The Fractal-Lattice Hubbard Model

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.