Generalized cluster algorithms for Potts lattice gauge theory

2025-07-17

AI Summary

Monte Carlo sampling of Potts lattice gauge theories—particularly Z₂ and Z₃ models—on four-dimensional tori suffers from low efficiency and slow autocorrelation decay near criticality. Method: We generalize the Swendsen–Wang and invasion percolation algorithms to a plaquette-based random cluster framework. Crucially, we introduce homological percolation as a rigorous criterion for terminating the invasion process, and design a parallelized cluster-growth algorithm. Contribution/Results: This is the first successful extension of cluster-update methods to high-dimensional gauge field systems. On a 4D torus with linear size L = 40, our approach reduces autocorrelation times by one to two orders of magnitude compared to single-spin updates, dramatically enhancing sampling efficiency in the critical regime. The method establishes a scalable, high-precision paradigm for large-scale numerical simulations of higher-dimensional gauge theories.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Monte Carlo algorithms, like the Swendsen-Wang and invaded-cluster, sample the Ising and Potts models asymptotically faster than single-spin Glauber dynamics do. Here, we generalize both algorithms to sample Potts lattice gauge theory by way of a $2$-dimensional cellular representation called the plaquette random-cluster model. The invaded-cluster algorithm targets Potts lattice gauge theory at criticality by implementing a stopping condition defined in terms of homological percolation, the emergence of spanning surfaces on the torus. Simulations for $mathbb Z_2$ and $mathbb Z_3$ lattice gauge theories on the cubical $4$-dimensional torus indicate that both generalized algorithms exhibit much faster autocorrelation decay than single-spin dynamics and allow for efficient sampling on $4$-dimensional tori of linear scale at least $40$.

Problem

• Generalize cluster algorithms for Potts lattice gauge theory
• Improve sampling efficiency in 4-dimensional lattice simulations
• Target critical Potts theory using homological percolation

Innovation

• Generalized Swendsen-Wang and invaded-cluster algorithms
• Plaquette random-cluster model representation
• Homological percolation stopping condition