Fictitious gauge fields in bilayer graphene

We discuss the effect of elastic deformations on the electronic properties of bilayer graphene membranes. Distortions of the lattice translate into fictitious gauge fields in the electronic Dirac Hamiltonian which are explicitly derived here for arbitrary elastic deformations. We include gauge fields associated to intra- as well as inter-layer hopping terms and discuss their effects in different contexts. As a first application, we use the gauge fields in order to study the recently predicted strain-induced Lifshitz transition for the Fermi surface at low energy. As a second application, we discuss the electron-phonon coupling induced by the fictitious gauge fields and analyse its contribution to the electrical resistivity of suspended bilayer membranes. Of special interest is the appearance of a linear coupling for flexural modes, in stark contrast to the case of monolayer graphene. This new coupling channel is shown to dominate the temperature-dependent resistivity in suspended samples with low tension.