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Phonon dispersions generically display non-analytic points, known as Kohnanomalies, due to electron-phonon interactions. We analyze this phenomenon fora zone boundary phonon in undoped graphene. When electron-electron interactionswith coupling constant $\beta$ are taken into account, one observes behaviordemonstrating that the electrons are in a critical phase: the phonon dispersionand lifetime develop power law behavior with $\beta$ dependent exponents. Theobservation of this signature would allow experimental access to the criticalproperties of the electron state, and would provide a measure of its proximityto an excitonic insulating phase.

Power law Kohn anomaly in undoped graphene induced by Coulomb interactions