Interaction Induced Quantum Valley Hall Effect in Graphene

We use pseudo-quantum electrodynamics in order to describe the full electromagnetic interaction of the $p$ electrons in graphene in a consistent 2D formulation. We first consider the effect of this interaction in the vacuum polarization tensor or, equivalently, in the current correlator. This allows us to obtain the $T\to 0$ conductivity after a smooth zero-frequency limit is taken in Kubo’s formula. Thereby, we obtain the usual expression for the minimal conductivity plus corrections due to the interaction that bring it closer to the experimental value. We then predict the onset of an interaction-driven spontaneous quantum valley Hall effect below an activation temperature of the order of 2 K. The transverse (Hall) valley conductivity is evaluated exactly and shown to coincide with the one in the usual quantum Hall effect. Finally, by considering the effects of pseudo-quantum electrodynamics, we show that the electron self-energy is such that a set of $P$- and $T$-symmetric gapped electron energy eigenstates are dynamically generated, in association with the quantum valley Hall effect.

Popular Summary

The discovery of the quantum Hall effect in semiconductor quantum wells subject to a perpendicular magnetic field highlights systems that are insulating in the bulk but carry quantized currents at their edges. The recent synthesis of graphene suggested that it may be possible to realize quantized spin currents at a material’s edges. However, this idea turned out to be impossible because the spin-orbit coupling is too weak. Here, we show that at low temperatures, another type of edge current, due to interactions, may actually occur in graphene.

We use a two-dimensional formulation to analyze conductivity in graphene as the temperature approaches zero. This formulation resolves the dimensional mismatch between the electrons in graphene, which are constrained to move in the plane, and the quanta of the electromagnetic field (photons) mediating their interaction, which are allowed to move in three dimensions. We theoretically apply an external electric field to determine how the conductivity of the material responds, assuming that the electrons act as Dirac fermions. We find that the well-known quantum Hall effect does not occur, but a variant of it does. Electrons carrying the valley index $K$ move in the opposite direction of electrons carrying the valley index $K^{\prime }$. Time-reversal symmetry is dynamically broken for each type of carrier, although it is not broken for both together. Contrary to the case of silicene, where similar currents arise because of the broken inversion symmetry of the lattice, here these currents emerge because of interactions. The transverse component of the valley conductivity survives, and the longitudinal component cancels below an activation temperature of approximately 2 K. We suggest that this effect can be probed experimentally in the future using light to separate the effect of each valley.

This discovery represents an important step forward from previously established quantum electronics and spintronics toward quantum valleytronics in graphene.

Figure 1

Graphical representation for solutions of the transcendental equation. The value of $\gamma$ is artificial, to facilitate the visualization.

Figure 2

Activation temperature $T^{*}$ as a function of the coupling $\alpha$ for $N_f=4$ and $\mathrm{\Lambda }\sim 3.0\mathrm{eV}$.