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Ultracold Fermi gases of spin-3/2 atoms provide a clean platform to realiseSO(5) models of 4-Fermi interactions in the laboratory. By confining the atomsin a two-dimensional Raman lattice, we show how this system can be used as aflexible quantum simulator of Dirac quantum field theories (QFTs) that combineGross-Neveu and Thirring interactions with a higher-order topological twist. Weshow that the lattice model corresponds to a regularization of this QFT with ananisotropic twisted Wilson mass. This allows us to access higher-ordertopological states protected by a hidden SO(5) symmetry, a remnant of theoriginal rotational symmetry of the 4-Fermi interactions that is not explicitlybroken by the lattice discretization. Using large-$N$ methods, we show that the4-Fermi interactions lead to a rich phase diagram with various competingfermion condensates. Our work opens a route for the implementation ofcorrelated higher-order topological states with tunable interactions that hasinteresting connections to non-trivial relativistic QFTs of Dirac fermions in$D = 2 + 1$ dimensions.

A higher-order topological twist on cold-atom SO(5) Dirac fields