  Three-dimensional theories with cubic symmetry are studied using the
machinery of the numerical conformal bootstrap. Crossing symmetry and unitarity
are imposed on a set of mixed correlators, and various aspects of the parameter
space are probed for consistency. An isolated allowed region in parameter space
is found under certain assumptions involving pushing operator dimensions above
marginality, indicating the existence of a conformal field theory in this
region. The obtained results have possible applications for ferromagnetic phase
transitions as well as structural phase transitions in crystals. They are in
tension with previous $\varepsilon$ expansion results, as noticed already in
earlier work.
