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: {\it quasistatic} and purely {\it 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.