In regards to the posts I’ve made about plasmons in the past (see here and here for instance), it seems like the plasmon in a metal will always exist at a finite energy at due to the long-ranged nature of the Coulomb interaction. Back in 1956, D. Pines published a paper, where in collaboration with P. Nozieres, he proposed a method by which an acoustic plasmon could indeed exist.
The idea is actually quite simple from a conceptual standpoint, so a cartoony description should suffice in describing how this is possible. The first important ingredient in realizing an acoustic plasmon is two types of charge carriers. Pines, in his original paper, chose -electrons and -electrons from two separate bands to illustrate his point. However, electrons from one band and holes from another could also suffice. The second important ingredient in realizing the acoustic plasmon is that the masses of the two types of carriers must be very different (which is why Pines chose light -electrons and heavy -electrons).
So why are these two features necessary? Well, simply put, the light charge carriers can screen the heavy charge carriers, effectively reducing the range of the Coulomb interaction (see image above). Such a phenomenon is very familiar to all of us who study solids. If, for instance, the interaction between the ions on the lattice sites in a simple 3D monatomic solid were not screened by the electrons, the longitudinal acoustic phonon would necessarily be gapped because of the Coulomb interaction (forgetting, for the moment, about what the lack of charge neutrality would do to the solid!). In some sense, therefore, the longitudinal acoustic phonon is indeed such an acoustic plasmon. The ion acoustic wave in a classical plasma is similarly a manifestation of an acoustic plasmon.
This isn’t necessarily the kind of acoustic plasmon that has been so elusive to solid-state physicists, though. The original proposal and the subsequent search was conducted on systems where light electrons (or holes) would screen heavy electrons (or holes). Indeed, it was suspected that Landau damping into the particle-hole continuum was preventing the acoustic plasmon from being an observable excitation in a solid. However, there have been a few papers suggesting that the acoustic plasmon has indeed been observed at solid surfaces. Here is one paper from 2007 claiming that an acoustic plasmon exists on the surface of beryllium and here is another showing a similar phenomenon on the surface of gold.
To my knowledge, it is still an open question as to whether such a plasmon can exist in the bulk of a 3D solid. This has not stopped researchers from suggesting that electron-acoustic plasmon coupling could lead to the formation of Cooper pairs and superconductvity in the cuprates. Varma has suggested that a good place to look would be in mixed-valence compounds, where -electron masses can get very heavy.
On the experimental side, the search continues…
A helpful picture: If one imagines light electrons and heavy holes in a compensated semimetal for instance, the in-phase motion of the electrons and holes would result in an acoustic plasmon while the out-of-phase motion would result in the gapped plasmon.