Electron self-energy effects on chiral symmetry breaking in graphene

We investigate the dynamical breakdown of the chiral symmetry in the theory of Dirac fermions in graphene with long-range Coulomb interaction. We analyze the electron-hole vertex relevant for the dynamical gap generation in the ladder approximation, showing that it blows up at a critical value ${\alpha }_c$ in the graphene fine structure constant, which is quite sensitive to many-body corrections. Under static random phase approximation (RPA) screening of the interaction potential, we find that taking into account electron self-energy corrections to the vertex increases the critical coupling to ${\alpha }_c\approx 4.9$, for a number $N=4$ of two-component Dirac fermions. When dynamical screening of the interaction is instead considered, the effect of Fermi velocity renormalization in the electron and hole states leads to the value ${\alpha }_c\approx 1.75$ for $N=4$, substantially larger than that obtained without electron self-energy corrections ($\approx$0.99), but still below the nominal value of the interaction coupling in isolated free-standing graphene.

Figure 1

Self-consistent diagrammatic equation for the vertex $\langle {\rho }_m(\mathbf{q},{\omega }_q)\psi (\mathbf{k}+\mathbf{q},{\omega }_k+{\omega }_q){\psi }^{†}(\mathbf{k},{\omega }_k)\rangle$, equivalent to the sum of ladder diagrams built from the iteration of the Coulomb interaction (wavy line) between electron and hole states (arrow lines).

Figure 2

Electron self-energy correction leading to a divergent renormalization of the Fermi velocity $v_F$.

Figure 3

Plot of the absolute value of the coefficients $c_1^{\left(n\right)}$ in the expansion of $c_1\left(\lambda \right)$ as a power series of the coupling $\lambda$.

Figure 4

Phase diagram showing the boundary between the metallic phase and the phase with dynamical gap generation ($\langle {\rho }_m\rangle \ne 0$) in the ladder approximation. The thin dashed (solid) line represents the phase boundary obtained with static (dynamic) RPA screening of the interaction potential and no electron self-energy corrections. The thick dashed (solid) line represents the boundary after including the effect of the electron self-energy corrections on top of the static (dynamic) RPA screening of the interaction in the ladder series.