Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction

Motivated by the physics of graphene, we consider a model of $N$ species of $2+1$ dimensional four-component massless Dirac fermions interacting through a three dimensional instantaneous Coulomb interaction. We show that in the limit of infinitely strong Coulomb interaction, the system approaches a quantum critical point, at least for sufficiently large fermion degeneracy. In this regime, the system exhibits invariance under scale transformations in which time and space scale by different factors. The elementary excitations are fermions with dispersion relation $\omega \sim p^z$, where the dynamic critical exponent $z$ depends on $N$. In the limit of large $N$, we find $z=1-4∕\left({\pi }^{2}N\right)+O\left(N^{-2}\right)$. We argue that due to the numerically large Coulomb coupling, graphene (freely suspended) in a vacuum stays near the scale-invariant regime in a large momentum window, before eventually flowing to the trivial fixed point at very low momentum scales.

Figure 1

The functions $f_1\left(\lambda \right)$ (solid line) and $f_0\left(\lambda \right)$ (dashed line) and their difference $f_1\left(\lambda \right)-f_0\left(\lambda \right)$ (dotted line).

Figure 2

The simplest phase diagram. The line $g=\infty$ is a line of infrared unstable fixed points. From any finite value of $g$, the system flows to an infrared stable fixed point at $g=0$.

Figure 3

A slightly more complicated phase diagram. For $N>N_{\mathrm{crit}}$, the system is always in the semimetal phase. For $N<N_{\mathrm{crit}}$, it can be in the semimetal phase [for $g<g_c\left(N\right)$] or in the insulating phase [for $g>g_c\left(N\right)$].

Figure 4

The photon polarization diagram.

Figure 5

The fermion self-energy diagram.