I. Identify the Broken Symmetry

https://sethna.lassp.cornell.edu/OrderParameters/BrokenSymmetry.html

What is it which distinguishes the hundreds of different states of matter? Why do we say that water and olive oil are in the same state (the liquid phase), while we say aluminum and (magnetized) iron are in different states? Through long experience, we've discovered that most phases differ in their symmetry.

The cube has many symmetries. It can be rotated by 90 degrees, 180 degrees, or 270 degrees about any of the three axes passing through the faces. It can be rotated by 120 degrees or 240 degrees about the corners, and by 180 degrees about an axis passing from the center through any of the 12 edges. The sphere, though, can be rotated by any angle. The sphere respects rotational invariance: all directions are equal. The cube is an object which breaks rotational symmetry: once the cube is there, some directions are more equal than others.

Consider figure 2, showing a cube and a sphere. Which is more symmetric? Clearly, the sphere has many more symmetries than the cube. One can rotate the cube by 90 degrees in various directions and not change its appearance, but one can rotate the sphere by any angle and keep it unchanged.

II. Define the Order Parameter

We're always looking for the important variables, the important degrees of freedom. In a crystal, the important variables are the motions of the atoms away from their lattice positions. In a magnet, the important variable is the local direction of the magnetization (an arrow pointing to the ``north'' end of the local magnet). The local magnetization comes from complicated interactions between the electrons, and is partly due to the little magnets attached to each electron and partly due to the way the electrons dance around in the material: these details are for many purposes unimportant.

The important variables are combined into an order parameter field. In figure 4, we see the order parameter field for a magnet. At each position x=(x,y,z) we have a direction for the local magnetization M(x). The length of M is pretty much fixed by the material, but the direction of the magnetization is undetermined. By becoming a magnet, this material has broken the rotational symmetry. The order parameter M labels which of the various broken symmetry directions the material has chosen.

https://math.stackexchange.com/questions/2539181/meaning-of-inverse-temperature

https://en.wikipedia.org/wiki/Critical_exponent

Cheeger inequality