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.)
I discovered that second-order phase transitions are characterized by scale invariance but not necessarily Lorentz invariance. For example, any critical point whose low-lying excitation spectrum is quadratic in momentum rather than linear will be scale invariant, but neither Lorentz nor conformal symmetry will emerge at the phase transition. Examples of phase transitions that are scale-invariant but not conformal are free fermions with the chemical potential tuned to the metal-band-insulator transition and Heisenberg antiferromagnets at the saturation applied field.