We calculate the behaviour of the momentum distribution function n ( k ) for the ground state of the one-dimensional Hubbard model with arbitrary band filling in the weak-coupling regime. For the incommensurate case and U > 0 we derive the variation of n ( k ) in the vicinity of the Fermi wave vector k F and find a power law behaviour with a maximum exponent of one instead of a discontinuous jump. We also make a comparison with the n ( k ) obtained from recent Monte Carlo simulations and conclude that these numerical results are still seriously affected by finite-size effects. We finally consider the half-filled band and find a linear variation of n ( k ) close to k F , the scale being set by the charge gap of the Hubbard model.
The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby
kF.
Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction
U at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (
t being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For
U>Uc2, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For
Uc1<U<Uc2, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the
K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing
U and thus the superconductivity is unlikely in the intermediate phase.
The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.
The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (
ty=
tx); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=
ty/
tx, is different from the unity and zero the ground state is a conductor up to a critical value (U/t
)c1, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t
)c2\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.