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ORIGINAL RESEARCH article

Front. Acoust., 29 September 2025

Sec. Acoustic Metamaterials

Volume 3 - 2025 | https://doi.org/10.3389/facou.2025.1615210

This article is part of the Research TopicAcoustic Topological Insulators: Envisioned Applications and Technology IntegrationView all 4 articles

The acoustic Dirac equation as a model of topological insulators

Abhirup Basu,
Abhirup Basu1,2*Keith Runge,Keith Runge1,3Pierre A. Deymier,Pierre A. Deymier1,3
  • 1New Frontiers of Sound Science and Technology Center, The University of Arizona, Tucson, AZ, United States
  • 2Department of Mathematics, The University of Arizona, Tucson, AZ, United States
  • 3Department of Materials Science and Engineering, The University of Arizona, Tucson, AZ, United States

The dynamical equations of motion for a discrete, one-dimensional harmonic chain with side restoring forces are analogous to the relativistic Klein–Gordon equation. Dirac factorization of the discrete Klein–Gordon equation introduces two equations with time reversal (T) and parity (P) symmetry-breaking conditions. The Dirac-factored equations enable the exploration of the properties of the solutions of the dynamic equations under PT symmetry-breaking conditions. The spinor solutions of the Dirac factored equations describe two types of acoustic waves: one with a conventional topology (Berry phase equal to 0) and the other with a non-conventional topology (Berry phase of π). In the latter case, the acoustic wave is isomorphic to the quantum spin of an electron, also known as an “acoustic pseudospin,” which requires a closed path, corresponding to two Brillouin zones (BZs), to restore the original spinor. We also investigate the topology of evanescent waves supported by the Dirac-factored equations. The interface between topologically conventional and non-conventional chains exhibits topological surface states. The Dirac-factored equations of motion of the one-dimensional harmonic chain with side springs can serve as a model for the investigation of the properties of acoustic topological insulators.

1 Introduction

A topological insulator cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state (Qi and Zhang, 2011). While the bulk system is insulating, the surface can support conduction that is topologically protected, meaning that the surface states are insensitive to local perturbations (Hasan and Kane, 2010). Topological insulators can exhibit quantum mechanical properties such as the quantum spin Hall (QSH) effect (Qi and Zhang, 2010) and the anomalous quantum Hall (QH) effect, which occurs in the absence of an external magnetic field due to the breaking of time-reversal symmetry (Chang et al., 2023). Acoustic analogs of the QSH effect have been implemented (a) by tuning an accidental double Dirac cone in graphene-like lattices (He et al., 2016; Mei et al., 2016) and (b) via a BZ folding mechanism (Zhang et al., 2017; Yves et al., 2017; Xia et al., 2017; Deng et al., 2017). QH-related effects have been realized in acoustics by arranging circulating flows into periodic settings to form an acoustic lattice that breaks time reversal symmetry (Yang et al., 2015; Ni et al., 2015; Khanikaev et al., 2015).

Phononic structures can support elastic waves with non-conventional topology by breaking symmetry (Xue et al., 2022). Dirac factorization of the wave equation reveals potential topological properties that may result from symmetry breaking (which might be brought about by structural or external perturbations). For instance, the equations of motion of two coupled one-dimensional harmonic systems (Deymier and Runge, 2016) can be factored in the long wavelength limit as a product of two Dirac-like equations, each of which breaks parity symmetry and time reversal symmetry. Propagative wave solutions to each Dirac equation can satisfy two possible dispersion relations, giving rise to symmetric and anti-symmetric eigenmodes. The former exhibit the conventional character of Boson-like phonons, while the latter exhibit Fermion-like behavior of phonons (Deymier et al., 2015; Deymier et al., 2014). The topological properties of evanescent waves in product parity-time symmetry have also been investigated near exceptional points (Chen et al., 2024).

The wave equation for a two-dimensional plate coupled to a rigid substrate can also be subject to Dirac factorization (Deymier and Runge, 2022). These factors are analogous to the long-wavelength limit of the Qi, Wu, and Zhang (QWZ) model of the anomalous quantum Hall effect (Qi et al., 2006). The Dirac factorization reveals waves with spin-like degrees of freedom that have a gapped band structure, which is similar to the spin Hall effect. Kane and Lubensky (2014) demonstrate a method inspired by the Dirac factorization of the Klein–Gordon equation to establish a connection between topological mechanical modes and the topological band theory in electronic systems. This leads to the prediction of new topological bulk mechanical phases with distinct boundary modes. Topological phonons can also be classified using local symmetries (Süsstrunk and Huber, 2016) by adapting the classification of non-interacting electron systems to mechanical systems.

