Loop-Cluster Monte Carlo Algorithm for Classical Statistical Models

We introduce a joint model of bond-occupation variables interacting with so-called $q$-flow variables. It leads us to formulate a Loop-Cluster (LC) Monte Carlo algorithm which passes back and forth between the Fortuin-Kasteleyn (FK) bond representation of the $q$-state Potts model and the so-called $q$-flow representation. Together with the Swendsen-Wang (SW) cluster method, the LC algorithm couples the spin, FK and $q$-flow representations of the Potts model. As a result, a single Markov-chain simulation can use an arbitrary combination of the SW, worm, LC and other algorithms, and simultaneously sample physical quantities in any representation. Generalizations to real value $q \geq 1$ and to a single-cluster version are also obtained. Investigation of dynamic properties is performed for $q=2$, $3$ on both the complete graph and toroidal grids of dimension $2\leq d\leq 5$. Our numerical results suggest that the LC algorithm and its single-cluster version are in the same dynamic universality class as the SW and the Wolff algorithm, respectively. Finally, it is shown, for the Potts model undergoing continuous phase transition, the $q$-flow clusters, defined as sets of vertices connected via non-zero flow variables, are fractal objects.