Roughness at the depinning threshold for a long-range elastic string

In this paper, we compute to high precision the roughness exponent zeta of a long-range elastic string, at the depinning threshold, in a random medium. Our numerical method exploits the analytic structure of the problem ("no-passing" theorem), but avoids direct simulation of the evolution equations. The roughness exponent has recently been studied by simulations, functional renormalization-group calculations, and by experiments (fracture of solids, liquid meniscus in 4He). Our result zeta=0.388 +/- 0.002 is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization-group calculation. Furthermore, the data are incompatible with the experimental results for crack propagation in solids and for a 4He contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasistatic limit.