We will present how 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 will 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.
