Origin of the Roughness Exponent in Elastic Strings at the Depinning Threshold

Within a recently developed framework of dynamical Monte Carlo algorithms, we compute the roughness exponent $\ensuremath{\zeta}$ of driven elastic strings at the depinning threshold in $1+1$ dimensions for different functional forms of the (short-range) elastic energy. A purely harmonic elastic energy leads to an unphysical value for $\ensuremath{\zeta}$. We include supplementary terms in the elastic energy of at least quartic order in the local extension. We then find a roughness exponent of $\ensuremath{\zeta}\ensuremath{\simeq}0.63$, which coincides with the one obtained for different cellular automaton models of directed percolation depinning. We discuss the implications of our analysis for higher-dimensional elastic manifolds in disordered media.