Its amazing how slow human beings are. The atoms inside your eyelash collide with one another a million million times during each time you blink your eye. It's not surprising, then, that we spend most of our time in condensed--matter physics studying those things in materials that happen slowly. Typically only vast conspiracies of immense numbers of atoms can produce the slow behavior that humans can perceive.
Figure 8. One dimensional crystal: phonons.
The order parameter field for a one--dimensional crystal is the local displacement u(x). Long-wavelength waves in u(x) have low frequencies, and cause sound.
Crystals are rigid because of the broken translational symmetry. Because they are rigid, they fight displacements. Because there is an underlying translational symmetry, a uniform displacement costs no energy. A nearly uniform displacement, thus, will cost little energy, and thus will have a low frequency. These low-frequency elementary excitations are the sound waves in crystals.
A good example is given by sound waves. We won't talk about sound waves in air: air doesn't have any broken symmetries, so it doesn't belong in this lecture. Consider instead sound in the one-dimensional crystal shown in figure 8. We describe the material with an order parameter field u(x), where here x is the position within the material and x - u(x) is the position of the reference atom within the ideal crystal.
Now, there must be an energy cost for deforming the ideal crystal. There won't be any cost, though, for a uniform translation: u(x)u0 has the same energy as the ideal crystal. (Shoving all the atoms to the right doesn't cost any energy.) So, the energy will depend only on derivatives of the function u(x). The simplest energy that one can write looks like
(Higher derivatives won't be important for the low frequencies that humans can hear.) Now, you may remember Newton's law F=m a. The force here is given by the derivative of the energy F=-(d/du). The mass is represented by the density of the material . Working out the math (a variational derivative and an integration by parts, for those who are interested) gives us the equation
The solutions to this equation
represent phonons or sound waves. The wavelength of the sound waves is , and the frequency is . Plugging (4) into (3) gives us the relation
The frequency gets small only when the wavelength gets large. This is the vast conspiracy: only huge sloshings of many atoms can happen slowly. Why does the frequency get small? Well, there is no cost to a uniform translation, which is what (4) looks like for infinite wavelength. Why is there no energy cost for a uniform displacement? Well, there is a translational symmetry: moving all the atoms the same amount doesn't change their interactions. But haven't we broken that symmetry? That is precisely the point.
Long after phonons were understood, Jeffrey Goldstone started to think about broken symmetries and order parameters in the abstract. He found a rather general argument that, whenever a continuous symmetry (rotations, translations, SU(3), ...) is broken, long-wavelength modulations in the symmetry direction should have low frequencies. The fact that the lowest energy state has a broken symmetry means that the system is stiff: modulating the order parameter will cost an energy rather like that in equation equation (2). In crystals, the broken translational order introduces a rigidity to shear deformations, and low frequency phonons (figure 8). In magnets, the broken rotational symmetry leads to a magnetic stiffness and spin waves (figure 9a). In nematic liquid crystals, the broken rotational symmetry introduces an orientational elastic stiffness (it pours, but resists bending!) and rotational waves (figure 9b).
Figure 9A. Magnets: spin waves.
Magnets break the rotational invariance of space. Because they resist twisting the magnetization locally, but don't resist a uniform twist, they have low energy spin wave excitations.
Figure 9B. Nematic liquid crystals: rotational waves.
Nematic liquid crystals also have low-frequency rotational waves.
In superfluids, the broken gauge symmetry leads to a stiffness which results in the superfluidity. Superfluidity and superconductivity really aren't any more amazing than the rigidity of solids. Isn't it amazing that chairs are rigid? Push on a few atoms on one side, and atoms away atoms will move in lock-step. In the same way, decreasing the flow in a superfluid must involve a cooperative change in a macroscopic number of atoms, and thus never happens spontaneously any more than two parts of the chair ever drift apart.
The low-frequency Goldstone modes in superfluids are heat waves! (Don't be jealous: liquid helium has rather cold heat waves.) This is often called second sound, but is really a periodic modulation of the temperature which passes through the material like sound does through a metal.
O.K., now we're getting the idea. Just to round things out, what about superconductors? They've got a broken gauge symmetry, and have a stiffness to decays in the superconducting current. What is the low energy excitation? It doesn't have one. But what about Goldstone's theorem? Well, you know about physicists and theorems ...
That's actually quite unfair: Goldstone surely had conditions on his theorem which excluded superconductors. It's just that everybody forgot the extra conditions, and just remembered that you always got a low frequency mode when you broke a continuous symmetry. We of course understood all along why there isn't a Goldstone mode for superconductors: my advisor, Phillip W. Anderson, showed that it's related to the Meissner effect. The high energy physicists forgot, though, and had to rediscover it for themselves. Now we all call the loophole in Goldstone's theorem the Higgs mechanism, because (to be truthful) Higgs and his high-energy friends found a much simpler and more elegant explanation than we had. We'll discuss Meissner effects and the Higgs mechanism in the next lecture.
I'd like to end this section, though, by bringing up another exception to Goldstone's theorem: one we've known about even longer, but which we don't have a nice explanation for. What about the orientational order in crystals? Crystals break both the continuous translational order and the continuous orientational order. The phonons are the Goldstone modes for the translations, but there are no orientational Goldstone modes. We'll discuss this further in the next lecture, but I think this is one of the most interesting unsolved basic questions in the subject.
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).