Probing Local Topology in a Disordered Higher-Order Topological Insulator

Abstract

Higher-order topology is prized for its ability to realize lower-dimensional boundary states which are stable beyond fine-tuning. However, disorder presents a failure mechanism that can destroy topological in-gap states. Here, we investigate a disordered two-dimensional polariton lattice and employ the spectral localizer framework to define a real-space topological index rooted in crystalline spatial symmetries. This framework enables direct real-space mapping of topology beyond conventional momentum-space classifications, confirming the presence of corner and edge modes in this generalized Su-Schrieffer-Heeger model. Furthermore, it can directly quantify topological protection of a state. We leverage the versatility of our platform to experimentally realize normally distributed, random disorder and find that the corner states persist until the spectral gap closes. Experimentally, this corresponds to a disorder strength of approximately one quarter of the spectral gap. The spectral localizer accurately identifies the disorder strength at which the bandgap closes, establishing the framework as a predictive tool for every finite size system. Our results broaden the design principles for higher-order topological insulators and open the way towards imple menting disorder-resilient devices for robust lasing, light-routing, and quantum computation.

Figures (3)

• Figure 1: a, Sketch of a generalized 2D SSH Lattice, featuring a boundary between a compressed bulk unit cell (green solid square) and a topological stretched bulk unit cell (blue dotted square). b, Brillouin zone of the stretched bulk. c, Measurement of the spectrum along the path $\boldsymbol{\Gamma}$-$\mathbf{X'}$-$\mathbf{M}$-$\boldsymbol{\Gamma}$. Dotted lines are calculated Bloch bands. d, Real-space spectrum along the horizontal black dashed line in (a), cutting through the corners (orange) and edge (red). At specific energies, corner states (green arrows) and edge states (blue arrows) are visible. e, and f, Local density of states of the lower-energy corner state and edge state, respectively. g-k, Simulations based on a tight-binding model and the spectral localizer. g, Spectrum based on a tight-binding model along the high symmetry path $\boldsymbol{\Gamma}$-$\mathbf{X'}$-$\mathbf{M}$-$\boldsymbol{\Gamma}$. h, Integrated density of states. i, 1D local gap at the middle of the topological bulk. j, Local index of the $\mathcal{M}_{xy}$ symmetry. It shows a jump of 2 at the energy of the corner, coinciding with a zero in the local gap in (i). k, 2D local gap at $E_\mathrm{corner}$, the inset serves as a reference to the underlying lattice.
• Figure 2: a, AFM measurement of a polariton lattice with broken, where only the $M_{xy}$ symmetry is retained. b, c, Real-space intensity distributions of the lower energy edge- and corner state, respectively. d, Spectrum along the diagonal, white dashed line in (a), projected onto the $x$-axis. There is only a corner state at the lower left site (green arrow). e, Simulated local density of states. f, Local gap at the position of the lower left corner with a zero crossing at the energy of the corner state. g, Local index for the only remaining symmetry $\mathcal{M}_{xy}$, at the position of the lower left corner site. It changes by one at the energy of the corner state.
• Figure 3: a, Spectrum of a disordered 2D-SSH lattice along the lower horizontal interface. b,c, Iso-energy cut at the energy of the edge- and corner state, respectively. The corner states are marked with green circles for better visibility. Panel (a) to (c) use a disorder of $\sigma_\mathrm{s}=60\,$nm. d ,Analysis of the eigenmode energies at different levels of spatial disorder. For higher disorder strengths, the bulk bandgap closes. e ,Calculated mean local gap at the position of the corner atoms in dependence on energy for 300 different configurations each. Shaded regions are the sample standard deviations. f, Mean local gap (300 configurations) at the position of the corner atoms, at the energy of the corner (red diamonds), at the energy between the corner and edge state (black triangles), and the energy between the corner and second bulk band (blue crosses). Arrows denote the robustness against disorder for the corner (blue arrow) and edge state (black arrow) predicted by the spectral localizer framework.