Linear relaxation in large two-dimensional Ising models

Critical dynamics in two-dimension Ising lattices up to $2048\times 2048$ is simulated on field-programmable-gate-array- based computing devices. Linear relaxation times are measured from extremely long Monte Carlo simulations. The longest simulation has $7.1\times {10}^{16}$ spin updates, which would take over 37 years to simulate on a general purpose computer. The linear relaxation time of the Ising lattices is found to follow the dynamic scaling law for correlation lengths as long as 2048. The dynamic exponent $z$ of the system is found to be 2.179(12), which is consistent with previous studies of Ising lattices with shorter correlation lengths. It is also found that Monte Carlo simulations of critical dynamics in Ising lattices larger than $512\times 512$ are very sensitive to the statistical correlations between pseudorandom numbers, making it even more difficult to study such large systems.

Figure 1

Processing units (colored squares) that update nonneighboring spins concurrently. The arrows indicate the direction of the movements of the data after each updating operation.

Figure 2

The log of susceptibility $\chi$ over FSS theory ${\chi }_{\mathrm{FSS}}\left(L\right)$ from RNG No. 1 (diamonds), RNG No. 2 (circles), and RNG No. 3 (squares). The error bars are the standard deviations of the mean of the multiple runs at each point

Figure 3

Magnetization $m$ of a $2048\times 2048$ Ising lattice as a function of simulation time $t$.

Figure 4

The log of the autocorrelation of the magnetization in a $2048\times 2048$ Ising lattice as a function of time delay $t$. The straight line is a linear fit to the data.

Figure 5

The log-log of relaxation time τ and lattice size $L$ from RNG No. 3.

Figure 6

The log of relaxation time $\tau$ over FSS theory ${\tau }_{\mathrm{FSS}}\left(L\right)$ from RNG No. 1 (diamonds), RNG No. 2 (circles), and RNG No. 3 (squares). The error bars are the standard deviations of the mean of the multiple runs at each point.