Interactive demosSymmetry Breaking in the Landau Free EnergyOne of the coolest things we discovered in the meanfield Ising model is how the magnetization suddenly becomes nonzero once we cool the system below its critical temperature . In this applet, you'll visualize this behavior firsthand. As we learned in our lesson about probe fields, the equilibrium magnetization is the value of that minimizes the Landau free energy . And since is so small near the critical point, we can do a Taylor expansion of about and just keep the first few terms. Here, I've kept the quadratic and quartic terms. As we vary the temperature, the shape of will change, because the coefficients in front of each of the terms will change. Above the critical temperature , when the coefficient of the term is positive, there is a unique minimum at , meaning that the magnet is disordered and the spins are pointing any which way. However, once we've cooled the system below , the coefficient of the term becomes negative, and we find minima at . The spins begin to align, and the system develops a spontaneous magnetization even though there's no external field! We say that spinflipsymmetry has been broken in the ordered lowtemperature phase. Try it out yourself below. On the left panel, I've plotted as a function of , and I've labeled the minimum value . On the right panel, I show how varies as you drag around the temperature. Turning on the External FieldThis applet demonstrates the phase diagram in the plane! Wow! Smells like part of our homework for this week ;) On the left panel, I've plotted the free energy as a function of the magnetization , and I've labeled the equilibrium value of magnetization as . On the right panel, you can adjust the values of the external field and the parameter , which you can think about as the temperature. Note that this panel has been flipped 90 degrees compared to the previous applet! Here are a few pointers to wrap your head around the plane
Try dragging the ball around the plane in various paths. See if you can answer the following questions!
Leave a Comment Below!Comment Box is loading comments...
