Loop-Cluster Coupling and Algorithm for Classical Statistical Models

Potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin, the Fortuin–Kasteleyn (FK) bond or the $q$-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with $q$-flow variables and formulate an LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen–Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the $q$-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model but also brings new insights into the rich geometric structures of the FK clusters.

Figure 1

Representations and algorithms for the Potts model. The spin, $q$-flow, and FK representations are coupled by the combination of the Swendsen–Wang and the Loop-Cluster algorithm.

Figure 2

Illustration of a cluster-to-loop update for $q=3$, with an up and right orientation, as shown in (d). From an FK bond configuration (a), a spanning tree is constructed from a root vertex, as marked by the green color (b). Each occupied edge missing from the tree defines an independent cycle and is assigned a random flow variable $f\in \{0,1,\dots ,q-1\}$ (c). Finally, the $q$-flow variables for all the other edges are obtained by backtracking vertices and applying the $q$-modular Kirchhoff conservation law for each vertex, yielding a $q$-flow configuration (d).

Figure 3

Convergence of the fractal dimension $D_{\text{qF}}$ of $q_F$-flow clusters from $D_{\mathrm{EP}}=4/3$ to the backbone dimension $D_{\mathrm{bb}}=1.64336$ with $1/(q_F-q_0),q_0=0.937$ for the 2D percolation. Inset: scaling of the size $F_1$ of the largest $q_F$-flow clusters for increasing value of $q_F$.