Rigorous solution of a Hubbard model extended by nearest-neighbour Coulomb and exchange interaction on a triangle and tetrahedron

Abstract

The Hubbard model extended by either nearest-neighbour Coulomb correlation and/or nearest neighbour Heisenberg exchange is solved analytically for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed form. The groundstate crossings and degeneracies are discussed both for the canonical and grand-canonical energy levels. The grand canonical potential (, T, h) and the electron occupation N (, T, h) of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. In the pure Hubbard model we found various steps in N (, T = 0, h) higher than one. It is shown that the various degeneracies of the grand-canonical energy levels are partially lifted by an antiferromagnetic exchange interaction, whereas a moderate ferromagnetic exchange modifies only slightly the results of the pure Hubbard model. A repulsive nn Coulomb correlation lifts these degeneracies completely. The relation of the cluster gas results to extended systems is discussed.