Dynamic critical exponent $z$ of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model

We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain $z=2.0245\left(15\right)$ for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of $z$ is fully consistent with recent field theoretic results.

Figure 1

Results for the critical exponent $\eta$ obtained by fitting our numerical estimates of the magnetic susceptibility $\chi$ at $D=0.655$ and $\beta =0.387721735$ by using the Ansätze (24) and (25). The solid line indicates the result obtained from conformal bootstrap $\eta =0.0362978\left(20\right)$, Ref. [1]. All data with $L\ge L_{\text{min}}$ are taken into account in the fit.

Figure 2

Results for the dynamic critical exponent $z$ obtained from fitting our numerical results for the integrated autocorrelation time of the magnetic susceptibility ${\tau }_{\text{int},\chi }$ at $D=0.655$ and $\beta =0.387721735$ by using the Ansätze (26) and (27). All data with $L\ge L_{\text{min}}$ are taken into account in the fit. In the caption, the Metropolis and the heat bath algorithm are indicated by M and HB, respectively. For better readability we have slightly shifted the values of $L_{\text{min}}$. The solid line gives the central value of our estimate $z=2.0245\left(15\right)$, while the dashed lines indicate the error.

Figure 3

Simulations with the Metropolis algorithm. We plot the ratios $m_{L_1}\left(t\right)/m_{L_2}\left(t\right)$ for $L_1=50$ and 100 and $L_2=300$ as a function of $t$. For the readability of the figure we only give a fraction of the $t$ values.

Figure 4

Simulations with the Metropolis algorithm. The effective exponent $z_{\text{eff}}$ as defined by Eq. (30) for $L=300$ and $t_0=-2.1$. The dashed line indicates the preliminary result $z=2.0245$ of this section.

Figure 5

Simulations with the heat bath algorithm. The effective exponent $z_{\mathrm{eff}}$ as defined by Eq. (30) for $L=300$ and $t_0=0$. The dashed line indicates the preliminary result $z=2.024$ of this section.

Figure 6

We plot the Binder cumulant $U_4=\frac{\langle m^4\rangle }{{\langle m^2\rangle }^2}$ at the critical temperature for the Ising model and the Blume-Capel model at $D=0.655$ and 1.15. Note that the error bars are clearly smaller than the symbol size. The dashed line gives the estimate $U_4^{*}=1.6036\left(1\right)$ of the fixed point value [37].