## 2 Interface Modes through Material Composition Tuning  
The concept of inverted band structures is of paramount importance in the context of topological interface modes. In two-dimensional materials, the concept of inverted band structures usually refers to systems where the conduction and valence band symmetries are inverted [34]. In the context of this work, we denote that two one-dimensional systems present inverted bands when: (1) they have a common band gap; and (2) the modes at the Brillouin zone edge present opposite spatial symmetries. The concatenation of two superlattices with an inverted band gap typically leads to the formation of a confined mode at the interface. The presence of this interface state is unaffected by any perturbation which does not affect the topological invariants associated to the underlying lattices. In the case of a periodic one-dimensional system, the most commonly considered topological invariant is the Zak phase [28; 29; 30; 35]. In a previous work [27], using the relative layer thickness ratio in a GaAs/AlAs superlattice, a band inversion [36] was achieved for the second minigap. By changing the thickness ratio, a band inversion can, however, only be achieved for the second and higher order minigaps [37]. In contrast, the first minigap at the Brillouin zone edge remains open for any thickness ratio, and thus a band inversion cannot be implemented in the same manner. We will show how to implement the band inversion principle around the first minigap through material composition tuning and how this strategy enables us to establish a link between Fabry-Perot resonators and a one-dimensional topological model. The Su-Schrieffer-Heeger (SSH) model [32; 33; 38] is one of the paradigms in one-dimensional topology. It describes an electron in the potential of a carbon chain with alternating single and double chemical bonds, i.e., a polyacetylene molecule. The order of the single and double bonds leads to two energetically degenerate isomers. In the tight binding approximation, the periodic potential gives rise to a simple band structure, as shown in Figure 1c. Let us denote the strength of a single bond as \(t\) and that of a double bond as \(1-t\). For the case \(t=0.4\) (left panel), we observe an open minigap at the Brillouin zone edge. For (right panel) the weak and strong coupling terms are exchanged with respect to \(t=0.4\) giving rise to exactly the same band structure. For the particular case \(t=0.5\), however, all hopping terms become identical and the minigap closes (central panel). By looking at the symmetries of the corresponding electronic wave function at the band edges bounding the minigap (shown in the insets of panel d), it becomes evident that an inversion of the symmetries occurs when crossing the point \(t=0.5\). We denote (anti-)symmetric electron wave functions with a violet (red) dot. By varying the value of \(t\) continuously, we trace out the evolution of the spectral width of the electronic minigap as shown in Figure 1d. The violet and red lines indicate the normalized energies of the two band edges enclosing the minigap. In grey, the span of the minigap is indicated. We now establish a direct relationship between the elements of an acoustic superlattice and the electronic potential of a polyacetylene molecule. Note that the chemical bonds are mirror planes of the infinite carbon chain, whereas the centers of the material layers are mirror planes of the infinite  
Figure 1: Band inversion principle shown for the nanophononic superlattice case (left) and the SSH model (right). (**a,c**) The band structure diagram for the nanophononic and polyacetylene cases. Three conditions are illustrated: open minigap, closed minigap, and reopened minigap with the symmetries of the band edge modes inverted. The symmetries of the Bloch modes are indicated with violet (symmetric) and red (antisymmetric) dots, respectively; (**b,d**) The span of the minigap as a function of the material composition (left) or SSH hopping parameter (right).