Roughness and critical force for depinning at 3-loop order

A d-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the functional renormalization group (FRG). Here, we analyze this theory to 3-loop order, equivalent to third order in ε=4-d, where d is the internal dimension. The critical exponent reads ζ=ε3+0.04777ε²-0.068354ε³+O(ε⁴). Using that ζ(d=0)=2^-, we estimate ζ(d=1)=1.266(20), ζ(d=2)=0.752(1), and ζ(d=3)=0.357(1). For Gaussian disorder, the pinning force per site is estimated as fc=Bm²ρₘ+fc⁰, where m² is the strength of the confining potential, B a universal amplitude, ρₘ the correlation length of the disorder, and fc⁰ a nonuniversal lattice-dependent term. For charge-density waves, we find a mapping to the standard φ⁴ theory with $O(n)$ symmetry in the limit of n→-2. This gives fc= ̃ \~A(d)m²ln(m)+fc⁰, with ̃ \~A(d)=-∂ₙ[ν(d,n)^-1+η(d,n)]_n=-2, reminiscent of logarithmic conformal field theories.