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 $\zeta =\frac{ϵ}{3}+0.04777{\epsilon }^2-0.068354{\epsilon }^3+\mathcal{O}({\epsilon }^4)$. 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 $f_{\mathrm{c}}=\mathcal{B}{m}^2{\rho }_m+f_{\mathrm{c}}^0$, where $m^2$ is the strength of the confining potential, $\mathcal{B}$ a universal amplitude, ${\rho }_m$ the correlation length of the disorder, and $f_{\mathrm{c}}^0$ 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 $f_{\mathrm{c}}=\tilde{\mathcal{A}}(d)m^2\ln (m)+f_{\mathrm{c}}^0$, with $\tilde{\mathcal{A}}(d)=-{\mathrm{\partial }}_n[\nu (d,n{)}^{-1}+\eta (d,n){]}_{n=-2}$, reminiscent of log-CFTs.