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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-statistical lattice modes at the critical point. Do they all have emergent conformal symmetry and if not is there any counterexamples?

mr.no
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1 Answers1

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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.

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?

With that said, it appears that an enormous number of relevant examples with emergent scale+rotational symmetry also have emergent conformal symmetry, so there has been a lot of work trying to understand whether one can get conformal symmetry by adding some other reasonable assumptions.

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?

For classical models, you will at least want emergent scale and rotational symmetry at the critical point before you can ask if conformal symmetry emerges (this is the analog to requiring Lorentz invariance in the quantum case). For example, the Pokrovsky-Talapov critical point between incommensurate and commensurate phases in two dimensions involves highly anisotropic scaling between the two dimensions, so it is scale invariant but not conformal.

However, if you have rotational symmetry and scale invariance, then in two dimensions you also have conformal invariance.

Seth Whitsitt
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