Scale invariance implies conformal invariance for the three-dimensional Ising model

Using the Wilson renormalization group, we show that if no integrated vector operator of scaling dimension $-1$ exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the three-dimensional Ising model.

Figure 1

Possible behavior of the lowest eigenvalue $D_V$ associated with a vector perturbation as a function of dimension. Left panel: (a) and (b) correspond to typical behavior, while (c) corresponds to the exceptional case in which $D_V=-1$ right in $d=3$. In the three cases, conformal invariance holds. Right panel: the shaded area represents a continuum of eigenvalues, and the curve denotes an eigenvalue $D_V$ having a plateau at $-1$ around $d=3$. In these cases, conformal invariance can be broken.