# Do condensed-matter field theories with multiple fields generically have multiple speeds of sound?

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.

• The simplest example I can think of is graphene. The speed of sound for the dirac fermions is much larger than the speed of sound of the phonon field. Apr 18, 2019 at 6:26
• Generically if you have fermions of different bands, then each one has their own fermi velocity, this is the case for metals like iron. Apr 18, 2019 at 6:30
• I totally agree with what you're saying here. I imagine the reason for the discrepancy is the usual theorist's bias of just omitting (or perhaps not even knowing about) any details that would spoil the nice symmetries. Apr 21, 2019 at 10:31