Approaching the conformal window of O(n) x O(m) symmetric Landau-Ginzburg models using the conformal bootstrap

O(n) x O(m) symmetric Landau-Ginzburg models in d = 3 dimensions possess a rich structure of the renormalization group, which offers a theoretical prediction of the phase diagram in frustrated spin models with noncollinear order. Depending on n and m, they may show chiral/antichiral/Heisenberg/Gaussian fixed points within the same universality class. We approach all the fixed points in the conformal bootstrap program by examining the bound on the conformal dimensions for scalar operators as well as nonconserved current operators with consistency cross-checks. For large n/m, we show strong evidence for the existence of four fixed points by comparing the operator spectrum obtained from the conformal bootstrap program with that from the large-n/m analysis. We propose a novel nonperturbative approach to the determination of the conformal window in these models based on the conformal bootstrap program. From our numerical results, we predict that for m = 3, n = 7 similar to 8 is the edge of the conformal window for the antichiral fixed points.