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

Abstract

We calculate the fractal dimension d_{f} of critical curves in the O(n)-symmetric (ϕ[over ⃗]^{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 d_{f}^{tot} of all lines, i.e., backbone plus the surrounding loops, identical to d_{f}^{tot}=1/ν. The combination ϕ_{c}=d_{f}/d_{f}^{tot}=νd_{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.