Numerical Studies of Correlated Topological Systems

In this thesis, we study the interplay of Hubbard U correlation and topological effects in two different bipartite lattices: the dice and the Lieb lattices. Both these lattices are unique as they contain a flat energy band at E = 0, even in the absence of Coulombic interaction. When interactions are introduced both these lattices display an unexpected multitude of topological phases in our U -λ phase diagram, where λ is the spin-orbit coupling strength. We also study ribbons of the dice lattice and observed that they qualitative display all properties of their two-dimensional counterpart. This includes flat bands near the Fermi level, edge currents when open boundary conditions are used, two chiral edge modes (because the planar Chern number is 2), and a nonzero Hall conductance. This opens the possibility of studies of these systems using powerful techniques such as the density matrix renormalization group that work well for ribbons. Finally, we study a multi-orbital Hubbard model for two and three orbitals per site on a two-site cluster and mapped our results into a higher-order effective spin Heisenberg model to know the limitations in the value of the couplings that can be used with those effective models. Numerical techniques, such as the Hartree-Fock approximation, Lanczos and density matrix renormalization group, were used to carry out these studies at half-filling.