Restoring isotropy in a three-dimensional lattice model: The Ising universality class

We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest-neighbor coupling there is a next-to-next-to-nearest-neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high-temperature phase of the model for three different spatial directions. It turns out that the spatial anisotropy depends very little on the dilution or crystal-field parameter $D$ of the model and is essentially determined by the ratio of the nearest-neighbor and the next-to-next-to-nearest-neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite-size scaling (FSS) study to tune $D$ such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\nu =0.62998\left(5\right)$ and $\eta =0.036284\left(40\right)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore, we provide accurate results for fixed-point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class.

Figure 1

We plot $(r_2-1){\xi }^{{\omega }_{NR}}$ (upper part) and $(r_3-1){\xi }^{{\omega }_{NR}}$ (lower part) for the Ising model and the Blume-Capel model at $D=0.655$, both at $q_3=0$, versus the correlation length $\xi$. Note that the values on the $x$ axis are slightly shifted to reduce the overlap of the Ising and the Blume-Capel data points. The two parts of the figure share the legend and the labeling of the $x$ axis. Note the different scales on the $y$ axis.

Figure 2

We plot $(r_2-1){\xi }^{2.022665}$ (upper part) and $(r_3-1){\xi }^{2.022665}$ (lower part) versus the correlation length $\xi$ for the Ising model and the Blume-Capel model at $D=-0.43$ at $q_3=\frac{2}{15}$ and $\frac{1}{8}$. Note that the values on the $x$ axis are slightly shifted to reduce the overlap of the Ising and the Blume-Capel model data points. The two parts of the figure share the legend and the labeling of the $x$ axis. Note the different scales on the $y$ axis.

Figure 3

Numerical estimates of the amplitude $d_{Z_a/Z_p}$ of corrections $\propto L^{-2.022665}$ in $Z_a/Z_p$ as a function of the minimal lattice size $L_{\text{min}}$. These estimates are obtained from the fits 1, 2, and 3, which are discussed in the text. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 4

Estimates of $-D^{*}$ plotted versus the minimal lattice size $L_{\text{min}}$ taken into account in the fit. The numerical estimates of $-D^{*}$ are obtained from the fits 1, 2 and 3, which are discussed in the text. The solid black line gives our final estimate of $-D^{*}$, while the dashed lines indicate the error bar. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 5

Numerical estimates of ${(Z_a/Z_p)}^{*}$ obtained from the fits 1, 2 and 3, which are discussed in the text. These estimates are plotted versus the minimal lattice size $L_{\text{min}}$ taken into account in the fit. The solid black line gives our final estimate of ${(Z_a/Z_p)}^{*}$, while the dashed lines indicate the error bar. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 6

We plot estimates of $-D^{*}$ obtained by fitting $U_4$ at $Z_a/Z_p=0.54253$ and ${\xi }_{2nd}/L=0.64312$ by using the Ansatz (28) versus the minimal lattice size $L_{\min }$ taken into account in the fit. In the legend the Ansatz (28) is indicated by F1. In the case of fixing ${\xi }_{2nd}/L=0.64312$ we fitted in addition by using the Ansatz (29), which is indicated by F2. The solid and the dashed lines give the final result of the previous section and the corresponding error bar. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 7

We plot estimates of $\omega$ obtained by fitting $U_4$ at $Z_a/Z_p=0.54253$ and ${\xi }_{2nd}/L=0.64312$ by using the Ansatz (28) versus the minimal lattice size $L_{\min }$ taken into account in the fit. In the legend the Ansatz (28) is indicated by F1. In the case of fixing ${\xi }_{2nd}/L=0.64312$ we fitted in addition by using the Ansatz (29), which is indicated by F2. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols. The dashed-dotted line gives the result of Ref. [17].

Figure 8

We plot estimates of amplitudes of subleading corrections obtained by fitting $U_4$ at ${\xi }_{2nd}/L=0.64312$ by using the Ansatz (31) versus the minimal lattice size $L_{\text{min}}$ taken into account in the fit. We give the amplitude $\overline{c}$ of $L^{-{\epsilon }_1}$ and $\overline{d}$ of the difference $L^{-{\epsilon }_1}-L^{-{\epsilon }_2}$. In the legend we give either the value of the correction exponent ${\epsilon }_1$ or “difference.” Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 9

Numerical estimates of $\eta$ for $D=-0.38$ plotted versus the minimal lattice size $L_{\text{min}}$ taken into account in the fit. We have fitted the nonimproved $\chi$ by using the Ansatz (33) for both fixing $Z_a/Z_p$ and ${\xi }_{2nd}/L$. These fits are denoted by F1 in the legend. In the case of fixing ${\xi }_{2nd}/L$, we give results obtained by using the Ansatz (34) in addition. These fits are denoted by F2 in the legend. The values on the $x$ axis are slightly shifted to reduce overlap of the symbols. The solid black line gives our final estimate of $\eta$, while the dashed lines indicate the error bar. The dashed-dotted line gives the estimate of Refs. [17, 47]. Note that the error bar of the CB result is by a factor of 20 smaller than the one obtained here.

Figure 10

Numerical estimates of $\eta$ obtained by fitting $\chi$ at $Z_a/Z_p=0.54253$ by using the Ansatz (33) versus $D$. We compare results obtained for the standard and the improved, Eq. (32), version of the magnetic susceptibility. In all cases we give the estimate obtained with $L_{\text{min}}=12$. The values on the $x$ axis are slightly shifted to reduce overlap of the symbols. The dashed-dotted line gives the estimate of Refs. [17, 47].

Figure 11

Numerical estimates of $y_t$ for $D=-0.38$ obtained by fitting the slopes of dimensionless quantities at $Z_a/Z_p=0.54253$ by using the Ansatz (36) versus the minimal lattice size $L_{\text{min}}$ that is taken into account. The dashed-dotted line gives the estimate of Refs. [17, 47]. The legend refers to the quantity that is analyzed.

Figure 12

Numerical estimates of $y_t$ for $D=-0.38$ obtained by fitting the slopes of dimensionless quantities with the Ansatz (37) versus the minimal lattice size $L_{\text{min}}$ that is taken into account. The dashed-dotted line gives the estimate of Refs. [17, 47]. The legend refers to the quantity that is analyzed in the fit.

Figure 13

Numerical estimates of $y_t$ for $D=-0.35$, $-0.38$, $-0.4$, and $-0.42$ obtained by fitting the slope of $Z_a/Z_p$ by using the Ansatz (37) with $\epsilon =y_t+\omega$ as correction exponent. These estimates are plotted versus the minimal lattice size $L_{\text{min}}$ taken into account in the fit. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols. The dashed-dotted line gives the estimate of Refs. [17, 47].

Figure 14

Numerical estimates of $y_t$ for $D=-0.38$ obtained by fitting the slopes of all four dimensionless quantities jointly with the Ansatz (37) or Ansatz (38). These estimates are plotted versus the minimal lattice size $L_{\text{min}}$ that is taken into account in the fit. In the legend, these Ansätze are denoted by fit 1 and fit 2, respectively. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols.

Figure 15

Numerical estimates of $y_t$ for $D=-0.35$, $-0.38$, $-0.4$, and $-0.42$ obtained by fitting the slopes of all four dimensionless quantities jointly with the Ansatz (38) versus the minimal lattice size $L_{\text{min}}$ taken into account. Note that the values on the $x$ axis are slightly shifted to reduce overlap of the symbols for different values of $D$.