Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries

Abstract: In this work, we analyze the macroscopic observables in the 1+1D gapped phases with Abelian onsite symmetries and show that the spacetime observables for each gapped phase form a clear structure that can be mathematically described by enriched fusion categories, which uncovers the behavior of nonlocal excitations that were blurry in traditional Landau paradigm. These categorical descriptions not only generate the known classification results for symmetry preserving and breaking phases, but also unifies lattice dualities in a broader picture. After analyzing the general lattice model together with their boundaries, we give explicit examples including nontrivial SPT phase, where nontrivial boundaries can be given directly through our classification. Using enriched categorical descriptions, the lattice dualities and their gapped phases are unified under a holographic duality between an 2d. topological order with gapped 1d boundaries and 1+1D gapped quantum liquids with a categorical symmetry, which shed light on a unified definition of all quantum phases.