  Strongly correlated electrons on an Apollonian network are studied using the
Hubbard model. Ground-state and thermodynamic properties, including specific
heat, magnetic susceptibility, spin-spin correlation function, double occupancy
and one-electron transfer, are evaluated applying direct diagonalization and
quantum Monte Carlo. The results support several types of magnetic behavior. In
the strong-coupling limit, the quantum anisotropic spin 1/2 Heisenberg model is
used and the phase diagram is discussed using the renormalization group method.
For ferromagnetic coupling, we always observe the existence of long-range
order. For antiferromagnetic coupling, we find a paramagnetic phase for all
finite temperatures.
