Why do we need numerical methods for constrained fermion systems?

Approximative methods such as perturbation theory, mean-field calculations or variational treatments are not reliable for many strongly interacting fermion models. Numerical methods are thus essential in the investigation of these strongly correlated fermion systems. A strong Coulomb repulsion present in these systems can often be replaced by a constraint to no-double occupancy, leading to an effective model for the low-energy excitations. These constrained systems are often easier to investigate. We review the open-quotes exactclose quotes algorithms mainly used to simulate constrained fermion systems, exact diagonalization, quantum Carlo and the quantum transfer matrix algorithm (QTM). We report on a new improved QTM algorithm, obtained by a combination of the QTM with the recently developed look-ahead Lanczos algorithm for non-hermitian matrices. These methods give reliable results and can answer many open questions in the field of strongly correlated electron systems