Marginal pinning of vortices at high temperature

We analyze the competition between thermal fluctuations and pinning of vortices in bulk type II superconductors subject to point-like disorder and derive an expression for the temperature dependence of the pinning length L_c(T) which separates different types of single vortex wandering. Given a disorder potential with a basic scale \xi and a correlator K_0(u) \sim K_0 (u/xi)^{-\beta} ln^alpha (u/xi) we determine the dependence of L_c(T) on the correlator range: correlators with \beta > 2 (short-range) and \beta <2 (long-range) lead to the known results L_c(T) \sim L_c(0) exp[C T^3] and L_c(T) \sim L_c(0) (C T)^{(4+beta)/(2-beta)}, respectively. Using functional renormalization group we show that for \beta =2 the result takes the interpolating form L_c(T) \sim L_c(0) exp[C T^{3/(2+alpha)}]. Pinning of vortices in bulk type II superconductors involves a long-range correlator with \beta=2, \alpha=1 on intermediate scales \xi<u<\lambda, with \xi and \lambda the coherence length and London penetration depth, hence L_c(T) \sim L_c(0) exp[C T]; at large distances L_c(T) crosses over to the usual short-range behavior.