Renormalization group for the $\varphi^4$-theory with long-range interaction and the critical exponent $\eta$ of the Ising model

Abstract

We calculate the critical exponent $\eta$ of the $D$-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long-range to short-range; the numerical value of $\eta$ can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result $\eta = 0.03651$ is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions $D$ and obtain the resulting critical exponent $\eta(D)$ between two and four dimensions. For dimensions $2\leq D \leq 3$ our result for $\eta$ is consistent with previous calculations.