## 2 Interface Modes through Material Composition Tuning
This accounts for the fact that high frequency phonons do not propagate in air and are hence completely reflected. The analogy to the edge mode in the SSH model is evident. Maxima of the mechanical displacement (the electronic wavefunction) are observed at every other interface (on every other carbon atom) and both modes decay evanescently away from the surface (the end of the carbon chain). Perhaps one of the most important predictions of the SSH model is the generation of an interface mode when two carbon chains with inverted bands are concatenated. This interface mode appears at the center of the minigap as shown in Figure 3c,d. Two possible ways exist to connect two semi-infinite carbon chains: either by adding an additional single (weak hopping) bond, or a double (strong hopping) bond. The strength of the additional bond determines whether the confined state will be monomer-like (with a single maximum) or a trimer-like state (with two maxima). In Figure 3c, we show the representation of these two concatenated carbon chains with an additional weak link and the corresponding electronic density confined around the central atom. Following this construction principle, two acoustic superlattices with inverted bands can be concatenated by a weak link (i.e., an AlAs layer) as well. This generates the interface mode shown in Figure 3a. As predicted by the SSH model, the acoustic mode appears at the center of the first acoustic minigap. The resulting nanophononic structure presents two identical \(\lambda/4\) layers at the interface forming a \(\lambda/2\) spacer. Therefore, the system can be effectively decomposed into a resonant spacer enclosed by two identical DBRs, resulting in the widely used Fabry-Perot resonator. By using the composition as a design parameter inducing a topological phase transition, Fabry-Perot resonators can hence be explained in terms of symmetry considerations, in particular the inversion of the Bloch mode symmetries at the Brillouin zone edge. In Figure 3b, we show the nanophononic case resulting in a trimer like confined state. Here, the two DBRs are connected by an additional \(\lambda/4\) GaAs layer generating a \(\lambda/2\) GaAs spacer. The GaAs spacer generates an anti-guiding mode, i.e., a mode with maxima of the field at the interfaces of the DBRs, contrary to what happens with an AlAs spacer that presents minima at these interfaces. The equivalent trimer state in the framework of the SSH model is shown in Figure 3d. Figure 3: Interface mode resulting from the concatenation of two semi-infinite systems with different topological phases: (**a**,**b**) the nanophononic case; and (**c**,**d**) the polyacetylene–SSH model. There are two possibilities to connect the two semi-infinite systems with different topological phases: (**a**,**c**) the monomer case, in which the two chains are connected with a single bond, i.e., a weak hopping link; and (**c**,**d**) the trimer case, i.e., when the two chains are connect with a double bond, resulting in a strong hopping link. Figure 2: Edge mode observed in a semi-infinite system in the case of: a nanophononic GaAs/AlAs superlattice (**a**); and a polyacetylene molecule (**b**). This mode only exists when the last bond in the chain is the weaker among the two types: in the acoustic case it corresponds to finishing the chain with an AlAs layer, in the case of the polyacetylene it corresponds to a single (weak hopping) bond at the end of the chain. At high phononic frequencies, usually only modes around the Brillouin zone center are accessible through Raman scattering and pump probe experiments. A confined mode at the edge of the Brillouin zone, as the one presented in Figure 3, could be experimentally measured in _resonant_ Raman scattering or in _resonant_ pump probe coherent phonon generation experiments. In these cases, the excitation laser should be resonant with the excitonic transition of the acoustic spacer, as described in Refs. [13; 40].