Recently the conformal bootstrap program has turned out to be a quite powerful way to study d>2 conformal field theories (CFTs) in a completely non-perturbative fashion. Indeed the “solution” obtained by the method offers the most precise estimate for the 3d Ising model critical exponents. In this talk, I will discuss the application of the bootstrap program to more controversial yet physically important problems, namely, the d=3 O(n)xO(2) symmetric Landau-Ginzburg models.

These models realize wide variety of physical systems at criticality, including anti-ferromagnetic spin systems placed on triangular lattices and 2-flavor QCD chiral phase transition (provided the axial anomaly is negligible). Despite their obvious physical relevance, the study of their renormalization group (RG) flow are notoriously hard and there are serious controversies over the nature of their phase transitions: depending on the methods (e.g. perturbative RG series, functional RG equation, lattice Monte-Carlo, etc) one obtains different result regarding the presence of IR-stable fixed points.

I will propose the resolution of this long-standing controversies using the conformal bootstrap program, based on our papers arXiv:1404.0489 and arXiv:1407.6195 with Yu Nakayama.