Here, we study solutions of the Dirac factorizations of the discrete one-dimensional harmonic chain with side restoring forces and investigate the appearance of edge modes at the interface of conventional and non-conventional topologies. In Section 2, we introduce the Dirac factorization of the discrete Klein–Gordon equation, and in Section 3, we find the dispersion relation and amplitude vectors corresponding to propagative wave solutions of the Dirac equations. Section 4 addresses evanescent wave solutions and their dispersion relation. In Sections 5 and 6, we compute the respective Berry phases of the amplitude vectors of the propagative and evanescent waves. Section 7 addresses the existence of edge modes at the interface between two topologically different semi-infinite media that obey the acoustic Dirac equation. Finally, in Section 8, we summarize and draw conclusions.

2 Model system and equation of motion for the discrete harmonic chain

We consider the model of the one-dimensional harmonic chain illustrated in Figure 1. We assume that the chain lies along the x-axis. The chain is composed of identical masses, M, interacting with their neighbors via linear spring forces with a spring constant K; each of the masses is connected to a rigid substrate through harmonic springs with a spring constant K. The coordinates of the nth mass at rest are x0n=na, where a is the spacing between adjacent masses at rest.

Figure 1
Schematic of a one-dimensional mass-spring lattice model with alternating spring constants $K$ and $K'$, where identical masses $M$ are evenly spaced by distance $a$, connected both to adjacent masses and to a fixed top boundary.

Figure 1. Schematic illustration of the one-dimensional harmonic chain attached elastically to a rigid substrate (top gray box). The side springs restore longitudinal motion.

We denote the displacement of the nth mass by un=xnx0n. Newton’s equation for the nth mass is

Md2undt2=Kun+1unKunun1Kun(1)

Taking β2=KM and α2=KM in Equation 1, we can rewrite the above equation as

d2undt2β2un+12un+un1+α2un=0(2)

The quantity un+12un+un1 can be identified as a discrete second derivative with respect to position. Thus, Equation 2 can be thought of as a discrete version of the Klein–Gordon equation 2ut2β22ux2+α2u=0. It can be shown that the continuous Klein–Gordon equation (when interpreted as an operator acting on vectors with two components) can be factorized as a product of Dirac equations:

σ1t±iβσ2xiαIσ1t±iβσ2x+iαI=I2ut2β2I2ux2+α2I(3)

—where σ1=0110, and σ2=0ii0 are the Pauli matrices and I is the 2 × 2 identity matrix (Deymier and Runge, 2016).

Now, let us provide a physical interpretation of amplitude vectors with two components. We note that if α=0, the equation corresponding to the first operator in the square bracket of Equation 3 becomes σ1t±iβσ2xψ=0. Using a plane wave solution with ψ=a1a2eikxeiωt, this equation reduces to the system ω±βka2=0ωβka1=0. We obtain two solutions for the angular velocity of the plane wave ω=±βkh. These correspond to plane waves propagating in the positive and negative directions. In this case, the components a1 and a2 are now independent of each other and of the wave number. The amplitude of the plane wave propagating in the positive direction is independent of that of the wave propagating in the opposite direction. When α0, the plane wave solutions of the equation σ1t±iβσ2xiαIψ=0 obey the system of linear equations ω±βka2αa1=0ωβka1αa2=0 with the dispersion relation ω=±βk2+α2. The amplitude components a1 and a2 are given by a1a2ω±βkωβk. The components of the 2 × 1 amplitude vector are not independent of each other. This indicates that the directions of propagation are not independent of each other anymore; it is parameter α that couples those directions. The wave function ψ has the character of quasi-standing waves, which are composed of forward and backward waves with a very specific proportion of their respective amplitudes a1 and a2.

In order to achieve a factorization of the discrete Klein–Gordon equation as a product of Dirac-like equations, we will interpret the operator d2dt2β2Δ+Δ+α2 as acting on vectors with four components. Here, Δ+ is the forward difference operator defined by Δ+un=un+1un, and Δ is the backward difference operator defined by Δun=unun1. These difference operators are linear (note that Δ+Δun=Δ+unun1=Δ+unΔ+un1=un+12un+un1). The operators Δ+ and Δ can also be interpreted as operators on vectors where the difference operations are carried out component-wise.

