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On some modified spherical models
1Department of Mathematics, University of Wisconsin, Madison, Wisc. 53706.
Summary
This study introduces a new family of models related to the spherical model. For positive parameter values, these models exhibit Ising-like symmetry and are conjectured to share the same phase transition critical temperature.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
Background:
- The spherical model is a fundamental model in statistical mechanics.
- Understanding phase transitions is crucial for various physical systems.
Purpose of the Study:
- To investigate a one-parameter family of models generalizing the spherical model.
- To explore the phase transition behavior of these generalized models.
Main Methods:
- Consideration of a one-parameter family of models, where alpha=0 corresponds to the spherical model.
- Heuristic discussion of phase transitions and symmetry properties (close to Ising model for alpha > 0).
- Checking the conjecture on a mean-field model.
Main Results:
- Models with alpha > 0 display symmetry akin to the Ising model.
- A conjecture is proposed: if the spherical model (alpha=0) has a phase transition, so do models with alpha > 0.
- The critical temperature is conjectured to remain the same for small alpha > 0.
Conclusions:
- The generalized models offer insights into phase transitions beyond the standard spherical model.
- The conjecture suggests a robust nature of phase transitions under small perturbations.
- Mean-field analysis supports the conjecture regarding critical temperature preservation.