  Conformal field theories (CFTs) with cubic global symmetry in 3D are relevant
in a variety of condensed matter systems and have been studied extensively with
the use of perturbative methods like the $\varepsilon$ expansion. In an earlier
work, we used the nonperturbative numerical conformal bootstrap to provide
evidence for the existence of a previously unknown 3D CFT with cubic symmetry,
dubbed "Platonic CFT". In this work, we make further use of the numerical
conformal bootstrap to perform a three-dimensional scan in the space of scaling
dimensions of three low-lying operators. We find a three-dimensional isolated
allowed region in parameter space, which includes both the 3D (decoupled) Ising
model and the Platonic CFT. The essential assumptions on the spectrum of
operators used to provide the isolated allowed region include the existence of
a stress-energy tensor and the irrelevance of certain operators (in the
renormalization group sense).
