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

Clement Delcamp1,2 and Norbert Schuch1,2,3,4

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

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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.

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