Let e1=0100 and e2=0010. We can reformulate Equation 2 in terms of a product of two operators:

σ1It+iβσ2e1++e2+iαIIσ1It+iβσ2e1++e2iαII

It can be verified that this product is the same as I2t2Iβ2++α2I (with I being the 4 × 4 identity matrix in this equation). Note that the matrices in the above product are tensor products of 2 × 2 matrices and hence are 4 × 4 matrices. Thus, the operators in the factorization are acting on vectors with four components, and the product of the operators is a four-dimensional version of the discrete Klein–Gordon equation:

σ1It+iβσ2e1++e2±iαIIψn=I2t2Iβ2++α2Iψn(4)

—where ψn=ψ1nψ2nψ3nψ4n.

Now the solution to Equation 4 ψn is a 4×1 vector. We have seen that the long wavelength limit (continuous limit) of the operators in the square brackets of Equation 3 leads to solutions with two components, corresponding to the mutually dependent amplitudes of the forward and backward waves. The four components of the discrete system reflect the fact that the forward difference Δ+ and backward difference Δ operators act differently on the forward and backward amplitudes of a quasi-standing wave. However, this solution form does not correspond to any straightforward physical interpretation.

The tensor products of the matrices appearing in the above equation are as follows:

Taking C=σ1I, A=iσ2e1, and B=iσ2e2, the Dirac factorization of the discrete Klein–Gordon equation becomes

Ct+βA++BiαIψn=0(5)

—where α=±α.

3 Eigenvectors and dispersion relation

3.1 Dispersion relation

We will now find propagative solutions to the Dirac equations in Equation 5. Let us consider an ansatz taking the form of a plane wave with a 4×1 amplitude vector, ζk:

ψn=ζkeiωteikna=a1a2a3a4eiωteikna(6)

We have the matrices

C=0010000110000100, A=0001000001000000,B=0000001000001000,andI=1000010000100001. Then, with the ansatz in Equation 6, the Dirac equations become the following system of equations:

iαa1+iωa3+βeika1a4=0iαa2+β1eikaa3+iωa4=0iωa1βeika1a2iαa3=0β1eikaa1+iωa2iαa4=0(7)

In matrix form, Equation 7 becomes

iα0iωβeika10iαβ1eikaiωiωβeika1iα0β1eikaiω0iαa1a2a3a4=0(8)

The determinant of this 4 × 4 matrix is

α2β2eika11eikaω22(9)

and there exist non-zero solutions to Equation 8 if the determinant is zero. Equation 9 gives us

ω=±α2β2eika11eika=±α2β2eika2+eika.(10)

Equation 10 gives us the dispersion relations

ω=±α2+4β2sin2ka2(11)

The dispersion relation is illustrated in Figure 2.

Figure 2
Plot showing angular frequency versus wave number, representing the dispersion relation. The lower curve is the negative of the upper curve.

Figure 2. Schematic illustration of the discrete system dispersion relation ωk=±α2+4β2sinka22 for =1, β=1, and α=1.

3.2 Eigenvectors

We now solve Equation 8 for the components of the amplitude vector a1, a2, a3, and a4, satisfying the dispersion relations given by Equation 11. Let us redefine ka=2θ such that eika1=eiθeiθeiθ, 1eika=eiθeiθeiθ, and eika11eika=eiθeiθ2. For the sake of simplifying the notation, we also define X=eiθeiθ. With this notation, Equation 8 becomes the system of four linear equations:

iαa1+iωa3+βeiθXa4=0(12a)
iαa2+βeiθXa3+iωa4=0(12b)
iωa1βeiθXa2iαa3=0(12c)
βeiθXa1+iωa2iαa4=0(12d)

To find solutions to the system of equations given by Equations 12 ad, we hypothesize that

a1=ia4eiθ(13a)
a2=ia3eiθ(13b)
a3=iωeiθ2(13c)
a4=eiθ2αβX(13d)

Recognizing that ω=±α+βXαβX, αβX=αβXαβX, and α+βX=α+βXα+βX, the normalized amplitude eigenvector is obtained in the form

ζ^k=12α2β2X2ieiθ2αβX±eiθ2α+βX±ieiθ2α+βXeiθ2αβX(14)

4 Evanescent waves

In order to find non-propagative solutions of the Dirac equations (Equation 5), we consider the ansatz

ψn=ξkeiωteik+ikna=b1b2b3b4eiωteik+ikna(15)

We follow the procedure used in Section 3 to find the dispersion relation for waves of the form given by Equation 15:

ω=±α24β2sinhka22(16)

When k=0, then ω=α, and ω=0 when α24β2sinhka22 = 0; that is, sinhka2=±α2β or equivalently k=2asinh1±α2β. We denote this value of k by k0. The points ±k0 are illustrated schematically in Figure 3.

