### Abstract

We study the entanglement entropy of gapped phases of matter in three spatial dimensions. We focus in particular on size-independent contributions to the entropy across entanglement surfaces of arbitrary topologies. We show that for low energy fixed-point theories, the constant part of the entanglement entropy across any surface can be reduced to a linear combination of the entropies across a sphere and a torus. We first derive our results using strong sub-additivity inequalities along with assumptions about the entanglement entropy of fixed-point models, and identify the topological contribution by considering the renormalization group flow; in this way we give an explicit definition of topological entanglement entropy Stopo in (3+1)D, which sharpens previous results. We illustrate our results using several concrete examples and independent calculations, and show adding "twist" terms to the Lagrangian can change Stopo in (3+1)D. For the generalized Walker-Wang models, we find that the ground state degeneracy on a 3-torus is given by exp(-3Stopo[T2]) in terms of the topological entanglement entropy across a 2-torus. We conjecture that a similar relationship holds for Abelian theories in (d+1) dimensional spacetime, with the ground state degeneracy on the d-torus given by exp(-dStopo[Td-1]).

Original language | English (US) |
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Article number | 195118 |

Journal | Physical Review B |

Volume | 97 |

Issue number | 19 |

DOIs | |

State | Published - May 10 2018 |

### All Science Journal Classification (ASJC) codes

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics

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## Cite this

*Physical Review B*,

*97*(19), [195118]. https://doi.org/10.1103/PhysRevB.97.195118