  The study of correlation effects in topological phases of matter can benefit
from a multidisciplinary approach that combines techniques drawn from condensed
matter, high-energy physics and quantum information science. In this work, we
exploit these connections to study the strongly-interacting limit of certain
lattice Hubbard models of topological insulators, which map onto four-Fermi
quantum field theories with a Wilson-type discretization, and have been
recently shown to be at reach of cold-atom quantum simulators based on
synthetic spin-orbit coupling. We combine large-S and tensor-network techniques
to explore the possible spontaneous symmetry-breaking phases that appear when
the interactions of the topological insulators are sufficiently large. In
particular, we show that varying the Wilson parameter $r$ of the lattice
discretizations leads to a novel Heisenberg-Ising compass model with critical
lines that flow with the value of $r$.
