  Materials with non-trivial lattice geometries allow for the creation of
exotic states of matter like topologically insulating states. Therefore
searching for such materials is an important aspect of current research in
solid-state physics. In the field of ultracold gases there are ongoing studies
aiming to create non-trivial lattices using optical means. In this paper we
study two species of fermions trapped in a square optical lattice and show how
non-trivial lattices can emerge due to strong interaction between atoms. We
theoretically investigate regimes of tunable parameters in which such
self-assembly may take place and describe the necessary experimental
conditions. Moreover we discuss the possibility of such emergent lattices
hosting topologically insulating states.
