Multi-fixed point numerical conformal bootstrap: a case study with structured global symmetry

The numerical conformal bootstrap offers the possibility of finding and classifying conformal field theories of quantum critical systems in 2+1 dimensions, and thus obtaining non-perturbative information about their phase diagrams. However, its future utility depends on its ability to accurately predict the existence of hitherto unknown non-trivial conformal field theories (CFTs). Here we investigate the extent to which this is possible in the case where the global symmetry group has a product structure.

As a case study, we select the space of CFTs with product-group symmetry O(15)xO(3) in d=3 dimensions, where we find a single 'allowed' region in our chosen space of scaling dimensions. The scaling dimensions corresponding to two known large-N critical theories, the Heisenberg and the chiral ones, lie on or very near the boundary of this region. The large-N antichiral point lies well outside the 'allowed' region, which is consistent with the expectation that the antichiral theory is unstable, and thus has an additional relevant scalar singlet operator. We also find a sharp kink in the boundary of the 'allowed' region at values of the scaling dimensions that do not correspond to the (N,M)=(15,3) instance of any large-N-predicted O(N)xO(M) critical theory.