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We provide a generalization of the Symmetry Topological Field Theory (SymTFT)framework to characterize phase transitions and gapless phases with categoricalsymmetries. The central tool is the club sandwich, which extends the SymTFTsetup to include an interface between two topological orders: there is asymmetry boundary, which is gapped, and a physical boundary that may begapless, but in addition, there is also a gapped interface in the middle. Theclub sandwich generalizes so-called Kennedy-Tasaki (KT) transformations.Building on the results in [1, 2] on gapped phases with categorical symmetries,we construct gapless theories describing phase transitions with non-invertiblesymmetries by applying suitable KT transformations on known phase transitionsprovided by the critical Ising model and the 3-state Potts model. We alsodescribe in detail the order parameters in these gapless theoriescharacterizing the phase transitions, which are generally mixtures ofconventional and string-type order parameters mixed together by the action ofcategorical symmetries. Additionally, removing the physical boundary from theclub sandwiches results in club quiches, which characterize all possible gappedboundary phases with (possibly non-invertible) symmetries that can arise on theboundary of a bulk gapped phase. We also provide a mathematicalcharacterization of gapped boundary phases with symmetries as pivotal tensorfunctors whose targets are pivotal multi-fusion categories.

The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries