On tensor network representations of the (3+1)d toric code

Clement Delcamp; Norbert Schuch

doi:10.22331/q-2021-12-16-604

On tensor network representations of the (3+1)d toric code

Clement Delcamp^1,2 and Norbert Schuch^1,2,3,4

¹Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
²Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
³University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria
⁴University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Published:	2021-12-16, volume 5, page 604
Eprint:	arXiv:2012.15631v2
Doi:	https://doi.org/10.22331/q-2021-12-16-604
Citation:	Quantum 5, 604 (2021).

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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 $\mathbb Z_2$-symmetry.

► BibTeX data

@article{Delcamp2021tensornetwork,
  doi = {10.22331/q-2021-12-16-604},
  url = {https://doi.org/10.22331/q-2021-12-16-604},
  title = {On tensor network representations of the (3+1)d toric code},
  author = {Delcamp, Clement and Schuch, Norbert},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {604},
  month = dec,
  year = {2021}
}

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