The Hubbard model at high dimensions: some exact results and weak coupling theory

Dai S. Hirashima

B. Menge, E. M�ller-Hartmann

V. Zlatić, Berislav Horvatić

Walter Metzner, D. Vollhardt

The correlation energy of the Hubbard model on simple-cubic lattices is calculated in the weak-coupling regime for dimensions d=1,2,3. It is shown that, in spite of the nonanalytic dependences known to arise and in contrast to earlier expectations, the exact second-order contribution to the ground-state energy in d=1 can be obtained by ordinary perturbation theory. This result is argued to hold for any finite order and, in particular, for any dimension.

P. G. J. van Dongen

The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.

D. M. Esterling

Exact spectral moment relations are derived and presented for the low temperature, strong coupling limit of the Hubbard model, for any dimension, lattice structure and electron density. These results generate an exact, rigorous and quantitative test for proposed solutions to the Hubbard model that claim to be valid in the strong coupling, low temperature region. The test complements analyses for weak coupling and high temperatures, offering insights into the essential physics of the model, specifically the quasiparticle energy (centroid) and lifetime (width) of the spectral weight function in the infinite coupling, finite hopping limit. Numerical results are given for nearest neighbour hopping in 1, 2 and 3D.