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Tensor network renormalization with fusion charges: applications to 3d lattice gauge theory

Tensor network methods are powerful and efficient tools to study the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods were applied to lattice gauge theories, yet these theories remain a challenge in $(2+1)$ dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages: Firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentangles, that remove short-scale entanglement. We apply this new algorithm to lattice gauge theories defined for the quantum group $\text{SU}(2)_{\rm k}$ and identify a weak and a strong coupling phase for various levels $\rm k$. As we increase the level $\rm k$, the critical coupling $g_c$ decreases linearly, suggesting the absence of a deconfining phase for the continuous group $\text{SU}(2)$. Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.