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Kac limit for finite-range spin glasses.

Silvio Franz1, Fabio Lucio Toninelli

  • 1The Abdus Salam International Center for Theoretical Physics, Condensed Matter Group, Strada Costiera 11, P.O. Box 586, I-34100 Trieste, Italy. franz@ictp.trieste.it

Physical Review Letters
|February 3, 2004
PubMed
Summary

We studied a finite-range spin glass model and found its free energy converges to the Sherrington-Kirkpatrick model's in a specific limit. This work may advance mean-field theory for spin glass systems.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Mathematical Physics

Background:

  • Spin glass models are complex systems with disordered magnetic interactions.
  • Understanding their behavior, especially in the thermodynamic limit, is a key challenge.
  • Finite-range interactions are more realistic but analytically harder than infinite-range models.

Purpose of the Study:

  • To investigate the thermodynamic limit of a finite-range spin glass model.
  • To establish a connection between finite-range spin glasses and the exactly solvable Sherrington-Kirkpatrick model.
  • To explore potential theoretical expansions around mean-field theory for spin glass systems.

Main Methods:

  • Analysis of a finite-range spin glass model in arbitrary dimensions.

Related Experiment Videos

  • Mathematical treatment of the interaction potential decay with distance (gamma^-1).
  • Application of the Kac limit (gamma --> 0) to analyze the infinite-volume free energy.

    • Main Results:

    • Demonstrated convergence of the system's infinite-volume free energy to the Sherrington-Kirkpatrick model's free energy.
    • Established this convergence under mild assumptions on the interaction potential.
    • Showed the result holds in the Kac limit (gamma --> 0).

    Conclusions:

    • The finite-range spin glass model effectively reduces to the Sherrington-Kirkpatrick model in the Kac limit.
    • This finding provides a theoretical bridge between realistic finite-range interactions and idealized mean-field models.
    • Represents a potential foundational step for developing expansions around mean-field theory in spin glass physics.