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

Ultracold Fermi gases of spin-3/2 atoms provide a clean platform to realise SO(55) 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 discrete SO(55) group, a remnant of the continuous rotational symmetry of the 4-Fermi interactions that is not explicitly broken by the lattice discretization. Using large-NN 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+1D=2+1 dimensions.