  We demonstrate that Dirac fermions self-interacting or coupled to dynamic
scalar fields can emerge in the low energy sector of designed bosonic and
fermionic cold atom systems. We illustrate this with two examples defined in
two spacetime dimensions. The first one is the self-interacting Thirring model.
The second one is a model of Dirac fermions coupled to a dynamic scalar field
that gives rise to the Gross-Neveu model. The proposed cold atom experiments
can be used to probe spectral or correlation properties of interacting quantum
field theories thereby presenting an alternative to lattice gauge theory
simulations.
