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

TL;DR

The paper applies Functional Renormalization Group techniques to a dually weighted Ising–CDT matrix model, demonstrating the existence of continuum limits via fixed points and recovering known gravity points. Importantly, it identifies a novel fixed point with three relevant directions, aligning with the three primary fields of the Ising CFT, indicating the emergence of Ising CFT content in CDT. The analysis also reveals fixed-point segments, suggesting a rich structure in theory space and highlighting the role of truncation choices. The work lays groundwork for more exhaustive truncations and numerical explorations to sharpen quantitative predictions for the coupled matter–gravity system.

Abstract

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.

Figures (2)

• Figure 1: An example of a CDT strip and the dual ribbon graph. The red lines are the propagators for the $A$ matrix and the gray dashed lines are the $B$ propagators.
• Figure 2: Projection of the phase space of the theory to the $(g_4^{(1)}, g_4^{(4)}, g_4^{(5)})$ subspace. This is composed by solutions A, B, C, D (blue), and critical segments R (red) and S (green).