Perturbation theory for ac-driven interfaces in random media

We study $D$-dimensional elastic manifolds driven by ac forces in a disordered environment using a perturbation expansion in the disorder strength and the mean-field approximation. We find that for $D\le 4$, perturbation theory produces nonregular terms that grow unboundedly in time. The origin of these nonregular terms is explained. By using a graphical representation we argue that the perturbation expansion is regular to all orders for $D>4$. Moreover, for the corresponding mean-field problem we prove that ill-behaved diagrams can be resummed in a way that their unbounded parts mutually cancel. Our analytical results are supported by numerical investigations. Furthermore, we conjecture the scaling of the Fourier coefficients of the mean velocity with the amplitude of the driving force $h$.

Figure 1

(Color online) Plot of the first nonvanishing perturbative correction $⟨v_2⟩\left(t\right)$ to the disorder average $⟨v\left(x,t\right)⟩$ for different interface dimensions. For the plot we used $h=1$ and the units are chosen such that $\omega =\ell =1$.

Figure 2

(Color online) Numerical solution of Eq. (47) for different driving field strengths and $c=1.0$, $\eta =2.5$. For the simulation, $t$ and $z$ are measured in units such that $\omega =\ell =1$.

Figure 3

(Color online) Numerical solution of Eq. (47) for $h=6.0$, $c=1.0$, and $\eta =2.5$ for different frequencies, $t$ and $z$ being measured in units such that ${\omega }_0=\ell =1$.

Figure 4

(Color online) Numerical solution of Eq. (47) for different elastic constants and $h=40.0$, $\eta =10.0$. The units of $t$ and $z$ are chosen such that $\omega =\ell =1$.

Figure 5

(Color online) Comparison of the full numerical solution of Eq. (47) with the result obtained from the first nonvanishing perturbative order Eq. (67) for (a) $h=3.0$, $c=3.0$, $\eta =1.5$ and (b) $h=1.0$, $c=1.0$, $\eta =0.6$. The units of $t$ and $z$ are chosen such that $\omega =\ell =1$.

Figure 6

(Color online) Plotting the logarithms of $\left|A_N\right|$ and $\left|B_N\right|$ reveals the exponential decay with $N$. In the regime where numerical errors do not dominate the result, a linear regression seems appropriate.

Figure 7

(Color online) Performing the linear regression for many $h$ yields slopes $\alpha$ and $\beta$ appearing to depend on $h$ in a power-law fashion.

Figure 8

(Color online) Plot of the change of the prefactor $C\left(c\right)=\left(C_{\alpha }+C_{\beta }\right)/2$ in Eq. (73) on $c$. The linear fit yields a fairly tiny slope.