Bulk-boundary correspondence of (1+1)D symmetric gapped phases

Abstract

We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$, an indecomposable semisimple right $\mathcal{C}$-module category $\mathcal{M}$, a Q-system $Q\in \mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\in {\mathcal{M}}_Q$, regarded as an object of ${\mathcal{M}}_Q^{\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor ${\mathcal{M}}_Q^{\mathrm{op}}\to \mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of ${\mathcal{M}}_Q$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $({\mathcal{C}}_{\mathcal{M}}^{\vee }{)}^{\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1({\mathcal{C}}^{\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $({\mathcal{C}}_{\mathcal{M}}^{\vee }{)}^{\mathrm{rev}}$ on ${\mathcal{M}}_Q^{\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.

56 pages