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 $\epsilon=4-d$, where $d$ is the internal dimension. The critical exponent reads $\zeta = \frac \epsilon3 + 0.04777 \epsilon^2 -0.068354 \epsilon^3 + {\cal O}(\epsilon^4)$. Using that $\zeta(d=0)=2^-$, we estimate $\zeta(d=1)=1.266(20)$, $\zeta(d=2)=0.752(1)$ and $\zeta(d=3)=0.357(1)$. For Gaussian disorder, the pinning force per site is estimated as $f_{\rm c}= {\cal B} m^{2}\rho_m + f_{\rm c}^0$, where $m^2$ is the strength of the confining potential, $\cal B$ a universal amplitude, $\rho_m$ 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 $\phi^4$-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 $\tilde {\cal A}(d) = -\partial_n \big[\nu(d,n)^{-1}+\eta(d,n)\big]_{n=-2}$, reminiscent of log-CFTs.