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 $\zeta =\frac{\epsilon }{3}+0.04777{\epsilon }^2-0.068354{\epsilon }^3+O\left({\epsilon }^4\right)$. Using that $\zeta (d=0)=2^{-}$, we estimate $\zeta (d=1)=1.266\left(20\right)$, $\zeta (d=2)=0.752\left(1\right)$, and $\zeta (d=3)=0.357\left(1\right)$. 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 nonuniversal lattice-dependent term. For charge-density waves, we find a mapping to the standard ${\varphi }^4$ theory with $O\left(n\right)$ symmetry in the limit of $n\to -2$. This gives $f_{\mathrm{c}}=\tilde{\mathcal{A}}\left(d\right)m^{2}ln\left(m\right)+f_{\mathrm{c}}^0$, with $\tilde{\mathcal{A}}\left(d\right)=-{\partial }_n{[\nu {(d,n)}^{-1}+\eta (d,n)]}_{n=-2}$, reminiscent of logarithmic conformal field theories.

Figure 1

$\tilde{\mathrm{\Delta }}\left(w\right)$ for $mL=4$ (blue, top curve) and $mL=6$ (red).

Figure 2

$\mathrm{\Delta }\left(w\right)$ for $mL=6$ (blue) and $L=1024$. The tangent at $w=0$ defines the correlation length ${\rho }_m$.

Figure 3

Diagrams at 3-loop order (without insertion of lower-order counterterms).

Figure 4

All diagrams correcting the disorder up to 3-loop order. All $\mathrm{\Delta }$ are bare ${\mathrm{\Delta }}_0$, with the index suppressed for compactness of notation. While we calculated the corrections to $\mathrm{\Delta }\left(w\right)$, we report its integrated form $\delta R\left(w\right):=-{\int }_0^w\mathrm{d}{w}^{\prime }{\int }_0^{w^{\prime }}\mathrm{d}{w}^{″}\delta \mathrm{\Delta }\left(w\right)$ for compactness. This is the correction to the potential correlator $R\left(w\right)$. The nonunderlined terms are present in the statics [66]; the underlined ones are additional contributions at depinning. We note that the following expressions are proportional to each other: $\left(o\right)\sim \left(l\right)$ and $\left(h\right)\sim \left(j\right)$. The momentum integrals, which correspond to the icons in the same line, are given in Appendix pp1.

Figure 5

$\zeta \left(\epsilon \right)$ in different schemes: 1-loop (black, dashed), direct 2-loop (cyan), direct 3-loop (green), as well as two Padé appoximants: ${\text{Padé}}_{1,2}$ (red) and ${\text{Padé}}_{2,1}$ (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 $\zeta =\frac{5}{4}$ in $d=1$ as well as $\zeta =2$ in $d=0$.

Figure 6

All spatial diagrams for corrections of $F_{\mathrm{c}}$ and $\eta$; the first three diagrams (without label) are the 1- and 2-loop contributions. The remaining nine diagrams $\left(r\right)$ to $\left(z\right)$ are 3-loop contributions.

Figure 7

The amplitude $\mathcal{B}$ in Eq. (108). 1-loop (blue solid), 2-loop (red, dashed), and 3-loop (green, dot-dashed).

Figure 8

The combination $\mathcal{B}\zeta \equiv \tilde{\mathcal{A}}/\tilde{\rho }$. 1-loop (blue solid), 2-loop (red, dashed), and 3-loop (green, dot-dashed).

Figure 9

${\tilde{\mathcal{A}}}_{\mathrm{c}}/\epsilon$ for charge density wave (CDW), given by Eq. (125). The error bars are probably an underestimation, as they do not catch the singularity at $d=2$.

Figure 10

$u\left(x\right)-u_0$ for $L=64$, and $mL=8$. Between successive samples, the control parameter $w$ is increased by ${\rho }_m$, starting at $w=w_0$. We see that augmenting $w$ by the correlation length ${\rho }_m$ (in the $u$ direction), $u\left(x\right)$ takes a different configuration at a substantial fraction of sites.

Figure 11

$\tilde{\mathrm{\Delta }}\left(0\right)-\tilde{\mathrm{\Delta }}\left(w\right)$ for $mL=4$.

Figure 12

$\tilde{\mathrm{\Delta }}\left(0\right)-\tilde{\mathrm{\Delta }}\left(w\right)$ for $mL=2$, 4, and 6, in a system of size $L=1024$.

Figure 13

Shape comparison: 1-loop functional renormalization group (FRG; black, bottom curve), exact solution in dimension $d=0$ (blue solid, top curve), Padé resummed 2-loop result from Eq. (145) in $d=0$ for $\alpha =0.35$ (cyan, dotted), the same Padé in $d=1$ (magenta, dashed), the same Padé in $d=2$ (green, dashed), our simulations in dimension $d=1$ (magenta, solid), and the same Padé in $d=3$ (gray, dot-dashed).

Figure 14

Shape comparison of $\mathrm{\Delta }\left(w\right)-{\mathrm{\Delta }}_{\text{1-loop}}\left(w\right)$, with colors as in Fig. 13.

Figure 15

$\widehat{\rho }$ as a function of $m^{-\zeta }$, for different $mL$, and different system sizes. The larger systems are to the right.

Figure 16

$f_{\mathrm{c}}$ as a function of $m^{2-\zeta }$ for $mL=4$. The fit used to extrapolate to $m=0$ is via an exponential function $f_{\mathrm{c}}=f_{\mathrm{c}}^{0}exp(-b{m}^{2-\zeta })$, with the two fit parameters $f_{\mathrm{c}}^0$ and $b$. The slope indicated with a dashed line is $-b\equiv \mathcal{B}\widehat{\rho }$, as given in Eq. (150).

Figure 17

Shape comparison of 1-loop functional renormalization group (FRG) result (black, thick, solid), improved 2-loop FRG result as given by Eq. (145) (magenta dashed), simulation in dimension $d=1$ (magenta), exact solution in dimension $d=0$ (blue), and 3-loop improved FRG results as given by Eq. (F1) for values of dimension $d=0,1,\cdots ,3$. The last four curves (orange, dot-dashed) are indistinguishable.

Figure 18

Dependence of $\widehat{\rho }$ on $c$.