Roughness and critical force for depinning at 3-loop order

TL;DR

This work advances the field-theoretic understanding of depinning by pushing the FRG analysis to 3-loop order in the ε=4−d expansion, yielding a precise ζ(ε) up to ε^3 and detailing the renormalized disorder correlator Δ̃(w). It derives a universal expression for the depinning critical force, fc, in terms of disorder derivatives and a universal amplitude B, and demonstrates a logarithmic driving-scale dependence for CDWs via an O(n) mapping at n→−2, linking fc to log-CFT structures. The authors validate key predictions with 1D numerical simulations, confirming universal fc behavior and providing insights into Δ(w) shape across dimensions, while highlighting limitations of ε-expansion resummations for certain observables. Collectively, the results connect high-order RG theory, universal amplitudes, and logarithmic conformal-field-theory concepts to pinned elastic manifolds and CDWs, with implications for interpreting experiments and simulations of depinning phenomena.

Abstract

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.

Figures (18)

• Figure 1: $\Delta(w)$ for $mL = 6$ (blue) and $L=1024$. The tangent at $w=0$ defines the correlation length $\rho_m$.
• Figure 2: Diagrams at 3-loop order (without insertion of lower order counter-terms)
• Figure 3: All diagrams correcting the disorder up to 3-loop order. All $\Delta$ are bare $\Delta_0$, with the index suppressed for compactness of notation. While we calculated the corrections to $\Delta(w)$, we report its integrated form $\delta R(w):=-\int_0^w {\mathrm{d}} w' \int_0^{w'}{\mathrm{d}} w"\, \delta \Delta(w)$ for compactness. This is the correction to the potential correlator $R(w)$. The non-underlined terms are present in the statics WieseHusemannLeDoussal2018, the underlined ones are additional contributions at depinning. We note that the following expressions are proportional to each other, $(o)\sim (l)$, and $(h) \sim (j)$. The momentum integrals, which correspond to the icons in the same line, are given in appendix \ref{['app:Integrals']}.
• Figure 4: $\zeta(\epsilon)$ in different schemes: 1-loop (black, dashed), direct 2-loop (cyan), direct 3-loop (green), as well as two Padé appoximants, $\hbox{Pad'e}_{1,2}$ (red) and $\hbox{Pad'e}_{2,1}$ (blue). For the latter, we also show the Padé-Borel resummation as explained in the main text. The black dots are the result of numerical simulations for $d=2,3$, and the exact values $\zeta=5/4$ in $d=1$ as well as $\zeta=2$ in $d=0$.
• Figure 5: All spatial diagrams for corrections of $F_{\rm c}$ and $\eta$; the first three diagrams (without label) are the 1- and 2-loop contributions. The remaining nine diagrams $(r)$ to $(z)$ are 3-loop contributions.