Measuring Topological Invariants in a Polaritonic Analog of Graphene

Topological materials rely on engineering global properties of their bulk energy bands called topological invariants. These invariants, usually defined over the entire Brillouin zone, are related to the existence of protected edge states. However, for an important class of Hamiltonians corresponding to 2D lattices with time-reversal and chiral symmetry (e.g., graphene), the existence of edge states is linked to invariants that are not defined over the full 2D Brillouin zone, but on reduced 1D subspaces. Here, we demonstrate a novel scheme based on a combined real- and momentum-space measurement to directly access these 1D topological invariants in lattices of semiconductor microcavities confining exciton polaritons. We extract these invariants in arrays emulating the physics of regular and critically compressed graphene where Dirac cones have merged. Our scheme provides a direct evidence of the bulk-edge correspondence in these systems and opens the door to the exploration of more complex topological effects, e.g., involving disorder and interactions.

Figure 1

(a) Definition of the unit cell for bearded and zigzag terminations. ${\mathbf{a}}_{1,2}=\{\pm 3/\sqrt{2},3/2\}a$ are the primitive vectors. (b) Schematic representation of the two topological phases of the SSH model. The dotted lines indicate the boundaries of unit cells. (c) Evolution of the winding number in graphene as $k_x$ spans the BZ for bearded and zigzag terminations. White and blue regions correspond, respectively, to ${\mathcal{W}}_{\mathrm{HC}}=0$ and ${\mathcal{W}}_{\mathrm{HC}}=1$. Red dashed line indicates the position of $k_x=0$.

Figure 2

(a) Scanning electron microscopy (SEM) image of a honeycomb lattice of coupled micropillars. (b) Momentum-resolved emission spectra of a polaritonic graphene lattice. $E_0=1.571\mathrm{eV}$. The cut in the BZ along which the image is taken is depicted in the inset. (c) Schematic representation of the setup. Upper and lower panels depict top and side views. (d) Spatially resolved steady-state emission spectra (along $y$) for a value of $k_x\sim 0$. The position of the different sites of the effective SSH lattice is indicated above, where the pumped pillar is in red. (e) Emission intensity integrated over both bands as a function of spatial position. (f) Definition of the chiral displacement $\mathrm{\Gamma }Y$ as a function of spatial position. The blue (red) curves correspond to definition of the unit cell compatible with zigzag (bearded) edges. The values of the mean chiral displacements ${\mathcal{C}}_z$ and ${\mathcal{C}}_b$ extracted from the spectrum presented in (e) are given on the right-hand side of this panel.

Figure 3

(a),(b) Spatially resolved (along $y$) emission spectra for two distinct momentum components $k_x$ schematically depicted on the right. Values of $\mathcal{C}$ for zigzag and bearded edges are provided above each panel. $E_0=1.571\mathrm{eV}$. (c) Evolution of the mean chiral displacement as a function of $k_x$. Blue and red circles correspond to values associated to the two possible definitions of the unit cell: respectively, bearded and zigzag. Blue (red) shaded areas correspond to regions where ${\mathcal{W}}_{\mathrm{HC}}=1$ for a unit cell definition compatible with zigzag (bearded) terminations. The solid line represents a theoretical calculation of the mean chiral displacement including losses, which is presented (along with error analysis) in Supplemental Material [30].

Figure 4

(a) Schematic representation of the effect of compression on the lattice. The compression factor is defined as $\beta =j^{\prime }/j$. (b) Top view SEM image of the compressed honeycomb lattice. White circles are added to indicate the positions of the pillars. (c) Evolution of the mean chiral displacement for both definitions of the unit cell. Blue circles correspond to the definition depicted in Fig. 1. (d),(e) Momentum-resolved PL spectra measured at the boundary of the lattice, along a bearded (d) and a zigzag (e) termination. The cut in the BZ along which both images are taken is depicted on the right. The blue arrow highlights the presence of an edge state in the zigzag termination case.