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We propose a general framework to characterize gapped infra-red (IR) phasesof theories with non-invertible (or categorical) symmetries. In this paper wefocus on (1+1)d gapped phases with fusion category symmetries. The approachthat we propose uses the Symmetry Topological Field Theory (SymTFT) as a keyinput: associated to a field theory in d spacetime dimensions, the SymTFT livesin one dimension higher and admits a gapped boundary, which realizes thecategorical symmetries. It also admits a second, physical, boundary, which isgenerically not gapped. Upon interval compactification of the SymTFT bycolliding the gapped and physical boundaries, we regain the original theory. Inthis paper, we realize gapped symmetric phases by choosing the physicalboundary to be a gapped boundary condition as well. This set-up providescomputational power to determine the number of vacua, the symmetry breakingpattern, and the action of the symmetry on the vacua. The SymTFT alsomanifestly encodes the order parameters for these gapped phases, thus providinga generalized, categorical Landau paradigm for (1+1)d gapped phases. We findthat for non-invertible symmetries the order parameters involve multipletscontaining both untwisted and twisted sector local operators, and hence can beinterpreted as mixtures of conventional and string order parameters. We alsoobserve that spontaneous breaking of non-invertible symmetries can lead tovacua that are physically distinguishable: unlike the standard symmetriesdescribed by groups, non-invertible symmetries can have different actions ondifferent vacua of an irreducible gapped phase. This leads to the presence ofrelative Euler terms between physically distinct vacua. We also provide amathematical description of symmetric gapped phases as 2-functors fromdelooping of fusion category characterizing the symmetry to Euler completion of2-vector spaces.

Gapped Phases with Non-Invertible Symmetries: (1+1)d