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Journal of Physics Condensed Matter

Some exact solutions to the Hubbard model with infinite-range hopping

M W Long

Published May 13, 1991• 0 citations
Publisher

Abstract

For the Hubbard model with infinite-range hopping, a sensible competition between hopping and Hubbard repulsion can only be found when U/t scales with the volume of the system. Only two electrons carry all the kinetic energy and so the variations in Hubbard repulsion are bounded by only 2U for an eigenstate. To be comparable with the kinetic energy this must be extensive. For this case the author derives some exact eigenstates for the model. The lowest energy solution is paramagnetic although the spin degeneracy energy scale is very small. The behaviour of this system appears to be very different to that for the shorter-range hopping systems and so little can be deduced about more physical models.

DOI
10.1088/0953-8984/3/19/006

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For the Hubbard model with infinite-range hopping, a sensible competition between hopping and Hubbard repulsion can only be found when U/t scales with the volume of the system. Only two electrons carry all the kinetic energy and so the variations in Hubbard repulsion are bounded by only 2U for an eigenstate. To be comparable with the kinetic energy this must be extensive. For this case the author derives some exact eigenstates for the model. The lowest energy solution is paramagnetic although the spin degeneracy energy scale is very small. The behaviour of this system appears to be very different to that for the shorter-range hopping systems and so little can be deduced about more physical models.
The t - J model with constant t and J between any pair of sites is studied by exploiting the symmetry of the Hamiltonian with respect to site permutations. For a given number of electrons and a given total spin the exchange term simply yields an additive constant. Therefore the real problem is to diagonalize the ` t model', or equivalently the infinite U Hubbard Hamiltonian. Using extensively the properties of the permutation group, we are able to find explicitly both the energy eigenvalues and eigenstates, labelled according to spin quantum numbers and Young diagrams. As a corollary we also obtain the degenerate ground states of the finite U Hubbard model with infinite-range hopping -t>0 .
The ground-state wave function and energy of the Hubbard model with unlimited hopping can be obtained for arbitrary fillings using the transformation recently introduced by Brandt and Giesekus. A simple linear behavior is obtained for the ground-state energy dependence on the number of particles: EGS{\mathit{E}}_{\mathrm{GS}}EGS​=-2t(L-N), where -t is the hopping energy (t>0), L is the number of sites, and N is the number of electrons. The ground state shows a very large total spin degeneracy. The system is shown to behave as a Fermi liquid with a spectral weight Z=1-N/L for all fillings less than 1 (N\ensuremath{\le}L).
In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S -k .S k ) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.
The Hubbard model is one of the most important models of many-body physics. It is a simplified model for how electrons move and interact in a crystal. It consists of a hopping term, where electrons can move between neighboring sites (sometimes more distant neighbors are allowed). When two electrons (of opposite spin) occupy the same lattice site, they feel a Coulomb repulsion. This model can be exactly solved only in one dimension, in the limit of infinite dimensions, or for small systems via exact diagonalization. It has been studied by thousands of physicists over the years. While much is known about the model, even more remains unknown, so it is still a great frontier to examine further. In this chapter, we focus on the known solutions and provide concrete examples and rigorous theorems about the behavior of the model. This includes the symmetries of spin, particle number, and pseudospin, the Nagaoka theorem for ferromagnetism, spin-reflection positivity, and spin and pseudospin towers.