(No) e/2 States in Graphene

Electron Fractionalization in Two-Dimensional Graphenelike Structures

C.Y. Hou, Claudio Channon, and Christopher Mudry

PRL 98, 186809 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e186809

This is an interesting paper that probably has no "practical application" whatsoever. The authors show how the order parameter of a Kekule phonon interaction in graphene can lead to fractionally charged states.

The Kekule texture, as the authors call it, is a periodic variation in the hopping amplitudes of the tight-binding model. One of my colleagues has done some work on this type of perturbation. It expands the unit cell in real space to include 4 lattice sites, and reduces the Brillouin zone accordingly. The order parameter that describes the Kekule phonons can be a complex number. However, the authors point out that a constant phase can be removed by an appropriate chiral transformation of the Hamiltonian.

Interesting things happen when the phase becomes a local parameter. Randy, one of my profs who taught a great course on liquid crystals, emphasized this point repeatedly. The central idea behind Goldstone modes is that gapless excitations arise when a global symmetry operation is applied locally. Phonons were the first example. In a lattice, the crystal is the same if the entire lattice is translated by any amount. However, if we make the translation a local operation --- i.e., each lattice site is translated by a different amount --- it leads to a gapless excitation in the system: acoustic phonons.

The work of the authors is similar. They introduce vortices in the Kekule parameter. The Hamiltonian can be solved to give a single-valued normalizable wave function for a zero mode --- a mode that occurs precisely in the middle of the band gap. (The Kekule parameter introduces a gap into the graphene energy surface.)

Next, they calculate the charge bound to a vertex by taking the difference of the local density of states when there is one vortex and no vortices. Conservation of the number of electrons says that 2 times the number of electrons associated with a vortex plus the integral of the zero mode --- i.e. 1 --- must vanish. Ergo, there is a charge of -e/2 associated with each vortex. This would seem to violate charge conservation --- adding one vortex changes the charge of the system by half an electron charge. The authors point out, however, that vortices come in pairs, so there is no contradiction.

The disappointed disclaimer of this work is that when you factor in the degeneracy associated with spin, the fractionally charged states disappear. Only scalar electrons permit fractionally charged excitations. Too bad. The authors go on to discuss how one might introduce local variations in the Kekule parameter, but the point seems moot.

The authors made a few statements I would like to know more about. They claim that the time-reversal symmetry of the tight-binding Hamiltonian follows from the fact that the hopping parameters are real. Moreover, they say that states come in pairs of energy +E and -E because there exists a gauge transformation that can flip the sign of the ladder operators on the A sublattice without affecting those on the B sublattices. This sounds like a 2 pi rotation of a spin-1/2 particle, but I don't understand how it works for a tight-binding model.

In a four page paper, the authors managed to explain quite a lot!