High-resolution Monte Carlo study of the order-parameter distribution of the three-dimensional Ising model

We apply extensive Monte Carlo simulations to study the probability distribution $P\left(m\right)$ of the order parameter $m$ for the simple cubic Ising model with periodic boundary condition at the transition point. Sampling is performed with the Wolff cluster flipping algorithm, and histogram reweighting together with finite-size scaling analyses are then used to extract a precise functional form for the probability distribution of the magnetization, $P\left(m\right)$, in the thermodynamic limit. This form should serve as a benchmark for other models in the three-dimensional Ising universality class.

Figure 1

Scaled probability distribution $P\left(m\right)L^{-\beta /\nu }$ as a function of $m{L}^{\beta /\nu }$ at the critical point $K_c=0.221654626$ [7]. Curves from the top to the bottom in the inset correspond to lattice sizes $L=16$, 32, 96, and 256, respectively.

Figure 2

The red (dark gray) line is the difference between the Monte Carlo data and the fit corresponding to Eq. (4), while the blue (light gray) line is the error bar for the Monte Carlo data (top: $L=16$; middle: $L=96$; bottom: $L=1024$).

Figure 3

Analogous to Fig. 2, but the fit is corresponding to Eq. (6) (top: $L=16$; middle: $L=96$; bottom: $L=1024$).

Figure 4

Variation of the fit parameters $a, b$, and $c$ for Ansatz Eq. (6) as functions of $L^{-\omega }$. The abscissa is chosen so that the leading corrections to scaling are linearized [1], where $\omega =0.82968\left(23\right)$ [17]. The solid lines show extrapolations to $L=\infty$ for $L\ge 32$.

Figure 5

Logarithm of the tail of the probability distribution of the order parameter (average of the left and right tails), where $m{L}^{\beta /\nu }\ge 2.25$, for different lattice sizes $L$. The lines are the best fits to Eq. (6). Curves from the top to the bottom correspond to lattice sizes $L=256$, 96, 32, and 16 respectively.