The Random-Bond Ising Model in 2.01 and 3 Dimensions

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.

43 pages, 2 figures; v2 fixed typos

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