We study the properties of strongly correlated systems, calculating the self-energy of the electronic propagator of small-size clusters. We focus our attention on the one-dimensional Hubbard model. It is shown that for adequate sizes and boundary conditions, the self-energy of such a small cluster possesses the correct singular behaviour in the vicinity of the Fermi level for all values of the intra-atomic electron-electron interaction parameter U . The charge-transfer gap and other physical properties of the system are obtained at the thermodynamic limit. The survival of a single-pole singularity at w = 0 for the infinite-system self-energy constitutes a criterion for establishing whether the system is an insulator or a metal.
When electron-electron correlations are important, it is often necessary to use exact numerical methods, such as Lanczos diagonalization, to study the full many-body Hamiltonian. Unfortunately, such exact diagonalization methods are restricted to small system sizes. We show that if the Hubbard
U term is replaced by a ``periodic Hubbard" term, the full many body Hamiltonian may be numerically exactly solved, even for very large systems (even $>$100 sites), though only for low fillings. However, for half-filled systems and large
U this approach is not only no longer exact, it no longer improves extrapolation to larger systems. We discuss how generalized ``randomized variable averaging'' (RVA) or ``phase randomization'' schemes can be reliably employed to improve extrapolation to large system sizes in this regime. This general approach can be combined with any many-body method and is thus of broad interest and applicability.
We examine the existence of the interaction-generated quantum anomalous Hall phase on the honeycomb lattice. For the spinless model at half-filling, the existence of a quantum anomalous Hall phase (Chern insulator phase) has been predicted using mean-field methods. However, recent exact diagonalization studies for small clusters with periodic boundary conditions have not found a clear sign of an interaction-driven Chern insulator phase. We use the exact diagonalization method to study properties of small clusters with open boundary conditions, and contrary to previous studies, we find clear signatures of the topological phase transition for finite-size clusters. We also examine the applicability of the entangled-plaquette-state (correlator-product-state) ansatz to describe the ground states of the system. Within this approach the lattice is covered with plaquettes and the ground-state wave function is written in terms of the plaquette coefficients. Configurational weights can then be optimized using a variational Monte Carlo algorithm. Using the entangled-plaquette-state ansatz we study the ground-state properties of the system for larger system sizes and show that the results agree with the exact diagonalization results for small clusters. This confirms the validity of the entangled-plaquette-state ansatz to describe the ground states of the system and provides further confirmation of the existence of the quantum anomalous Hall phase in the thermodynamic limit, as predicted by mean-field theory calculations.