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The energy gap of correlated Hubbard clusters is well studied for
one-dimensional systems using analytical methods and density-matrix-
renormalization-group (DMRG) simulations. Beyond 1D, however, exact results
are available only for small systems by quantum Monte Carlo. For this reason
and, due to the problems of DMRG in simulating 2D and 3D systems, alternative
methods such as Green functions combined with many-body approximations
(GFMBA), that do not have this restriction, are highly important. However, it
has remained open whether the approximate character of GFMBA simulations
prevents the computation of the Hubbard gap. Here we present new GFMBA
results that demonstrate that GFMBA simulations are capable of producing
reliable data for the gap which agrees well with the DMRG benchmarks in 1D.
An interesting observation is that the accuracy of the gap can be significantly
increased when the simulations give up certain symmetry restriction of the
exact system, such as spin symmetry and spatial homogeneity. This is seen as
manifestation and generalization of the “symmetry dilemma” introduced by
Löwdin for Hartree–Fock wave function calculations
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