Fractal dimension of critical curves in the $O(n)$-symmetric $\phi^4$-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 $d_{\rm f}$ of critical curves in the $O(n)$ symmetric $(\vec \phi^2)^2$-theory in $d=4-\varepsilon$ 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 $d_{\rm f}^{\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\rm f}^{\rm tot} = 1/\nu$. The combination $\phi_{\rm c}= d_{\rm f}/d_{\rm f}^{\rm tot} = \nu d_{\rm f}$ is the crossover exponent, describing a system with mass anisotropy. Introducing a novel 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.