Nobel Prize Season

Update: I definitely wasn’t expecting this! It’s nice to see something applied win the prize.

So Nobel Prize season is upon us once again. It’s always fun to try to guess who the winners will be — and because I know nothing about Medicine, Peace, or Economics, I’ll stick to making predictions on the Physics and Chemistry prizes. I happened to pick 2007’s Physics winners correctly by sheer luck: maybe it’ll happen again?

Actually, I don’t know Chemistry as a field broadly enough to make any reasonable predictions, either. Just for fun, I’m going to guess Richard Zare will win this year, for the extensive work he has done using lasers for all sorts of spectroscopy — in particular, he developed laser-induced fluorescence spectroscopy. The only problem with picking Zare is that his work might (?) be too closely related to that of last year’s Chemistry picks (Shimomura, Chalfie and Tsien) for their discovery and development of green fluorescent protein.

This year I’m guessing that the Physics prize will go to Aharanov and Berry for their work on quantum topological and geometrical phases (see, for example, the Aharonov-Bohm effect or Berry phase). There are a number of reasons for this guess. Every physics student learns about how a charged particle moving in a region of zero electric and magnetic field is still affected by the potentially non-zero electromagnetic vector potential A – that is, its wave function picks up a phase shift given by integrating A along its path. This is an incredibly deep and fundamental result of quantum mechanics: unlike in classical electrodynamics, in quantum electrodynamics the effects of this potential can be felt. For example, as Aharonov and Bohm proposed in 1959, two charged particles going in opposite directions around a circle encircling a solenoidal magnetic field interfere when they are recombined — in particular, their phase difference (which can be measured) is directly proportional to the magnetic flux penetrating the circle, even though they feel zero magnetic field along the path they traverse!

Indeed, this effect has been verified in numerous measurements since. The earliest example that I am aware of is this elegant experiment by Chambers in 1960, using an electrostatic “biprism” consisting of an aluminized quartz fiber flanked by two grounded metal plates (schematic here) to interfere two beams of electrons.  This was followed up by further electron holography measurements using toroidal ferromagnets, as well as by work studying oscillations in the resistance of tiny metal rings as a function of the magnetic field being applied through their core. More recently, these magnetoresistance oscillations have been observed in individual carbon nanotubes with the field applied parallel to the tube axis, which I think is pretty cool. I remember when I first learned about this effect: it was one of the first times I was truly, genuinely, acutely thrown by quantum mechanics. And it has profound consequences — namely, it suggests that the electromagnetic vector potential is in some sense more “real” than the electric or magnetic fields on their own.

in 1984, Berry went one step further, pointing out that the Aharonov-Bohm effect is a particular example of geometric phase, and that a geometric phase often arises in many quantum situations. In particular, if a quantum system is changed very slowly (that is, adiabatically) such that it is eventually brought back to its initial conditions in parameter space, it turns out that it remembers the path it took: it picks up a phase factor that depends on the geometry of the path it took through parameter space. For example, if you subject a fixed electron to a constant magnetic field that changes in direction — say the magnetic field vector sweeps out an arbitrary closed loop on the surface of a sphere centered on the electron — it turns out that the electron state picks up a Berry’s phase proportional to the solid angle subtended by the path relative to the origin. That’s it. Isn’t that crazy?

The idea of a Berry phase (and the way in which it links physical effects to topological quantities) is quite general, and has found applications in many physical systems. For example, the quantum Hall effect can be understood as an example of Berry’s phase applied to 2D electronic systems, while the anomalous Hall effect for dilute magnetic semiconductors has recently been linked to Berry’s phase, as well. Graphene is a nice recent experimental system for studying Berry’s phase for electrons in two dimensions: electrons in graphene can be understood using the Dirac equation for spin-1/2 particles, and are characterized by “pseudospin”. Just as in the Berry phase example I gave earlier, an electron in graphene that completes a cyclotron orbit in an applied magnetic field has its pseudospin rotated by 360 degrees, and thus picks up a phase shift of pi in its wavefunction. The consequences of this have recently been observed in quantum Hall measurements of monolayer and bilayer graphene.  In related work, topological insulators and the quantum spin Hall effect have recently begun receiving a huge amount of attention from the physics community, because of their unusual properties — while they are insulating in the bulk, they can support unique “surface states”. I don’t fully understand the theory of these, but the main framework within which they appear to be studied is by describing them as topologically ordered states, characterized by topological invariants such as Chern numbers and a non-trivial Berry phase.

An interesting side note: in all of these Aharonov-Bohm/Berry phase experiments, the quantum phase is measured through some kind of interference process. Recently, Manoharan’s group at Stanford has done some pretty cool STM experiments to directly measure quantum phase information, by comparing the STM signal of physically different but electronically identical quantum corral-type nanostructures.

One potential problem: the Aharonov-Bohm effect was apparently previously predicted by Ehrenberg and Siday ten years earlier, and Berry phase was apparently discussed by Pancharatnam some 28 years before Berry’s paper! On the other hand, history suggests that this may not be enough to deter the prize committee.

