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Duality relation for frustrated spin models.

D-H Lee1, F Y Wu

  • 1Department of Physics, University of California, Berkeley, CA 94720, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 15, 2003
PubMed
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Frustrated spin models on planar graphs have dual models with external fields. This algebraic approach simplifies analysis of the Ising and Potts spin glass models, particularly when all plaquettes are frustrated.

Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Graph Theory

Background:

  • Frustrated interactions in discrete spin models pose significant analytical challenges.
  • Planar graphs and lattices are common frameworks for studying spin systems.

Purpose of the Study:

  • To develop an algebraic method for analyzing Ising and Potts spin models with frustrated interactions on planar graphs.
  • To establish a duality relationship between frustrated spin models and models with external fields.

Main Methods:

  • An algebraic approach was employed to analyze the Ising model with frustrated plaquettes.
  • The concept of dual models was utilized to map frustrated systems to systems with external fields.
  • The analysis was extended to the Potts spin glass model.

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Main Results:

  • A dual model was derived for an Ising model with frustrated plaquettes, featuring an external field at dual sites.
  • For Ising models with all plaquettes frustrated, the dual model exhibits a uniform external field, allowing for partition function evaluation in the thermodynamic limit.
  • Analogous results were obtained for the Potts spin glass model.

Conclusions:

  • The algebraic approach provides an effective method for simplifying the analysis of frustrated spin models.
  • Duality offers a powerful tool for understanding the behavior of complex spin systems on planar graphs.
  • The findings are applicable to both Ising and Potts spin glass systems.