# Value of Mexican hat potential

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• spaghetti3451
In summary: Yes, here are a few steps:1. Show that the field configuration ##\phi(x) = 0## is a local maximum of the free energy.2. Show that this local maximum is unstable.
spaghetti3451
Consider the Lagrangian

$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}{\partial_{\mu}}-y\phi)\psi +\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi),$$

where the potential energy ##V(\phi)## is given by

$$V(\phi) = \frac{1}{2}\kappa^{2}\phi^{2} + \frac{\lambda}{24}\phi^{4}.$$

For ##\kappa^{2}=-|\kappa^{2}|##, we obtain a mexican hat potential.

The stationary points of this potential give us the true vacua ##\phi = \pm \sqrt{\frac{6|\kappa^{2}|}{\lambda}}## of this potential as well as the unstable field configuration ##\phi = 0##.

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Is there an alternative way, for example, by using the classical equations of motion of ##\phi## (setting ##\psi## to zero) to show that the
field configuration ##\phi(x) = 0## is unstable?

Last edited:
You want to choose the solution that minimizes the free energy which you can just compute from the path integral. The classical limit means that you are evaluating it on the saddle point i.e., the equations of motion.

You want to choose the solution that minimizes the free energy which you can just compute from the path integral. The classical limit means that you are evaluating it on the saddle point i.e., the equations of motion.

Computing the free energy (from the path integral) and then minimising it and then evaluating it on the saddle point is overkill as I already have the classical equation of motion.

Do you have some way to use the classical equations of motion of ##\phi## (setting ##\psi## to zero) to show that the field configuration ##\phi(x) = 0## is unstable?

Yes, what I wrote is how you would do it.

In general, you show that something is unstable by showing that it is actually a local maximum of the free energy. In this case case you evaluate the free energy from the path integral on the equations of motion in the classical limit, so this is how you would see that there is an instability using the equations of motion.

Can you provide a few steps to show how to do this?

## 1. What is the "Mexican hat potential" in the context of science?

The Mexican hat potential, also known as the sombrero potential, is a concept in theoretical physics and mathematics that describes the shape of a potential energy function. It resembles a sombrero or Mexican hat, with a flat circular base and a steep circular wall around it.

## 2. How is the Mexican hat potential used in scientific research?

The Mexican hat potential is used in various fields of science, such as particle physics, cosmology, and neuroscience. It is often used to model and explain phenomena that exhibit symmetry-breaking, such as the Higgs mechanism in particle physics.

## 3. What is the significance of the Mexican hat potential in understanding the behavior of physical systems?

The Mexican hat potential plays a crucial role in understanding the behavior of physical systems because it helps scientists to visualize and predict the equilibrium points and stability of a system. It also helps to explain how systems transition from one state to another.

## 4. Can you explain the concept of "Mexican hat potential" in simple terms?

The Mexican hat potential is like a hill with a flat top and steep slopes on the sides. It is often used to describe how energy is distributed in a system and how it affects the behavior of that system.

## 5. Are there any real-life examples that demonstrate the concept of Mexican hat potential?

Yes, there are many real-life examples of the Mexican hat potential, such as the behavior of ferromagnets, where the spins of atoms align themselves in a pattern resembling a Mexican hat potential. Another example is the shape of a soap bubble, which is formed due to the surface tension of the soap film, which can be described by a Mexican hat potential.

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