Ground-State Energy of the One-Dimensional Hubbard Model in a Simple Self-Consistent Version of the Ladder Approximation

Abstract

The ground-state energy of the one-dimensional Hubbard model is calculated within the ladder approximation; from the comparison with the exact results in the repulsive case, it follows that the approximation is good at low densities or small couplings. The ladder approximation can be improved by imposing a self-consistency condition; using a simple assumption, the results become close to the exact ones in a large range of the model parameters.

The Composite Operator Method (COM)

“…1.4 (right panel) for U = 4. For comparison, we also report the BA results [80] and LA and SCLA approaches [78,79]. COM agrees reasonably with the BA, reaching the best agreement at half filling.…”

“…The agreement with BA is excellent. Such a good agreement is not reached by any other analytical approach, neither by the Gutzwiller or the ladder and the self-consistent ladder approximations [77][78][79]. In particular, these approaches fail to reproduce the correct asymptotic behavior.…”

“…COM agrees reasonably with the BA, reaching the best agreement at half filling. The ladder approximation [78] deviates more and more from the BA as approaching half filling; the self-consistent ladder approximation [78] probes excellently at any doping. In Fig.…”

“…1.4 (left panel). The results obtained by means of the BA [80] and other analytical approaches [77][78][79] are also reported. The agreement between COM and BA is excellent.…”

“…The self-consistent Ladder approximation (SCLA) [78] shows also a very good agreement for all values of the coupling, but it does not have the correct behavior for an infinite value of the Coulomb interaction. Moreover, both the Ladder (LA) [78] and the Gutzwiller (GWA and GWF) [77] approximations go to zero at finite U , whereas the Renormal- ization Group (RG) [79] has the right asymptotic behavior for U → ∞, but it does not reproduce the non-interacting limit. The doping dependence of the internal energy is shown in Fig.…”