Towards a phase diagram for spin foams

One of the most pressing issues for loop quantum gravity and spin foams is the construction of the continuum limit. In this paper, we propose a systematic coarse-graining scheme for three-dimensional lattice gauge models including spin foams. This scheme is based on the concept of decorated tensor networks, which have been introduced recently. Here we develop an algorithm applicable to gauge theories with non-Abelian groups, which for the first time allows for the application of tensor network coarse-graining techniques to proper spin foams. The procedure deals efficiently with the large redundancy of degrees of freedom resulting from gauge invariance. The algorithm is applied to 3D spin foams defined on a cubical lattice which, in contrast to a proper triangulation, allows for non-trivial simplicity constraints. This mimics the construction of spin foams for 4D gravity. For lattice gauge models based on a finite group we use the algorithm to obtain phase diagrams, encoding the continuum limit of a wide range of these models. We find phase transitions for various families of models carrying non-trivial simplicity constraints.