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.

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 n t h mass at rest are x 0 n = n a , where a is the spacing between adjacent masses at rest.

We denote the displacement of the n t h mass by u n = x n − x 0 n . Newton’s equation for the n t h mass is M d 2 u n d t 2 = K u n + 1 − u n − K u n − u n − 1 − K ′ u n (1)

Taking β 2 = K M and α 2 = K ′ M in Equation 1, we can rewrite the above equation as d 2 u n d t 2 − β 2 u n + 1 − 2 u n + u n − 1 + α 2 u n = 0 (2)

The quantity u n + 1 − 2 u n + u n − 1 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 ∂ 2 u ∂ t 2 − β 2 ∂ 2 u ∂ x 2 + α 2 u = 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: σ 1 ∂ ∂ t ± i β σ 2 ∂ ∂ x − i α I σ 1 ∂ ∂ t ± i β σ 2 ∂ ∂ x + i α I = I ∂ 2 u ∂ t 2 − β 2 I ∂ 2 u ∂ x 2 + α 2 I (3) —where σ 1 = 0 1 1 0 , and σ 2 = 0 − i i 0 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 σ 1 ∂ ∂ t ± i β σ 2 ∂ ∂ x ψ = 0 . Using a plane wave solution with ψ = a 1 a 2 e i k x e i ω t , this equation reduces to the system ω ± β k a 2 = 0 ω ∓ β k a 1 = 0 . We obtain two solutions for the angular velocity of the plane wave ω = ± β k h . These correspond to plane waves propagating in the positive and negative directions. In this case, the components a 1 and a 2 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 α ≠ 0 , the plane wave solutions of the equation σ 1 ∂ ∂ t ± i β σ 2 ∂ ∂ x − i α I ψ = 0 obey the system of linear equations ω ± β k a 2 − α a 1 = 0 ω ∓ β k a 1 − α a 2 = 0 with the dispersion relation ω = ± β k 2 + α 2 . The amplitude components a 1 and a 2 are given by a 1 a 2 ∝ ω ± β k ω ∓ β k . 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 a 1 and a 2 .

In order to achieve a factorization of the discrete Klein–Gordon equation as a product of Dirac-like equations, we will interpret the operator d 2 d t 2 − β 2 Δ +   Δ − + α 2 as acting on vectors with four components. Here,   Δ + is the forward difference operator defined by   Δ + u n = u n + 1 − u n , and   Δ − is the backward difference operator defined by   Δ − u n = u n − u n − 1 . These difference operators are linear (note that Δ +   Δ − u n = Δ + u n − u n − 1 = Δ + u n − Δ + u n − 1 = u n + 1 − 2 u n + u n − 1 ). The operators Δ + and Δ − can also be interpreted as operators on vectors where the difference operations are carried out component-wise.

Let e 1 = 0 1 0 0 and e 2 = 0 0 1 0 . We can reformulate Equation 2 in terms of a product of two operators: σ 1 ⨂ I ∂ ∂ t + i β σ 2 ⨂ e 1 ∆ + + e 2 ∆ − + i α I ⨂ I σ 1 ⨂ I ∂ ∂ t + i β σ 2 ⨂ e 1 ∆ + + e 2 ∆ − − i α I ⨂ I

It can be verified that this product is the same as I ∂ 2 ∂ t 2 − I β 2 ∆ + ∆ − + α 2 I (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: σ 1 ⨂ I ∂ ∂ t + i β σ 2 ⨂ e 1 ∆ + + e 2 ∆ − ± i α I ⨂ I ψ n = I ∂ 2 ∂ t 2 − I β 2 ∆ + ∆ − + α 2 I ψ n (4) —where ψ n = ψ 1 n ψ 2 n ψ 3 n ψ 4 n .

Now the solution to Equation 4 ψ n 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 = σ 1 ⨂ I , A = i σ 2 ⨂ e 1 , and B = i σ 2 ⨂ e 2 , the Dirac factorization of the discrete Klein–Gordon equation becomes C ∂ ∂ t + β A ∆ + + B ∆ − − i α ′ I ψ n = 0 (5) —where α ′ = ± α .

In order to find non-propagative solutions of the Dirac equations (Equation 5), we consider the ansatz ψ n = ξ k ′ e i ω t e i k + i k ′ n a = b 1 b 2 b 3 b 4 e i ω t e i k + i k ′ n a (15)

We follow the procedure used in Section 3 to find the dispersion relation for waves of the form given by Equation 15: ω = ± α ′ 2 − 4 β 2 sinh k ′ a 2 2 (16)

When k ′ = 0 , then ω = α ′ , and ω = 0 when α ′ 2 − 4 β 2 sinh k ′ a 2 2 = 0; that is, sinh k ′ a 2 = ± α 2 β or equivalently k ′ = 2 a sinh − 1 ± α 2 β . We denote this value of k ′ by k 0 ′ . The points ± k 0 ′ are illustrated schematically in Figure 3.

The normalized amplitude eigenvector is of the form ξ ^ k ′ = 1 2 α ′ e θ ′ + e − θ ′ i e − θ ′ 2 α ′ − β X ′   ± e θ ′ 2 α ′ + β X ′ ± i e − θ ′ 2 α ′ + β X ′ e θ ′ 2 α ′ − β X ′ (17) —where X ′ = e − θ ′ − e θ ′ with k ′ a = 2 θ ′ .

