  We unveil an interesting connection of Lorentz-violating quantum field
theories, studied in the context of the standard model extension, and
Hubbard-type models of topological crystalline phases. These models can be
interpreted as a regularisation of the former and, as hereby discussed,
explored with current quantum simulators based on ultra-cold atoms in optical
Raman lattices. In particular, we present a complete analysis of the
Creutz-Hubbard ladder under a generic magnetic flux, which regularises a
Gross-Neveu model extension, and presents a characteristic circulating chiral
current whose non-zero value arises from a specific violation of Lorentz
invariance. We present a complete phase diagram with trivial insulators,
ferromagnetic and anti-ferromagnetic phases, and current-carrying topological
crystalline phases. These predictions are benchmarked using tools from
condensed matter and quantum-information science, showing that self-consistent
Hartree-Fock and strong-coupling Dzyaloshinskii-Moriya methods capture the
essence of the phase diagram in different regimes, which is further explored
using extensive numerical simulations based on matrix-product states.
