Organizing transitions and their cascades: Generalized symmetry enforcement in massless flows or Higgs transitions

We study the role of generalized symmetry in massless renormalization group flows or Higgs transitions. In particular, we revisit the massless renormalization group flows in unitary minimal models, M(p,p+1) to preserving the fusion ring symmetry FR (SU(2) p-2) M(p,p+1) where p is an integer satisfying p>3. In this series of flows, we demonstrate that the unbroken fusion ring symmetry FR (SU(2) p-2) eliminates all relevant perturbations in the model. Hence, the infrared theory is stable at the level of the scaling analysis and can be interpreted as a (weak-)symmetry-enforced gapless phase in contemporary theoretical physics. Phenomenologically, by the folding trick, the unbroken fusion ring symmetry corresponds to a (half-)integer spin nonsimple current, a variant of the Cooper pair involving nonabelian anyons generated from the coset or level-rank duality structures. Moreover, we demonstrate that the structure of (half-)integer spin nonsimple current plays a fundamental role in causing the resonance effect of relevant and dangerously irrelevant perturbations. This resonance effect may result in the cascade of phase transitions (or the system flows to unexplored fixed points), and the symmetry can be a stopper of such unconventional flows.