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 stage for the calculation is the Kiselev black hole, an exact solution of Einstein’s equations found by V. V. Kiselev in 2003 that describes a static, uncharged black hole wrapped in what is often called a quintessence field. Quintessence is a proposed dynamical form of dark energy, a scalar field carrying negative pressure that could be driving the accelerated expansion of the cosmos. As the authors emphasize, following a definitive analysis by Matt Visser, the Kiselev spacetime is intrinsically anisotropic and cannot be treated as a perfect fluid, so it is best understood as a toy model: a black hole surrounded by a quintessence-like anisotropic fluid rather than a realistic portrait of cosmic dark energy. Even as a toy model, it is a powerful laboratory, because the strength of the dark-energy influence is controlled by a single parameter, alpha, which can be dialed up and down to see how the quantum behavior of the black hole responds.

The Brazilian duo attacked the problem with the Klein–Gordon equation, the relativistic wave equation for spinless particles, written in the curved geometry of the Kiselev metric. The metric carries a correction term that depends on alpha and on an equation-of-state parameter omega-zero, which characterizes how the pressure of the dark-energy fluid relates to its density. The researchers examined two representative choices: omega-zero equal to minus one-third and omega-zero equal to minus two-thirds, both lying in the range allowed for quintessence-like behavior. In each case, the geometry modifies the event horizon. For the first case the horizon sits at twice the black hole mass divided by one minus alpha, while for the second case two horizons emerge, an inner physical horizon and an outer quintessence-like horizon, provided alpha does not exceed one over eight times the mass. If alpha grows beyond that critical value, the horizons vanish entirely and the solution degenerates into a naked singularity, a configuration that general relativity’s cosmic censorship conjecture suggests should be forbidden.

Solving the Klein–Gordon equation in such a background is notoriously difficult, because the radial part of the wave equation becomes a complicated second-order differential equation with position-dependent coefficients. By separating the wave function into a radial piece, the familiar spherical harmonics, and a time-dependent oscillation, the authors reduced the problem to a radial equation whose solutions turn out to be special functions of a rare breed: Heun functions. For the omega-zero equals minus-one-third case, the radial wave function is expressed through the confluent Heun function, a power series whose coefficients obey a three-term recurrence relation. For the omega-zero equals minus-two-thirds case, the solution involves the generalized local Heun function, governed by a more demanding four-term recurrence. These functions are the curved-spacetime cousins of the Legendre and Bessel functions that dominate textbook quantum mechanics, and their appearance here is typical of wave equations in black hole backgrounds.

The real payoff comes when the infinite series is forced to terminate. Demanding that the wave function remain physically acceptable, finite at the horizon and well-behaved at infinity, truncates the Heun series into a polynomial, and that truncation can only happen at special, discrete values of the particle’s energy. These resonant frequencies form what physicists call the quasispectrum, and they are purely imaginary, meaning they describe modes that decay in time rather than oscillate forever. For the first model, the quasispectrum takes the form of the imaginary quantity proportional to n plus one, scaled by factors involving alpha and the black hole mass, and it reduces exactly to the known Schwarzschild result when alpha goes to zero. Intriguingly, the truncation conditions also quantize the orbital angular momentum quantum number l, which in general becomes complex, a hallmark of quasinormal-mode physics around black holes.

For the second model, the analysis produced a surprise: the truncation conditions can only be satisfied if the scalar particle is massless. The quasispectrum in that case again collapses to the Schwarzschild value of i times n plus one over four M in the limit of vanishing quintessence. The authors also verified that the radial wave function remains finite far from the black hole, where it behaves as an oscillating sine of a logarithmic phase with a calculable phase shift, a result that opens the door to studying the scattering of both massive and massless scalar fields off the Kiselev geometry using asymptotic methods developed by Abramov and collaborators.

With the exact wave functions in hand, the team then computed the quantity that connects this abstract mathematics to observable physics: the Hawking radiation itself. Following the standard tunneling-style analysis, they examined how outgoing modes behave just inside and just outside the event horizon, performed an analytic continuation around the pole at the horizon, and transformed to Eddington–Finkelstein coordinates, the coordinate system in which light rays travel along simple straight lines. Using Sannan’s formulation, they obtained the decay rate of particles escaping the horizon and, by matching to the Bose–Einstein distribution, extracted the Hawking temperature. For the omega-zero equals minus-one-third case, the temperature is proportional to one minus alpha squared divided by eight pi k-B times M, while for the omega-zero equals minus-two-thirds case it is proportional to alpha times the separation between the two horizons divided by four pi k-B times the inner horizon radius.

Two features of these temperatures stand out. First, both formulas agree exactly with the temperatures derived independently from the surface gravity approach, in which the temperature is proportional to the derivative of the metric function evaluated at the horizon. This cross-check is a strong consistency test, showing that the full quantum wave-function calculation reproduces the thermodynamic result obtained by purely geometric reasoning. Second, when alpha is set to zero, both cases recover the canonical Hawking temperature of a Schwarzschild black hole, one over eight pi k-B M, confirming that the entire dark-energy apparatus smoothly interpolates with known physics. Notably, in the second model the temperature is completely independent of the energy of the escaping particle, a clean separation between the thermodynamics of the hole and the quantum state of its radiation.

The most striking conclusion, however, concerns what dark energy does to the glow. Graphical analysis of the radiation spectrum shows that in both models, the larger the value of alpha, the weaker the Hawking radiation observed outside the event horizon. In the extreme limit of maximal quintessence influence in the first model, the radiation and the temperature both vanish entirely. The authors suggest this suppression may offer a tantalizing theoretical clue: since observations reveal very little matter streaming away from black hole horizons, a universe permeated by dark energy of the kind modeled here would naturally produce quieter black holes, consistent with what astronomers actually see. In other words, the cosmic acceleration that stretches the universe might simultaneously be muffling the quantum whispers of its most extreme objects.

The work, published as an open-access article in The European Physical Journal C and supported by the Brazilian funding agencies CNPq and FAPERJ, adds a precisely solved entry to the growing catalog of quantum fields in exotic spacetimes. Because quasinormal modes, black hole shadows, photon orbits, and thermodynamic stability are all known to shift in the presence of quintessence-like fluids, exact solutions of this kind provide the analytical scaffolding on which future phenomenological predictions can be built. As gravitational-wave detectors and black hole imaging campaigns grow ever more sensitive, distinguishing the fingerprint of dark energy in the behavior of black holes may move from a theorist’s daydream to a testable proposition, and calculations like this one map out exactly where that fingerprint should appear.

Subject of Research: Exact solutions of the Klein–Gordon equation for scalar particles in the spacetime of a Kiselev black hole surrounded by a quintessence-like anisotropic fluid

Article Title: Exact solution of the Klein–Gordon equation in a Kiselev black hole background

Article References: de Oliveira, M. D., & Schmidt, A. G. M. (2026). Exact solution of the Klein–Gordon equation in a Kiselev black hole background. The European Physical Journal C, 86(9), Article 1097. https://doi.org/10.1140/epjc/s10052-026-16325-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16325-4

Keywords: Kiselev black hole, Klein–Gordon equation, Hawking radiation, quintessence, dark energy, event horizon, Heun functions, quasinormal modes, general relativity, black hole thermodynamics, Schwarzschild limit, theoretical physics