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 (φ- 2)2 theory in d=4-ϵ 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 dftot of all lines, i.e., backbone plus the surrounding loops, identical to dftot=1/ν. The combination φc=df/dftot=νdf 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.