The energy level statistics of the Hubbard model for $L \times L$ squarelattices (L=3,4,5,6) at low filling (four electrons) is studied numerically fora wide range of the coupling strength. All known symmetries of the model(space, spin and pseudospin symmetry) have been taken into account explicitlyfrom the beginning of the calculation by projecting into symmetry invariantsubspaces. The details of this group theoretical treatment are presented withspecial attention to the nongeneric case of L=4, where a particular complicatedspace group appears. For all the lattices studied, a significant amount oflevels within each symmetry invariant subspaces remains degenerated, but exceptfor L=4 the ground state is nondegenerate. We explain the remainingdegeneracies, which occur only for very specific interaction independentstates, and we disregard these states in the statistical spectral analysis. Theintricate structure of the Hubbard spectra necessitates a careful unfoldingprocedure, which is thoroughly discussed. Finally, we present our results forthe level spacing distribution, the number variance $\Sigma^2$, and thespectral rigidity $\Delta_3$, which essentially all are close to thecorresponding statistics for random matrices of the Gaussian ensembleindependent of the lattice size and the coupling strength. Even very smallcoupling strengths approaching the integrable zero coupling limit lead to theGaussian ensemble statistics stressing the nonperturbative nature of theHubbard model.
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