Generalized Symmetries Phase Transitions with Local Quantum Fields

Symmetries are important guiding principle for phase transitions. We systematically construct field theory models with local quantum fields that exhibit the following phase transitions: (1) different symmetry protected topological (SPT) phases with generalized symmetries; (2) different symmetry enriched topological (SET) phases with generalized symmetries differ by symmetry fractionalizations; (3) spontaneously broken generalized symmetries, where the unbroken phases can have nontrivial SPT or SET. The models are ordinary gauge theories with bosons or fermions in 3+1d and 2+1d. We focus on one-form symmetries and symmetries generated by condensation defects, which do not act on local operators. The phase transitions are protected from local operator perturbations which do not change the asymptotic phases. In particular, we show that continuous gauge theories in 3+1d can have different phases distinguished by fractionalizations of unbroken one-form symmetries.