Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap

TL;DR

This paper uses conformal bootstrap techniques to determine the fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D, revealing features like kinks and minima in unitarity bounds that help locate critical points.

Contribution

It introduces a novel application of conformal bootstrap to compute fractal dimensions and critical exponents for 3D models, including predictions for supersymmetric critical points.

Findings

Identification of kink structures in unitarity bounds for N<1

Detection of asymmetric minima in current central charge C_J

Predictions of critical exponents related to supersymmetry

Abstract

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the $O(N)$ models from $N=1$ (Ising model) to $N=0$ (polymer). The unitarity bound below $N=1$ of the scaling dimension for the the $O(N)$-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as $N$ tends to zero, an emerging asymmetric minimum in the current central charge $C_J$ can be used to locate the CFT. It is pointed out that certain level degeneracies at the $O(N)$ CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the $\mathcal{N}=1$ supersymmetry, which could be relevant for locating the correspoinding fixed point in the phase diagram.