Acoustic Dirac Equation Turns Sound Waves Into a Model of Topological Insulators

Physicists have long borrowed the mathematics of quantum mechanics to describe how sound and vibrations move through engineered materials. Now a team at the University of Arizona has shown that one of the most famous equations in physics, the Dirac equation, can be written entirely in the language of a simple mass-and-spring chain, opening a new mathematical doorway into the world of acoustic topological insulators. The study, published in Frontiers in Acoustics, demonstrates that ordinary mechanical vibrations can carry the same topological fingerprints that protect exotic quantum states in electronic materials.

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