Excitonic effects in two-dimensional massless Dirac fermions

We study excitonic effects in two-dimensional massless Dirac fermions with Coulomb interactions by solving the ladder approximation to the Bethe-Salpeter equation. It is found that the general four-leg vertex has a power-law behavior with the exponent going from real to complex as the coupling constant is increased. This change of behavior is manifested in the antisymmetric response, which displays power-law behavior at small wave vectors reminiscent of a critical state, and a change in this power law from real to complex that is accompanied by poles in the response function for finite-size systems, suggesting a phase transition for strong enough interactions. The density-density response is also calculated, for which no critical behavior is found. We demonstrate that exciton correlations enhance the cusp in the irreducible polarizability at 2k_F, leading to a strong increase in the amplitude of Friedel oscillations around a charged impurity.