Square-root topology is a conceptually elegant trick. Instead of designing a topological insulator directly, one starts from a known topological model, such as the celebrated Su–Schrieffer–Heeger (SSH) chain, and inserts additional sites into the lattice. The resulting ‘square-root’ system is described by a Hamiltonian whose square decomposes into a direct sum of the original parent Hamiltonian and a trivial companion. Because of this algebraic relationship, the child system inherits the topology of its parent, yet its band structure becomes richer: eigenvalues appear in symmetric positive and negative pairs, extra gaps open, and new varieties of boundary states emerge. Theoretical work over recent years has shown that square-root procedures can generate higher-order topological insulators with corner-localized modes, fractionalized topological invariants, and even connections to non-Hermitian and Floquet physics.

Existing photonic demonstrations, however, have faced practical limitations. Most square-root topological states realized so far have lived in microwave or low-frequency terahertz structures such as photonic crystals and coupled waveguides. Plasmonic nanoparticle arrays can push the physics into the visible range, but their unit cells are only nanometers across, making them difficult to fabricate, and long-range dipolar couplings distort their band structures away from the ideal tight-binding picture. The new work circumvents both problems by turning to plasmon-polaritonic systems, in which a chain of circular air cavities is embedded in a metallic background. These structures support designer surface modes whose effective couplings depend on lattice spacing in a controlled way, while long-range interactions decay exponentially and can be safely neglected.

The one-dimensional design consists of circular air cavities, each 120 nanometers in radius, arranged in unit cells of four cavities with a lattice constant of 1500 nanometers—dimensions comfortably accessible to modern nanofabrication. The metallic background is modeled with a Drude-like dielectric function tuned to visible-range operation, using parameters characteristic of gold. The four-cavity unit cell can be understood as a composite of a standard dimerized SSH chain and a uniformly spaced cavity chain. Mathematically, the square of the resulting Hamiltonian equals the direct sum of the cavity-chain Hamiltonian and the parent SSH Hamiltonian, which means the square-root system inherits the eigenvalues of the SSH model with an added sign-symmetric partner, and its band gaps track those of the original chain. When the intracell and intercell spacings are equal the gaps close; making them unequal opens two gaps, exactly as in the parent SSH physics.

The topological character follows from the quantized Zak phase, the standard one-dimensional invariant computed from the Berry connection across the Brillouin zone. When the intracell coupling is weaker than the intercell coupling, the Zak phase equals π and edge states appear inside the gap—a hallmark inherited by the square-root descendant. Because the inserted sites are themselves topologically trivial, the child system’s invariants are not strictly quantized, but the protected boundary physics survives. Simulations of finite chains show edge modes emerging symmetrically about zero energy whenever the dimerization ratio exceeds unity, confirming that the square-root construction transfers the parent’s boundary states intact.

The researchers then extended the idea to two dimensions, building a square-root lattice from a two-dimensional SSH model combined with a uniformly spaced cavity array, giving six cavities per unit cell. Here the inheritance relation becomes particularly striking: the two-dimensional square-root eigenvalues consist of a zero band together with plus-and-minus pairs of the parent model’s eigenvalues, producing a band structure perfectly symmetric about the horizontal axis. Because long-range couplings are negligible in this plasmon-polaritonic platform, the simulated band structures match the ideal theoretical predictions far more cleanly than in nanoparticle systems, where such couplings blur the picture.

Two dimensions also bring higher-order topology into play. In a conventional two-dimensional SSH lattice, inversion and chiral symmetries enforce degenerate edge states, while corner states are often buried within the bulk bands, limiting their usefulness. In the square-root lattice, the inserted sublattice breaks inversion symmetry even though chiral symmetry is preserved. This symmetry reduction lifts the degeneracy of mid-gap states, yielding spectrally isolated, non-degenerate corner modes alongside multiple edge channels. Numerical spectra of finite two-dimensional lattices reveal single corner states sharply localized at the structure’s corners and multiple, doubly degenerate edge states running along its sides—a combination absent from the parent model. Direct excitation simulations, using a point source inside a finite lattice with a dimerization ratio of 1.1, successfully launch both edge and corner modes at distinct visible-range frequencies near 679 terahertz.

Robustness, the defining virtue of topological states, was tested against two kinds of realistic imperfections: random displacement of cavity positions and outright deletion of sites. Position deviations of 30 and 60 nanometers shifted the corner-mode frequencies by only 2.58 and 3.18 gigahertz respectively, while deleting 5 and 15 cavities produced shifts of 6.52 and 4.12 gigahertz. In every case the in-gap edge and corner states persisted, demonstrating that the square-root topology withstands geometric disorder. Material loss, an inevitable feature of plasmonics at visible frequencies, was also included using gold’s attenuation coefficient; the topological modes survived, with the corner mode exhibiting a quality factor of 19.2—modest, but characteristic of plasmonic platforms in this spectral range.

The authors emphasize that the proposed structures are experimentally achievable with current techniques. Feature sizes of 1500-nanometer lattice constants and 120-nanometer cavity radii fall well within the resolution of electron-beam lithography, and electron-beam evaporation of gold produces nanofilms in the 20-to-50-nanometer range with thickness deviations of roughly two nanometers. Fabrication tolerance is further helped by the fact that the topological phase depends only on the dimerization condition, which random position errors do not overturn. Taken together, the results deliver square-root topological edge and corner states at visible frequencies in a platform that is both easier to prepare than nanoparticle arrays and cleaner in its band structure than microwave analogues, expanding the practical toolkit for topological photonics and pointing toward mode-selective, defect-tolerant nanophotonic devices.

Subject of Research: Square-root topological states realized in visible-range plasmon-polaritonic cavity systems

Article Title: Realization of square-root topology in plasmon-polaritonic system

Article References: Fan, Y. (2026). Realization of square-root topology in plasmon-polaritonic system. Results in Optics, 25, Article 101144. https://doi.org/10.1016/j.rio.2026.101144

Image Credits: AI Generated

DOI: 10.1016/j.rio.2026.101144

Keywords: topological photonics, square-root topology, plasmon-polaritons, SSH model, corner states, edge states, Zak phase, higher-order topology, gold nanocavities, visible light, topological protection, nanofabrication