Lifted worm algorithm for the Ising model

We design an irreversible worm algorithm for the zero-field ferromagnetic Ising model by using the lifting technique. We study the dynamic critical behavior of an energylike observable on both the complete graph and toroidal grids, and compare our findings with reversible algorithms such as the Prokof'ev-Svistunov worm algorithm. Our results show that the lifted worm algorithm improves the dynamic exponent of the energylike observable on the complete graph and leads to a significant constant improvement on toroidal grids.

Figure 1

Worm configuration $\omega \in {\mathcal{C}}_2$ where $x,y$ are the vertices with odd degree, $N_{\omega }(x,-)=\{x{o}_1,x{o}_2,x{o}_3\}$, and $N_{\omega }(x,+)=\left\{x{v}_1\right\}$. If $x$ is selected as the mobile vertex and one proposes to increase the number of edges in the B-S-type worm algorithm, the only possible transition is $\omega \to \omega \mathrm{\Delta }x{v}_1$ where $\omega \mathrm{\Delta }x{v}_1\in {\mathcal{C}}_2$. The corresponding proposal and acceptance probabilities are stated in Eqs. (6) and (7).

Figure 2

Comparison of the integrated autocorrelation time among the P-S, B-S-type, and lifted worm algorithms. The black diamonds show the ratio of the integrated autocorrelation time of the P-S and B-S-type algorithms, while the blue circles compare ${\tau }_{\mathrm{int}}^{\left(\mathcal{N}\right)}$ among the lifted and the B-S-type worm algorithms. The asymptotic improvement factors can be found in Table 1. The lines correspond to the fits in Sec. 5a.

Figure 3

Normalized autocorrelation function ${\rho }_{\mathrm{irre}}^{\left(\mathcal{N}\right)}\left(t\right)$ ($t$ in MC hits) for the irreversible worm algorithm in five dimensions. As stated in Sec. 5a, ${\rho }_{\mathrm{irre}}^{\left(\mathcal{N}\right)}\left(t\right)$ is well described by the ansatz in Eq. (15).

Figure 4

Finite-size scaling of ${\tau }_{\text{int}}^{\left(\mathcal{N}\right)}/n$ for the B-S-type worm algorithm (upper panel) and lifted worm algorithm (lower panel) on the complete graph with $n$ vertices. The dashed lines correspond to the fits in Sec. 5b.

Figure 5

Normalized autocorrelation function ${\rho }_{\mathrm{irre}}^{\left(\mathcal{N}\right)}\left(t\right)$ ($t$ in MC hits) for the lifted worm algorithm on the complete graph with $n$ vertices. The blue (left) curve corresponds to $n=10\times {10}^3$, the green (middle) curve corresponds $n=30\times {10}^3$, and the red (right) curve corresponds to $n=100\times {10}^3$. As stated in Sec. 5b, ${\rho }_{\mathrm{irre}}^{\left(\mathcal{N}\right)}\left(t\right)$ is well described by the ansatz in Eq. (16).

Figure 6

Dependence on the cutoff time $n$ of the truncated integration autocorrelation time ${\widehat{\tau }}_{\mathrm{int}}^{\left(\mathcal{N}\right)}\left(n\right)$ defined in (A1) for the lifted worm algorithm in five dimensions and $L=56$ at criticality. The thick red line shows ${\widehat{\tau }}_{\mathrm{int}}^{\left(\mathcal{N}\right)}\left(n\right)$ averaged over 100 independent runs, whereas the surrounding region corresponds to $\pm 1\sigma$. The thin lines correspond to the curves $n/c$ with increasing $c$ from left ($c=6$) to right ($c=200$). One clearly sees how $n<100$ underestimates ${\widehat{\tau }}_{\mathrm{int}}^{\left(\mathcal{N}\right)}\left(n\right)$ and much larger $n$ are required to capture contributions corresponding to the suppressed mode.