Figure 3
Plot similar to Fig. 2 with dispersion relation of evanescent waves added. A tilted ellipse intersects the sinusoidal curves, and a diagonal labeled

Figure 3. Schematic illustration of the discrete system dispersion relation for propagative waves (blue and yellow lines): ωk=±α2+4β2sinka22 for a =1, β=1, and α=1 and evanescent waves (green line) at k=0: ωk=±α24β2sinhka22.

The normalized amplitude eigenvector is of the form

ξ^k=12αeθ+eθieθ2αβX±eθ2α+βX±ieθ2α+βXeθ2αβX(17)

—where X=eθeθ with ka=2θ.

5 Berry phase for propagative waves

5.1 Continuous contribution to the Berry phase

The contribution of the continuous part of the function, ζ^k to the Berry connection (Berry, 1984) is given by

BCk=iζ^k*Tζ^kk(18)

Note that ζ^kk=ζ^kθθk=a2ζ^kθ since ka=2θ, and therefore, Equation 18 gives us BCk=0 for all kπa,πa for which ζ^k is continuous. So, the continuous contribution to the Berry connection is zero.

5.2 Contribution of discontinuities in the eigenvectors to the Berry connection

The eigenvector given by Equation 14 contains components in the form of square roots—α+βX and αβX—which are square roots of complex numbers.

The square root of a complex number is given by the formula

A+iB=±A2+12A2+B2+iBB12A2+B2A2

Here, A=α and B=2βsinθ. Near the origin on both sides (positive and negative) of the Brillouin zone: θ±=0±, so B2θ±=0±. We therefore have α+βX±(A2+12A2+iBB12A2A2)=(A2+12A+iBB12AA2).

The phase of A+iB in the vicinity of the origin of the first Brillouin zone satisfies tanφsgnBA+AA+A.

If A=+α—that is A>0—then A=A and tanφsgnBA+AA+A=0. The quantity α+βX remains continuous at the origin of the Brillouin zone; the same is true for the quantity αβX. Therefore, in this case, there is no discontinuity in the complex amplitude, which leads to the Berry phase being equal to zero, as discussed in in Section 5.1.

If A=α, then A<0—that is, A=A and tanφsgnBA+AAA=sgnB. On the positive side of the origin of the Brillouin zone B=0+, sgnB>0 and φ=π2. On the negative side of the origin of the Brillouin zone B=0, sgnB<0 and φ=π2. The quantity α+βX undergoes a π phase discontinuity at the origin of the Brillouin zone.

Considering the component αβX, we still have A=α but B=2βsinθ. We still have a discontinuity when A=α, but then tanφsgnB. On the positive side of the origin of the Brillouin zone B=0, sgnB<0 and φ=π2. On the negative side of the origin of the Brillouin zone B=0+, sgnB>0 and φ=+π2. The quantity αβX undergoes a -π phase discontinuity at the origin of the Brillouin zone.

In the case of A<0, to calculate the discontinuity contribution of the components at 2θ=ka=0 to the Berry phase, consider two amplitude vectors on both sides of the origin of the Brillouin zone:

ζ^k0+12ieiφθ+±eiφθ+±ieiφθ+eiφθ+ and ζ^k012ieiφθ±eiφθ±ieiφθeiφθ

Their inner product gives ζ^k0+*Tζ^k0cosπ. Therefore, the change in geometric phase as the amplitude vector is crossing the origin of the first Brillouin zone is Δη=π. Now, the Berry phase is therefore the sum of the contributions from the continuous and the discontinuous parts of the amplitude vector, so the Berry phase is equal to π. Note that this phase is independent of the specific value of α as long as we consider the Dirac-factored equation with α.

The Berry phase is a topological invariant of the system. Dirac-factored equations Ct+βA++Bi±αIψn=0 describe two types of acoustic waves: one with a conventional topology (Berry phase equal to 0) and the other one with an unconventional topology (Berry phase of π). In the latter case, the acoustic waves are isomorphic to the quantum spin of an electron, which requires a closed path, corresponding to two Brillouin zones to recover the original eigen amplitude vector. This is an example of an acoustic pseudospin.

