Logarithmic finite-size scaling of the self-avoiding walk at four dimensions

The $n$-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the $n\to 0$ limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size $L=768$. From an unwrapped end-to-end distance, we obtain the critical fugacity as $z_c=0.147622380\left(2\right)$, improving over the existing result $z_c=0.1476223\left(1\right)$ by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near $z_c$, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as $\chi \sim L^2{(lnL)}^{2{\widehat{y}}_h}$ and $C\sim {(lnL)}^{2{\widehat{y}}_t}$, and the logarithmic exponents are determined as ${\widehat{y}}_h=0.251\left(2\right)$ and ${\widehat{y}}_t=0.25\left(3\right)$, in excellent agreement with the field theoretical prediction ${\widehat{y}}_h={\widehat{y}}_t=1/4$. Our results provide a strong support for the recently conjectured finite-size scaling form for the $\mathrm{O}\left(n\right)$ universality classes at 4D.

Figure 1

Plot of the unwrapped end-to-end distance ${\xi }_u$, rescaled by $L{(lnL)}^{1/4}$, near the critical point and for various system sizes. The red vertical line indicates the new estimate of $z_c$, while the dashed vertical line is the central value of the previously best estimate of $z_c=0.1476223$. The inset zooms in the intersection region, to clearly show their deviation.

Figure 2

(a) Plot of the scaling function ${\tilde{\xi }}_u\left(x\right)$, where $x=(z-z_c)L^2{(lnL)}^{1/4}$. The red curve corresponds to our preferred fit of the ${\xi }_u$ data to the ansatz Eq. (19). (b) Plot of ${\xi }_u/\left[L{(lnL)}^{1/4}\right]$ versus $L$ for fixed values of $z$. The curves correspond to our preferred fit of the data by the ansatz Eq. (19). We note that, the central value of the previously best estimate of $z_c$ deviates away from our estimate about 40 times of our quoted error bar.

Figure 3

The scaling behavior of susceptibility $\chi$ (a) and its Fourier mode ${\chi }_{\mathbf{k}}$ (b), at the critical point. The line in the top figure has slope $1/2$, and the shadow in the bottom figure indicates our estimate of the value that ${\chi }_{\mathbf{k}}/L^2$ converges to. All data points here have error bars which are smaller than the size of data points.

Figure 4

The scaling behavior of the mean walk length $N$(a) and its Fourier mode $N_{\mathbf{k}}$(b) at the critical point $z_c$. The line in the top figure has slope $1/4$, and the shadow in the bottom figure indicates our estimate of the value that $N_{\mathbf{k}}/L^2$ converges to. All data points here have error bars which are smaller than the size of data points.

Figure 5

The scaling behavior of the specific heat $C$(a) and its Fourier mode $C_{\mathbf{k}}$(b) at the critical point $z_c$. The line in the top figure has slope $1/2$, and the shadow in the bottom figure indicates our estimate of the value that $C_{\mathbf{k}}$ converges to. All data points here have error bars which are smaller than the size of data points.

Figure 6

Plot of the Binder cumulant $Q_N$ near the critical point, for various system sizes. The shadow column shows our estimate of $z_c$.

Figure 7

Plot of the Binder cumulant $Q_N$ versus $L$ at the critical point. The curve corresponds to our preferred fit of the $Q_N$ data to the ansatz $Q_N=Q_0+b_1{L}^{y_1}$. The left and right insets plot $Q_N$ versus $L^{-1/2}$ and ${(lnL)}^{-1/2}$, respectively. The two red lines are to guide the eyes.

Figure 8

Plot of the critical Binder cumulant $Q_N$ for various dimensions (a) $d=2$, (b) $d=3$, and (c) $d=5$.

Figure 9

Plot of the critical Binder cumulant $Q_0$ for $d=1,2,3,4,5$. The gray dashed line shows the value $\pi /2$. Both the two possible scenarios, that $Q_0$ is consistent with $\pi /2$ (red) or less than $\pi /2$ (blue), are shown for the 4D case. The horizontal blue line indicates that $Q_0$ takes the complete-graph value $\pi /2$ for $d>4$.