The research, carried out by Abhirup Basu, Keith Runge, and Pierre A. Deymier at the New Frontiers of Sound Science and Technology Center, begins with a deceptively simple model: a one-dimensional chain of identical masses connected by linear springs, with each mass also tethered to a rigid substrate by an additional side spring. When the equations of motion for this discrete chain are written down, they take the same form as the Klein–Gordon equation, the relativistic wave equation that governs spinless quantum particles. This analogy is not merely cosmetic; it means that the full machinery of relativistic quantum field theory can be brought to bear on purely classical sound waves.

The crucial step is what mathematicians call factorization. Just as the Klein–Gordon equation can be split into two first-order Dirac equations using the Pauli matrices, the discrete acoustic version can be factored into a pair of Dirac-like equations acting on four-component amplitude vectors. The factorization is achieved with forward and backward difference operators, which act differently on waves traveling in opposite directions. Each of the resulting Dirac equations carries built-in conditions that break two fundamental symmetries: time reversal, denoted T, and parity, denoted P. In the quantum world, breaking these symmetries is what gives rise to some of the most striking phenomena in condensed matter physics, including the quantum Hall effect and the quantum spin Hall effect.

Solving the factored equations for traveling waves yields dispersion relations that split into two branches, one for each Dirac equation. The eigenvectors associated with these branches reveal something remarkable. One family of solutions behaves like an ordinary phonon, the boson-like quantum of sound, with a conventional topology characterized by a Berry phase of zero. The other family behaves like a fermion, the class of particles that includes electrons, and carries a Berry phase of π. The Berry phase, a geometric quantity that accumulates as a wave’s amplitude vector is transported around a closed loop in momentum space, serves as a topological invariant: it cannot change under smooth deformations of the system, which is precisely what makes it a robust label for distinguishing different phases of waves.

The π-Berry-phase solutions are what the researchers call acoustic pseudospins. A true quantum spinor, describing an electron’s intrinsic angular momentum, picks up a minus sign when carried around a full loop in momentum space, and it needs two complete Brillouin zones to return to its original form. The acoustic analog behaves in exactly the same way. The forward and backward components of the wave become locked together in fixed proportions, forming quasi-standing waves whose amplitude vectors undergo a π phase discontinuity at the origin of the Brillouin zone. This behavior, previously explored experimentally in ultrasonic systems, shows that sound waves can mimic one of the defining features of quantum particles without any quantum mechanics at all.

The analysis does not stop at propagating waves. The team also examined evanescent waves, solutions that decay exponentially in space rather than propagating, which arise at frequencies outside the allowed band of traveling waves. Computing the Berry phase along the closed loop traced by the evanescent dispersion relation, they found that both Dirac equations, whether with the plus or minus sign, yield a Berry phase of π for the evanescent branch. This contrasts sharply with the propagating case, where the two branches split between zero and π. The result means that evanescent acoustic waves in this system are intrinsically non-conventional in their topology, regardless of which of the factored equations they satisfy.

The most consequential finding concerns what happens when two chains with different topologies are joined together. The researchers considered two semi-infinite chains governed by the acoustic Dirac equation but differing in a single parameter that flips the sign of the symmetry-breaking term. At the interface, they imposed continuity conditions on the wave amplitudes and sought solutions in the form of evanescent waves decaying away from the junction on both sides. The existence of such solutions reduces to whether an eight-by-eight matrix of interface conditions has a nonzero determinant. The calculation shows that the determinant is nonzero for nearly all frequencies in the evanescent band, meaning that localized interface modes exist generically between the topologically conventional and non-conventional chains.

These interface modes are the acoustic counterpart of the protected surface states that make electronic topological insulators so valuable. In an electronic topological insulator, the bulk refuses to conduct while the surface conducts with impunity, immune to defects and disorder because the conducting states cannot be removed without closing the bulk energy gap. The acoustic version promises the same robustness for sound: waves confined to the interface between two regions of different topology should persist even when the interface is imperfect. The researchers note two special frequencies at which the determinant vanishes, where real-frequency solutions disappear, although solutions with complex frequencies, corresponding to waves that decay in time, may still exist.

The work builds on a rich history of topological acoustics. Acoustic analogs of the quantum spin Hall effect have been realized by tuning accidental double Dirac cones in graphene-like lattices and by folding the Brillouin zone in carefully designed phononic crystals. Analogues of the quantum Hall effect have been achieved by arranging circulating flows of air into periodic lattices that break time-reversal symmetry, producing one-way sound transport that scatters around obstacles rather than reflecting. Kane and Lubensky’s landmark 2014 work connected topological mechanical modes to electronic band theory through a Dirac-factorization-inspired approach, and subsequent classifications have extended the periodic table of topological phases to phonons in linear mechanical metamaterials.

