Depinning free of the elastic approximation

We model the isotropic depinning transition of a domain wall using a two-dimensional Ginzburg-Landau scalar field instead of a directed elastic string in a random media. An exact algorithm accurately targets both the critical depinning field and the critical configuration for each sample. For random bond disorder of weak strength $\mathrm{\Delta }$, the critical field scales as ${\mathrm{\Delta }}^{4/3}$ in agreement with the predictions for the quenched Edwards-Wilkinson elastic model. However, critical configurations display overhangs beyond a characteristic length $l_{\mathtt{0}}\sim {\mathrm{\Delta }}^{-\alpha }$, with $\alpha \approx 2.2$, indicating a finite-size crossover. At large scales, overhangs recover the orientational symmetry which is broken by directed elastic interfaces. We obtain quenched Edwards-Wilkinson exponents below $l_{\mathtt{0}}$ and invasion percolation depinning exponents above $l_{\mathtt{0}}$. A full picture of domain-wall isotropic depinning in two dimensions is hence proposed.

Figure 1

Typical critical configurations of the domain wall for different values of the disorder strength $\mathrm{\Delta }$. System size is $L=4096$ (the aspect ratio is one). Red shaded regions highlight overhangs and/or pinch-off loops.

Figure 2

Magnetization jump per site $\langle \varphi \rangle$ as a function of $h$, for different values of the disorder strength $\mathrm{\Delta }$. Data correspond to averages over 50 samples, $L=256$. Dashed-lines are guides to the eye.

Figure 3

Phase diagram of the domain-wall dynamics. A critical field $h_{\mathtt{d}}$ as a function of disorder strength $\mathrm{\Delta }$ separates the static and moving phases. It displays a $h_{\mathtt{d}}\sim {\mathrm{\Delta }}^{1.33}$ in agreement with the predicted behavior $h_{\mathtt{d}}\sim {\mathrm{\Delta }}^{4/(4-d)}{\left(\sigma \delta \right)}^{-d/(4-d)}$ of weak collective pinning theory (dotted line). We display it for different system sizes $L$, finite-size effects are only observed for very small $\mathrm{\Delta }$.

Figure 4

Size dependence of the sample-to-sample fluctuations of the critical field for different disorder strengths $\mathrm{\Delta }$. Data is fairly described $\overline{{[h_{\mathtt{d}}-\overline{h_{\mathtt{d}}}]}^2}\sim {\mathrm{\Delta }}^2{L}^{1/1.27}$ (solid line), as can be appreciated by fitting the rescaled data. For comparison, we plot the scaling expected for the qEW model (dashed line) and for the qKPZ model (dotted line).

Figure 5

Fraction $\rho$ of multivalued domain walls at the depinning transition. (a) Raw data as a function of disorder strength $\mathrm{\Delta }$ and transverse size $L$. (b) Scaled data, showing that $\rho (\mathrm{\Delta },L)\approx \tilde{\rho }(L/l_{\mathtt{o}})$ with $l_{\mathtt{o}}\sim {\mathrm{\Delta }}^{-1/0.45}$. The inset shows typical domain walls for $L<l_{\mathtt{o}}$ (left) and $L>l_{\mathtt{o}}$ (right).

Figure 6

Disorder-averaged structure factor $S\left(q\right)$ of domain-wall critical configurations. Data obtained for a system size $L=8192$ (corresponding to a grid of ${8192}^2\simeq 6.7\times {10}^7$ sites). (a) $S\left(q\right)$ averaged over configurations with overhangs, without overhangs, and averaged over all configurations (180 samples) for a fixed disorder strength $\mathrm{\Delta }=0.2$. Lines are guides to the eye and display pure power laws for $\zeta =1.2$ (solid line) $\zeta =1$ (dotted line), and $\zeta =0.9$ (dashed line). (b) $S\left(q\right)$ averaged over all sampled critical configurations for increasing values of disorder strength $\mathrm{\Delta }$ (180 samples for $\mathrm{\Delta }=0.2$, 50 samples for $\mathrm{\Delta }>0.2$). Guides-to-the-eye lines display pure power laws for $\zeta =1.2$ (solid line) and $\zeta =0.5$ (dashed line).

Figure 7

Typical linear size of overhangs $u_{\mathtt{o}}$ [Eq. (11)] in the displacement direction as a function of the system size $L$ for $\mathrm{\Delta }=1$. The dashed line is a guide to the eye with slope $\propto L$.

Figure 8

Blowing an initially circular domain with a constant field $h=0.1$ for ${\epsilon }_0=1$ and different values of $c$ as indicated, corresponding to different values of the DW width $\delta \approx \sqrt{c/{\epsilon }_0}$. Lines of different colors corresponds to DW configurations at regularly distributed times.

Figure 9

Maximum depinning field $h_{\mathtt{d}}^{\text{max}}$ in the absence of disorder ($\mathrm{\Delta }=0$) due to the effect of the square numerical mesh versus the order parameter elastic constant $c$. The dashed-line indicates $h_{\mathtt{d}}^{\text{max}}$ for the parameters used to obtain our results in the main text.