Condensed-matter physics has now opened a different route into this territory, one that does not require finding a fundamental particle at all. Topological magnetoelectric theory predicts that the surface electrons of a topological insulator respond to an electric field in a remarkable way: when an active electric field is applied near the boundary between a topological insulator and a topologically trivial material such as vacuum, the electrons generate a magnetic response that can be mathematically described as an image magnetic monopole. In effect, the surface behaves as though a magnetic monopole were lurking just beneath it, mirroring the applied electric field the way an ordinary image charge mirrors an electric charge. The prediction is elegant, but realizing it in the laboratory demands two difficult things at once: a way to create an electric field that is intense and localized on nanometre scales, and a way to detect the faint magnetic response it induces. A new experimental study suggests that both requirements can be met with a single, well-established instrument.

Scanning tunnelling microscopy, or STM, turns out to be a natural engine for this kind of physics. When a sufficiently large bias voltage is applied between the microscope’s metallic tip and the sample, the instrument enters the field-emission regime: electrons are torn from the tip apex and travel across the tunnelling junction toward the sample surface. This process produces an intense, highly localized active electric field confined to the tiny gap between tip and surface, exactly the kind of field that topological magnetoelectric theory calls for. Crucially, the field in the junction has more structure than a simple perpendicular pull. It comprises the tip-induced field together with a transient field generated by the moving, field-emitted electrons themselves, and the tip-induced field possesses a radial component, Er, that points outward from the junction axis. The strength of this radial component depends inversely on the dielectric constant of the substrate and on the tip-sample distance, a dependence that would soon prove decisive in interpreting the experiment.

The same field-emission conditions create a second phenomenon that serves as the experiment’s readout mechanism. An electron that has been emitted but has not yet joined the sample remains outside the surface, where it is attracted by the image charge it induces in the material beneath it. This attractive potential can briefly trap the electron just outside the surface, binding it into a ladder of Rydberg-like electronic states known as image potential states, or IPSs. These states hover nanometres above the surface, exquisitely sensitive to anything that perturbs the electrostatic landscape there, including the active electric field produced by the tip. On ordinary, topologically trivial surfaces, IPSs form a conventional, predictable series of spectroscopic peaks. That predictability is precisely what makes them powerful: any deviation from the expected pattern signals that something unusual is happening at the surface, and the details of the deviation encode what that something might be.

A research team led by Lan Chen at the Institute of Physics of the Chinese Academy of Sciences exploited this sensitivity to examine IPSs on Bi(111), the (111) surface of bismuth, which possesses higher-order topological character. Higher-order topological insulators, or HOTIs, are materials whose unusual topology manifests not in conducting edge channels on every boundary, as in conventional topological insulators, but in more subtle surface electronic structure that emerges under specific conditions. Using low-temperature STM, the researchers first established a baseline: on topologically trivial reference surfaces, the spectra displayed the conventional series of IPS peaks, exactly as textbook physics predicts. When they turned to Bi(111), however, the spectra revealed a strikingly unexpected splitting of the IPS features, an anomaly with no place in the standard description of image-potential states and an immediate hint that the higher-order topology of the surface was somehow involved.

To trace the origin of the splitting, the researchers systematically varied two experimental parameters: the thickness of the Bi(111) films and the dielectric properties of the substrate supporting them. The results were revealing. In the thinnest Bi(111) films, the anomalous splitting was entirely absent; it appeared only as the films grew thick enough for the characteristic Bi(111) surface electronic structure to develop. This thickness dependence tied the anomaly directly to the emergence of the topological surface states rather than to any bulk property of bismuth or to a generic artifact of the measurement. The surface electronic structure, in other words, was not a bystander but an active participant in producing the splitting, consistent with the idea that the effect involves the special response of topological surface electrons to the local field environment created by the STM junction.

The substrate experiments delivered the second, and arguably most telling, clue. When Bi(111) was supported on semiconducting Si(111), which screens electric fields only weakly, the splitting grew as the tunnelling current was raised and the STM tip moved closer to the surface, intensifying the field in the junction. When Bi(111) rested instead on metallic Sb(111), a far more effective screening substrate, the splitting remained nearly unchanged regardless of tip approach. This contrast indicates that the anomalous splitting is governed not by the perpendicular electric field in the junction, the component one might naively expect to dominate, but by the radial component of the active electric field, the component whose strength is controlled by substrate screening. A purely electrostatic explanation tied to the perpendicular field could not account for this behavior, pointing instead toward a coupling between the radial field and the surface’s electronic structure that carries magnetic-response character.

Building on these observations, the researchers proposed a phenomenological picture that connects all the pieces. In their interpretation, the radial active electric field couples to the Bi(111) surface electronic structure and induces an effective magnetic-response-like perturbation, one that is compatible with the monopole-like topological magnetoelectric response predicted by theory. Such a perturbation would lift the degeneracy of the orbital-like sublevels of the image potential states, splitting what should be a single peak into the multiple components observed in the spectra. The thickness dependence enters because the surface states that participate in this coupling only exist once the film is thick enough; the substrate dependence enters because screening controls the radial field strength. Within this framework, every experimental observation finds a natural place, and the anomalous splitting becomes a candidate spectroscopic fingerprint of a phenomenon that has otherwise been accessible only through theoretical formalism.

The broader significance of the work lies in the experimental method itself. Field-emission IPS spectroscopy, as demonstrated here, accomplishes in a single measurement what previously required separate capabilities: it simultaneously creates a highly localized active electric field through the STM tip and sensitively probes the surface’s response to that field through the trapped image-potential states. This dual role transforms the STM from a passive imaging tool into an active probe of magnetoelectric coupling at higher-order topological surfaces. For a field in which the central prediction, the image magnetic monopole, has remained a theoretical construct awaiting a practical detection strategy, the demonstration that a splitting of Rydberg-like surface states can serve as its signature represents a genuine methodological advance, opening a new experimental route for investigating active-field responses in topological materials.

Much work remains before the interpretation is settled. The study, published in National Science Review with DOI 10.1093/nsr/nwag548, reports an experimental observation together with a phenomenological framework, and the proposed connection between the splitting and a monopole-like magnetoelectric response is presented as a possible, physically consistent explanation rather than a definitive proof. Nevertheless, the findings give physicists something they have rarely had in this area: a reproducible, tunable spectroscopic anomaly whose dependence on film thickness, substrate screening and tip-sample distance can be systematically mapped, and whose theoretical meaning can be sharpened with each new measurement. If subsequent studies confirm the monopole-like interpretation, the humble image-potential states that have hovered above surfaces since the early days of surface physics will have become the medium through which one of Dirac’s most enduring ideas finally meets a laboratory analogue, not in the cosmos or a collider, but in the nanometre-scale gap beneath a scanning tunnelling microscope tip.

Subject of Research: Anomalous image-potential state splitting on higher-order topological insulator Bi(111) as a spectroscopic probe of monopole-like topological magnetoelectric response

Article Title: Anomalous splitting of image-potential states on HOTI Bi(111) provides a new spectroscopic window into possible monopole-like topological magnetoelectric responses

Article References: Anomalous splitting of image-potential states on HOTI Bi(111) provides a new spectroscopic window into possible monopole-like topological magnetoelectric responses. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: magnetic monopoles, topological insulators, bismuth, image-potential states, scanning tunnelling microscopy, magnetoelectric effect, field emission, higher-order topology, surface states, spectroscopy, condensed matter physics, National Science Review

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