Renormalization of pinned elastic systems: how does it work beyond one loop ?

Abstract

We study the field theories for pinned elastic systems at equilibrium and at depinning. Their $β$-functions differ to two loops by novel ``anomalous'' terms. At equilibrium we find a roughness $ζ=0.20829804 ε+ 0.006858 ε^2$ (random bond), $ζ=ε/3$ (random field). At depinning we prove two-loop renormalizability and that random field attracts shorter range disorder. We find $ζ=\fracε{3}(1 + 0.14331 ε)$, $ε=4-d$, in violation of the conjecture $ζ=ε/3$, solving the discrepancy with simulations. For long range elasticity $ζ=\fracε{3}(1 + 0.39735 ε)$, $ε=2-d$, much closer to the experimental value ($\approx 0.5$ both for liquid helium contact line depinning and slow crack fronts) than the standard prediction 1/3.