  We study the half-filled Hubbard model on a one-parameter family of
vortex-full square lattices ranging from the isotropic case to weakly coupled
Hubbard dimers. The ground-state phase diagram consists of four phases: A
semi-metal and a band insulator which are connected to the weak-coupling limit,
and a magnetically ordered N\'eel phase and a valence bond crystal (VBC) which
are linked to the strong-coupling Mott limit. The phase diagram is obained by
quantum Monte Carlo (QMC) and continuous unitary transformations (CUTs). The
CUT is performed in a two-step process: Non-perturbative graph-based CUTs are
used in the Mott insulating phase to integrate out charge fluctuations. The
resulting effective spin model is tackled by perturbative CUTs about the
isolated dimer limit yielding the breakdown of the VBC by triplon condensation.
We find three scenarios when varying the interaction for a fixed anisotropy of
hopping amplitudes: i) one direct phase transition from N\'eel to semi-metal,
ii) two phase transitions VBC to N\'eel and N\'eel to semi-metal, or iii) a
smooth crossover from VBC to the band insultor. Our results are consistent with
the absence of spin-liquid phases in the whole phase diagram.
