Phonon heat transfer across a vacuum through quantum fluctuations
nature.com
nature.com
The concept of phonons is confusing because unlike other particles, phonons don’t have an accompanying fundamental field. They seem more like an emergent phenomenon. Why should phonons be considered particles?
But, I would guess that it has to do with wiggles in space-time, since they do say it's from the vibrating of the atoms. Energy is transferred through a medium via imparting of wiggles to that medium, so I think it would be reasonable to guess that it's either the vibrations moving through space-time or maybe some other field related to mass.
In the case of phonons, they can carry momentum, they travel like particles. A sibling comment is correct -- they are technically quasi-particles [1]. It is okay to think about a phonon field within a material. That field is not fundamental, in the sense that an electron is presently-believed to be fundamental, but that doesn't keep it from being a very useful way to think about condensed-matter physics.