Towards Non-Invertible Anomalies from Generalized Ising Models

The 1d transverse-field Ising model, when projected to the Z₂ symmetric sector, is known to have a noninvertible gravitational anomaly that can be compensated by the Z₂ toric code model in 2d. In this work, we study the generalization of this type of bulk-boundary correspondence in a large class of qubit lattice models in arbitrary dimensions, called the generalized Ising (GI) models. We provide a systematic construction of exactly solvable bulk models, where the GI models can terminate on their boundaries. In each bulk model, any ground state is robust against local perturbations. If the model has degenerate ground states with periodic boundary condition, the phase is topological and/or fracton ordered. The construction generates abundant examples, including not only prototype ones such as Z₂ toric code models in any dimensions no less than two, and the X-cube fracton model, but also more diverse ones. The boundary of the solvable model is potentially anomalous and corresponds to precisely only sectors of the GI model that host certain total symmetry charges and/or satisfy certain boundary conditions. We derive a concrete condition for such bulk-boundary correspondence. A generalized notion of Kramers-Wannier duality plays an important role in the construction. Also, utilizing the duality, we find an example where a single anomalous theory can be realized on the boundaries of two distinct bulk fracton models, a phenomenon not expected in the case of topological orders.