Moir\'e-pattern fluctuations and electron-phason coupling in twisted bilayer graphene

In twisted bilayer graphene, long-wavelength lattice fluctuations on the scale of the moir\'e period are dominated by phason modes, i.e., acoustic branches of the incommensurate lattice resulting from coherent superpositions of optical phonons. In the limit of small twist angles, these modes describe the sliding motion of stacking domain walls separating regions of partial commensuration. The resulting soliton network is a soft elastic manifold, whose reduced rigidity arises from the competition between intralayer (elastic) and interlayer (adhesion) forces governing lattice relaxation. Shear deformations of the beating pattern dominate the electron-phason coupling to the leading order in $t_{\perp}/t$, the ratio between interlayer and intralayer hopping parameters. This coupling lifts the layer degeneracy of the Dirac cones at the corners of the Moir\'e Brillouin zone, which could explain the observed 4-fold (instead of 8-fold) Landau level degeneracy. Electron-phason scattering gives rise to a linear-in-temperature contribution to the resistivity that increases with decreasing twist angle due to the reduction of the stiffness of the soliton network. This contribution alone, however, seems to be insufficient to explain the huge enhancement of the resistivity of the normal state close to the magic angle.