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## Abstract

We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge Z 2-symmetry.

Original language | English |
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Pages (from-to) | 604 |

Number of pages | 31 |

Journal | Quantum |

Volume | 5 |

DOIs | |

Publication status | Published - 16 Dec 2021 |

## Austrian Fields of Science 2012

- 103025 Quantum mechanics
- 103015 Condensed matter
- 103029 Statistical physics

## Keywords

- ANYONS
- DUALITY
- INVARIANTS
- MODELS
- QUANTUM

## Projects

- 1 Active