## 2 Interface Modes through Material Composition Tuning
The symmetries of the Bloch edge modes are inverted when crossing the value 0.5, i.e., equal material for all layers or equal hopping strength for all bonds, respectively. The insets in the bottom panels indicate the acoustic field distribution (left) and the electronic wavefunction (right) corresponding to the Bloch modes at the Brillouin zone edge. Note that the symmetries of the modes are exchanged between the upper and lower edge when crossing the topological phase transition at a value of \(x=t=0.5\). acoustic superlattice. The carbon atoms can thus be associated to acoustic interfaces and the bonds to material layers. Interestingly, if the atoms were associated to acoustic layers, this correspondence between mirror planes would be broken. In the following, we explore to which extent this mapping can account for the behavior of a nanophononic systems by relying on conclusions drawn from the SSH model. A method to realize the band inversion principle for the first minigap at the Brillouin zone edge is depicted in Figure 1a,b. We consider DBRs for longitudinal acoustic phonons, which are constituted by alternating layers of materials A and B with acoustic impedances \(Z_{A}=\rho_{A}v_{A}\) and \(Z_{B}=\rho_{B}v_{B}\) (\(\rho\) mass density and speed of sound). At a design frequency \(f_{0}=175\) GHz, the total acoustic path length of each layer is set to a quarter of a phonon wavelength \(\lambda/4\), i.e., the thicknesses \(d\) of the layers obey \(\frac{d_{A}}{v_{A}}=\frac{d_{B}}{v_{B}}=\frac{1}{4f_{0}}\). For GaAs (AlAs), \(\rho=3.35\) g/cm\({}^{3}\) (\(\rho=3.77\) g/cm\({}^{3}\)) and \(v=4780\) m/s (\(v=5660\) m/s) resulting in \(d=6.8\) nm (\(d=8.1\) nm). A phase of \(\pi\) is thus accumulated by a phonon at frequency \(f_{0}\) traversing both layers of the unit cell. Consequently, all band gaps of the DBR are centered at integer multiples of \(f_{0}\). For the sake of clarity, we consider that materials A and B are Al\({}_{\rm x}\)Ga\({}_{\rm 1-x}\)As and Al\({}_{\rm 1-x}\)Ga\({}_{\rm x}\)As where \(x\) parametrizes the composition. Let us analyze the band inversion in this system. In Figure 1a, we show three band structures for different values of \(x\), obtained by a transfer matrix method. First, for \(x=0\) the first minigap is open, presenting a symmetric (anti-symmetric) Bloch mode at the upper (lower) band edge. As before, we denote (anti-)symmetric modes with a violet (red) dot. Second, for \(x=0.5\), the minigap is closed and thus no symmetries can be assigned to the degenerate edge modes. Note that this case represents a homogeneous material and the folding of the dispersion relation in the Brillouin zone is merely an arbitrary choice. Third, for \(x=1\), we observe the same bandstructure as in the first case, but the spatial symmetries of the band edge modes are inverted. By varying the value of \(x\) continuously, we trace out the evolution of the spectral width of the first acoustic minigap, as shown in Figure 1b. The violet and red lines indicate the frequencies of the two band edges enclosing the considered gap. In grey, the span of the minigap is indicated. For our particular case, according to Refs. [27; 28; 29; 35; 39], the Zak phase corresponding to the lowest band is unequivocally related to the order of mode symmetries at the band edge. The SSH model predicts the appearance of an edge mode in the center of the minigap when the last bond in a terminated carbon chain is of the weaker (single bond) type. The electronic density distribution of this edge mode is shown in Figure 2b together with an illustration of the corresponding semi-infinite carbon chain. In Figure 2a, we show the nanophononic equivalent of this situation: a \(\lambda/4\) AlAs layer terminating a DBR. In this case, a surface mode appears, with a frequency at the center of the first acoustic minigap. The acoustic displacement corresponding to this confined mode is shown in the same figure. Note that for the DBR the standard zero-stress condition is applied at the left boundary.