Lecture 20: Spin Waves are the Goldstone Modes of Ferromagnets

Ferromagnets spontaneously break a continuous symmetry -- that is, when the net magnetization develops, it must choose a particular direction to point. But raise the temperature to disorder this, then lower it again, and -- surprise! -- the magnetization will now form in a different direction. You already know that when a continuous symmetry (here, the rotational symmetry) is broken, the system has Goldstone modes. (See Lectures 1 and 3.) The Goldstone modes of a ferromagnet are called spin waves. These are waves of precession of the magnetic moments.