  Quantum Monte Carlo simulations are used to study the magnetic and transport
properties of the Hubbard Model, and its strong coupling Heisenberg limit, on a
one-third depleted square lattice. This is the geometry occupied, after charge
ordering, by the spin-$\frac{1}{2}$ Ni$^{1+}$ atoms in a single layer of the
nickelate materials La$_4$Ni$_3$O$_8$ and (predicted) La$_3$Ni$_2$O$_6$. Our
model is also a description of strained graphene, where a honeycomb lattice has
bond strengths which are inequivalent. For the Heisenberg case, we determine
the location of the quantum critical point (QCP) where there is an onset of
long range antiferromagnetic order (LRAFO), and the magnitude of the order
parameter, and then compare with results of spin wave theory. An ordered phase
also exists when electrons are itinerant. In this case, the growth in the
antiferromagnetic structure factor coincides with the transition from band
insulator to metal in the absence of interactions.
