They claim that phonons might have negative mass and also have negative gravity.
It is usually said that sound waves do not transport mass. They carry momentum and energy, and lead to temporary oscillations of the local mass density of any region they happen to pass through, but it is an accepted fact that the net mass transported by a sound wave vanishes.
A first indication that sound waves can in fact carry a nonzero net mass is contained in the results: using an effective point-particle theory, it was shown that phonons in zero-temperature superfluids have an effective coupling to gravity, which depends solely on their energy (or momentum) and on the superfluid's equation of state. For ordinary equations of state, such a coupling corresponds to a negative effective gravitational mass: in the presence of an external gravitational field, such as that of Earth, a phonon's trajectory bends upwards.