Luttinger Liquid at the Edge of Undoped Graphene in a Strong Magnetic Field

We demonstrate that an undoped two-dimensional carbon plane (graphene) whose bulk is in the integer quantum Hall regime supports a nonchiral Luttinger liquid at an armchair edge. This behavior arises due to the unusual dispersion of the noninteracting edge states, causing a crossing of bands with different valley and spin indices at the edge. We demonstrate that this stabilizes a domain wall structure with a spontaneously ordered phase degree of freedom. This coherent domain wall supports gapless charged excitations, and has a power law tunneling $I\mathrm{\text{−}}V$ with a nonintegral exponent. In proximity to a bulk lead, the edge may undergo a quantum phase transition between the Luttinger liquid phase and a metallic state.

Figure 1

Energy bands for electrons in a graphene ribbon with armchair edges in a magnetic field from tight-binding calculations. $a_0=$ lattice constant, unit of energy ${\epsilon }_1=\sqrt{2}\gamma a_0/\ell$. $k_y=X/{\ell }^2$ with $X$ the guiding center coordinate. $B=100\mathrm{T}$; ribbon width is 460 Å. Solid black lines are spin up states, red dashed lines are spin down. Inset: Graphene ribbon with armchair edges.

Figure 2

Example of a domain wall configuration. $\cos [\theta (X)/2]$ ($\sin [\theta (X)/2]$) denotes the amplitude for an electron to occupy a spin up electronlike (spin down holelike) single particle state near the edge.

Figure 3

Geometry for tunneling into the coherent domain wall.