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Physical review. B, Condensed matter

Exact solution of the Hubbard model for a four-center tetrahedral cluster

L. M. Falicov, Randall H. Victora

Published Aug 15, 1984• 87 citations
Publisher

Abstract

The eigenvalues of a Hubbard Hamiltonian for a four-center tetrahedral cluster are calculated exactly. Full use is made of the symmetry of the problem, which is analyzed for an arbitary number of electrons, $0\ensuremath{\le}N\ensuremath{\le}8$. Comparison is made with the phenomenological Hund's-rule predictions for the ground states. The diversity of the low-energy states is surprising: magnetic and nonmagnetic solutions, single and degenerate representations, accidental degeneracies, and symmetry crossovers are all found for the ground states. Implications for three-dimensional lattices are discussed.

DOI
10.1103/physrevb.30.1695

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Journal of Physics Condensed Matter•Oct 21, 2008•2 citations
The eigenvalues of a Hubbard Hamiltonian for a four-center tetrahedral cluster are calculated exactly. Full use is made of the symmetry of the problem, which is analyzed for an arbitary number of electrons, 0\ensuremath≤N\ensuremath≤80\ensuremath{\le}N\ensuremath{\le}80\ensuremath≤N\ensuremath≤8. Comparison is made with the phenomenological Hund's-rule predictions for the ground states. The diversity of the low-energy states is surprising: magnetic and nonmagnetic solutions, single and degenerate representations, accidental degeneracies, and symmetry crossovers are all found for the ground states. Implications for three-dimensional lattices are discussed.
We present exact explicit analytical results describing the exact ground state of four electrons in a two-dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The Hamiltonian acting on these states generates a closed system of 85 linear equations, providing by its minimum eigenvalue, the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but also emphasize further characteristics of the spectrum. On this line (i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or (ii) explicitly given low-lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.
With the purpose of assessing the validity of commonly used approximations, best variational solutions are calculated for a four-site (tetrahedral-cluster) Hubbard model with periodic boundary conditions and compared with exact results. In particular, the optimum number-conserving Bardeen-Cooper-Schrieffer-type variational solution is obtained for two, four, and six electrons, and the optimal resonanting-valence-bond state for four electrons. For an attractive interaction U, the exact ground state of the system is obtained in all four cases. If the interaction U is repulsive, the exact ground state is obtained only when the cluster contains two and six electrons. For four electrons the BCS approximation is only satisfactory for relatively small U. For any positive U the resonating-valence-bond state on the other hand, gives very good (albeit not exact) energies for the ground state.
The ground states of bcc, fcc, hcp, and icosahedral clusters having N=12−−14N=12--14N=12−−14 atoms are calculated exactly in the framework of the Hubbard Hamiltonian by using Lanczos's numerical diagonalization method. The total spin S shows a diversity of remarkable, sometimes surprising behaviors as a function of Coulomb interaction strength U/tU/tU/t and band filling $\ensuremath{\nu}/N.$ The magnetic properties are analyzed, in particular, by relating them to the corresponding single-particle spectra. Changes of symmetry close to real-space shell closings at N=13N=13N=13 are discussed. The relative stability of the different cluster geometries is determined and the corresponding structural and magnetic diagrams are derived. Electron correlation effects close to half-band filling result in changes of structure that are often accompanied by important changes in the magnetic behavior. Ferromagnetism is found to be particularly stable for $\ensuremath{\nu}/N\ensuremath{\simeq}1.2--1.4.$ The main trends as a function of $N, \ensuremath{\nu}$ and U/tU/tU/t are compared with previous exact diagonalization studies on smaller clusters.
An exact solution of a modified Hubbard model for a tetrahedral cluster with periodic boundary conditions is presented. It includes a static electron-lattice interaction in which the intersite bond-hopping parameter is a function of the bond electron occupation. It is found that the intrasite Coulomb repulsion and the effective attractive electron-lattice interaction have, generally speaking, opposite (although not identical) effects. They cancel each other exactly along a line of singularities in parameter space. The state vectors are easily obtained analytically. A study is made of magnetic and superconducting static correlation functions as functions of the parameters. Various forms of anisotropic superconducting fluctuations are favored in the various regimes. There is an indication of possible states with coexisting nonuniform, nonisotropic superconductivity and antiferromagnetism caused by a ``compromise'' between the competing forces.
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.
We present a study of the two-orbital degenerate Hubbard model in which the exact numerical solution on a regular tetrahedron is obtained via suitable implementation of the symmetries generated by the spin, the pairing and the orbital pseudospin operators. In particular, we show that a large variety of high-spin magnetic ground states can develop away from half filling, depending on the values of the electron density and the parameters of the model. As the tetrahedron is the simplest finite-size cluster where hopping processes connect all pairs of sites with constant probability, the study is extended by providing the exact analytical solution of the model on an infinite lattice in the unconstrained hopping limit.