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Peer-reviewedCC BY 4.0SciPost Physics

Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions

  • Kansei Inamura1,2
  1. 1Rudolf Peierls Centre for Theoretical Physics
  2. 2University of Oxford
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Abstract

We propose an index of non-invertible symmetry operators in 1+1 dimensions and discuss its relation to the realizability of non-invertible symmetries on the tensor product of finite dimensional on-site Hilbert spaces on the lattice. Our index generalizes the Gross-Nesme-Vogts-Werner index of invertible symmetry operators represented by quantum cellular automata (QCAs). Assuming that all fusion channels of symmetry operators have the same index, we show that the fusion rules of finitely many symmetry operators on a tensor product Hilbert space can agree, up to QCAs, only with those of weakly integral fusion categories. We also discuss an attempt to establish an index theory for non-invertible symmetries within the framework of tensor networks. To this end, we first propose a general class of matrix product operators (MPOs) that describe non-invertible symmetries on a tensor product Hilbert space. These MPOs, which we refer to as topological injective MPOs, include all invertible symmetries, non-anomalous fusion category symmetries, and the Kramers-Wannier symmetries for finite abelian groups. For topological injective MPOs, we construct the defect Hilbert spaces and the corresponding sequential quantum circuit representations. We also show that all fusion channels of topological injective MPOs have the same index if there exist fusion and splitting tensors that satisfy appropriate conditions. The existence of such fusion and splitting tensors has not been proven in general, although we construct them explicitly for all examples of topological injective MPOs listed above.

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Cite

BibTeXRIS
BibTeX
@article{inamura2026remarks,
  title = {{Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions}},
  author = {Kansei Inamura},
  journal = {SciPost Physics},
  year = {2026},
  month = oct,
  doi = {10.21468/scipostphys.21.4.088},
  url = {https://scipost.org/10.21468/SciPostPhys.21.4.088},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions
AU  - Kansei Inamura
PY  - 2026
DA  - 2026/10/09
JO  - SciPost Physics
DO  - 10.21468/scipostphys.21.4.088
UR  - https://scipost.org/10.21468/SciPostPhys.21.4.088
AB  - We propose an index of non-invertible symmetry operators in 1+1 dimensions and discuss its relation to the realizability of non-invertible symmetries on the tensor product of finite dimensional on-site Hilbert spaces on the lattice. Our index generalizes the Gross-Nesme-Vogts-Werner index of invertible symmetry operators represented by quantum cellular automata (QCAs). Assuming that all fusion channels of symmetry operators have the same index, we show that the fusion rules of finitely many symmetry operators on a tensor product Hilbert space can agree, up to QCAs, only with those of weakly integral fusion categories. We also discuss an attempt to establish an index theory for non-invertible symmetries within the framework of tensor networks. To this end, we first propose a general class of matrix product operators (MPOs) that describe non-invertible symmetries on a tensor product Hilbert space. These MPOs, which we refer to as topological injective MPOs, include all invertible symmetries, non-anomalous fusion category symmetries, and the Kramers-Wannier symmetries for finite abelian groups. For topological injective MPOs, we construct the defect Hilbert spaces and the corresponding sequential quantum circuit representations. We also show that all fusion channels of topological injective MPOs have the same index if there exist fusion and splitting tensors that satisfy appropriate conditions. The existence of such fusion and splitting tensors has not been proven in general, although we construct them explicitly for all examples of topological injective MPOs listed above.
SN  - 2542-4653
ER  -