Ising exponents from the functional renormalization group

We study the 3d Ising universality class using the functional renormalization group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and antisymmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross correlations of scaling exponents, their dependence on dimensionality, and the numerical convergence of the derivative expansion. Collecting all available data from functional renormalization group studies to date, we estimate that systematic errors are in good agreement with findings from Monte Carlo simulations, $ϵ$-expansion techniques, and resummed perturbation theory.

Figure 1

Wilson-Fisher fixed point in $D=3$ for (a) the first derivative of the potential $v_{*}^{\prime }(x)$, (b) the deviation of the wave function renormalization from its classical value $10(z_{*}(x)-1)$, and (c) the four-derivative operator ${10}^3{w}_{*}(x)$. Coding: local potential approximation (black, dotted curve), 2nd order derivative expansion (red, dashed curves), 4th order derivative expansion (blue, continuous curves).

Figure 2

Eigenperturbations $\Phi (x)$ with eigenvalue $\nu$ at the Wilson-Fisher fixed point for (a) the potential $v_{*}(x)$, (b) $z_{*}(x)$ (rescaled by a factor 10), and for (c) $w_{*}(x)$ (rescaled by a factor ${10}^3$). Coding as in Fig. 1.

Figure 3

Eigenperturbations $\Phi (x)$ with eigenvalue $\omega$ at the Wilson-Fisher fixed point for (a) the potential $v_{*}(x)$, (b) $z_{*}(x)$ (rescaled by a factor 10), and for (c) $w_{*}(x)$ (rescaled by a factor ${10}^3$). Coding as in Fig. 1.