Collective excitations in a large-d model for graphene

We consider a model of Dirac fermions coupled to flexural phonons to describe a graphene sheet fluctuating in dimension $2+d$. We derive the self-consistent screening equations for the quantum problem, exact in the limit of large $d$. We first treat the membrane alone, and work out the quantum to classical, and harmonic to anharmonic crossover. For the coupled electron-membrane problem we calculate the dressed two-particle propagators of the elastic and electron interactions and find that it exhibits a collective mode which becomes unstable at some wave-vector $q_{\rm c}$ for large enough coupling $g$. The saddle point analysis, exact at large $d$, indicates that this instability corresponds to spontaneous and simultaneous appearance of gaussian curvature and electron puddles. The relevance to ripples in graphene is discussed.