Roughness and critical force for depinning at 3-loop order
A $d$-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the Functional Renormalization Group (FRG). Here we analyze this theory to 3-loop order, equivalent to third order in $ε=4-d$, where $d$ is the internal dimension. The critical exponent reads $ζ= \frac \epsilon3 + 0.04777 ε^2 -0.068354 ε^3 + {\cal O}(ε^4)$. Using that $ζ(d=0)=2^-$, we estimate $ζ(d=1)=1.266(20)$, $ζ(d=2)=0.752(1)$ and $ζ(d=3)=0.357(1)$. For Gaussian disorder, the pinning force per site is estimated as $f_{\rm c}= {\cal B} m^{2}ρ_m + f_{\rm c}^0$, where $m^2$ is the strength of the confining potential, $\cal B$ a universal amplitude, $ρ_m$ the correlation length of the disorder, and $f_{\rm c}^0$ a non-universal lattice dependent term. For charge-density waves, we find a mapping to the standard $ϕ^4$-theory with $O(n)$ symmetry in the limit of $n\to -2$. This gives $f_{\rm c} = \tilde {\cal A}(d) m^2 \ln (m) + f_{\rm c}^0 $, with $\tilde {\cal A}(d) = -\partial_n \big[ν(d,n)^{-1}+η(d,n)\big]_{n=-2}$, reminiscent of log-CFTs.