Effects of different 3D QED vertex Ansätze on the critical coupling

Abstract

We study the semimetal-insulator phase transition in graphene using a Schwinger-Dyson approach. We consider various forms of vertex Ans\"atze to truncate the hierarchy of Schwinger-Dyson equations. We define a Ball-Chiu-type vertex that truncates the equations without violating gauge invariance. We show that there is a family of these vertices, parametrized by a continuous parameter that we call $a$, all of which satisfy the Ward identity. We have calculated the critical coupling of the phase transition using different values of $a$. We have also tested a common approximation in which only the first term in the Ball-Chiu Ansatz is included. This vertex is independent of $a$, and, although it is not gauge invariant, it has been used many times in the literature because of the numerical simplifications it provides. We have found that, with a one-loop photon polarization tensor, the results obtained for the critical coupling from the truncated vertex and the full vertex with $a=1$ agree very well, but other values of $a$ give significantly different results. We have also done a fully self-consistent calculation, in which the photons are backcoupled to the fermion degrees of freedom, for one choice $a=1$. Our results show that when photon dynamics are correctly taken into account, it is no longer true that the truncated vertex and the full Ball-Chiu vertex with $a=1$ agree well. The conclusion is that traditional vertex truncations do not really make sense in a system that does not respect Lorentz invariance, like graphene, and the need to include vertex contributions self-consistently is likely inescapable.