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....

String theory (!) and "bad metals"

I saw a remarkable talk today by Hong Liu from MIT, about quantum gravity and what it has to say about high temperature superconductivity.  Yes, you read that correctly.  It was (at least for a nonexpert) a reasonably accessible look at a genuinely useful physics result to come from string theory.  I doubt I can do it justice, so I'll just give the bare-bones idea.  Within string theory, Maldacena (and others following) showed that there is a duality (that is, a precise mathematical correspondence) between some [quantum theories of gravity in some volume of d+1 dimensions] and some [quantum field theories w/o gravity on the d-dimensional boundary of that volume].  This sounds esoteric - what could it be good for?  Well, we know what we think the classical limit of quantum gravity should be:  Einstein's general relativity, and we know a decent number of solutions to the Einstein equations.  The duality means that it is possible to take what could be a very painful interacting many-body quantum mechanics problem (say, the quantum field theory approach to dealing with a large number of interacting electrons), and instead of solving it directly, we could convert it into a (mathematically equivalent) general relativity problem that might be much simpler with a known solution.  People have already used this approach to make predictions about the strongly-interacting quark-gluon plasma produced at RHIC, for example.


I'd known about this basic idea, but I always assumed that it would be of very limited utility in general.  After all, there are a whole lot of possible hard many-body problems in solid state physics, and it seemed like we'd have to be very lucky for the duals of those problems to turn out to be easy to find or solve.  Well, perhaps I was wrong.  Prof. Liu showed an example (or at least the results), in which a particular general relativity solution (an extremal charged blackhole) turns out to give deep insights into a long-standing issue in the strongly-correlated electron community.  Some conducting materials are said to be "bad metals".  While they conduct electricity moderately well, and their conductivity improves as temperature goes down (one definition of metal), the way that the conductivity improves is weird.  Copper, a good metal, has an electrical resistance that scales like T² at low temperatures.  This is well understood, and is a consequence of the fact that the low-energy excitations of the electrons in copper act basically like noninteracting electrons.  A bad metal, in contrast, has a resistance that scales like T, which implies that the low energy excitations in the bad metal are very complex, rather than electron-like.  Well, looking at the dual to the extremal black hole problem actually seems to explain the properties of this funny metallic state.  A version of Prof. Liu's talk is online at the KITP.  Wild stuff!  It's amazing to me that we're so fortunate that this particular correspondence exists.

Climate change talk

This afternoon we were fortunate enough to have our annual Rorschach Lecture, delivered by Ralph Cicerone, president of the US National Academy of Sciences.  The subject was climate change and its interaction with energy policy, and unsurprisingly to anyone who isn't willfully ignorant, this was a scary talk.  The atmospheric CO₂ data, the satellite-based measurements of accelerating Greenland and Antarctic ice loss, the amazing pace at which China is building coal-fire power plants (roughly 1 GW of electric generating capacity from coal coming on line every 10 days), are all very sobering.  The planet doesn't care, of course, but it sure looks like the human species had better get its act together, and the only way that's going to happen is if we come up with an energy approach that is cheap compared to coal (that includes the possibility of making coal more expensive, of course, but how do you persuade China and India not to burn their cheap, abundant coal?). 

Ahh, Air China

Posting from International Check-in at Beijing International Airport....

I was actually supposed to get home last night, but Air China had other plans.  At least I have quite the story out of it.  I'd originally booked a 2 hour 45 min layover in Beijing, figuring that would be plenty of time.  However, our Hangzhou-Beijing flight was delayed 2 hours.  Then, the pilot made two go-arounds at Beijing, very bumpy (cue the airsick bags and retching noises from fellow passengers), each time getting w/in about 30 feet of the ground, before giving up (due to high winds, I guess), and we diverted to Tianjin.  In Tianjin they kept us on the plane on the tarmac out at the end of their runway for close to 4 hours.  At least the AC worked there.  They ran out of water, and then orange juice.  Finally, they refueled and flew the plane back to Beijing, arriving only 8 hours late.  At least I wasn't alone (two other americans on the flight in the same situation as me), and Air China did, after some convincing, spring for a hotel for the night.

Clearly the simplest possible explanation for this is that I'm destined to make some universe-shattering discovery in the future, the echoes of which are rippling backward in time to try to prevent my return to the US. 

