One such method, the Lanczos algorithm, can produce static and dynamic properties of the system. Ground state calculations using this method require the storage of three vectors of the size of the number of states. The number of states scales exponentially with the size of the system, which limits the number of sites in the lattice to about 20 on 21st century hardware. With projector and finite-temperature auxiliary-field Monte Carlo, two statistical methods exist that can obtain certain properties of the system. For low temperatures, convergence problems appear that lead to an exponential computational effort with decreasing temperature due to the so-called fermion sign problem.