An SPT-LSM theorem for weak SPTs with non-invertible symmetry

Like ordinary symmetries, non-invertible symmetries can characterize Symmetry-Protected Topological (SPT) phases. In this talk, we will discuss weak SPTs protected by projective non-invertible symmetries. Projective symmetries are ubiquitous in quantum spin models and can be leveraged to constrain their phase diagram and entanglement structure, e.g., Lieb-Schultz-Mattis (LSM) theorems. We will show how, surprisingly, projective non-invertible symmetries do not always imply LSM theorems. We will first discuss a simple, exactly solvable 1+1D quantum spin model in an SPT phase protected by both translation and non-invertible symmetries forming a non-trivial projective algebra. We will then generalize this example to a class of projective non-invertible Rep(G) x G x translation symmetries. For some finite groups G, this projectivity implies an LSM theorem. When it does not, we prove it still provides a constraint through an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak SPT state with non-trivial entanglement. [This talk is based on arXiv:2409.18113]