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, the critical field scaling with disorder is in agreement with the predictions for the
quenched Edwards-Wilkinson elastic model. However, critical configurations display overhangs beyond a
characteristic length l0 that diverges only when the disorder vanishes, 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 l0 and invasion percolation depinning
exponents above l0. A full picture of domain-wall isotropic depinning in two dimensions is hence proposed.