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The effect of vertex corrections on the possibility of chiral symmetry breaking, induced by long-range Coulomb repulsion in graphene

In this paper we consider the possibility of chiral (charge or spin density wave) symmetry breaking in graphene due to long-range Coulomb interaction by comparing the results of the Bethe-Salpeter and functional renormalization-group approaches. The former approach performs summation of ladder diagrams in the particle-hole channel, and reproduces the results of the Schwinger-Dyson approach for the critical interaction strength of the quantum phase transition. The renormalization-group approach combines the effect of different channels and allows to study the role of vertex corrections. The critical interaction strength, which is necessary to induce the symmetry breaking in the latter approach is found in the static approximation to be $α_{c}=e^{2}/(εv_{F})\approx 1.05$ without considering the Fermi velocity renormalization, and $α_{c}=3.7$ with accounting the renormailzation of the Fermi velocity. The latter value is expected to be however reduced, when the dynamic screening effects are taken into account, yielding the critical interaction, which is comparable to the one in freely suspended graphene. We show that the vertex corrections are crucially important to obtain the mentioned values of critical interactions.