Direct experimental access to the bulk band inversion in a topological metamaterial

TL;DR

The work directly probes bulk topology in SSH exciton-polariton chains by accessing momentum-resolved sublattice phases, circumventing reliance on edge states. It leverages two-dimensional $k$-space mapping and displacement of sublattices to induce an effective sublattice phase, yielding measurements of $I_0$ and $I_\pi$ whose difference $I_0-I_\pi$ traces $\langle\sigma_x\rangle$ and reveals bulk band inversion at the Brillouin-zone edge. By extracting band energies $E_±(k_y)$ and the relative sublattice phase, the authors reconstruct the effective $2\times2$ Hamiltonian in the Pauli basis, with $d_z\approx0$ and a winding of $(d_x,d_y)$ around the origin in the topological phase. The approach demonstrates a momentum-space, bulk-topology diagnostic that complements edge-state measurements and is transferable to other photonic/polaritonic platforms and beyond.

Abstract

Topological phases in exciton-polaritons and other metamaterial platforms have attracted significant attention due to their flexibility as Hamiltonian simulators. In previous works, signatures of topology have mainly been investigated from the perspective of edge states - strongly localised modes with exponentially decaying intensity into the bulk. While these edge states have become the hallmark of topological systems as they can facilitate non-reciprocal transport in potential applications, the topology is fundamentally encoded in the bulk band structure. In particular, the momentum-dependence of the eigenstates, i.e., the wave functions, determines the topology, usually reflected in a bulk band inversion. We present a band inversion in the paradigmatic Su-Schrieffer-Heeger (SSH) model, characterised by a reversal of the sublattice symmetry, quantified by the expectation value $\langle σ_\mathrm{x} \rangle$, when going from the centre of the Brillouin zone to the zone boundary. Here, we show direct experimental access to this bulk band inversion in SSH exciton-polariton chains, using two-dimensional momentum-space ($k$-space) mapping - without the need for real-space imaging. This technique enables the direct observation of the momentum-dependent inversion of the sublattice symmetry in the bulk bands, providing a unique perspective on topological phases beyond conventional edge state measurements. Our approach establishes effective momentum-resolved sublattice phase measurements as a powerful tool for accessing the wave function and bulk topology in photonic and polaritonic systems and beyond.

Figures (6)

• Figure 1: Band inversion and parametrisation of the Hamiltonian. a-c, Band structures of the one-dimensional Su-Schrieffer-Heeger (SSH) Hamiltonian for different intra and inter-cell hoppings $v$ and $w$. The different chains are in the topological (a), metallic (b) and trivial insulator phase (c). The bands are overlaid with the sublattice symmetry $\langle \sigma_\mathrm{x} \rangle$ that distinctly inverts at the edge of the Brillouin zone in the topological case (a). d-f, Parametrisation of the Hamiltonian in the Pauli matrix basis $\hat{H}(k_\mathrm{y})=d_\mathrm{x}(k_\mathrm{y})\sigma_\mathrm{x}+d_\mathrm{y}(k_\mathrm{y})\sigma_\mathrm{y}$ when passing through the Brillouin zone. The winding number $\nu$ of this circle is the topological invariant of the SSH model Asboth2016.
• Figure 2: Real and reciprocal space signatures of topology. a, Real-space distribution of a topologically non--trivial polariton SSH chain, energy integrated over the full $s$--band. The chain is oriented in $y$--direction, while the sublattices $A$ and $B$ are displaced in $x$--direction. b, Constant energy cut through the topological edge states of the chain. c, Cuts along the dashed grey lines through the two sublattices indicated in b. The intensity in the lattice sites decays when moving away from the respective edge. d, Band structure of the chain shown in a with visible discretisation. The edge state is visible in the band gap at $E_\mathrm{topo}$. e, Band structure of a chain consisting of 20 unit cells.
• Figure 3: Experimental access to the bulk band inversion. a-c, Band structures and difference spectra for $\Delta \theta = 0$ and $\Delta \theta = \pi$ of chains comprising 20 unit cells. The chains are in a topological (a), metallic (b) and trivial configuration (c). The experimental band inversion can be seen for the topological chain. d-f, Reconstructed Hamiltonian in a Pauli matrix basis from the sublattice phase fitting procedure. The closed curves define the winding number. For $|v|<|w|$, the curve contains the origin of the $d_\mathrm{x}$--$d_\mathrm{y}$ coordinate system, hinting to nontrivial topology. For each of the chain configurations, the real-space location of the mesas is sketched. The sublattice relative $y$-shift of the gapped chains is exaggerated for illustrative purposes.
• Figure 4: Reciprocal space cuts. Cuts through the far field of an SSH chain. Optical access at large angles is limited by the objective numerical aperture. The cuts corresponding to $\Delta \theta=0$ (zero effective sublattice phase shift) are displayed in blue, while the $\Delta \theta=\pi$ cuts are coloured red. The Brillouin zones (up to second order) are highlighted with horizontal dashed lines. A real-space sketch of the chain is included to show the connection between the vectors: $\Delta \vec{k}_\mathrm{BZ} \cdot \vec{a}=2\pi$ and $\Delta \vec{k}_\mathrm{\pi} \cdot \vec{a}_\mathrm{subl}=2\pi$. The cuts labeled a-c ($\Delta \theta=0$) and e-h ($\Delta \theta=\pi$) are shown individually in Fig. \ref{['fig:methods:indivCuts']}.
• Figure 5: Individual cuts of the topological configuration. a-c, Cuts through the far field along the $\Delta \theta = 0$ lines displayed in Fig. \ref{['fig:methods:kSpaceCuts']}. d, Sum of the three zero effective sublattice phase spectra. e-h,$\Delta \theta = \pi$ spectra displayed in Fig. \ref{['fig:methods:kSpaceCuts']}. i, Sum of the four $\pi$ effective sublattice phase spectra. k, Resulting difference spectrum (positive $k_\mathrm{y}$, showing the band inversion. The difference spectra presented in the main part are symmetrised with respect to $k_\mathrm{y}=0$.