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

The motion of driven interfaces in random media at finite temperature $T$ and small 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 the creep formula as $V(F,T)\sim e^{-K(T)\mathrm{/}{F}^{\mu }}$ for $F\to 0$. In this paper, we question this picture on the specific example of the directed polymer in a two 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 strong disorder renormalization procedure, where the distribution of renormalized barriers flows towards some “infinite disorder fixed point”. In the present paper, we obtain that for small $F$, this “infinite disorder fixed point” becomes a “strong disorder fixed point” with an exponential distribution of renormalized barriers. The corresponding distribution of trapping times then only decays as a power-law $P(\tau )\sim 1\mathrm{/}{\tau }^{1+\alpha }$, where the exponent $\alpha (F,T)$ vanishes as $\alpha (F,T)\propto F^{\mu }$ as $F\to 0$. Our conclusion is that in the small force region $\alpha (F,T)<1$, the divergence of the averaged trapping time $\bar{\tau }=+\mathrm{\infty }$ induces strong non-self-averaging effects that invalidate the usual creep formula obtained by replacing all trapping times by the typical value. We find instead that the motion is only sub-linearly in time $h_G(t)\sim t^{\alpha (F,T)}$, i.e. the asymptotic velocity vanishes V=0. This analysis is confirmed by numerical simulations of a directed polymer with a metric constraint driven in a traps landscape. We moreover obtain that the roughness exponent, which is governed by the equilibrium value ${\zeta }_{eq}=2\mathrm{/}3$ up to some large scale, becomes equal to $\zeta =1$ at the largest scales.