Figure 1. Example of a dimer chain that posses the particle-hole symmetry without any fine-tuning.

The dimers can rotate in the plane and the centre of the dimers can move along the longitudinal axis of the chain. The dimers are interconnected by two springs attached directly to the point masses. In (a), the two springs are identical and the system has both the inversion and reflection symmetries relative to the marked inversion centres and reflection planes, respectively. In this case, the phonon spectrum, shown (b), is ungapped. In (d), the two springs are different and only the inversion symmetry remains. In this case, the phonon spectrum, shown in (e), is gapped and the chain is in a topological phase from the BDI symmetry class, with bulk winding number ν=1. If the frequency is replaced by ω² (pulsation squared), then the particle-hole symmetry becomes explicit in (c,f), relative to the dotted lines. The phonon spectrum in (e) was computed with theoretical parameters that fit the experimental phonon spectrum.