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 gaugeZ2-symmetry.
Highlights
Tensor networks have proven very powerful as a numerical as well as analytical framework for the study of strongly correlated quantum many-body systems
Using the properties of the Projected Entangled-Pair Operators (PEPOs) tensors, we find that the tensor T3Zd satisfies the following symmetry property that the stabilizer constraints are satisfied at every plaquette
We have shown that the toric code admits two canonical tensor network representations that differ in the order according to which the two families of stabilizer constraints are enforced
Summary
Tensor networks have proven very powerful as a numerical as well as analytical framework for the study of strongly correlated quantum many-body systems. We recover this self-duality at the boundary—in essence, the (1+1)d Ising self-duality—such that the mapping preserves the symmetry but exchanges the symmetric and symmetrybroken phases These two representations respond dually under uniform perturbations of the tensors that break the virtual symmetry, leading to a doping with electric or magnetic excitations, respectively, which breaks the topological order either way. Given that the model is no longer self-dual, such that electric and (bulk) magnetic excitations are point-like and loop-like, respectively, we expect these two tensor network representations to be inequivalent and characterized by distinct symmetry conditions This raises the question, how will the difference in symmetry structure manifests itself in the study of the topological model? The tensor network analysis of the (3+1)d toric code is presented in sec. 4, where both representations and the corresponding duality mapping are discussed in detail
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