Mexican hat potential around spin site (TOPIC: spin models)

In summary: In the ordered phase of the xy-model, the free energy has the shape of the bottom of a champagne bottle, with a minimum achieved on a circle on the base of the bottle. Rotations around this circle do not alter the free energy, but nonuniform changes in the angle θ do increase the free energy.
  • #1
giann_tee
133
1
I am not sure if I am reading well the following article about KT phase transition in spin systems.
http://www.itp.uzh.ch/~llucas/projects/proseminar.pdf

Page 5 quotes a theory regarding the Helmholtz free energy function. The curve surrounding a spin site or an atom looks like everyone's best guess on the shape of potential energy.

The Free Energy in the xy-Model

Considering Ginzburg-Landau theory, we find that, in the ordered phase of the xy-model, the free energy F has the shape of the bottom of a champagne bottle (Fig. 1). Its minimum is achieved on a circle on the base of the bottle. Points on this circle are depicted by the angle θ and the radius specified by the magnitude of the order parameter. Rotations around the circle correspond to spatially uniform changes in θ, and thus do not alter the free energy. However, spatially nonuniform changes in θ increase the free energy.

Are they saying that Ising spin model with 2 possible spin states or Potts model with several states would have greater free energy?
 
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  • #2
No, they are not saying that Ising spin model or Potts model would have greater free energy. The article is discussing the free energy of the xy-model, which is a spin system with continuous spin states.
 
  • #3


No, the article is specifically discussing the xy-model, which is a type of spin model that has continuous spin states instead of just two or several states. The article suggests that in the ordered phase of the xy-model, the free energy has a specific shape resembling the bottom of a champagne bottle. This shape is determined by the angle θ and the magnitude of the order parameter. The article is not comparing the free energy of the xy-model to other types of spin models.
 

1. What is the Mexican hat potential in spin models?

The Mexican hat potential is a mathematical concept used in spin models to describe the energy landscape around a spin site. It is named after its shape, which resembles a traditional Mexican hat. It represents the energy levels of the spin site and its neighboring sites, and is used to study phase transitions and critical behavior in spin systems.

2. How is the Mexican hat potential related to spin symmetry?

The Mexican hat potential is closely related to spin symmetry. In spin models, the symmetry of the system is often broken at a critical point, causing the energy landscape to shift from a symmetric to an asymmetric shape, resembling a Mexican hat. This symmetry breaking is a key factor in understanding the behavior of spin systems and phase transitions.

3. What is the significance of the Mexican hat potential in condensed matter physics?

The Mexican hat potential is a fundamental concept in condensed matter physics, as it helps to explain the behavior of materials at the atomic level. Understanding the energy landscape of spin systems is crucial in predicting and controlling the properties of materials, such as magnetism and superconductivity.

4. How is the Mexican hat potential used in statistical mechanics?

The Mexican hat potential is a key tool in statistical mechanics, as it allows for the calculation of the partition function and other thermodynamic quantities. It is used to study the behavior of spin systems at different temperatures and to predict phase transitions and critical points.

5. Can the Mexican hat potential be applied to other systems besides spin models?

While the Mexican hat potential was originally developed for spin systems, it has also been applied to other systems in physics, such as the Higgs field in particle physics and the potential energy of molecules in chemistry. Its usefulness lies in its ability to describe the energy landscape of a system and its phase transitions, making it a versatile tool in various fields of science.

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