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 − α 2 β ≤ sinh k ′ a 2 ≤ α 2 β , which is equivalent to − α β ≤ X ′ ≤ α β —that is, 0 ≤ α + β X ′ and 0 ≤ α − β X ′ .

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 ′ = 1 2 α e θ ′ + e − θ ′ i e − θ ′ 2 α − β X ′   ± e θ ′ 2 α + β X ′ ± i e − θ ′ 2 α + β X ′ e θ ′ 2 α − β X ′ . 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 B C k ′ = − i ξ ^ k ′ *T ∂ ξ ^ k ′ ∂ k ′

We note that ∂ ξ ^ k ′ ∂ k ′ = ∂ ξ ^ k ′ ∂ θ ′ ∂ θ ′ ∂ k ′ = a 2 ∂ ξ ^ k ′ ∂ θ ′ since k ′ a = 2 θ ′ , which leads to B C k ′ = 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 ′ = − 1 2 α e θ ′ + e − θ ′ i e − θ ′ 2 α + β X ′   ± e θ ′ 2 α − β X ′ ± i e − θ ′ 2 α − β X ′ e θ ′ 2 α + β X ′ . Again, the quantities inside the square root are non-negative. The Berry connection is thus given by − i ξ ^ k ′ *T ∂ ξ ^ k ′ ∂ k ′ . A similar calculation to that in case α ′ = α gives us B C k ′ = 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 ± k 0 ′ , 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 − = 2 β α ′ sinh ⁡ θ ′ .

As k ′ → k 0 ′ , note that ω → 0 . So in that limit, the dispersion relation given by Equation 16 gives us 2 β α ′ = 1 sinh ⁡ θ ′ . Therefore, when α ′ = α , we have ξ ^ k ′ , ω +   * ·       ξ ^ k ′ , ω − = sinh ⁡ θ ′ sinh ⁡ θ ′ as k ′ → k 0 ′ . When α ′ = − α , we have ξ ^ k ′ , ω +   * ·       ξ ^ k ′ , ω − = sinh ⁡ θ ′ sinh ⁡ θ ′ as k ′ → k 0 ′ . So, when α ′ = α , ξ ^ k ′ , ω + ·     * ·   ξ ^ k ′ , ω − →   sinh ⁡ θ ′ sinh ⁡ θ ′ = 1 as k ′ → k 0 ′ . Therefore, ξ ^ k ′ , ω +   * and ξ ^ k ′ , ω − are parallel. When α ′ = − α , ξ ^ k ′ , ω +   * ·   ξ ^ k ′ , ω − → − sinh ⁡ θ ′ sinh ⁡ θ ′ = − 1 as k ′ → k 0 ′ . So, ξ ^ k ′ , ω +   * and ξ ^ k ′ , ω − are anti-parallel.

Similarly, when k ′ → − k 0 ′ , 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 .

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.

At the interface, the Dirac equations are C ∂ ∂ t + β A ∆ + + B ∆ − − i + α I ψ n = − 1 = 0 (19a) C ∂ ∂ t + β A ∆ + + B ∆ − − i − α I ψ n = 0 = 0 (19b)

The Equations 19 a,b can be expanded as C ∂ ∂ t ψ − 1 + β A ψ 0 − ψ − 1 + B ψ − 1 − ψ − 2 − i + α I ψ − 1 = 0 (20a) C ∂ ∂ t ψ 0 + β A ψ 1 − ψ 0 + B ψ 0 − ψ − 1 − i − α I ψ 0 = 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 ψ n = b 1 b 2 b 3 b 4 + α , k ′ > 0 e i ω t e k ′ n a   i f   n < − 1 (21a) ψ n = b 1 b 2 b 3 b 4 − α , − k ′ < 0 e i ω t e − k ′ n a   i f   n > 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 ψ − 1 = W − 1 X − 1 Y − 1 Z − 1 + α , k ′ > 0 e i ω t e − k ′ a (22a) ψ 0 = W 0 X 0 Y 0 Z 0 − α , − k ′ < 0 e i ω t (22b)

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

Inserting Equations 22a,b into Equations 20a,b gives us a system of eight linear equations in the eight unknowns W − 1 , X − 1 , Y − 1 , Z − 1 , W 0 , X 0 , Y 0 , and Z 0 : − i + α 0 i ω − β 0 0 0 β e k ′ a 0 − i + α β i ω 0 0 0 0 i ω β − i + α 0 0 − β e k ′ a 0 0 − β i ω 0 − i + α 0 0 0 0 0 0 0 0 − i − α 0 i ω − β 0 0 − β e − k ′ a 0 0 − i − α β i ω 0 0 0 0 i ω β − i − α 0 β e − k ′ a 0 0 0 − β i ω 0 − i − α W − 1 X − 1 Y − 1 Z − 1 W 0 X 0 Y 0 Z 0 = 0 β b 3 + e − k ′ a 0 − β b 1 + e − k ′ a − β b 4 − e − k ′ a 0 β b 2 − e − k ′ a 0 (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 = i ω / β , and c = 1 , we calculate the determinant of that matrix to be det ⁡ ω = b 8 + b 6 − 4 a 2 + 6 + b 4 6 a 4 − 14 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.

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.