# Are second-order phase transitions always scale/Lorentz invariant?

I know that both scale invariance and Lorentz invariance typically emerge at second-order phase transitions, but is there a proof or a counterexample? (I know that it's believed that any theory that is both scale- and Lorentz-invariant must also be conformally invariant, and that Joe Polchinski has proven this to be true in two dimensions, but that's not my question.) In fact, have any of the many qualitative differences between first- and second-order phase transitions been proven to hold in general? (I'm defining the order of a phase transition to be the lowest discontinuous derivative of the thermodynamic-limit free energy density.)

• @PeterKravchuk There are two different contexts in which a Lorentz-like symmetry can emerge. In statistical field theory, as you say the time coordinate is imaginary, so in systems like the $\varphi^4$ scalar field theory that describes the Ising model, the emergent symmetry group is $SO(4)$. But independently of equilibrium statistical field theory, you can also consider the real-time dynamics of the excitations about the ground state. (You can do this for any system, e.g. lattice systems, not just field theories.) If the excitations have a linear dispersion of the form ... – tparker Jul 30 '16 at 23:44
• ... $\omega = c_s k$ for small $k$, then the dynamics of the long-wavelength excitations will show true $SO(3,1)$ Lorentz invariance, with a "speed of light" given by the speed of sound $c_S$. This happens all the time in, e.g., systems that spontaneously break a continuous symmetry, like acoustic phonons in a superfluid or spin waves in an antiferromagnet. See physics.stackexchange.com/questions/63507/… for further discussion. – tparker Jul 30 '16 at 23:48