Noninvertible symmetry-enriched quantum critical point

Noninvertible symmetry generalizes traditional group symmetries, advancing our understanding of quantum matter, especially one-dimensional gapped quantum systems. In critical lattice models, it is usually realized as emergent symmetries in the corresponding low-energy conformal field theories. In this Letter, we study critical lattice models with the noninvertible Rep(D8) symmetry in one dimension. This leads us to another class of quantum critical points (QCPs), noninvertible symmetry-enriched QCPs, as a generalization of known group symmetry-enriched QCPs. They are realized as phase transitions between one noninvertible symmetry-protected topological phase and another different one or spontaneous symmetry-breaking phase. We identify their low-energy properties and topological features through the Kennedy-Tasaki duality transformation. We argue that distinct noninvertible symmetry-enriched QCPs cannot be smoothly connected without a phase transition or a multicritical point.