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

TL;DR

This paper introduces two dual tensor network representations of the (3+1)d toric code ground state, highlighting their different virtual symmetries and topological properties, and relating boundary entanglement phases to (2+1)d models with distinct symmetries.

Contribution

It presents novel dual tensor network representations of the (3+1)d toric code and analyzes their topological and boundary entanglement properties.

Findings

Different virtual symmetries characterize the two representations.

Boundary entanglement phases relate to (2+1)d models with global or gauge symmetries.

The representations reveal distinct topological features of the (3+1)d toric code.

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.