Update: apparently Thomson Reuters agrees with my pick for Physics…

Kepler’s Year

As some readers may already know, 2009 is the International Year of Astronomy, commemorating 400 years since Galileo raised his telescope to the stars and Kepler put forward his laws of planetary motion in his treatise Astronomia Nova. This put the Copernican idea of heliocentrism on a more rigorous scientific footing, and would eventually pave the way for Newton to formulate his theory of gravity.

Even aside from these achievements, Kepler is particularly fascinating character in the history of science. His work always seemed to be strongly guided by a Platonic view of a perfectly structured, geometrically harmonic universe. For example, in his Mysterium Cosmographicum – a lesser-known precursor to Astronomia Nova – Kepler devised an elegant, if incorrect, model of the six known planets as following orbits along spheres inscribed within and circumscribed around five platonic solids. Another example is his extensive exploration of tesselations and sphere-packings, which he often used to explain the structure of various materials, such as snowflakes.

A well-known result of this is Kepler’s conjecture on sphere packings – namely, that the maximum possible volume fraction of a container that can be filled by equally-sized spheres is that of face-centered cubic packing, slightly more than 74%. This was only proved by Thomas Hales recently, using a good deal of computational machinery.

A related problem – part of Hilbert’s list of some of the most important problems in mathematics – involves finding the densest packings of regular polyhedra. While cubes and truncated octahedra can tile space (so their densest packing has a volume fraction of 100%), it turns out that none of the other five Platonic solids or thirteen Archimedean solids can. A good deal of recent work has focused on trying to find the densest possible packing of regular tetrahedra, mainly by construction. In two very nice recent papers, however, Sal Torquato and his student Yang Jiao have come up with a computational scheme for generating very, very dense packings of polyhedra, calculating the densest known packings of the Platonic and Archimedean solids. I won’t go into the details of the algorithm – the follow-up paper does a very nice job of explaining it. The basic idea is to start with a randomized ‘dilute’ configuration of the polyhedra in a box of some shape and either randomly move a randomly chosen polyhedron by a small amount, or deform the box by some amount, only allowing changes that increase the volume fraction while still preventing polyhedra from overlapping. One can imagine that iterating this many times would result in very, very dense packings — the hope, of course, is that these are the densest possible polyhedral packings. This remains to be proven.

The cool thing that Torquato and Jiao found is that for all the Platonic and Archimedean solids possessing central symmetry, the densest possible packings they found had volume fractions equal to the volume fractions of Bravais lattice packings of the same solids, to within a few hundredths of a percent. A nice Kepler-style argument using inscribed spheres gives upper bounds on the volume fractions of these densest possible packings – these are larger by only ~3-13%. Taken together, these results hint at a possible “Kepler Conjecture” for polyhedral packings: namely, that the densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings. Very cool — this suggests that quite complicated polyhedral packings might be able to be understood using some very simple rules.

On the other hand, Torquato and Jiao found that for the two polyhedra not possessing central symmetry — the tetrahedron and the truncated tetrahedron — the volume fractions of the densest lattice packings grossly underestimated the volume fractions of the densest packings they found using simulations. This leads to a converse conjecture: in particular, that the densest possible packing of any convex, congruent polyhedron without central symmetry is not a Bravais lattice packing — rather, it is significantly more complicated. (A nice side result is that the densest packing they found possesses no long-range order — more on this later.) It is not clear at this point what rules, if any, would dictate what the densest possible packing of these polyhedra look like.

One word of caution: as I mentioned before, because this approach is computational, none of these densest-known packings has been proved to be the densest-possible. Because the algorithm requires some choice of the starting ‘dilute’ configuration, it is possible that this choice will influence the final structure the algorithm settles at. In fact, Torquato and Jiao already found this to be the case in their search for the densest possible tetrahedral packing, as they note in their second paper — using a different initial condition, they found a densest packing ~4% more dense than the one they initially reported in the first paper.

While they don’t mention it, I think these results are particularly interesting in the context of amorphous systems, such as glasses. For example,why can a simple liquid metal be supercooled below its freezing point without crystallizing, potentially forming a glass? F. C. Frank put forward a very nice explanation for this. Considering the atoms of the liquid metal as spheres interacting via a non-directional Lennard-Jones potential, it turns out that the local energy density can be minimized by forming “locally-preferred” tetrahedral clusters. These then come together to form polytetrahedral Frank-Kasper phases, because forming these clusters requires less energy than forming crystalline clusters. The only problem with these phases is that they cannot tile space and are geometrically ‘frustrated‘ — the system is not in the crystalline state of lowest possible free energy, but is rather trapped a local free energy minimum in phase space. A significant amount of work has focused on trying to understand these kinds of structures, and connecting the geometric frustration inherent in these phases to the physical properties of supercooled liquids and glasses. It is not too surprising, then, that definitively finding the densest possible packing of tetrahedra (and analyzing the physical properties of such a structure) could help flesh out these connections — and Torquato and Jiao’s work seems to point the way.