Igor Herbut: "Coulomb interactions, ripples, and minimal conductivity of graphene"

Herbut started the presentation by stressing that, in his view, there are two problems in graphene physics which might be worth the while of a theoretical physicist interested in correlated systems. First, QHE, where he argued the interaction effects are essential and, second, the problem of minimum conductivity, where the interactions might be a part of the ultimate solution. He then proceeded to define an experimental puzzle: the measured minimum conductivity of graphene sheets tuned to Dirac point is about 300% larger than what one would compute for the clean or weakly disordered system.

Herbut then set up the theoretical background. The symmetry of the honeycomb lattice of graphene sheets guarantees two Dirac points under rather general conditions. Near these points the electron-hole spectrum has an appearance of a relativistic massless Dirac fermion and the gate voltage can be tuned so that the Fermi surface passes right through them. When this is the case, we can view the problem as a nice example of a fermionic quantum criticality. The universality of the familiar result, \sigma_0 = \pi/2 e^2/h, derived by Fradkin and others, is a manifestation of such criticality.

Next, Herbut introduced interaction. This is just the unscreened 1/r Coulomb interaction, which, one can show using the technology of quantum critical phenomena, turns marginally irrelevant at low energies. He demonstrated this by computing the correction to \sigma_0 arising from such interaction. Indeed, the correction due to the Coulomb interaction was found to fall of logarithmically, as one moves to low frequencies. Importantly, however, this correction was positive – the conductivity at some low but finite frequency was enhanced relative to \sigma_0. This set the stage for an intriguing piece of physics: in a typical experiment, the frequency scaling of conductivity will generically be cut off by temperature, disorder or some other effect. It could be that the observed access conductivity is actually due to such a phenomenon.

The specific example worked out by Herbut, Juricic and Vafek is due to a disorder effect arising from rippling of graphene bonds. When graphene sheet is fixed onto a substrate, such ripples act as a gauge field frozen into a particular configuration – the effect arises through the modulation of hopping integrals on bonds. Such “magnetic field” disorder is precisely marginal and it acts on the interaction to arrest the logarithmic decline of its contribution to conductivity. The result is a line of fixed points along which the minimum conductivity takes on a non-universal value, set by the rippling disorder, but always larger than \sigma_0.
This picture supplies a rather attractive explanation for the available experiments. The details of their work can be found in http://www.arxiv.org/abs/0707.4171.

Several comments and questions were lobbed at Herbut by clearly animated audience members. Chubukov inquired about the work of Efetov and Aleiner and its relation to this presentation. Herbut answered that they were considering a “non-critical” case, where the gate voltage moves the chemical potential away from Dirac nodes and thus the system acquires a small Fermi surface. Several audience members, including Morr, Vekhter and Eschrig, wanted to know more about the ordinary potential disorder, resulting in a random variation of a chemical potential. Herbut replied by pointing out that, for the current experiments, he felt his picture of the rippling disorder was the most appropriate.