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Annalen der Physik

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

Rolf Schumann

Published Apr 15, 2009• 1 citation
Publisher

Abstract

In connection with the study of the extended Hubbard model on an isosceles triangular cluster, we discovered a factor two error in the coding of the Ising part of the Heisenberg term in the Hamiltonian. In consequence the rigorous results remain valid, but not for the isotropic Heisenberg model, but for an anisotropic Heisenberg model.
DOI
10.1002/andp.20095210408

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

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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 Ω(µ, T, h) and the electron occupation N (µ, 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 (µ, 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.
Abstract The Hubbard model extended by both nearest‐neighbour (nn) Coulomb correlation and nearest‐neighbour Heisenberg exchange is solved rigorously for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed analytical form. For fixed electron numbers we found a multitude of level crossings, both in the ground state and in the excited states in dependence on the various model parameters. By coupling an ensemble of clusters to an electron bath we get the cluster gas model or the cluster gas approximation, if an extended array of weak‐interacting clusters is considered. The grand‐canonical potential Ω (μ, T , h ) and the electron occupation N (μ, T , h ) of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. For the cluster gases without the additional interactions we found various steps in N (μ, T = 0, h = 0) higher than one. The reason is the degeneration of ground states differing in their electron occupation by more than one electron. For the triangular cluster gas we have one such degeneration point. For the tetrahedral cluster gas two. As a consequence, we do not find areas with one electron in the μ‐ U ground‐state phase diagram of the triangular cluster gas or with one, two and five electrons in the case of the tetrahedral cluster gas. The degeneration point of the triangular cluster gas can not be destroyed by an applied magnetic field. This holds also for the lower degeneration point of the tetrahedral cluster gas. Otherwise, the upper degeneration point breaks down at a critical magnetic field h c . The dependence of h c on U shows a maximum for strong on‐site correlation. The influence of nn‐exchange and nn‐Coulomb correlation on the ground‐state phase diagrams is calculated. Whereas antiferromagnetic nn‐exchange breaks the degeneration points of the tetrahedral cluster gas partially only, a repulsive nn‐Coulomb correlation lifts the underlying degeneracies completely. Otherwise both ferromagnetic nn‐exchange and attractive nn‐Coulomb interaction stabilise the degeneration points. The consequences of the cluster gas results for extended cluster arrays are discussed.
Abstract The Hubbard model on a cube was revisited and extended by both nearest‐neighbor Coulomb correlation W and nearest‐neighbor Heisenberg exchange J. The complete eigensystem was computed exactly for all electron occupancies and all model parameters ranging from minus infinity to plus infinity. For two electrons on the cluster the eigensystem is given in analytical form. For six electrons and infinite on‐site correlation U we determinded the groundstate and the groundstate energy of the pure Hubbard model analytically. For fixed electron numbers we found a multitude of ground state level crossings depending on the various model parameters. Furthermore the groundstates of the pure Hubbard model in dependence on a magnetic field h coupled to the spins are shown for the complete U‐h plane. The critical magnetic field, where the zero spin groundstate breaks down is given for four and six electrons. Suprisingly we found parameter regions, where the ground state spin does not depend monotonously on J in the extended model. For the cubic cluster gas, i.e. an ensemble of clusters coupled to an electron bath, we calculated the density n (μ, T, h) and the thermodynamical density of states from the grand potential. The ground states and the various spin‐spin correlation functions are studied for both attractive and repulsive values of the three interaction constants. We determined the various anomalous degeneration lines, where n (μ, T = 0, h = 0) shows steps higher than one, since in this parameter regions exotic phenomena as phase separation are to expect in extended models. For the cases where these lines end in triple points, i.e. groundstates of three different occupation numbers are degenerated, we give the related parameter values. Regarding the influence of the nn‐exchange and the nn‐Coulomb correlation onto the anomalous degeneration we find both lifting and inducing of degeneracies depending on the parameter values.
The eigenvalue problem of the two-site and three-site extended Hubbard Hamiltonian, which contains the intraatomic repulsion UUU, the nearest-neighbor repulsion VVV, the isotropic exchange JJJ, the anisotropic exchange ${J}^{\ensuremath{'}}$, the hopping term ttt, the semihopping term SSS, and the double hopping term DDD, has been solved exactly and the grand partition function is obtained. The magnetic and thermodynamic quantities such as the magnetic susceptibility and the specific heat are calculated for arbitrary electron density. Several features believed to have wider validity have been deduced from this cluster model.