Effective Hamiltonian approach for strongly correlated systems

We demonstrate the utility of effective Hamilonians for studying extended correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical contractor renormalization (CORE) method is applied in two steps (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut-off using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semi-quantitative numerical method. For ladder type geometries, we explicitly check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagome lattice as non-trivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagome lattice we find that the previously proposed twofold degenerate S = 1/2 basis can account for a large number of phenomena of the spin 1/2 kagome system. In general, we are able to simulate system sizes which correspond to an 8 x 8 lattice for the plaquette lattice or a 48-site kagome lattice, which are beyond the possibilities of a standard exact diagonalization approach.