Finite-size scaling of the correlation length above the upper critical dimension in the five-dimensional Ising model

We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size $L$ as $L^{5∕4}$, rather than proportional to $L$, as in standard finite-size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension $d$ greater than the upper critical dimension of 4 should have $L$ replaced by $L^{d∕4}$ for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in $d=4$.

Figure 1

Data for ${\xi }_L∕L$ in $d=5$. Clearly the data do not intersect at a common point, as would be expected if the conventional FSS expression [Eq. (5)] applied.

Figure 2

Data for ${\xi }_L∕L^{5∕4}$ in $d=5$. Clearly the data intersect close to a common point, as expected for the modified FSS expression in Eq. (9). The vertical line is at $T=8.7785$, which is our best estimate for $T_c$.

Figure 3

A scaling plot for the data in Fig. 2 according to the second expression in Eq. (9) with $T_c=8.7785$.

Figure 4

Data for the Binder ratio in $d=5$. The vertical dashed line corresponds to $T=8.7785$, which is our best estimate of $T_c$ from the correlation length data, see Fig. 3. The horizontal dashed line corresponds to $g=0.4058\dots$, the predicted (see Refs. 7, 12) universal value.

Figure 5

Data for ${\xi }_L∕L$ in $d=4$. According to conventional FSS [Eq. (5)], the data should have a common intersection. This is clearly not the case.

Figure 6

Data for ${\xi }_L∕L{\left(\ln L\right)}^{1∕4}$ in $d=4$. The data intersect at close to a common point.

Figure 7

Data for ${\xi }_L∕L{\left(\ln L\right)}^{1∕4}$ in $d=5$. Clearly the data do not intersect at a commont point, indicating that the assumption of logarithmic corrections to FSS does not work in $d=5$, although it is correct at $d=4$ (see Fig. 6).