Sweeny dynamics for the random-cluster model with small $Q$

The Sweeny algorithm for the $Q$-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As $Q$ decreases, the so-called critical speeding-up for non-local quantities becomes more and more pronounced. However, for some quantity of specific local pattern – e.g., the number of half faces on the square lattice, we observe that, as $Q\to 0$, the integrated autocorrelation time $\tau$ diverges as $Q^{-\zeta }$, with $\zeta \simeq 1\mathrm{/}2$, leading to the non-ergodicity of the Sweeny method for $Q\to 0$. Such $Q$-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight $v=\sqrt{Q}$, can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order $O(1)$.