Directed walk in probability space that locates mean field solutions to spin models

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Computational Physics

Background:

• Lattice spin models are fundamental in physics but often analytically intractable.
• Mean field theory offers simplification but struggles with nonlocal interactions and nonperiodic boundaries.
• Existing numerical methods like Monte Carlo sampling can be computationally intensive and slow to converge.

Purpose of the Study:

• To develop a new, efficient method for generating mean field solutions for classical continuous spin models.
• To address challenges posed by nonlocal interactions and nonperiodic boundary conditions.
• To demonstrate superior performance over conventional Monte Carlo sampling.

Main Methods:

• Utilized functional optimization to derive a closed-form optimality condition.
• Developed self-consistent mean field equations.
• Applied the method to a 1D dipolar chain and extended to higher dimensions and continuum field theories.

Main Results:

• The functional optimization approach significantly outperforms Monte Carlo sampling in convergence speed and accuracy.
• Demonstrated the method's efficacy on nonperiodic spin models of various dimensionalities.
• Showcased the approach's applicability to complex spin systems and continuum field theories.

Conclusions:

• The proposed functional optimization method provides an efficient and accurate alternative for solving challenging spin models.
• This approach offers a powerful tool for studying systems with nonlocal interactions and nonperiodic boundaries.
• The method's versatility allows for application across diverse spin systems and field theories.