6 Berry phase for evanescent waves

We now compute the Berry connection of the unit amplitude vector ξ^k given by Equation 17 along the loop given by the dispersion relation of the evanescent waves with k=0. Since ω is a real number, we have the restriction α2βsinhka2α2β , which is equivalent to αβXαβ —that is, 0α+βX and 0αβX.

We first compute the Berry connection along the top and the bottom halves of the loop, when ω>0 and ω<0.

Case 1. α=α with ω>0 or ω<0.

In this case, the unit amplitude vector is ξ^k=12αeθ+eθieθ2αβX±eθ2α+βX±ieθ2α+βXeθ2αβX. Note that the quantities inside the square root are non-negative because of the restriction on k, so the unit amplitude vector is a continuous function of k. Therefore, the Berry connection is given by

BCk=iξ^k*Tξ^kk

We note that ξ^kk=ξ^kθθk=a2ξ^kθ since ka=2θ, which leads to BCk=0.

Therefore, the Berry connection when α=α is zero along the top and bottom halves of the loop.

Case 2. α=α with ω>0 or ω<0.

In this case, the unit amplitude vector is ξ^k=12αeθ+eθieθ2α+βX±eθ2αβX±ieθ2αβXeθ2α+βX. Again, the quantities inside the square root are non-negative. The Berry connection is thus given by iξ^k*Tξ^kk. A similar calculation to that in case α=α gives us BCk=0 along the top and bottom halves of the loop.

Case 3. The Berry connection near the points where the loop intersects the x-axis.

We wish to determine whether near ±k0, ξ^k,ω+, the unit amplitude vector at k with positive ω, is parallel or anti-parallel to ξ^k,ω, the unit amplitude vector at k with negative ω. We calculate the dot product

ξ^k,ω+*·ξ^k,ω=2βαsinhθ.

As kk0, note that ω0. So in that limit, the dispersion relation given by Equation 16 gives us 2βα=1sinhθ. Therefore, when α=α, we have ξ^k,ω+*·ξ^k,ω=sinhθsinhθ as kk0. When α=α, we have ξ^k,ω+*·ξ^k,ω=sinhθsinhθ as kk0. So, when α=α, ξ^k,ω+·*·ξ^k,ωsinhθsinhθ=1 as kk0. Therefore, ξ^k,ω+* and ξ^k,ω are parallel. When α=α, ξ^k,ω+*·ξ^k,ωsinhθsinhθ=1 as kk0. So, ξ^k,ω+* and ξ^k,ω are anti-parallel.

Similarly, when kk0, if α=α, ξ^k,ω+* and ξ^k,ω are anti-parallel, and if α=α, ξ^k,ω+* and ξ^k,ω are parallel.

Therefore, along the closed loop over the evanescent mode, the Berry phase amounts to π for both α=±α.

7 Interface modes

We now consider a system composed of two semi-infinite chains described by the acoustic Dirac equation, but differing only in the value of the parameter α. Such a system may be realized by considering the mass spring system in Figure 1 (which is governed by the Klein–Gordon equation) (Calderin et al., 2019) and by considering evanescent waves corresponding to the +α Dirac equation for masses with a negative label and α Dirac equation for masses with a positive label (since solutions to the Dirac equations are solutions to the Klein–Gordon equation). The consideration of such waves is mathematical. Figure 4 describes the set up.

Figure 4
Modified mass-spring schematic with five labeled masses in a chain. The system is divided into two regions with different spring parameters $\alpha’ = +\alpha$ and $\alpha’ = −\alpha$, indicating a topological interface.

Figure 4. Schematic illustration of an interface between two topologically different semi-infinite media obeying the acoustic Dirac equation. The medium with α=α is topologically non-trivial, and the medium with α=+α is topologically trivial.

At the interface, the Dirac equations are

Ct+βA++Bi+αIψn=1=0(19a)
Ct+βA++BiαIψn=0=0(19b)

The Equations 19 a,b can be expanded as

Ctψ1+βAψ0ψ1+Bψ1ψ2i+αIψ1=0(20a)
Ctψ0+βAψ1ψ0+Bψ0ψ1iαIψ0=0(20b)

We now seek solutions to these equations that take the form of evanescent waves decaying on both sides of the interface. These are solutions of the form

ψn=b1b2b3b4+α,k>0eiωteknaifn<1(21a)
ψn=b1b2b3b4α,k<0eiωteknaifn>0.(21b)

Equations 21 a,b give are solutions to the bulk acoustic Dirac equation of infinite chains, each with its respective values of α. At the interface, we consider

ψ1=W1X1Y1Z1+α,k>0eiωteka(22a)
ψ0=W0X0Y0Z0α,k<0eiωt(22b)

The existence of solutions in the forms given by Equations 22a,bwhich satisfy Equations 20a,b implies the existence of interface modes between the two chains with different topologies.

