Finite-frequency anomaly-induced electromechanical response of Dirac fermions in deformed graphene

A deformation of a graphene sheet changes more than the positions of the atoms. In the low-energy Dirac theory it also produces geometric electron-phonon vertices. One of these vertices acts as an emergent phonon gauge field, $\calA_\mu$, which couples to the same Dirac current as the electromagnetic vector potential. This shared current vertex gives a direct route from mechanics to electronics: a moving deformation can generate a transverse electric current, and a deformation pattern with emergent phonon flux can bind electric charge. We show that the coefficient of this mixed electromechanical response is the parity-odd current-current correlator of a massive Dirac cone. For an insulating cone the coefficient is the one-cone Chern-Simons value, while for a doped cone in the local regime it is reduced by the Berry curvature factor $m/|\mu|$. We apply the response to explicit deformations. A traveling flexural wave generates a transverse second-harmonic current; a static ripple mixed with a dynamic phonon generates a transverse current at the drive frequency; and two non-collinear modes can generate charge modulation through the emergent phonon flux. We keep the spin and valley sum explicit, so the paper shows when the one-cone anomaly becomes a charge current in graphene and when it instead appears in a valley, spin, or spin-valley channel. For sublattice-gapped graphene with a valley-odd deformation gauge coupling, the two valleys add rather than cancel. The experimentally sharp signature is a transverse electrical signal at twice the flexural-wave frequency, with a phase fixed by the sign of the sublattice gap and a gate dependence that crosses over from a gap plateau to a $1/|\mu|$ decay. These direction, phase, frequency, and gate-voltage selection rules give clean tests of the anomaly-induced electromechanical channel in deformed graphene.