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Analytical Expressions for Ising Models on High Dimensional Lattices.

Boris Kryzhanovsky1, Leonid Litinskii1, Vladislav Egorov1

  • 1Center of Optical Neural Technologies, Scientific Research Institute for System Analysis RAS, Nakhimov Ave, 36-1, 117218 Moscow, Russia.

Entropy (Basel, Switzerland)
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PubMed
Summary

The m-vicinity method accurately estimates critical temperatures for high-dimensional Ising models (d≥3). This approach shows strong agreement with simulations, particularly for dimensions 5, 6, and 7.

Keywords:
Ising model on hypercubedensity of statesfree energym-vicinity method

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • Ising models are fundamental in statistical mechanics for understanding magnetism and phase transitions.
  • High-dimensional lattices present unique challenges for theoretical analysis and simulation.

Purpose of the Study:

  • To apply and validate the m-vicinity method for analyzing Ising models on hypercube lattices in dimensions d≥3.
  • To investigate the method's applicability for both short-range and long-range interactions.

Main Methods:

  • Utilized the m-vicinity method, introducing a parameter to determine its applicability in free energy calculations.
  • Derived an expression for critical temperature based on interaction constants.

Main Results:

  • The m-vicinity method provides accurate critical temperature estimates, aligning well with computer simulations for d=5, 6, 7.
  • For d=3 and d=4, the method is accurate for critical temperatures but predicts a finite heat capacity jump, which is debated for d=3 and d=4.

Conclusions:

  • The m-vicinity method is a viable tool for studying high-dimensional Ising models.
  • The method's predictions regarding heat capacity singularities warrant further investigation, especially for lower dimensions.