Inserting Equations 22a,b,b into Equations 20a,b,b gives us a system of eight linear equations in the eight unknowns W1, X1, Y1, Z1, W0, X0, Y0, and Z0:

i+α0iωβ000βeka0i+αβiω0000iωβi+α00βeka00βiω0i+α00000000iα0iωβ00βeka00iαβiω0000iωβiα0βeka000βiω0iαW1X1Y1Z1W0X0Y0Z0=0βb3+eka0βb1+ekaβb4eka0βb2eka0(23)

Equation 23 has solutions if the 8×8 matrix is invertible—that is, if its determinant is not equal to zero. By defining a=i+α/β, b=iω/β, and c=1, we calculate the determinant of that matrix to be

detω=b8+b64a2+6+b46a414 a 2 + 11 + b 2 4 a 6 + 10 a 4 14 a 2 + 6 + a 8 2 a 6 + 3 a 4 2 a 2 + 1 ( 24 )

Note that this determinant is independent of k’. To illustrate, let us set β = α = 1 and plot det ω = ω 8 10 ω 6 + 31 ω 4 34 ω 2 + 9 .

The det ω is non-zero for the majority of frequencies corresponding to evanescent waves, except for the two frequencies ω = ± 1 2 5 1 . There exist solutions to Equation 24 within this range of real frequencies. The interface between the chains with trivial and nontrivial topologies supports localized interfacial modes. These are topological interfacial modes. In Figure 5 there are two values of ω for which det( ω ) = 0. This implies that there are no real frequency solutions, although there may be solutions with complex frequencies. These waves may decay as a function of time.

Figure 5
Graph of a function plotted against angular frequency $\omega$. Vertical lines mark two key frequency regions, possibly indicating band gaps or critical thresholds.

Figure 5. Plot of det ω = ω 8 10 ω 6 + 31 ω 4 34 ω 2 + 9 . The region between the vertical lines corresponds to the frequency range of evanescent waves for α = 1 .

8 Conclusion

We have here demonstrated that the Dirac factorization of the equations of motion of a mass and spring model exposes the potential for topological insulator behavior in acoustic systems. In particular, for a 1-d harmonic mass and spring chain attached elastically to a rigid substrate, the equations of motion give rise to a discrete version of the Klein–Gordon equation that can be factorized into Dirac equations with broken time-reversal and parity symmetry. Propagative and evanescent wave solutions of the Dirac factored equations were obtained. For propagative modes, the Berry phase for the Dirac equation with the + sign was found to be zero (conventional topology), and that with the – sign was found to be π (non-conventional topology). In contrast, for the evanescent mode, the Berry phase was π for both the + and – equations. Using the distinction between topologies for the propagative waves, we demonstrated the existence of a topologically protected interface mode between conventional and non-conventional topologies.

Dirac factorization of classical wave equations exposes the possibility of topological insulators arising from broken symmetries. It reveals the possibilities offered by symmetry breaking in terms of the direction of wave propagation. However, additional physical conditions or mechanisms are needed to break T- or P-symmetry and realize one-way propagating waves in physical systems.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.

Author contributions

AB: Writing – original draft, Writing – review and editing. KR: Writing – original draft, Writing – review and editing. PD: Writing – original draft, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by the Science and Technology Center New Frontiers of Sound (NewFoS) through NSF cooperative agreement # 2242925.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

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Keywords: topological insulator, Berry phase, Dirac equation, Klein-Gordon equation, interface mode

Citation: Basu A, Runge K and Deymier PA (2025) The acoustic Dirac equation as a model of topological insulators. Front. Acoust. 3:1615210. doi: 10.3389/facou.2025.1615210

Received: 20 April 2025; Accepted: 29 August 2025;
Published: 29 September 2025.

Edited by:

Michael J. Leamy, Georgia Institute of Technology, United States

Reviewed by:

Stefano Laureti, University of Calabria, Italy
Zhang, Qicheng, Westlake University, China

Copyright © 2025 Basu, Runge and Deymier. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Abhirup Basu, YWJoaXJ1cC5iYXN1NTZAZ21haWwuY29t

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