Conformal invariance and the Ising model on a spheroid

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Conformal Field Theory

Background:

• Conformal mappings are crucial for transforming complex geometries.
• The Ising model is a fundamental model in statistical mechanics for studying phase transitions.
• Curved geometries present unique challenges for analyzing critical phenomena.

Purpose of the Study:

• To formulate conformal mappings between planar and spheroidal geometries.
• To apply these mappings to the critical Ising model.
• To investigate Ising models on curved surfaces using computational methods.

Main Methods:

• Formulation of conformal mappings for infinite and semi-infinite planes to spheroids and half spheroids.
• Analytical derivation of magnetization density moments and Binder cumulants for spherical and flat disc cases.
• Continuous cluster Monte Carlo simulations for Ising models on spheroids and half spheroids.
• Finite-size scaling analysis of Monte Carlo data.

Main Results:

• Analytical expressions for critical Ising model properties on spheres and flat discs.
• Precise agreement between Monte Carlo simulations and exact calculations for critical Binder cumulants.
• Accurate determination of magnetic and temperature scaling dimensions from simulations and conformal invariance theory.

Conclusions:

• Conformal mappings provide a powerful tool for studying critical phenomena in curved spaces.
• The Ising model on spheroidal geometries exhibits critical behavior consistent with theoretical predictions.
• Computational and analytical methods yield consistent results for scaling dimensions and critical properties.