# Do all the classical critical lattice models have emergent conformal invariance?

I understand that any quantum lattice model at the critical point which can be described by a massless relativistic quantum field theory has emergent conformal invariance. My question is what about classical lattice modes at the critical point. Do they all have such a property and if not is there any counterexamples?

This is not always true. For example, free quantum electrodynamics in $$(2+1)$$d is scale invariant and relativistic but not conformal. However, there is a proof that in $$(1+1)$$ dimensions, scale+Lorentz symmetry implies conformal symmetry. See the following Stack Exchange thread for more information: Does dilation/scale invariance imply conformal invariance?