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

17 pages, 18 figures, 7 tables. v2 with additional explanations

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