  In view of its physical importance in predicting the order of chiral phase
transitions in QCD and frustrated spin systems, we perform the conformal
bootstrap program of $O(n)\times O(2)$-symmetric conformal field theories in
$d=3$ dimensions with a special focus on $n=3$ and $4$. The existence of
renormalization group fixed points with these symmetries has been controversial
over years, but our conformal bootstrap program provides the non-perturbative
evidence. In both $n=3$ and $4$ cases, we find singular behaviors in the bounds
of scaling dimensions of operators in two different sectors, which we claim
correspond to chiral and collinear fixed points, respectively. In contrast to
the cases with larger values of $n$, we find no evidence for the anti-chiral
fixed point. Our results indicate the possibility that the chiral phase
transitions in QCD and frustrated spin systems are continuous with the critical
exponents that we predict from the conformal bootstrap program.
