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Extracting Infinite System Properties from Finite Size Clusters: `Phase Randomization/Boundary Condition Averaging'

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.