The unreasonable effectiveness of a toy model

As I've mentioned before, often theoretical physicists like to use "toy models" - mathematical representations of physical systems that are knowingly extremely simple, but are thought to contain the essential physics ingredients of interest.  One example of this that I've always found particularly impressive also happens to be closely related to my graduate work.  Undergraduate physicists that take a solid state class or a statistical physics class are usually taught about the Debye theory of heat capacity.  The Debye model counts up the allowed vibrational modes in a solid, and assumes that each one acts like an independent (quantum) harmonic oscillator.  It ends up predicting that the heat capacity of crystalline (insulating) solids should scale like T³ at low temperatures, independent of the details of the material, and this does seem to be a very good description of those systems.  Likewise, undergrads learn about Bloch waves and the single-particle picture of electrons in crystalline solids, which ends up predicting the existence of energy bands.  What most undergrads are not taught, however, is how to think about the vast majority of other solids, which are not perfect single crystals.  Glass, for example.

You might imagine that all such messy, disordered materials would be very different - after all, there's no obvious reason why glass (e.g., amorphous SiO₂) should have anything in common with a disordered polymer (e.g., photoresist).  They're very different systems.  Yet, amazingly, many, many disordered insulators do share common low temperature properties, including heat capacities that scale roughly like T^1.1, thermal conductivities that scale roughly like T^1.8, and particular temperature dependences of the speed of sound and the dielectric function.  To give you a flavor for how weird this is, think about a piece of crystalline quartz.  If you cool it down you'll find a heat capacity and a thermal conductivity that both obey the Debye expectations, varying like T³.  If you take that quartz, warm it up, melt it, and then cool it rapidly so that it forms a glass, if you remeasure the low temperature properties, you'll find the glassy power laws (!), and the heat capacity at 10 mK could be 500 times what it was when the material was a crystal (!!), and you haven't even broken any chemical bonds (!!!).

Back in the early 1970s, Anderson, Halperin, and Varma postulated a toy model to try and tackle this mysterious universality of disordered materials.  They assumed that, regardless of the details of the disorder, there must be lots of local, low-energy excitations in the material to give the increased heat capacity.  Further, since they didn't know the details, they assumed that these excitations could be approximated as two-level systems (TLSs), with an energy difference between the two levels that could range from zero up to some high energy cutoff with equal probability.  Such a distribution of splittings naturally gives you a heat capacity that goes like T¹.  Moreover, if you assume that these TLSs have some dipole-like coupling to phonons, you find a thermal conductivity that scales like T².  A few additional assumptions give you a pretty accurate description of the sound speed and dielectric function as well.  This is pretty damned amazing, and it seems to be a remarkably good description of a huge class of materials, ranging from real glasses to polycrystalline materials to polymers.  

The big mystery is, why is this toy model so good?!  Tony Leggett and Clare Yu worked on this back in the late 1980s, suggesting that perhaps it didn't matter what complicated microscopic degrees of freedom you started with.  Perhaps somehow when interactions between those degrees of freedom are accounted for, the final spectrum of (collective) excitations that results looks like the universal AHV result.  I did experiments as a grad student that seemed consistent with these ideas.  Most recently, I saw this paper on the arxiv, in which Moshe Schechter and P. C. E. Stamp summarizes the situation and seems to have made some very nice progress on these ideas, complete with some predictions that ought to be testable.  This kind of emergence of universality is pretty cool.

By the way, in case you were wondering, TLSs are also a major concern to the folks trying to do quantum computing, since they can lead to noise and decoherence, but that's a topic for another time....