  Simplicity constraints play a crucial role in the construction of spin foam
models, yet their effective behaviour on larger scales is scarcely explored. In
this article we introduce intertwiner and spin net models for the quantum group
$\text{SU}(2)_k \times \text{SU}(2)_k$, which implement the simplicity
constraints analogous to 4D Euclidean spin foam models, namely the
Barrett-Crane (BC) and the Engle-Pereira-Rovelli-Livine/Freidel-Krasnov
(EPRL/FK) model. These models are numerically coarse grained via tensor network
renormalization, allowing us to trace the flow of simplicity constraints to
larger scales. In order to perform these simulations we have substantially
adapted tensor network algorithms, which we discuss in detail.
  The BC and the EPRL/FK model behave very differently under coarse graining:
While the unique BC intertwiner model is a fixed point and therefore
constitutes a 2D topological phase, BC spin net models flow away from the
initial simplicity constraints and converge to several different topological
phases. Most of these phases correspond to decoupling spin foam vertices,
however we find also a new phase in which this is not the case, and in which a
non-trivial version of the simplicity constraints holds. The coarse graining
flow of the BC spin net models indicates furthermore that the phase transitions
are not of second order. The EPRL/FK model by contrast reveals a far more
intricate and complex dynamics. We observe an immediate flow away from the
original simplicity constraints, however, with the truncation employed here,
the models generically do not converge to a fixed point.
  The results show that the imposition of simplicity constraints can indeed
lead to interesting and complex dynamics. Thus we will need to further develop
coarse graining tools to efficiently study the large scale behaviour of spin
foam models, in particular for the EPRL/FK model.
