A Renormalization Group Analysis of the Ising Model Coupled to Causal Dynamical Triangulations

We analyze the matrix model characterizing the Ising model coupled to Causal Dynamical Triangulations (CDT) from the point of view of the Functional Renormalization Group Equation (FRGE). This model is a dually weighted matrix model, whose Feynman diagrams are in correspondence with discrete triangulations of two-dimensional geometries with a preferred time foliation. In particular, we find the fixed points of the beta-function equations, showing that the number of relevant directions in each case is compatible with the physical interpretation of the CFT at the fixed point. In addition to recovering the fixed points for topological gravity and pure gravity with a cosmological constant, we find a new fixed point featuring three relevant directions which matches the number of primary fields in the Ising CFT.