Duality between Spin networks and the 2D Ising model

TL;DR

This paper uncovers a deep duality between the partition function of the 2D Ising model on planar graphs and spin network evaluations, revealing new links between quantum gravity, statistical mechanics, and supersymmetry.

Contribution

It establishes a supersymmetric duality between spin networks and the Ising model, and explores applications in quantum gravity and phase transition analysis.

Findings

The inverse relationship between the two functions via supersymmetry.

Stationary points of spin network evaluations match Ising model critical points.

Mapping of Ising correlations to spin network observables elucidates phase transitions.

Abstract

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories which couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.