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The motion of driven interfaces in random media at finite temperature $T$ andsmall external force $F$ is usually described by a linear displacement $h_G(t)\sim V(F,T) t$ at large times, where the velocity vanishes according to thecreep formula as $V(F,T) \sim e^{-K(T)/F^{\mu}}$ for $F \to 0$. In this paper,we question this picture on the specific example of the directed polymer in atwo dimensional random medium. We have recently shown (C. Monthus and T. Garel,arxiv:0802.2502) that its dynamics for F=0 can be analyzed in terms of a strongdisorder renormalization procedure, where the distribution of renormalizedbarriers flows towards some "infinite disorder fixed point". In the presentpaper, we obtain that for small $F$, this "infinite disorder fixed point"becomes a "strong disorder fixed point" with an exponential distribution ofrenormalized barriers. The corresponding distribution of trapping times thenonly decays as a power-law $P(\tau) \sim 1/\tau^{1+\alpha}$, where the exponent$\alpha(F,T)$ vanishes as $\alpha(F,T) \propto F^{\mu}$ as $F \to 0$. Ourconclusion is that in the small force region $\alpha(F,T)<1$, the divergence ofthe averaged trapping time $\bar{\tau}=+\infty$ induces strongnon-self-averaging effects that invalidate the usual creep formula obtained byreplacing all trapping times by the typical value. We find instead that themotion is only sub-linearly in time $h_G(t) \sim t^{\alpha(F,T)}$, i.e. theasymptotic velocity vanishes V=0. This analysis is confirmed by numericalsimulations of a directed polymer with a metric constraint driven in a trapslandscape. We moreover obtain that the roughness exponent, which is governed bythe equilibrium value $\zeta_{eq}=2/3$ up to some large scale, becomes equal to$\zeta=1$ at the largest scales.

Driven interfaces in random media at finite temperature : is there an anomalous zero-velocity phase at small external force ?