Spontaneous Symmetry Breaking and Magnets

Suppose we have a system of particles that has a degree of freedom called “spin,” which can be either up or down. If you’re familiar with chemistry, you can think of this as electron spin. Recall also our discussion in Chapter 1 of electron’s spin and Dirac’s explanation of it. Let us further suppose that when two particles are brought together, they “like” to have the same spin (both up or both down), in the sense that these states have lower energy overall. (Ferromagnetic materials, for example, act like magnets when electron spins are lined up in the same direction. But these materials lose their magnetic properties if the electron spins are randomly aligned, in which case the magnetic effects cancel out). Same-spin configurations have energy , the former being the energy of same-spin and the latter being the energy of opposite-spin configurations. Imagine a lattice filled with such particles situated on a plane (Fig. 29 ). That is the model we are considering here, which is called the Ising model.

Illustration for Spontaneous Symmetry Breaking and Magnets

Figure 29. The Ising model involves up and down spins, where nearby spins prefer to align, to lower the total energy in the system.