Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Kansei Inamura

TL;DR

This work develops an index theory for non-invertible symmetries in 1+1 dimensions acting on tensor-product lattice Hilbert spaces. It presents two complementary frameworks: an axiomatic approach using defect Hilbert spaces and a tensor-network approach via topological injective MPOs, establishing that exact realizability of fusion rules without QCAs requires integral fusion categories, while allowing QCAs enforces weak integrality under a homogeneous index. A tensor-network formulation defines lattice quantum dimensions and an index that generalize the GNVW index to non-invertible operators, and introduces sufficient (though not universally proven) zipper conditions under which the index remains homogeneous. The paper provides concrete MPO examples (invertible symmetries, non-anomalous fusion categories, Kramers-Wannier dualities) illustrating the framework and demonstrating how defect Hilbert spaces and sequential quantum circuits encode symmetry actions. Overall, the results connect fusion category theory, lattice locality, and tensor-network formalisms, offering a path to characterize which generalized symmetries can be realized on a tensor-product Hilbert space and under which conditions such realizations are constrained to weak integrality.

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.

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

TL;DR

This work develops an index theory for non-invertible symmetries in 1+1 dimensions acting on tensor-product lattice Hilbert spaces. It presents two complementary frameworks: an axiomatic approach using defect Hilbert spaces and a tensor-network approach via topological injective MPOs, establishing that exact realizability of fusion rules without QCAs requires integral fusion categories, while allowing QCAs enforces weak integrality under a homogeneous index. A tensor-network formulation defines lattice quantum dimensions and an index that generalize the GNVW index to non-invertible operators, and introduces sufficient (though not universally proven) zipper conditions under which the index remains homogeneous. The paper provides concrete MPO examples (invertible symmetries, non-anomalous fusion categories, Kramers-Wannier dualities) illustrating the framework and demonstrating how defect Hilbert spaces and sequential quantum circuits encode symmetry actions. Overall, the results connect fusion category theory, lattice locality, and tensor-network formalisms, offering a path to characterize which generalized symmetries can be realized on a tensor-product Hilbert space and under which conditions such realizations are constrained to weak integrality.

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.
Paper Structure (36 sections, 1 theorem, 267 equations, 1 figure)

This paper contains 36 sections, 1 theorem, 267 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Physical assumptions
  3. Non-invertible symmetries without mixing with QCAs
  4. Left and right dimensions
  5. Exact fusion rules imply integrality
  6. Non-invertible symmetries mixed with QCAs
  7. Lattice quantum dimension and index
  8. Homogeneity of the index
  9. Homogeneity of the index implies weak integrality
  10. A comment on the case without tensor product decomposition
  11. Tensor network formulation of non-invertible symmetries
  12. Preliminaries
  13. Definitions
  14. General case
  15. Example: invertible symmetries
  16. ...and 21 more sections

Key Result

Corollary 1

If the answer to Question q: fusion and splitting tensors is affirmative, a unitary fusion category symmetry realized by topological injective MPOs must be weakly integral even if we allow mixing with non-trivial QCAs. More specifically, if there exists a finite set of topological injective MPOs who

Figures (1)

  • Figure 1: A schematic picture of defect Hilbert spaces $\mathcal{H}_{\mathsf{D}_X}^l$ (left) and $\mathcal{H}_{\mathsf{D}_X}^r$ (right).

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 4.1: Topological injective MPO
  • Definition 4.2: Equivalent definition of topological injective MPO
  • Definition 4.3: Left and right dimensions
  • Definition 4.4: Lattice quantum dimension and index
  • Definition 5.1: Fusion and splitting tensors
  • Definition 5.2: Broken zipper condition
  • Definition 5.3: Two-sided zipper condition
  • ...and 5 more