Dirac equation for cold atoms in artificial curved spacetimes

We argue that the Fermi-Hubbard Hamiltonian describing the physics of ultracold atoms on optical lattices in the presence of artificial non- Abelian gauge fields is exactly equivalent to the gauge theory Hamiltonian describing Dirac fermions in the lattice. We show that it is possible to couple the Dirac fermions to an 'artificial' gravitational field, i.e. to consider the Dirac physics in a curved spacetime.We identify the special class of spacetime metrics that admit a simple realization in terms of a Fermi-Hubbard model subjected to an artificial SU(2) field, corresponding to position-dependent hopping matrices. As an example, we discuss in more detail the physics of the 2+1D Rindler metric and its possible experimental realization and detection.