Fractal dimension of critical curves in the O(n)-symmetric φ4 model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY, and Heisenberg models

We calculate the fractal dimension df of critical curves in the O(n)-symmetric ((phi) over bar (2))(2) theory in d = 4 - epsilon dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at n = -2, self-avoiding walks (n = 0), Ising lines (n = 1), and XY lines (n = 2), in agreement with numerical simulations. It can be compared to the fractal dimension d(f)(tot) of all lines, i.e., backbone plus the surrounding loops, identical to d(f)(tot) = 1/nu. The combination phi c = df/d(f)(tot) = nu d(f) is the crossover exponent, describing a system with mass anisotropy. Introducing a self-consistent resummation procedure and combining it with analytic results in d = 2 allows us to give improved estimates in d = 3 for all relevant exponents at 6-loop order.