  The universal coupling of matter and gravity is one of the most important
features of general relativity. In quantum gravity, in particular spin foams,
matter couplings have been defined in the past, yet the mutual dynamics, in
particular if matter and gravity are strongly coupled, are hardly explored,
which is related to the definition of both matter and gravitational degrees of
freedom on the discretisation. However extracting this mutual dynamics is
crucial in testing the viability of the spin foam approach and also
establishing connections to other discrete approaches such as lattice gauge
theories.
  Therefore, we introduce a simple 2D toy model for Yang--Mills coupled to spin
foams, namely an Ising model coupled to so--called intertwiner models defined
for $\text{SU}(2)_k$. The two systems are coupled by choosing the Ising
coupling constant to depend on spin labels of the background, as these are
interpreted as the edge lengths of the discretisation. We coarse grain this toy
model via tensor network renormalization and uncover an interesting dynamics:
the Ising phase transition temperature turns out to be sensitive to the
background configurations and conversely, the Ising model can induce phase
transitions in the background. Moreover, we observe a strong coupling of both
systems if close to both phase transitions.
