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 ${\mathrm{\Delta }}^{4\mathrm{/}3}$ 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 the 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.