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

Understand

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.