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Loop-Cluster Coupling and Algorithm for Classical Statistical Models.

Lei Zhang1,2, Manon Michel3, Eren M Elçi4

  • 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.

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|December 1, 2020
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A new Loop-Cluster (LC) algorithm unifies representations of the Potts model, offering efficient computation across different frameworks. This method also reveals new geometric insights into Fortuin-Kasteleyn (FK) clusters.

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Area of Science:

  • Statistical mechanics
  • Quantum field theory
  • Computational physics

Background:

  • Potts spin systems are crucial in statistical mechanics and quantum field theory.
  • Existing studies utilize spin, Fortuin-Kasteleyn (FK) bond, or q-flow (loop) representations.
  • A unified approach and efficient algorithm for these diverse representations are lacking.

Purpose of the Study:

  • Introduce a novel Loop-Cluster (LC) joint model for Potts spin systems.
  • Develop an LC algorithm that unifies existing representations.
  • Explore the geometric structures within FK clusters and their fractal dimensions.

Main Methods:

  • Formulation of a Loop-Cluster (LC) joint model integrating bond-occupation and q-flow variables.
  • Development and analysis of the LC algorithm, comparing its dynamical universality to the Swendsen-Wang algorithm.
  • Construction of a hierarchy of geometric objects using the LC scheme.

Main Results:

  • The LC algorithm exhibits the same dynamical universality as the Swendsen-Wang algorithm.
  • Theoretical unification of spin, FK bond, and q-flow representations for the Potts model.
  • The LC scheme generates a hierarchy of geometric objects, including q-flow clusters and FK cluster backbones.

Conclusions:

  • The LC algorithm provides a unified framework and efficient computational tool for Potts models.
  • This approach allows for the application of the most efficient algorithm in one representation while measuring quantities in others.
  • New insights into the geometric structures of FK clusters and potential avenues for studying their fractal dimensions are presented.