  Here, we investigate the fractal-lattice Hubbard model using various
numerical methods: exact diagonalization, the self-consistent diagonalization
of a (mean-field) Hartree-Fock Hamiltonian and state-of-the-art Auxiliary-Field
Quantum Monte Carlo. We focus on the Sierpinski triangle with Hausdorff
dimension $1.58$ and consider several generations. In the tight-binding limit,
we find compact localised states, which are also explained in terms of symmetry
and linked to the formation of a ferrimagnetic phase at weak interaction.
Simulations at half-filling revealed the persistence of this type of magnetic
order for every value of interaction strength and a Mott transition for U/t
$\sim$ 4.5. In addition, we found a remarkable dependence on the Hausdorff
dimension regarding $i)$ the number of compact localised states in different
generations, $ii)$ the scaling of the total many-body ground-state energy in
the tight-binding limit, and $iii)$ the density of the states at the corners of
the lattice for specific values of electronic filling. Moreover, in the
presence of an intrinsic spin-orbit coupling, the zero-energy compact localized
states become entangled and give rise to inner and outer corner modes.
