It is well known that the low-energy physics of many non-relativistic condensed matter systems can be described by field theories that display emergent Lorentz symmetry. The heuristic way to figure out the effective "speed of light" (really more like a speed of sound) is to expand the dispersion relation $\epsilon(p) = \epsilon(\hbar k) = \hbar\, \omega(k) = \hbar c k + o(k^2)$ to linear order in $k$, and the coefficient of the linear term gives the speed of sound.
But what about a condensed-matter system whose low-energy physics is described by a field theory consisting of several massless (or low-mass) weakly interacting fields not related by symmetry? It seems to me that performing this procedure separately for each field would generically yield a different linear-order coefficient for each dispersion relation, and therefore a different effective speed of sound, for each field. Is this the case?
If so, then each field would be individually Lorentz-invariant when considered in isolation, but the interacting theory as a whole would not be, due to the different fields' different "speeds of light". And yet, while I've often seen lattice condensed-matter systems coarse-grained into field theories with multiple fields with different masses and coupling constants, I can't recall ever seeing one where the different fields have different speeds of sound.