What sets the new study apart is its completeness as a minimal model. By starting from nothing more than masses, springs, and a rigid substrate, and by factoring the resulting discrete Klein–Gordon equation, the authors expose the full topological structure of the acoustic system: two propagating branches with opposite topology, an evanescent branch that is always non-conventional, and protected interface states that emerge naturally where the topologies meet. The framework provides a clean theoretical laboratory for studying acoustic topological insulators, one in which every ingredient is transparent and every prediction can be checked against the underlying mechanical model.

The authors are careful to point out a limitation. The Dirac factorization reveals the potential for topology, but additional physical mechanisms are still needed to actually break time-reversal or parity symmetry in a real device and thereby realize one-way propagating sound waves. Structural design choices or external perturbations must supply the symmetry breaking that the factored equations assume. Nevertheless, the demonstration that a humble spring-mass chain can host fermion-like acoustic waves, Berry phases of π, and topologically protected interface modes marks a significant step toward sound-based technologies that exploit topology, from robust acoustic waveguides to information processing in which signals are carried by pseudospins rather than charges.

Subject of Research: Dirac factorization of acoustic wave equations as a model for topological insulators

Article Title: The acoustic Dirac equation as a model of topological insulators

Article References: Basu, A., Runge, K., & Deymier, P. A. (2025). The acoustic Dirac equation as a model of topological insulators. Frontiers in Acoustics, 3, Article 1615210. https://doi.org/10.3389/facou.2025.1615210

Image Credits: AI Generated

DOI: 10.3389/facou.2025.1615210

Keywords: acoustic topological insulator, Dirac equation, Klein-Gordon equation, Berry phase, acoustic pseudospin, topological interface modes, evanescent waves, phononics, symmetry breaking, mass-spring chain, time-reversal symmetry, parity symmetry

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

Polaritons arise when photons couple strongly to vibrations of the crystal lattice, forming so-called phonon polaritons. In certain van der Waals materials, these waves propagate hyperbolically, meaning their wavefronts sweep outward in open arcs rather than closed ellipses, a property that allows light to be confined and steered with extraordinary precision. Alpha-MoO3 has become a star of this field because its in-plane optical response is naturally anisotropic: light propagates differently along different crystallographic directions, producing hyperbolic dispersion in the mid-infrared range. Over the past several years, researchers have twisted stacked flakes of such crystals against one another to create topological transitions in polariton propagation, discovering everything from photonic magic angles to ghost hyperbolic surface waves. But all of these approaches share a fundamental limitation: the optical response of a natural material is fixed by its crystal symmetry, so reconfiguration has typically required physically restacking, rotating, or chemically modifying the sample.

Conventional strategies for controlling polaritons rely on refractive-index engineering. By placing a polariton-bearing crystal on top of another optical medium, engineers can coax the waves in the two layers to hybridize, blending their dispersion relations into something new. Yet this coherent hybridization demands strict wavevector matching between the coupled modes, a condition that is difficult to satisfy and sensitive to sample geometry, thickness, and frequency. It also ties the resulting behavior inseparably to the passive optical properties of the underlying layer. The team behind the new study, led by researchers at Shanghai Jiao Tong University and the National Center for Nanoscience and Technology, asked a different question: what if, instead of matching wavevectors, one simply filtered out the waves one did not want?

The answer came in the form of an unlikely partner for the molybdenum oxide flake: a film of aligned carbon nanotubes. These dense arrays of nanotubes, all pointing in the same direction, are electrically conductive along their axis but effectively insulating across it, making them a naturally anisotropic conductor. Crucially, the researchers found that the carrier dynamics within the nanotube film are overdamped, meaning that charge carriers lose their coherent momentum too quickly to sustain any resonant optical response of their own. As a result, the nanotubes refuse to hybridize with the polaritons in the overlying crystal. Instead of coherently mixing, the two layers interact through a purely resistive, proximity-induced coupling—essentially, the polaritons leaking into the nanotube film are simply dissipated as heat.

