Theory of electromechanical coupling in dynamical graphene

We study the coupling between mechanical motion and Dirac electrons in a dynamical sheet of graphene. We show that this coupling can be understood in terms of an effective gauge field acting on the electrons, which has two contributions: quasistatic and purely dynamic of the Berry-phase origin. As is well known, the static gauge potential is odd in the K and K' valley index, while we find the dynamic coupling to be even. In particular, the mechanical fluctuations can thus mediate an indirect coupling between charge and valley degrees of freedom.