Dynamic Critical Behavior of the Worm Algorithm for the Ising Model

We study the dynamic critical behavior of the worm algorithm for the two- and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than the Swendsen-Wang algorithm for simulating the two-point function of the three-dimensional Ising model.

Figure 1

Autocorrelation function ${\rho }_{\mathcal{N}}(t)$ versus $t/{\tau }_{\mathrm{int},\mathcal{N}}$ for the critical two-dimensional Ising model.

Figure 2

Autocorrelation function ${\rho }_{{\mathcal{F}}_{\mathrm{low}}}(t)$ versus $1+at/L^{d+z^{\prime }}$ with $a=2.5$, $z^{\prime }=0.315$ for the critical two-dimensional Ising model. The black line shows the initial decay with power $r\approx 3.05$.

Figure 3

Autocorrelation function ${\rho }_{{\mathcal{D}}_{\mathbf{0}}}(t)$ versus $t$ for the critical two-dimensional Ising model. The asymptotic straight line has power $s=0.75$. The dashed line has power $1-\eta /2=7/8$.

Figure 4

$t^s{\rho }_{{\mathcal{D}}_{\mathbf{0}}}(t)$ versus $t/{\tau }_{\mathrm{int},\mathcal{N}}$ for the critical two-dimensional Ising model, with $s=0.75$. The dashed line shows the decay rate ${\tau }_{\exp }$.

Figure 5

Autocorrelation function ${\rho }_{{\mathcal{F}}_{\mathrm{low}}}(t)$ versus $1+at/L^{d+z^{\prime }}$ with $a=1.0$, $z^{\prime }=-0.15$ for the critical three-dimensional Ising model.

Figure 6

Autocorrelation function ${\rho }_{{\mathcal{D}}_{\mathbf{0}}}(t)$ versus $t$ for the critical three-dimensional Ising model. The asymptotic straight line has power $s=0.66$. The dashed straight line has power $1-\eta /2=0.982$.