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.)
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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.
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1$\begingroup$ What precisely do you mean by Lorentz-invariance? The "Lorentz-invariance" which is a part of the conformal group which people talk about in phase transitions is simply the rotations, isn't it? $\endgroup$ Commented Jul 30, 2016 at 4:25
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$\begingroup$ @PeterKravchuk No, it's also the boosts, with the speed of light replaced by the speed of sound. $\endgroup$– tparkerCommented Jul 30, 2016 at 20:16
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$\begingroup$ Ok, that is interesting and not what I am used to think about. Do you have any references to situations when Lorentz-invariance emerges? I am a bit puzzled, because in equilibrium thermodynamic descriptions, time is either not a part of the description or is periodic and imaginary. Are you referring to some non-equilibrium properties (in the sense in which sound is not equilibrium)? $\endgroup$ Commented Jul 30, 2016 at 21:38
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$\begingroup$ @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 ... $\endgroup$– tparkerCommented Jul 30, 2016 at 23:44
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$\begingroup$ ... $\omega = c_s k$ 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. $\endgroup$– tparkerCommented Jul 30, 2016 at 23:48