ξqn.