Measuring Functional Renormalization Group Fixed-Point Functions for Pinned Manifolds

Exact numerical minimization of interface energies is used to test the functional renormalization group analysis for interfaces pinned by quenched disorder. The fixed-point function $R(u)$ (the correlator of the coarse-grained disorder) is computed. In dimensions $D=d+1$, a linear cusp in $R^{\prime \prime }(u)$ is confirmed for random bond ($d=1$, 2, 3), random field ($d=0$, 2, 3), and periodic ($d=2$, 3) disorders. The functional shocks that lead to this cusp are seen. Small, but significant, deviations from the 1-loop calculation are compared to 2-loop corrections and chaos is measured.

Figure 1

Filled symbols show numerical results for $Y(z)$, a normalized form of the interface displacement correlator $-R^{\prime \prime }(u)$ [Eq. (4)], for $D=2+1$ random field (RF) and $D=3+1$ random bond (RB) disorders. The inset plots the slope $Y^{\prime }(z)$, with $Y^{\prime }(0)\approx -0.807$ from a quadratic fit (dashed line), indicating a linear cusp. Open symbols plot the cross-correlator ratio $Y_s(z)={\Delta }_{12}(z)/{\Delta }_{11}(0)$ between two related RF disorders; it does not exhibit a cusp. The points are for confining wells with width given by $M^2=0.02$. Comparisons to 1-loop FRG predictions (curves) have no adjustable parameters.

Figure 2

The difference between the normalized correlator $Y(z)$ and the 1-loop prediction $Y_1(z)$ for RF disorder in $D=(1,2)+1$ and RB disorder in $D=(1,2,3)+1$. The dashed lines are the linear 2-loop correction $Y_2(z)=\frac{dY(z)}{dϵ}{|}_{ϵ=0}$ of Eq. (5). For each disorder class, the data are close to each other and to the $d=0$ and $ϵ=1$ linear two-loop estimates, but are distinct from the 1-loop result.

Figure 3

(a) Plots of the normalized pinning force correlator $Y(z)$ for RP disorder in $d=3$. For these values of $m$, the period $P=4$ points have mostly converged to the parabolic RP fixed-point function, while the $P=8$ curves are still crossing over from the RB to the RP universality class. (b) Residuals relative to the parabolic shape vanish, within error bars, for larger sizes and $P=4$.

Figure 4

A plot of the normalized cross-correlator $\frac{{\Delta }_{12}(0)}{{\Delta }_{11}(0)}$, computed for RF disorder in $d=2$, 3, showing the sensitivity to disorder of magnitude $\delta$, compared to the 1-loop prediction and to numerical $D=0+1$ computations (error bars not shown; $1\sigma$ errors are about $1/2$ of symbol size). Inset: the renormalized pinning forces $v-u_0(v)$ for a sample (solid line) and a $\delta =0.1$ perturbed sample (dashed line) in a typical sample; cross-correlations of such data give the main plot.