Collective excitations in a large-d model for graphene

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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 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.