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Relativistic condensed matter physics: chiral QFT inside a pencil

Relativistic condensed matter physics sounds as a contradiction, doesn't it? If it's condensed, it must be low-energy, and whatever is low-energy, must be non-relativistic, if not low-brow, many people would say. ;-)

Andre Geim gave a colloquium at Harvard University in which he convinced us that the converse is true. One can study relativistic quantum field theory in condensed matter systems.

Andre Geim is an Ig Nobel prize winner for his discovery of a levitating frog; yes, it is now the featured Ig Nobel discovery on Wikipedia. He has also made it to the fifth annual "Scientific American 50" even though he is only scientific but not American, as Charles Marcus has emphasized with a strong nationalist - but still decisively leftist - accent. ;-) Geim was born in Russia although he works in Holland right now.

OK. What material do you need in order to study chiral quantum field theories within condensed matter physics systems?


The material is called graphene and it was thought not to exist until 2004 or so when Geim et al. have made their breakthrough. It is a two-dimensional crystal of carbon filled with hexagons. This idealized material couldn't be observed in its two-dimensional incarnation for a long time but if had been observed in all other compactified editions (except for those with 4-10 large dimensions). We classify them according to the number of large dimensions:

  • 3D: graphite - the material inside your pencil - is a lot of layers of graphene
  • 2D: graphene - one layer - observed in 2004
  • 1D: carbon nanotubes may be viewed as graphene with one dimension compactified on a circle
  • 0D: carbon balls may be viewed as graphene compactified on a sphere
Fine. Geim et al. have simply used a better technology based on optics that allowed them to go to one layer of graphite: the graphene itself. Whenever you make a discovery, you should check literature and you will see that some Russian theorists or Japanese experimenters have done it 39 years ago. This was the case of nanotubes that were not discovered in 1991 but in 1952, we were told. Nevertheless, Geim et al. were lucky because they were indeed the first ones to isolate one layer of graphene.


The existence of purely two-dimensional crystals could have been surprising for many condensed matter physicists. It turns out that the substrate (sillicon) is important for stabilization of the two-dimensional shape of graphene. Without this background, graphene undergoes spontaneous compactification.

Relativistic dispersion relations

When you study electrons moving in this two-dimensional crystal, you will see that they have a different effective mass - the cyclotrone mass - and the effective speed of light is different, too. In their case, the speed is 1,000 km/s which is only 0.3% of the normal speed of light. Nevertheless, you find that the energy of the electrons grows linearly with the momentum:
  • E = PC = MC^2
where PC doesn't stand for political correctness and M,C in the equation above have unusual values. So electrons effectively follow the laws of 2+1-dimensional relativistic quantum field theory.

An entertaining intermezzo: Relativity with a different speed of light is analogous to "relativity of sound" discovered by Jan Fikáček, a philosopher who was a former chairman of Mensa Czechoslovakia and who was always a fun company to talk to. He argued that we only substitute the speed of light into Einstein's equations because we rely on vision. People who prefer hearing and music - for example the blind people - would end up with another type of special relativity that is based on the speed of sound. One of the striking predictions of Fikáček's theory that make it so attractive and so falsifiable - as well as quadruply special if not quintuply special - is that the blind people can't fly with supersonic airplanes. ;-)

They have a new type of isospin with two possible values. It's because the hexagon grid has two carbons in each fundamental cell. The electron can sit on either of them. It is helpful to write the wavefunctions of the electrons on the two types of carbons separately, as two components that depend on the continuous positions x,y. The most elementary transition is in between the two types of the cells which means that the Hamiltonian is an off-diagonal 2x2 matrix proportional to
  • p_x sigma_x + p_y sigma_y
Note that this differs from the non-relativistic Pauli equation where the Hamiltonian doesn't depend on the spin at all. The equation relevant for the electrons in graphene is more analogous to the Dirac equation for a Weyl spinor although the spin degree of freedom is "emergent". See their preprint. Once you identify this degree of freedom with the normal spin, you will find out that the precession of this spin agrees with the frequency of the rotation in a generic magnetic field.

Somewhat surprisingly, if you study a two-layer "graphene" instead of the single layer, the relativistic dispersion relation becomes non-relativistic again: the energy will scale like the momentum squared again. In this case, the emergent spin rotates with a frequency higher by a factor of two in comparison with the orbital angular momentum.


Geim has talked about various quantization rules for the resistivity. He also told us that it was already Mott, a Nobel prize winner who should be confused neither with your humble correspondent nor with the juice, who argued that the resistivity is always bounded from above and this law can be derived from a more fundamental fact that
  • the mean path is never shorter than the de Broglie wavelength.
In the recent article about the AdS/QCD correspondence, we mentioned that this fact can be reinterpreted in the dual gravitational variables as the Bekenstein bound. I guess that Geim doesn't realize this interpretation. He considers the inequality to be falsified in all systems where localization and interference play an important role, and I am somewhat confused by this remark.

He listed the authors of roughly 20 theories to explain these facts using condensed matter methods. Most of these theories end up with a missing factor of pi. Geim showed the conventional picture of a "missing pie", a very sweet one. ;-)


If you discover or explain QHE for the first time, you may get a Nobel prize. If you discover a new kind of IQHE, you get an Ig Nobel prize. ;-)

While the usual fractional quantum Hall effect gives you charges that are fractional multiples of the elementary charge with an odd denominator, Geim et al. have discovered different kinds of quantum Hall effects with graphene. One of them gives you half-integer multiples of the elementary charge which follows from the "relativistic" Dirac equation. Others give you something else, and so forth.


He believes that their work could turn out to be useful for a new generation of laptop batteries. Today, some laptop batteries are made of carbon nanotubes. In 2010, there could be laptop batteries made out of graphene that could hypothetically lead to a superior technology.

Also, carbon belongs to the same group as sillicon (remember the Czech verse: "Co Si Gertrudo Snědla? Olovo (Pb)" which means "What did you eat, Gertruda? Lead") and you could in principle imagine better semiconducting circuits than the sillicon circuits we use today. Carbon could in principle be more pure than sillicon. Purity is measured by the contrast between the minimum and maximum resistivity and they can get very high.

Stay tuned.

Andre Geim says that people should try to play with different materials than the graphene because there is too much competition there and you could see the condensed matter physicists' fists. ;-) Not all possibilities have been tried because the man-power was not sufficient so far. Nancy Hopkins didn't attend the colloquium so no one in the room could throw up because the offending term "man-power" was not replaced by the correct term "person-power".

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