Methods to determine the Hausdorff dimension of vortex loops in the three-dimensional $XY$ model

The geometric properties of critical fluctuations in the three-dimensional (3D) $XY$ model are analyzed. The 3D $XY$ model is a lattice model describing superfluids. We present a direct evaluation of the Hausdorff dimension $D_H$ of the vortex loops which are the critical fluctuations of the 3D $XY$ model. We also present analytical arguments for why $\vartheta$ in the scaling relation ${\eta }_{\varphi }+D_H=2+\vartheta$ between $D_H$ and the anomalous scaling dimension of the corresponding field theory must be zero.

Figure 1

(Color online) Percolation probability of sites as a function of concentration $p$ for a simple 3D cubic system. The concentration at which percolation takes place is given by the common inflection point of the different curves for increasing lattice size.

Figure 2

(Color online) Same as Fig. 1, but for a bond percolation.

Figure 3

(Color online) Percolation probability $O_L$ as function of temperature $T$ for the 3D $XY$ model at different lattice sizes. The critical temperature is given by the common inflection point of the different curves for increasing lattice size. The value obtained is $T_c=2.18727$.

Figure 4

Vortex-loop distribution $D\left(N\right)\sim N^{-\tau }$ as a function of perimeter length $N$ at the temperature $T=2.18727$, as explained in the text.

Figure 5

Loop length $N\left(R_F\right)$ as a function of the characteristic size $R_F$. The best fit has been performed only on the point between $4⩽R_F⩽47$.

Figure 6

Variation of $D_H$ with the lower cutoff. The saturation expected in the continuum limit is present but strongly effected by fluctuations. The finale value of $D_H=2.168\left(6\right)$ is obtained using a weighted fit over the last eight points.

Figure 7

Dependence of $D_{H-}$ and $D_{H+}$ on the lower cutoff $\left(R_F\right)$. The deviation between the two values is still too large even in the long-loop regime. The finale value of $D_H=2.14\left(7\right)$ is obtained averaging over the final values for $D_H^{-}$ and $D_H^{+}$.

Figure 8

Loop length $N\left(R_F\right)$ as a function of the characteristic size $R_F$. The best fit has been performed on the points between $4⩽R_F⩽150$, to be compared with that of Method 1, in Fig. 5.

Figure 9

Dependence of $D_H$ on the lower cutoff $\left(R_{F̱\mathit{min}}\right)$. The saturation region is, as can be seen, much clearer compared with previous methods. The finale value of $D_H=2.1693\left(4\right)$ is obtained using a weighted fit over the last three points.