Finite size scaling study of lattice models in the three-dimensional Ising universality class

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value $D^{\ast }=0.656\left(20\right)$ of the parameter $D$, where leading corrections to scaling vanish. We find $\omega =0.832\left(6\right)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\nu =0.63002\left(10\right)$ and $\eta =0.03627\left(10\right)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

Figure 1

(Color online) Results for the critical exponent $\eta$ obtained by fitting the standard and the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ for the Ising model and the Blume-Capel model at $D=1.15$ using the ansatz (49). $L_{min}$ is the minimal lattice size that is taken into account. In the case of the improved magnetic susceptibility, the results obtained from the two different models fall nicely on top of each other. The dashed lines should only guide the eyes.

Figure 2

(Color online) Results for the critical exponent $\eta$ obtained by fitting the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ and at ${\xi }_{2nd}/L=0.6431$ using the ansatz (57). Data for the Blume-Capel model at $D=0.641$ and $D=0.655$ are taken into account. $L_{min}$ is the minimal lattice size that is taken into account. The dashed lines should only guide the eyes. For a discussion see the text.

Figure 3

(Color online) Results for the critical exponent $\eta$ obtained by fitting the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ and at ${\xi }_{2nd}/L=0.6431$ using the ansatz (58). Data for the Blume-Capel model at $D=0.641$ and $D=0.655$ are taken into account. $L_{min}$ is the minimal lattice size that is taken into account. The dashed lines should only guide the eyes. For a discussion see the text.

Figure 4

(Color online) Results for the critical exponent $\nu$ obtained by fitting improved slopes of various phenomenological couplings at $Z_a/Z_p=0.5425$ as a function of $L_{min}^{-2}$, where $L_{min}$ is the minimal lattice size that is included into the fit. The dashed lines should only guide the eyes. For a discussion see the text.

Figure 5

(Color online) Results for the critical exponent $\nu$ obtained by fitting improved slopes of various phenomenological couplings at $Z_a/Z_p=0.5425$ with the ansatz (62) as a function of $L_{min}$. In the upper part of the figure the correction exponent is fixed to $ϵ=1.6$ and in the lower part it is fixed to $ϵ=2$. The dashed lines should only guide the eyes. For a discussion see the text.