We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

arXiv:2404.01369Physical Review LettersSep 10, 202469 citations

Critical entanglement spectrum of one-dimensional symmetry protected topological phases

Guang-Ming Zhang, Xin Wan

Bulletin of the American Physical SocietyMar 2, 20150 citations

Higher-order topological phases protected by noninvertible and subsystem symmetries

Aswin Parayil Mana, Yabo Li, Hiroki Sukeno +1 more

Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in $2+1$ dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a $d^{th}$ -order SPT phase in $d+1$ dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in $3+1$ dimensions exhibiting hinge modes protected by a noninvertible symmetry.Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in $2+1$ dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a $d^{th}$-order SPT phase in $d+1$ dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in $3+1$ dimensions exhibiting hinge modes protected by a noninvertible symmetry.

arXiv:2505.18119Physical review. B./Physical review. BApr 22, 20261 citation

Noninvertible symmetry protected topological phases from the duality method: Classification and lattice construction

Weiguang Cao

As the understanding of global symmetry has been expanded, quantum phases of matter can have a refined classification, protected or enriched by noninvertible symmetry, a type of generalized symmetry whose transformation may not have an inverse. In this paper, I use the duality method, proposed in Ref. 1 , to classify noninvertible symmetry protected topological (NISPT) phases. Under the duality transformation, different NISPT phases are mapped to different classes of spontaneous symmetry breaking phases with ordinary group symmetry. I demonstrate this method with the simplest Kramers–Wannier duality transformation, also known as gauging [Formula: see text] symmetry, to classify Rep([Formula: see text]) NISPT in [Formula: see text]. I also construct lattice models realizing these NISPT on tensor product Hilbert space and study the projective algebras at the interfaces between different classes of NISPTs.As the understanding of global symmetry has been expanded, quantum phases of matter can have a refined classification, protected or enriched by noninvertible symmetry, a type of generalized symmetry whose transformation may not have an inverse. In this paper, I use the duality method, proposed in Ref. 1 , to classify noninvertible symmetry protected topological (NISPT) phases. Under the duality transformation, different NISPT phases are mapped to different classes of spontaneous symmetry breaking phases with ordinary group symmetry. I demonstrate this method with the simplest Kramers–Wannier duality transformation, also known as gauging [Formula: see text] symmetry, to classify Rep([Formula: see text]) NISPT in [Formula: see text]. I also construct lattice models realizing these NISPT on tensor product Hilbert space and study the projective algebras at the interfaces between different classes of NISPTs.

International Journal of Modern Physics AAug 30, 20251 citation

Symmetry-Protected Topological Phases

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Quantum science and technologyJan 1, 20197 citations

2+1

dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a

d^{th}

-order SPT phase in

d+1

dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in

3+1

dimensions exhibiting hinge modes protected by a noninvertible symmetry.