General heatbath algorithm for pure lattice gauge theory

A heatbath algorithm is proposed for pure $\mathrm{SU}(N)$ lattice gauge theory based on the Manton action of the plaquette element for general gauge group $N$. Comparison is made to the Metropolis thermalization algorithm using both the Wilson and Manton actions. The heatbath algorithm is found to outperform the Metropolis algorithm in both execution speed and decorrelation rate. Results, mostly in $D=3$, for $N=2$ through 5 at several values for the inverse coupling are presented.

Figure 1

Displayed as $+$ is the ratio of the matrix and vector norms $\Vert Q{\Vert }_R/\Vert q{\Vert }_F$ averaged over 100 random unit vectors $\widehat{q}$ for various $N$. Also shown as $◇$ is the ratio of the Wilson and Manton actions $S_W(Q)/S_M(Q)$ which approaches unity as $\Vert q{\Vert }_F\to 0$. The injective limit $\Vert q{\Vert }_F=\pi$ corresponds to an element antipodal to the identity.

Figure 2

Distances are preserved under a unitary transformation of the group elements. The environment projection $\overline{G}$ minimizes the sum of the distances to the elements $G_k$, as shown by the dashed lines. The distances appearing in the action are shown by the solid lines.

Figure 3

Generalization of the CKP algorithm in terms of the group metric. The proposal $P^r$ is parametrized by its distance from the element which minimizes the action. These elements are related to those of Fig. 2 by $G_k=\overline{G}({\overline{G}}^{†}{G}_k)$.

Figure 4

Indirect heatbath updating generates $n_G$ proposals for plaquettes according to the canonical distribution whose projection is $\overline{P}$. Supposing $\overline{Q}\to \overline{P}$ given fixed $\overline{G}$ requires the updated link to take the value $U_{\mathrm{new}}=\overline{P}{\overline{G}}^{†}$.

Figure 5

Normalized frequency and cumulative functions for the gamma (solid lines) and Gaussian (dashed lines) distributions as described in the text. Also shown as $\times$ is the frequency distribution histogram of ${10}^4$ generated values.

Figure 6

Comparison of the mean Manton action value ${\overline{S}}_M$ for the Metropolis and indirect heatbath algorithms on a $L^D=4^3$ lattice as a function of the normalized coupling for $N\in \{2,3,4,5\}$ plotted as $\{\square ,◇,○,+\}$. The mean action for MP using $S_W$ is higher than for MP using $S_M$, as is the value for HB using $b_0$ compared to using $b_H$. The SU(2) values for HB are consistently higher than expected.

Figure 7

Frequency distribution of the Manton plaquette action for SU(3) at $\beta =24$ plotted as histograms $\times$ with either linear or logarithmically spaced bins. The regression function Eq. (14) is shown as the solid line. The MP and HB algorithms return virtually identical action distributions.

Figure 8

Running times $t_{10}$ for 10 thermal updating sweeps as a function of lattice volume $L^D$ for MP with $S_M$ shown as $\square$, for MP with $S_W$ as $◇$, and for HB with $b_H$ as $○$.

Figure 9

Link correlation $C(U,U^{\prime })$ as a function of the number of updating sweeps at $\beta /d=3$ on a $L^D=4^3$ lattice for the MP $\square$ and the HB $○$ algorithms.

Figure 10

Plaquette correlation $C(Q,Q^{\prime })$ as a function of the number of updating sweeps at $\beta /d=3$ on a $L^D=4^3$ lattice for the MP $\square$ and the HB $○$ algorithms.