Continuous depinning transition with an unusual hysteresis effect

We identify a strange hysteresis which occurs in models of the depinning transition with both elastic and transient overshoot stresses. This hysteresis occurs generically despite the fact that the phase transition is still second order. We calculate the size of the hysteresis gap exactly in a large class of models. Because it is caused by irrelevant perturbations, the hysteresis does not show up in the field theory of the continuum limit. But since it is a real phenomenon it may be a cause for the hysteresis observed in natural depinning. We therefore discuss its experimental and numerical signatures which include microscopic nucleation.