The Ising Model on Random Lattices in Arbitrary Dimensions

TL;DR

The paper studies the Ising model coupled to random lattices in dimensions $D\ge 3$ using colored random tensor models and the $1/N$ expansion, showing melonic graphs dominate and enabling a Schwinger-Dyson approach to the continuum limit. By solving closed SD equations for the two- point functions through melonic factorization, it analyzes the critical behavior at fixed temperature and finds the grand-canonical free energy is analytic for all $0\le c<1$, i.e., no finite-temperature phase transition in the continuum limit. In arbitrary dimension, the critical curve $g_c(c)$ can be described parametrically, and the infinite-temperature limit ($c\to 1$) exhibits a distinct scaling, indicating a transition only at infinite temperature. The work provides a general method to study critical behavior in colored tensor models and helps explain how fluctuating higher-dimensional geometries can suppress conventional Ising transitions, aligning with numerical results in related regimes.

Abstract

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.

Figures (2)

• Figure 1: Definition of melonic 1PI graphs. The self-energy $\Sigma$ factors into connected 2-point functions corresponding to the internal lines.
• Figure 2: Parametric curves on the critical line for $D=3,\dotsc,8$.

Theorems & Definitions (1)

• Proposition 1