Analysis of the New Unrestricted Hartree‐Fock Vortex Solution of the Hubbard Hamiltonian in Two‐Dimensional Systems A Small‐Cluster Study

Abstract

and J. FERRER (c)The properties of a recently discovered unrestricted Hartree-Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two-dimensional spin polarization of a vortex-like character. This and other several mean-field solutions are compared with the exact ground-state energy and wave function of two and four electrons on a four-site system (square). Results for two correlated electrons in large 3 x 3 and 4 x 4 clusters are also given and discussed. The vortex mean-field solution provides the best solution in almost all systems that deviate from half-filling.Die Eigenschaften einer kurzlich gefundenen, uneingeschrankten Hartree-Fock-Losung des HubbardHamiltonoperators wird fur ein quadratisches Gitter, das echt zweidimensionale Spinpolarisation mit Wirbelcharakter aufweist, analysiert. Diese und andere mean-field-Losungen werden mit der exakten Energie und Wellenfunktion des Grundzustandes von zwei und vier Elektronen in einem Vierplatzsystem (Quadrat) verglichen. Die Ergebnisse fur zwei korrelierte Elektronen in groDeren 3 x 3 und 4 x 4 Clustern werden ebenfalls mitgeteilt ,und diskutiert. In fast allen Systemen, die von einer Halbfullung abweichen, liefert die Wirbellosung die besten Ergebnisse.

Configuration-interaction approach to hole pairing in the two-dimensional Hubbard model

“…Hence, we do not expect that the lack of spin invariance is a serious drawback of the Hartree Fock solutions (this point is analyzed, in some detail in [21]). Results obtained for small clusters [16,26] show a slight improvement of the energy, which goes to zero as the cluster size is increased. On the other hand, translational invariance is expected to be present in the exact solution of clusters of any size.…”

Exact ground states for the four-electron problem in a two-dimensional finite Hubbard square system

“…In order to enhance further developments and enlighten further creative thinking, we provide, explicitly in Appendix A, the equations generating the ground state, together with the explicit form of the base vectors providing the ground state (see figures [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], hence describing the Hilbert space region in which this is placed. We mention that if the base vectors are defined only through the operation T, without subtracting two T components as shown in figures 4-18, a system containing 176 equations arises.…”

Hole pairs in the two-dimensional Hubbard model

“…In any case, spin isotropy can be restored, starting from the mean field wavefunction, by projecting out the components which do not have a predetermined spin. Results obtained for small clusters [18,19] show a slight improvement of the energy, which goes to zero as the cluster size is increased. On the other hand, translational invariance is expected to be present in the exact solution of clusters of any size.…”

“…Hence, we do not expect that the lack of spin invariance is a serious drawback of the Hartree-Fock solutions (this point is analyzed, in some detail, in [13]). Results obtained for small clusters [18,19] show a slight improvement of the energy, which goes to zero as the cluster size is increased. On the other hand, translational invariance is expected to be present in the exact solution of clusters of any size.…”