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 $\mathrm{Rep}\left(D_8\right)$ 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.

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• Quantum entanglement
• Quantum phase transitions
• Symmetry protected topological states
• Topological defects
• Conformal field theory
• Conformal symmetry
• Critical phenomena
• Duality
• Quantum spin chains