Many-body Signature of Coulomb Implosion in Graphene

We develop an asymptotic analysis for the scattering of a single electron off a Coulomb impurity in the Dirac equation description of (undoped) graphene, to demonstrate that the penetration of the centrifugal barrier that occurs in this problem may be assessed in a momentum representation. The method is directly generalizable to particle-hole scattering, which supports a similar phenomenon. We derive a Bethe-Salpeter equation for the 3-leg vertex for the sublattice antisymmetric response in the ladder approximation. We solve the integral equations for the lowest $(m=0)$ angular components numerically for both $q=0$ and $q\ll1$ but nonzero, where $q$ is the momentum transfer. In the $q=0$ case there is a clear power law behavior in the solution of the vertex function and the exponents become complex when the coupling constant is above a threshold. We also find that the response can have poles above a critical coupling constant, which we associate with a transition to an exciton condensate state. We can reproduce the poles by solving the integral equations with a model kernel analytically. However, we find that a small momentum cut-off is necessary for the existence of these poles in our analytical solution. In the case of nonzero but small $q$, we find that the correction to the antisymmetric response has a power law behavior in the coupling constant.