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 ζ 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 ζ. We include supplementary terms in the elastic energy of at least quartic order in the local extension. We then find a roughness exponent of ζ0.63, 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.