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.
Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron
LIPN, CNRS UMR 7030, Institut Galilée, Université Paris 13（巴黎第十三大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Université de Lyon, ENS de Lyon, Laboratoire de Physique, CNRS UMR 5672（里昂大学）
Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas（数学科学研究院）；Universität Tübingen（蒂宾根大学）；University of Cambridge（剑桥大学）；IBM Quantum, IBM T. J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）；Ghent University（根特大学）
