Fractal dimension of critical curves in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math> model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math>, and Heisenberg models
M. V. Kompaniets, Kay Jörg Wiese
Abstract
We calculate the fractal dimension ${d}_{\mathrm{f}}$ of critical curves in the $O(n)$-symmetric ${({\stackrel{P\vec}{\ensuremath{\phi}}}^{2})}^{2}$ theory in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=\ensuremath{-}2$, 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 ${d}_{\mathrm{f}}^{\mathrm{tot}}=1/\ensuremath{\nu}$. The combination ${\ensuremath{\phi}}_{c}={d}_{\mathrm{f}}/{d}_{\mathrm{f}}^{\mathrm{tot}}=\ensuremath{\nu}{d}_{\mathrm{f}}$ 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.