Bootstrapping the long-range Ising model in three dimensions

Abstract

The 3D Ising model and the generalized free scalar of dimension at least 0.75 belong to a continuous line of nonlocal fixed points, each referred to as a long-range Ising model. They can be distinguished by the dimension of the lightest spin-2 operator, which interpolates between 3 and 3.5 if we focus on the non-trivial part of the fixed line. A property common to all such theories is the presence of three relevant conformal primaries, two of which form a shadow pair. This pair is analogous to a superconformal multiplet in that it enforces relations between certain conformal blocks. By demanding that crossing symmetry and unitarity hold for a set of correlators involving the relevant operators, we compute numerical bounds on their scaling dimensions and OPE coefficients. Specifically, we raise the minimal spin-2 operator dimension to find successively smaller regions which eventually form a kink. Whenever a kink appears, its co-ordinates show good agreement with the epsilon expansion predictions for the critical exponents in the corresponding statistical model. As a byproduct, our results reveal an infinite tower of protected operators with odd spin.