The energy level statistics of the Hubbard model for L x L square lattices (L = 3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L = 4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L = 4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard those states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance Sigma(2), and the spectral rigidity Delta(3), which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model. [References: 32]
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