The starting point of the study is a principle that physicists have relied on for more than a century: the identity of a phase of matter is tied to its symmetries. The everyday example is water. Liquid water and steam look and behave very differently, yet at the level of fundamental symmetry they are the same. In both states the molecules have no preferred position and no preferred direction, so nothing in principle forbids one from being continuously deformed into the other. Indeed, if water is heated under high pressure, beyond a specific critical point the boundary between liquid and gas simply disappears, and the two states merge into a single thermodynamic phase. Ice is a different story. When water freezes, the molecules lock into a crystal lattice with fixed positions, breaking the translational and rotational symmetries that liquids and gases preserve. Because the symmetries differ, ice can never be smoothly transformed into liquid water; a genuine phase boundary must always separate them.

For decades, this symmetry-based picture has served as the organizing principle of condensed matter physics. But it leaves an uncomfortable question hanging over any pair of states that share the same symmetries yet look strikingly different. If symmetry cannot tell two states apart, is there any guarantee that they are actually connected by a continuous path, or could some subtler, non-symmetry-based mechanism keep them separated? Answering that question in a controlled and general way has proven remarkably difficult, and it is precisely the challenge that the HKUST-led team set out to confront.

Their testing ground was one of nature’s most counterintuitive substances: dense ice. Under extreme pressures, water freezes into a series of exotic crystalline forms with names like ice VII, ice VIII, and ice X. All of these share the same skeleton of oxygen atoms, arranged in an identical crystal framework. What distinguishes them is the placement of the hydrogen atoms that sit between the oxygens. In ice VII, the hydrogen atoms are distributed randomly among the available positions. In ice VIII, they settle into an ordered, periodic arrangement. In ice X, they drift to the midpoints of the bonds between neighboring oxygen atoms, becoming shared equally by two oxygens at once. The puzzle is that ice VII and ice X, when averaged over time, preserve exactly the same underlying crystal symmetries. By the standard symmetry criterion, there is no obvious reason they should be distinct phases, yet experiments and intuition have long suggested that something meaningful separates them.

The key insight of the new work is that this problem in high-pressure ice can be recast, exactly, as a problem in magnetism. The midpoints of the oxygen-oxygen bonds in these ice structures form a geometric pattern known as the pyrochlore lattice, a three-dimensional network of corner-sharing tetrahedra that is famous in condensed matter physics as the architecture underlying so-called spin ice materials. In those magnets, the magnetic moments behave as if they were hydrogen atoms in ice, obeying the same local rules. The correspondence runs deeper still: the three possible positions available to each hydrogen atom in ice map precisely onto the three allowed states of a quantum magnetic moment with spin equal to one. This mapping transforms a question about water under pressure into a question about magnets on a pyrochlore lattice, where a powerful theoretical toolkit becomes available.

In a companion study published in Physical Review B, the team first analyzed a simplified version of the problem in which the ordering of ice VIII is excluded. The result was striking. Ice VII and ice X are separated by a sharp, genuine phase transition only at absolute zero temperature. At any temperature above that point, thermally excited defects begin to roam the crystal, and these defects behave exactly like the celebrated magnetic monopoles of spin ice physics. The monopoles screen out the field that would otherwise hold the two states apart, dissolving the sharp transition into a smooth, gradual crossover. The conclusion is unambiguous: at any realistic temperature, ice VII and ice X are not distinct phases of matter but different manifestations of a single phase. Ice VIII, by contrast, remains genuinely separate, because its ordered hydrogen arrangement breaks a crystal symmetry that the other two forms preserve, placing it beyond the reach of any continuous connection.

The primary Physical Review X paper then generalizes this result into a comprehensive framework covering classical pyrochlore spin-ice models with arbitrary spin values, not just the spin-one case relevant to ice. The framework identifies what actually keeps two states apart in these systems: a network of string-like defects threading through the crystal. As long as those strings close on themselves, forming closed loops, the two states remain genuinely distinct phases. But the framework also exposes a sharp divide that earlier work could not see, one that depends on whether the spin of the magnetic moments is a whole number or a half-integer. Above absolute zero, thermally excited monopoles cut the strings open, and once the strings are severed, the topological distinction they carried is washed out. Almost every sharp phase boundary in the family then gives way to a smooth crossover, echoing the fate of the ice VII to ice X transition.

There is one dramatic exception to this rule. When the spin takes the value three-halves, the theory predicts that an abrupt transition between two distinct liquid-like magnetic states survives at finite temperature, ending at a critical point without any need for an external magnetic field to stabilize it. This makes spin three-half pyrochlore magnets an especially promising target for experimentalists searching for a phase transition that defies the conventional symmetry-based classification. Finding such a transition in the laboratory would demonstrate a form of order that persists at finite temperature through topology alone, something the new framework says should be impossible everywhere else in this family of models.

The implications reach well beyond dense ice and idealized magnets. Pyrochlore spin-ice physics appears in frustrated magnetic materials, in engineered nanomagnet arrays where artificial moments are arranged on the same lattice geometry, and in trapped cold-atom systems where researchers can simulate these models with exquisite control. For all of these platforms, the new theory provides a practical recipe: examine the symmetries, examine the string-like topological defects, and check whether the spin is integer or half-integer. The answers determine whether an observed change of behavior reflects a true phase boundary or merely a crossover, a distinction that governs what experiments can hope to observe and what theoretical surprises may remain hidden.

For Professor Watanabe, the deeper message of the work is a reaffirmation of a classical idea, sharpened by modern tools. As he reflected on the findings, our results ultimately support the conventional view that the phases of ordinary matter are classified by broken symmetry, and by nothing else. Wherever symmetry could not distinguish between two states, a continuous path connected them without crossing a phase boundary. Yet he is quick to point out that this conclusion is an invitation as much as an answer. It would be far more interesting, he notes, to discover a topological phase that genuinely survives at finite temperature beyond this classification, and that is exactly what the team is looking for now. In redrawing the boundary between phases of matter, the HKUST-led collaboration has also drawn a map of where the next discovery might lie.

Subject of Research: Classification of phases and phase transitions in pyrochlore spin-ice magnets and high-pressure ice

Article Title: HKUST researchers redraw the boundary between phases of matter

Article References: HKUST researchers redraw the boundary between phases of matter. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: phases of matter, phase transitions, pyrochlore lattice, spin ice, topological defects, magnetic monopoles, high-pressure ice, ice VII, ice X, symmetry breaking, condensed matter physics, HKUST

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