Conference observations so far

This is a nice gathering of people, and the organizers have done a very good job.  More discussion would be nice - the program is very dense.  A few (not very serious) observations:

• I used to think that I was the only condensed matter physicist not working on graphene.  Now I realize I'm the only condensed matter physicist not working on graphene, iron pnictide superconductors, or topological insulators. 
• Chinese ring tones are different than US or European ringtones.
• One speaker inadvertently stumbled on a great, subtle psychological trick:  he used a font for most of his talk that is identical to the font (some Helvetica variant) used by the Nature publishing group for their titles and subtitles.  That font makes everything seem important :-).  He blew this aura of profundity it at the end, though, by switching to comic sans.
• The Chinese groups that have been charging on the iron pnictides must have enormous resources in terms of people and equipment - the rate at which they are cranking out material and data is remarkable.  US materials growers seem very undersupported by comparison.
  
• Laser-based angle-resolved photoemission, in its appropriate regime, is damned impressive.
  

In China this week

I'm off tomorrow for a week-long trip to China, to go to this workshop.  I've never been to China before, so this should be an interesting experience!  I may try to blog a little, but I don't know how internet access will work during the conference.  Hopefully the trip will go more smoothly than the travel arrangements beforehand.  If I ever hear Expedia's "on hold" music again, I may snap.

Update: The trip in was long but problem-free. Blogger access only works through VPN, thanks to the Great Firewall....

Fiber and CCDs

As you've all no doubt read by now, the 2009 Nobel in Physics was awarded to Charles K. Kao, for the development of truly low loss fiber optics (a technology that you're all using right now, unless the internet backbone in your country consists of smoke signals or semaphore flags), and Willard Boyle + George Smith for the invention of the CCD (charge-coupled device, which is the basis for all digital cameras, and has revolutionized spectroscopy).  

The CCD portion makes a tremendous amount of sense.  CCDs work by using local gates on a doped semiconductor wafer to capture charge generated by the absorption of light.  The charge is then shifted to an amplifier and the resulting voltage pulses are converted into a digital signal that can be interpreted by a computer.  The description given in the supporting document (pdf) on the Nobel website is very good.  CCDs have revolutionized astronomy and spectroscopy as well as photography, and the physics that must be understood and controlled in order to get these things to work well is quite rich (not just the charge generation process, but the solid state physics of screening, transport, and carrier trapping).

The fiber optic portion is more tricky, since many people have worked on the development of fiber optic communications.  Still, Kao had the insight that the real limitation on light propagation in fiber came from particular types of impurities, understood the physics of those impurities, guided a program toward clean material, and had the vision to see where this could all lead.  

Certainly there will be grumbling from some that these are <sneer>engineering</sneer> accomplishments rather than essential physics, as if having a practical impact with your science that leads to technology and helps society is somehow dirty, second-rate, or a sign of intellectual inferiority.  That is a terrible attitude, and I'm not just saying that because my bachelor's degree is in engineering.  Trust me:  some engineers have just as much raw intellectual horsepower as high energy theoretical physicists.  Finding intellectual fulfillment in engineering is not some corruption of pure science - it's just how some very smart people prefer to spend their time.  Oh, by the way, the actual will of Alfred Nobel refers to accomplishments that "shall have conferred the greatest benefit on mankind", and specifically mentions "the person who shall have made the most important discovery or invention [my emphasis] within the field of physics".

Finally, this provides yet another data point on just how transformative Bell Labs (and other remarkable industrial R&D labs, including IBM, GE, and others) really was in the physical sciences.  The withering of long-term industrial research will be felt for a long, long time to come.


 

Single atoms in semiconductors

One last post before the obligatory Nobel post tomorrow.

Recently, there has been progress in examining the electronic transport properties of individual dopant atoms in semiconductors.  There are several motivations for this.  First and probably foremost, with increasing miniaturization we are rapidly approaching the limit when the active channel in semiconductor devices will contain, statistically, only a small number of dopants; it makes sense to figure out how these systems work and whether they have any intrinsically useful properties.  Second, these systems are the ultimate small-size limit of quantum dots, even smaller than single-molecule transistors.  Third, since the host materials are extremely well-studied, and quantum chemistry calculations can handle the relevant volumes of material, there is the possibility of realistic, detailed theoretical treatments.  This paper is a great example of treating an individual phosphorus donor in Si as a quantum dot.  This other paper looks at a single arsenic donor, and can see Kondo physics involving the unpaired electron on the donor site interacting with the (valley degenerate) Si conduction electrons.  Very cool stuff!