  Ultracold Fermi gases of spin-3/2 atoms provide a clean platform to realise
SO(5) models of 4-Fermi interactions in the laboratory. By confining the atoms
in a two-dimensional Raman lattice, we show how this system can be used as a
flexible quantum simulator of Dirac quantum field theories (QFTs) that combine
Gross-Neveu and Thirring interactions with a higher-order topological twist. We
show that the lattice model corresponds to a regularization of this QFT with an
anisotropic twisted Wilson mass. This allows us to access higher-order
topological states protected by a hidden SO(5) symmetry, a remnant of the
original rotational symmetry of the 4-Fermi interactions that is not explicitly
broken by the lattice discretization. Using large-$N$ methods, we show that the
4-Fermi interactions lead to a rich phase diagram with various competing
fermion condensates. Our work opens a route for the implementation of
correlated higher-order topological states with tunable interactions that has
interesting connections to non-trivial relativistic QFTs of Dirac fermions in
$D = 2 + 1$ dimensions.
