Loop-Cluster Coupling and Algorithm for Classical Statistical Models

Abstract

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 a 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 rich geometric structures of the FK clusters.

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Cited by in corpus (10)

• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Tensor network Monte Carlo simulations for the two-dimensional random-bond Ising model
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model
• Percolation of the two-dimensional XY model in the flow representation
• Geometric properties of the complete-graph Ising model in the loop representation
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Exploring the Nexus between Thermodynamic Phase Transitions and Geometric Fractals through Systematic Lattice Point Classification
• Universality of the complete-graph Potts model with $0<q\le 2$
• Percolation in the two-dimensional Ising model
• Phase transitions and crtical exponents in the six-vertex model on kagome lattices