The Random-Bond Ising Model in 2.01 and 3 Dimensions

TL;DR

This paper investigates the effects of quenched disorder on the Ising model in dimensions between 2 and 4, combining conformal perturbation theory, bootstrap results, and Monte Carlo simulations to analyze critical behavior and scaling exponents.

Contribution

It introduces a combined approach using conformal bootstrap, perturbation theory, and numerical methods to study disordered Ising models across dimensions 2 to 3.

Findings

Disorder is relevant for 2<d<4, leading to new fixed points.

Computed scaling exponents near d=2 using conformal methods.

Results in d=3 align with experimental and simulation data.

Abstract

We consider the Ising model between 2 and 4 dimensions perturbed by quenched disorder in the strength of the interaction between nearby spins. In the interval 2<d<4 this disorder is a relevant perturbation that drives the system to a new fixed point of the renormalization group. At d=2 such disorder is marginally irrelevant and can be studied using conformal perturbation theory. Combining conformal perturbation theory with recent results from the conformal bootstrap we compute some scaling exponents in an expansion around d=2. If one trusts these computations also in d=3, one finds results consistent with experimental data and Monte Carlo simulations. In addition, we perform a direct uncontrolled computation in d=3 using new results for low-lying operator dimensions and OPE coefficients in the 3d Ising model. We compare these new methods with previous studies. Finally, we comment about the $O(2)$ model in d=3, where we predict a large logarithmic correction to the infrared scaling of disorder.