Spin foam models are candidate theories of quantum gravity defined on a two-complex and can be understood as generalised lattice gauge theories. To construct them one discretises and quantises topological BF theory and implements the so-called simplicity constraints to reintroduce local degrees of freedom. However, due to the complexity of spin foam models, their dynamics is still poorly understood, in particular in the regime of many building blocks.

In order to make them accessible for numerical methods, developed e.g. in condensed matter theory, we introduce two simplifications: First one performs a dimensional reduction to a one-complex, i.e. a 2D lattice, and as a second step, one replaces the underlying symmetry group, e.g. SU(2), by a quantum group SU(2)_k, which comes with a natural cut-off on the representation labels. We call these analogue models spin nets and apply a coarse-graining scheme to them, which gives effective dynamics on larger scales. In particular we are interested in the fate of the simplicity constraints: As generalised lattice gauge theories, one would hope for new phases or fixed points (with realised simplicity constraints) beyond the topological phase and the degenerate phase. Indeed, we will show that spin net models have a very rich fixed point structure in the refinement limit and argue that they realise the simplicity constraints. This is significant new evidence that spin foam models have a continuous smooth phase, which allows for dynamics consistent with General Relativity.