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A sufficiently strong Coulomb interaction may open an excitonic fermion gapand thus drive a semimetal-insulator transition in graphene. In this paper, westudy the Eliashberg theory of excitonic transition by coupling the fermion gapequation self-consistently to the equation of vacuum polarization function.Including the fermion gap into polarization function increases the effectivestrength of Coulomb interaction because it reduces the screening effects due tothe collective particle-hole excitations. Although this procedure does notchange the critical point, it leads to a significant enhancement of thedynamical fermion gap in the excitonic insulating phase. The validity of theEliashberg theory is justified by showing that the vertex corrections aresuppressed at large $N$ limit.

Eliashberg theory of excitonic insulating transition in graphene