That dissipation, however, is anything but random. Because the nanotubes conduct only along their alignment axis, they absorb polariton energy selectively depending on the direction in which the wave propagates relative to the nanotube axis. The aligned carbon nanotube layer thus acts as what the researchers call a momentum-space loss filter: in the reciprocal space that describes the polariton dispersion, wavevectors aligned with the nanotube axis are strongly attenuated while others survive. The consequence in real space is dramatic. The intrinsic hyperbolic dispersion of the alpha-MoO3 flake, with its elegant pairs of symmetric arcs, is reshaped into a symmetry-broken dispersion in which the energy flow—the Poynting vector—is inhibited in some directions and concentrated in others. Full-wave electromagnetic simulations confirmed that an effective medium description of the heterostructure reproduces the observed field patterns, with the asymmetry gradually diminishing as the crystal flake becomes thicker and the resistive coupling weaker.

The experimental evidence came from scanning near-field optical microscopy, a technique that drags a nanoscale antenna across the sample surface to image polariton fringes with resolution far below the diffraction limit. By preparing heterostructures in which the crystal’s [100] axis was rotated at various angles relative to the nanotube axis—from zero degrees through 90 degrees—the team observed a smooth, continuous transformation of the polariton wavefronts. At certain twist angles the familiar symmetric hyperbolas remained essentially intact, while at others the fringes warped into highly asymmetric, shear-like patterns reminiscent of the hyperbolic shear polaritons previously seen only in exotic low-symmetry crystals such as beta-gallium oxide or monoclinic semiconductors. Here, however, the shear was not baked into the crystal lattice; it was imposed by an external, deliberately engineered dissipative layer whose orientation could be chosen freely at fabrication time.

The most striking result is that the entire transformation is electrically reversible. Because the dissipative loss in the nanotube film is governed by its Drude response—the same free-carrier absorption that limits the conductivity of any metal—applying a gate voltage changes the carrier density in the nanotubes and thereby tunes the strength of the momentum-space filter. In the experiments, sweeping the gate voltage from positive to negative values continuously steered the polariton propagation from symmetric hyperbolic wavefronts to strongly sheared, asymmetric ones and back again. Unlike approaches that require physically altering the sample, this means the polariton topology can be reprogrammed in real time, with a single device cycling through a whole family of optical responses that a static crystal could never display.

The work also carries conceptual weight for the broader field of non-Hermitian physics. In standard quantum and optical systems, the Hamiltonian describing the dynamics is Hermitian, guaranteeing energy conservation; loss must then be treated as a nuisance or, at best, compensated with gain. Physicists have increasingly recognized that deliberately engineered loss—described by non-Hermitian formalisms—can be a resource in its own right, enabling exceptional points, parity-time symmetry effects, and non-Hermitian skin phenomena. The new study extends that philosophy into topological nanophotonics in a particularly clean way: rather than balancing gain and loss or exploiting fine-tuned couplings, the researchers show that a simple, gate-tunable anisotropic resistor is enough to break symmetry and reshape topology, establishing dissipation itself as a programmable degree of freedom in device design.

The implications reach across several technologies. Mid-infrared polaritons are already being explored for molecular sensing, thermal management, and sub-diffraction imaging, and recent demonstrations of hyperbolic electroluminescence suggest routes toward electrically driven polariton sources. A platform in which the directionality, asymmetry, and topology of those waves can be switched by a voltage opens the door to dynamically reconfigurable nanophotonic circuits, electrically steered thermal emitters, modulators, and beam-shaping elements operating at wavelengths far below what conventional optics allows. The aligned carbon nanotube films used in the study are compatible with wafer-scale growth and high-performance electronics, hinting that the integration of polaritonic and electronic functionality on a single chip may be practical rather than aspirational. More broadly, the study suggests that the road to programmable nanophotonics may run not only through what materials let light do, but through what they quietly prevent it from doing—a reminder that in modern optics, sometimes the most powerful design tool is controlled loss.

Subject of Research: Electrically programmable polariton symmetry breaking via non-Hermitian dissipation engineering in van der Waals heterostructures

Article Title: Electrically programmable polariton symmetry breaking via non-Hermitian dissipation engineering

Article References: Chen, N., Teng, H., Sun, Y., Yang, Y., Xue, Z., You, O., Chen, K., Jiang, C., Wang, J., Zhou, S., Liu, X., Wang, C., Li, Z.-Z., Meng, S., Zhu, M., Hu, H., & Dai, Q. (2026). Electrically programmable polariton symmetry breaking via non-Hermitian dissipation engineering. Nature Materials. https://doi.org/10.1038/s41563-026-02734-z

Image Credits: AI Generated

DOI: 10.1038/s41563-026-02734-z

Keywords: phonon polaritons, alpha-MoO3, aligned carbon nanotubes, non-Hermitian photonics, dissipation engineering, symmetry breaking, hyperbolic dispersion, shear polaritons, electrical gating, nanophotonics, van der Waals heterostructures, topological photonics