Analog gravity and continuum effective theory of the graphene tight-binding lattice model

We consider the nearest-neighbor lattice tight-binding model of graphene with slowly spatially varying hopping functions. We develop a systematic low-energy approximation as a derivative expansion in a Dirac spinor field that is perturbative in the strength of the hopping function deformation. The well-known leading description in both the derivative and perturbative expansions is the Dirac equation in flat $2+1$-dimensional spacetime with magnetic (strain-)gauge field. Prior work has considered subleading corrections written as nontrivial frame and spin connection terms. We have argued previously that such corrections cannot be considered consistently without taking all the terms at the same order of approximation, which due to the unconventional power counting originating from the large gauge field, involve also higher covariant derivative terms. Here we confirm this, explicitly computing subleading terms in this approximation. To the order we explore, the theory is elegantly determined by the nontrivial gauge field and frame, both given by the slowly varying hopping functions, the torsion free spin connection of the frame, together with coefficients for the higher derivative terms derived from lattice invariants. We stress the importance of the local frame and gauge symmetries that are inherited from matching to the lattice model. For the first time we compute the metric that the Dirac field sees---the ``electrometric''---to quadratic order in the hopping function deformation. This allows us to describe the subleading corrections to the dispersion relation for inhomogeneous deformations that originate from corrections to the frame. Focussing on purely in-plane inhomogeneous strain, we use a simple model to relate the hopping functions to the strain field, finding the electrometric becomes curved at this quadratic order. Thus, this lattice model yields an effective ``analog gravity'' description as a curved space Dirac theory, with large magnetic field, and Lorentz violating higher covariant derivative terms. We check our calculations by a simple numerical diagonalization of the lattice model for a periodic arm-chair deformation, confirming that frame corrections contribute at the same order of approximation as higher covariant derivative terms. Finally, based on our lattice model derivation, we conjecture a form for the effective theory for monolayer graphene, written in terms of the strain tensor, and consistent up to quadratic order in the electrometric deformation.