We calculate the fractal dimension $d_{\mathrm{f}}$ of critical curves in the $O\left(n\right)$-symmetric ${\left({\vec{\varphi }}^2\right)}^2$ 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 $d_{\mathrm{f}}^{\mathrm{tot}}$ of all lines, i.e., backbone plus the surrounding loops, identical to $d_{\mathrm{f}}^{\mathrm{tot}}=1/\nu$. The combination ${\varphi }_c=d_{\mathrm{f}}/d_{\mathrm{f}}^{\mathrm{tot}}=\nu d_{\mathrm{f}}$ 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.

• Received 28 September 2019

DOI:https://doi.org/10.1103/PhysRevE.101.012104