Critical Behavior in Graphene with Coulomb Interactions

We demonstrate that, in the presence of Coulomb interactions, electrons in graphene behave like a critical system, supporting power law correlations with interaction-dependent exponents. An asymptotic analysis shows that the origin of this behavior lies in particle-hole scattering, for which the Coulomb interaction induces anomalously close approaches. With increasing interaction strength the relevant power law changes from real to complex, leading to an unusual instability characterized by a complex-valued susceptibility in the thermodynamic limit. Measurable quantities, as well as the connection to classical two-dimensional systems, are discussed.

Figure 1

(a) Diagrammatical equation for the 3-leg vertex ${\tilde{\Gamma }}_{\alpha \beta }^M(\overrightarrow{k},\overrightarrow{q})$ with ${\sigma }^z$ as the zeroth order vertex (shaded cross). (b) Diagram for $M(\overrightarrow{q})$.

Figure 2

Solutions of Eq. (11) with $k_0/\Lambda ={10}^{-10}$. (a) $\beta =0.5$. Because the plotted quantity is $|F|$, the oscillations appear as cusps. Note the amplitude of the oscillation scales roughly as $1/\sqrt{k}$. Inset: $F$ for $\beta =0.3$. It is clearly a power law except for $k$ close to $k_0$. (b) The antisymmetric response $M$ as a function of the interaction strength $\beta$.

Figure 3

The exponent in $\Delta M(q)$ as a function of $\beta$. Inset: $\Delta M(q)$ for $\beta =0.3$.