Statistical mechanics approach to the holographic renormalization group: Bethe lattice Ising model and p-adic AdS/CFT

Abstract

The Bethe lattice Ising model - a classical model of statistical mechanics for the phase transition - provides a novel and intuitive understanding of the prototypical relationship between tensor networks and the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary-spin correlations by introducing the relation between the lattice network and an effective Poincaré metric on a unit disk. We compare the Bethe lattice model in the high-temperature region with a scalar field in AdS2, and then discuss its more direct connection to the p-adic AdS/CFT. In addition, we find that the phase transition in the interior induces a crossover behavior of boundary-spin correlations, depending on the depth of the corresponding correlation path.